On some bidimensional versions of the generalized Benjamin-Ono equation

Aniura Milanés Barrientos

Contents

0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.2 Notation and preliminaries . . . . . . . . . . . . . . . . . . . 7

1 Local Theory in Sobolev Spaces 13

2 Decay properties 24 2.0.1 Isotropic weights . . . . . . . . . . . . . . . . . . . . . 24 2.0.2 Non isotropic weights . . . . . . . . . . . . . . . . . . 33

3 Long time behavior of small solutions and nonlinear data scattering 38

4 Solitary waves 44

5 Remarks and conclusions 50 5.1 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

0.1 Introduction

There are situations where internal waves arise, due to the gravitational effects, at the interface of two layers in a stratified fluid. Several theoretical models exist which govern the evolution of long internal waves with small amplitudes in such cases. When the order of magnitude of the relative depth of the two-fluids system (denoted below as δ) is finite, the motion is described by the so called intermediate long wave equation (ILW) (c.f. [27], [35]):

u_{t} − T u_{xx} + \frac{1}{δ}u_{x} + uu_{x} = 0, δ > 0.  (1)

Here the operator T is defined by:

T f(x) = p.v.\frac{1}{2δ} ∫ coth (\frac{π(x − y)}{2δ}) f(y)dy, δ > 0.

The Benjamin-Ono equation (BO)^{1},

u_{t} − Hu_{xx} + uu_{x} = 0,  (2)

describes the evolution of long internal waves in stratified fluids when the height of the heavier fluid is much larger than that of the upper layer (c.f. [10],[39]).

When the total depth of the fluid is very small, the motion is governed by the equation (suitably rescaled)

u_{t} + u_{xxx} + uu_{x} = 0.  (3)

This is the Korteweg-de Vries equation (KdV). It was derived in [34] as a model for the propagation of long one dimensional surface gravity waves with small amplitude in a shallow channel of water. KdV has a very rich structure from the mathematical point of view and it has also been derived in several other physical contexts (see [3]).

Corresponding to the phenomena that they model, KdV and BO can be obtained from ILW through suitable limit processes as δ → 0 and as δ → ∞, respectively (see [2] and [5] for the precise formulation of this assertion).

When transversal effects must also be considered, the resulting equations are bidimensional.

The most accepted generalizations of the KdV to two dimensions are the equation of Kadomtsev and Petviashvili (KP I for σ = −1, KP II for σ = 1):

(u_{t} + u_{xxx} + uu_{x})_{x} + σu_{yy} = 0 σ = ±1  (4)

1Here H denotes the Hilbert transform, see Section 2

and the equation of Zakharov and Kuznetsov (ZK):

u_{t} + Δu_{x} + uu_{x} = 0  (5)

where Δ denotes the bidimensional Laplacian; i.e. Δ = ∂_{x}^{2} + ∂_{y}^{2}.

The KP equation was first derived in [28], where the authors studied the stability of the soliton solutions of the KdV, with respect to transverse perturbations. It describes the evolution of weakly two dimensional long water waves of small amplitude.

The ZK equation models ion-acoustic waves propagating in a low-pressure magnetized plasma. It was obtained in [55], where were also established the existence and stability for circularly symmetric soliton solutions.

For BO there have also been developed some 2D versions. The following are analogous to KP and ZK respectively,

(u_{t} − H^{(x)}u_{xx} + uu_{x})_{x} + σu_{yy} = 0, σ = ±1,  (6)
u_{t} − D_{x,y}u_{x} + uu_{x} = 0.  (7)

Here H^{(x)} means the Hilbert transform with respect to the variable x and Ds_{x,y} = (−Δ)^{s/2}, s ∈ ℝ denotes the (bidimensional) homogeneous derivative of order s and we have simply written D_{x,y} for D1_{x,y} . We will refer to Eqns.(6)-(7) as KP-BO and ZK-BO, respectively. (This notation is certainly not standard, but we find it appropriate in this context).

The KP-BO equation governs the two dimensional evolution of long internal waves in fluids of great depth. It was obtained in [6] where the stability of the soliton solutions of BO, with respect to transverse perturbations was studied. Similarity solutions for this equation and also for BO were studied in [43] and [49].

In [53] a 2D-ILW version of KP type was obtained. It describes the interactions between solitary waves in stratified fluids. In the shallow water and deep fluid limits, this equation reduces to KP and KP-BO, respectively.

The ZK-BO equation describes the dynamics of 3-D, slightly nonlinear disturbances in boundary-layer shear flow. In its derivation (see [45]) it is not assumed (as it was for KP and KP-BO) that the scale of the disturbance is much smaller along than across the flow. Soliton solutions for this equation were studied from a numerical standpoint in [7].

We will also be interested in the equation:

u_{t} − H^{(x)}u_{xy} + uu_{y} = 0.  (8)

It can be obtained from

(u_{t} − H^{(x)}u_{xx} + uu_{x})_{x} = (−H^{(x)}u_{xx} + uu_{x})_{y},  (9)

by assuming enough regularity and decay at infinity and performing the change of variables (x, y) ↦ (x + y, −y). We will refer to Eqn.(8) as 2D-BO.

In the present work we intend to examine some properties of the following generalizations of 2D-BO and ZK-BO,

u_{t} − H^{(x)}u_{xy} + u^{p}u_{y} = 0,  (g2D-BO)  (10)
u_{t} − D_{x,y}u_{x} + u^{p}u_{x} = 0,  (gZK-BO)  (11)

with p ∈ ℕ. They are natural bidimensional versions of the generalized BO (see [21], [22], [32]),

u_{t} − Hu_{xx} + u^{p}u_{x} = 0, p ∈ ℕ (gBO).  (12)

Observe that both g2D-BO and gZK-BO posess a Hamiltonian structure, that is, they can be written in the form:

u_{t} = J H′(u),  (13)

where J is a skew symmetric operator and H′ denotes the Gateaux derivative of the Hamiltonian functional H. This structure is related with the existence of conserved quantities that play, in many cases, a fundamental role in the derivation of global estimates for the solutions. In particular, it is not difficult to see that H(u) is always a conserved quantity (see [36] for a discussion of KdV as an infinite dimensional Hamiltonian system). As a consequence we obtain that

H_{1}(u) = ∫ (\frac{1}{2}uH^{(x)}u_{x} − \frac{1}{(p + 1)(p + 2)}u^{p+2}) dxdy

and

H_{2}(u) = ∫ (\frac{1}{2}uD_{x,y}u − \frac{1}{(p + 1)(p + 2)}u^{p+2}) dxdy,

are conserved under the flux associated to g2D-BO and gZK-BO, respectively. Moreover, at least formally, ∫ u dxdy and ∫ u^{2}dxdy are also conserved quantities for both equations.

This work is organized as follows. In the next Section we present the basic notations and results that we will need. In Chapter 1 we obtain the local well-posedness for g2D-BO and gZK-BO in H^{s}, s > 2 using the method of parabolic regularization. In Chapter 2, we consider weighted Sobolev spaces of the type F_{r}^{s}(ℝ^{2}) = H^{s}(ℝ^{2}) ∩ L^{2}((1 + x^{2} + y^{2})^{r}dxdy), we prove local well-posedness for g2D-BO in F_{r}^{s}(ℝ^{2}), s > 2, r ∈ [0, 1] and that for r = 2, persistence fails to occur for odd values of p. As a consequence we obtain unique continuation principles. We also show local well-posedness in F_{1,k}^{k}(ℝ^{2}) = H^{k}(ℝ^{2}) ∩ L^{2}((1 + x^{2} + y^{2k})dxdy), k ≥ 3. In Chapter 3, we prove global existence of solutions in H^{s}(ℝ^{2}), s > 3 for initial data with small L_{1}^{1}(ℝ^{2}) ∩ H^{s}(ℝ^{2}) norm and nonlinear scattering for such solutions of g2D-BO. We show in Chapter 4, nonexistence of square integrable solitary waves for 2D-BO. A similar conclusion follows for both g2D-BO and gZK-BO with p = 2 by variational arguments. We also use a result established by Weinstein in [52] to obtain existence and further properties of solitary waves for ZK-BO. In the last Chapter, we present some comments on ill-posedness and on interesting questions about g2D-BO and gZK-BO that remain open.

0.2 Notation and preliminaries

Let Δ_{n} = ∑_{j=1}^{n} \frac{∂^{2}}{∂x_{j}^{2}} denote the n-dimensional Laplacian and S′(ℝ^{n}) the

space of tempered distributions in ℝ^{n}, dual to the Schwartz space S(ℝ^{n}). Define the (real) Lebesgue spaces by:

L_{s}^{p}(ℝ^{n}) = J^{−s}L^{p}(ℝ^{n}) = {f ∈ S′(ℝ^{n}) : Imf = 0, J^{s}f ∈ L^{p}(ℝ^{n})},

where s ∈ ℝ, p ∈ [1, +∞] and J^{s} = (1 − Δ_{n})^{s/2} is the Bessel potential of order −s. The norm of these spaces will be denoted by ∥ · ∥_{L_{s}^{p}} = | · |_{s,p}. For p = 2 we will write L_{s}^{2}(ℝ^{n}) = H^{s}(ℝ^{n}) (the usual L^{2}-type Sobolev spaces), ∥ · ∥_{H^{s}} = ∥ · ∥_{s} and (·, ·)_{H^{s}} = (·, ·)_{s}. In the case s = 0 the corresponding norm will be denoted as ∥ · ∥ and the inner product (·, ·). Observe that the equality L_{0}^{p}(ℝ^{n}) = L^{p}(ℝ^{n}) is trivial. The notation ∥ · ∥_{L^{p}} = | · |_{p} will also be used. For these spaces, we have the following theorem.

Theorem 0.2.1 (Embedding Theorem) Let s, r ∈ ℝ, k ∈ ℕ, p ∈ [1, +∞). We have:

(i) if s ≥ r + n(\frac{1}{p} − \frac{1}{q}), with 1 < p ≤ q < ∞, then L_{s}^{p}(ℝ^{n}) ↪ L_{r}^{q}(ℝ^{n}); (ii) if k = \frac{n}{p}, then L_{k}^{p}(ℝ^{n}) ↪ L^{q}(ℝ^{n}), for q ∈ [p, +∞); (iii) if k > \frac{n}{p}, then L_{k}^{p}(ℝ^{n}) ↪ L^{∞}(ℝ^{n}).

Proof: The items (ii) − (iii) are assertions of the Sobolev Embedding Theorem, since for k ∈ ℕ, we have L_{k}^{p}(ℝ^{n}) = W^{k,p}(ℝ^{n}, ℝ), where W^{k,p}(ℝ^{n}, ℝ) = {f ∈ L^{p}(ℝ^{n}, ℝ) : D^{α}f ∈L^{p}(ℝ^{n}, ℝ), |α| ≤ k}. It is proved for example in [13]. The item (i) is proved in ([11], Chapter 6). □

Proposition 0.2.2 For any k ∈ ℕ, we have

L_{k}^{p}(ℝ^{n}) = {f ∈ L^{p}(ℝ^{n}, ℝ) : ∂_{x_{j}}^{k}f ∈L^{p}(ℝ^{n}, ℝ), 1 ≤ j ≤ N}.

Lemma 0.2.3 Let f ∈ S(ℝ^{n}), p_{0}, p_{1} ∈ (1, +∞) and s_{0}, s_{1} ∈ ℝ. Then:

|J^{s}f|_{p} ≤ c|J^{s_{0}}f|_{p_{0}}^{θ}|J^{s_{1}}f|_{p_{1}}^{1−θ}  (1)

where

s = θs_{0} + (1 − θ)s_{1},
\frac{1}{p} = \frac{θ}{p_{0}} + \frac{1 − θ}{p_{1}} .

Lemma 0.2.4 If s > 0 and p ∈ (1, +∞), then

|[J^{s}, M_{f}]g|_{p} ≤ c(|∇f|_{∞}|J^{s−1}g|_{p} + |J^{s}f|_{p}|g|_{∞})  (2)

where M_{f} denotes the operator of multiplication by f, and f, g ∈ S(ℝ^{n}).

Lemma 0.2.5 With the assumptions of Lemma 0.2.4, L_{s}^{p} ∩ L^{∞} is an algebra. Moreover,

|fg|_{s,p} ≤ c(|f|_{∞}|g|_{s,p} + |f|_{s,p}|g|_{∞})  (3)

Proposition 0.2.2 and Lemma 0.2.3 are proved in ([11], Chapter 6). The proofs of Lemmas 0.2.4 and 0.2.5 can be found in [30].

The following Sobolev type inequality will be useful. It is obtained in ([44], Lemma 6):

Lemma 0.2.6 For any f ∈ H^{n/2+ϵ}(ℝ^{n}), ϵ ∈ (0, 1/2] we have:

|f|_{∞} ≤ c(n)ϵ^{−1/2}∥f∥_{n/2+ϵ}.  (4)

Recall that the Hilbert Transform, defined by

Hf(x) = p.v.\frac{1}{π} ∫ \frac{f(y)}{x − y} dy = (−isgn(.)f̂(·))ˇ (x),

is a unitary operator on L^{2}(ℝ), such that H^{2} = −I and that H∂_{x} = ∂_{x}H is a positive operator on L^{2}(ℝ) with domain H^{1}(ℝ). Here the symbols ˆ , ˇ denote the Fourier transform and its inverse, respectively. For the sake of clarity, since we deal with two spatial variables, we will denote H^{(x)} the Hilbert transform with respect to the variable x.

Theorem 0.2.7 (Calderón’s Commutator Theorem) Let A : ℝ → ℝ be Lipschitz, so that A′ ∈ L^{∞}(ℝ). Then for any f ∈ S(ℝ) we have:

∥[H, A]f′∥ ≤ C|A′|_{∞}∥f∥.  (5)

Proof: See [17] or [18]. □

Following Triebel [50] we introduce:

Definition 0.2.8 A positive function ρ ∈ C^{∞}(ℝ^{2}; ℝ) will be called an admissible weight if the following conditions hold:

a) ρ(x, y) ≥ 1 , ∀(x, y) ∈ ℝ^{2};

b) |ρ(x, y)| → ∞, as |(x, y)| → +∞; c)|∇ρ(x, y)| ≤ cρ^{2}(x, y), ∀(x, y) ∈ ℝ^{2}.

It is easy to see that the functions ρ defined by ρ(x, y) = (1 + x^{2} + y^{2})^{r/ν} and ρ(x, y) = (1 + x^{2r_{1}} + y^{2r_{2}})^{1/ν}, with ν > 0, r > 0, r_{j} > 0, j = 1, 2, are admissible weights.

In what follows ρ will always denote an admissible weight and we will write H^{s} for H^{s}(ℝ^{2}). Consider:

H^{s}(ρ^{ν}) = H^{s} ∩ L^{2}(ρ^{ν}), s ≥ 0, ν > 0

where L^{2}(ρ^{ν}) denotes the space of all real valued measurable functions such that:

∥f∥_{L^{2}(ρ^{ν})} = (∫ f^{2}(x, y)ρ^{ν}(x, y)dxdy)^{1/2} < ∞.

They are Hilbert spaces when provided with the inner product:

(f, g)_{H^{s}(ρ^{ν})} = (f, g)_{s} + (f, g)_{L^{2}(ρ^{ν})}

and they possess the following interpolation property:

Theorem 0.2.9 Let s ≥ 0, ν ≥ 2s, 0 ≤ j ≤ s, j ∈ ℤ and χ_{j} = ν(1 − \frac{j}{s}). Then there exists a positive number c such that for all f ∈ H^{s}(ρ^{ν})

∑_{α∈(ℤ^{+})^{2},|α|=j} ∫ ρ^{χ_{j}}(x, y)|D^{α}f(x, y)|^{2}dxdy ≤ c∥f∥_{H^{s}(ρ^{ν})}^{2}.

Proof: See [50], Chapter 3. □

In case ρ^{ν}(x, y) = (1+x^{2} +y^{2})^{r}, we will write H^{s}(ρ^{ν}) = F_{r}^{s}, ∥·∥_{H^{s}(ρ^{ν})} = ∥ · ∥_{s,r}. Similarly for ρ^{ν}(x, y) = (1 + x^{2r_{1}} + y^{2r_{2}}) we set H^{s}(ρ^{ν}) = F_{r_{1},r_{2}}^{s}. The following propositions are easy consequences of the preceding theorem.

Proposition 0.2.10 Let x = (x, y), α, β ∈ (ℤ^{+})^{2}, |α| + |β| ≤ k, k ∈ ℕ and f ∈ F_{k}^{k}. Then

∥x^{β}D^{α}f∥ ≤ c(|α|)∥f∥_{k,k}.

Proposition 0.2.11 Let α ∈ (ℤ^{+})^{2}, β ∈ ℤ^{+}, |α| + β ≤ k, k ∈ ℕ and f ∈ F_{1,k}^{k}. Then

∥y^{β}D^{α}f∥ ≤ c(|α|)∥f∥_{F_{1,k}^{k}}.

For these spaces we also have:

Proposition 0.2.12 Let r_{j} > 0, j = 1, 2, s > 1, then F_{r_{1},r_{2}}^{s} is a Banach algebra.

Proof: Observe that by the Sobolev’s Embedding Theorem:

∥fg∥_{F_{r_{1},r_{2}}^{s}} ≤ c_{s}∥f∥_{s}∥g∥_{F_{r_{1},r_{2}}^{s}} .

for all f, g ∈ F_{r_{1},r_{2}}^{s}. Then the result follows since ∥f∥_{s} ≤ ∥f∥_{F_{r_{1},r_{2}}^{s}}. □

For X, Y Banach spaces, we will denote B(X, Y) and B(X × X, Y) the space of bounded linear operators from X to Y and the space of bounded bilinear maps from X to Y, respectively. When X = Y we will simply write B(X) = B(X, X) and Bil(X) = B(X×X, X). Let I be an interval of the real line, C(I) will represent the set of (real) valued continuous functions on I, with the uniform topology. We will set C(I; X) = {f : I ↦ X, continuous}. This is a Banach space provided with the L^{∞} norm:

∥f∥_{∞} = sup_{t∈I} ∥f(t)∥_{X}.

For X a Hilbert space, we will also use the topological space defined by

C_{w}(I; X) = {f : I ↦ X : (f(·), φ)_{X} continuous, ∀φ ∈ X},

where (·, ·)_{X} represents the inner product in X. A sequence (f_{n}) converges in C_{w}(I; X) to an element f if and only if (f_{n}(·), φ)_{X} → (f(·), φ)_{X}, ∀φ ∈ X, uniformly in I.

For any α ∈ L^{∞}(ℝ^{n}), the operador defined in S(ℝ^{n}) by

L̂f = αf̂,  (6)

can be extended to a bounded operator in L^{2}(ℝ^{n}). The same holds in L^{p}(ℝ^{n}), p ∈ (1, ∞), under suitable conditions for α.

Theorem 0.2.13 (Mihlin-Hörmander multiplier theorem) Let α ∈ C^{k}(ℝ^{n} \ \{0\}), with k = [\frac{n}{2}] + 1 and assume there exists c > 0 such that

|ξ|^{|β|}|D^{β}α(ξ)| ≤ c, |β| ≤ k, ξ ∈ ℝ^{n} \ {0}.

Then L can be extended to a bounded operator in L^{p}, p ∈ (1, +∞).

Proof: See [11]. □

We recall the notion of an operator to be positivity preserving (see [42]).

Definition 0.2.14 Let < M, μ > be a σ-finite measure space. f ∈ L^{2}(M, dμ) is called positive if f(x) ≥ 0 x-a.e. in M. A ∈ B(L^{2}(M, dμ)) is positivity preserving if Af is positive whenever f is positive.

Now we consider the operator L defined in (6) as an operator in L^{2}(ℝ^{2}) so that D(L) = {f ∈ L^{2}(ℝ^{2}) : αf̂ ∈ L^{2}(ℝ^{2})}. Suppose that there exist real positive constants γ, m and M such that for all (ξ, η) ∈ ℝ^{2},

m(ξ^{2} + η^{2})^{γ} ≤ α(ξ, η) ≤ M(1 + ξ^{2} + η^{2})^{γ}.  (7)

Then L is a positive self-adjoint operator in L^{2}(ℝ^{2}) with D(L) ⊃ H^{2γ}. We have,

Theorem 0.2.15 If γ > \frac{p}{p+2} and p ≥ 0, then there exists a nonvanishing ϕ ∈ H^{γ} that satisfies,

cϕ + Lϕ − \frac{1}{p + 1}ϕ^{p+1} = 0.  (8)

If in addition e^{−tL} is positivity preserving and possesses a convolution kernel that is a decreasing function of |(x, y)|, then ϕ can be taken to be a nonnegative decreasing function of |(x, y)|.

Proof: This is a shortened version of Theorem 7.1 of [52], that we have reformulated only for n = 2. The solutions of (8) are obtained there by the method of concentration-compactness as minimizers of a constrained minimization problem in H^{γ}. The rest of the theorem is proved using some properties of decreasing symmetric rearrangements and the first Beurling-Deny criterion for the characterization of positivity preserving semigroups generated by positive self-adjoint operators (see [42]). □

We will explain now the notion of well-posedness that will be used. Let X and Y be Banach spaces such that X ↪ Y and suppose that f ∈ C(X, Y). Consider the initial value problem (IVP):

\frac{du}{dt} = f(u)  (9)
u(0) = ϕ.  (10)

Definition 0.2.16 We will say that (9)-(10) is locally well-posed in X if for any ϕ ∈ X there exists T > 0 such that the following conditions hold,

i) there exists a unique u ∈ C([0, T]; X) ∩ C^{1}([0, T]; Y) that satisfies (9)- (10); ii) u depends continuously on ϕ in the sense that if ϕ_{n} → ϕ in X, then for n large enough u_{n} ∈ C([0, T]; X) and u_{n} → u in C([0, T]; X). If T can be taken arbitrarily large, then the IVP is called globally well-posed.

Observe that i) expresses the persistence of u(t) in the space X. This is a rather strong hypothesis and in Chapter 3 we will see an example where it does not hold. On the other hand, ii) says that the local flow defined by (9)-(10) is continuous. This hypothesis can be strengthened and in fact, in Chapter 5 there are examples where we have ill-posedness when ii) is replaced by suitable alternative hypotheses requiring more smoothness of the flow.

In what follows different constants may be denoted by the same letter when their precise values are of no relevance to our arguments. When necessary, dependence on other quantities will be indicated.

Chapter 1

Local Theory in Sobolev Spaces

A natural first question is about well-posedness. We adopt the notion of well-posedness of a Cauchy problem as in [29], that is: existence, uniqueness, persistence in the sense that the solution describes a curve in the space which contains the initial data and continuous dependence on the initial data.

In order to investigate the local well-posedness in H^{s} of the IVP associated to g2D-BO, we will use the method of parabolic regularization as in [23] and [25]. For any μ ≥ 0, p ∈ ℕ we consider the equation:

u_{t} − H^{(x)}u_{xy} + u^{p}u_{y} = μΔu.  (1.1)

Using the properties of the solutions to these equations with μ > 0 we will be able to obtain the solution for μ = 0.

The linear problem associated to Eqn.(1.1) is:

u_{t} − H^{(x)}u_{xy} = μΔu  (1.2)
u(0) = ϕ  (1.3)

where ϕ ∈ H^{s}, s ∈ ℝ. Its (unique) solution u^{μ}(t) = E_{μ}(t)ϕ, is given by:

E_{μ}(t)ϕ = (e^{t(i|ξ|η−μ(ξ^{2}+η^{2}))}ϕ̂)ˇ = e^{t(H^{(x)}∂_{xy}^{2}+μΔ)}ϕ.  (1.4)

It is convenient to define:

G_{μ}(t, ξ, η) = e^{t(i|ξ|η−μ(ξ^{2}+η^{2}))},  (1.5)

then (E_{μ}(t)ϕ)ˆ = G_{μ}(t, ξ, η)ϕ̂.

The following lemma provides a very useful inequality.

Lemma 1.0.17 For any a > 0, b > 0 and r ≥ 0, we have:

h (r) = r^{a} exp (−br^{2}) ≤ (\frac{\sqrt{2}}{2}\sqrt{\frac{a}{b}})^{a} exp (−\frac{a}{2}) .  (1.6)

Proof: We have,

\frac{d}{dr} h (r) = r^{a−1}ae^{−br^{2}} − 2r^{a+1}be^{−br^{2}}.  (1.7)

Set

r_{max} = \frac{1}{2}\sqrt{2}\sqrt{(\frac{a}{b})}.

Then it is easy to see that \frac{d}{dr} h(r_{max}) = 0 and that h attains an absolute maximum at r_{max}. Therefore

h (r) ≤ r_{max}^{a} exp (−br_{max}^{2}) ,  (1.8)

and the result follows. □

Some interesting properties of E_{μ} can now be derived.

Lemma 1.0.18 Let s ∈ ℝ, μ ≥ 0. Then:

(a) E_{μ} defines a contraction semigroup in H^{s} that can be extended, when μ = 0 to a strongly continuous unitary group.

(b) Let λ ≥ 0, μ > 0, then for all t > 0:

E_{μ}(t) ∈ B(H^{s}, H^{s+λ})  (1.9)

and satisfies the estimate:

∥E_{μ}(t)f∥_{s+λ} ≤ K_{λ}(1 + (2μt)^{−λ})^{1/2}∥f∥_{s},  (1.10)

for any f ∈ H^{s}.

Proof: The result is completely analogous to Theorem 2.1 of [23]. Lemma 1.0.17 with a = 2λ, r = (ξ^{2} + η^{2})^{1/2} and b = 2μt gives

(ξ^{2} + η^{2})^{λ}e^{−2μt(ξ^{2}+η^{2})} ≤ \frac{λ^{λ}e^{−λ}}{(2μt)^{λ}} .  (1.11)

The proof is based on the above inequality. □

Note that (b) expresses a regularizing property of the semigroup E_{μ}(t), μ > 0. Now we state our result on local well-posedness in H^{s}.

Theorem 1.0.19 Given an initial condition ϕ ∈ H^{s}, s > 2, there exists T̃_{s}^{μ} = T̃(μ, ∥ϕ∥_{s}) such that the IVP for Eqn.(1.1) with μ ≥ 0 has a unique solution u^{μ} in the class:

C([0, T̃_{s}^{μ}]; H^{s}) ∩ C^{1}([0, T̃_{s}^{μ}]; H^{s−2})

that satisfies

∥u^{μ}(t)∥_{s} ≤ ∥ϕ∥_{s} exp {c ∫_{0}^{t}|u^{μ}(τ)|_{∞}^{p−1}|∇u^{μ}(τ)|_{∞}dτ}, ∀t ∈ [0, T̃_{s}^{μ}]. (1.12)

We present below some results that will lead to the proof of this theorem.

Lemma 1.0.20 Let ϕ ∈ H^{s}, s > 2, μ > 0, then there exists T̃_{s}^{μ} > 0 such that

Af(t) = E_{μ}(t)ϕ − ∫_{0}^{t} E_{μ}(t − τ)f^{p}f_{y}(τ)dτ  (1.13)

is a contraction in the complete metric space

X_{s}(T̃_{s}^{μ}) = {f ∈ C([0, T̃_{s}^{μ}]; H^{s}) : sup_{t∈[0,T̃_{s}^{μ}]} ∥f(t) − E_{μ}(t)ϕ∥_{s} ≤ ∥ϕ∥_{s}}. (1.14)

Its fixed point is a solution of Eqn.(1.1) with ϕ as initial value.

Proof: Using (1.10) one gets,

∥Af(t) − E_{μ}(t)ϕ∥_{s} ≤ \frac{1}{p + 1} ∫_{0}^{t} ∥E_{μ}(t − τ)f^{p+1}(τ)∥_{s+1}dτ
≤ c ∫_{0}^{t}(1 + (2μ(t − τ))^{−1})^{1/2}∥f(τ)∥_{s}^{p+1}dτ
≤ c∥ϕ∥_{s}^{p+1} ∫_{0}^{t}(1 + (2μ(t − τ))^{−1})^{1/2}dτ,

Then Af(t) ∈ X_{s}(T̃_{s}^{μ}) for T̃_{s}^{μ} small enough. A similar computation implies that A is a contraction (choosing a smaller T̃_{s}^{μ} if necessary). The last assertion of the theorem follows from the fact that the integral equation:

f(t) = E_{μ}(t)ϕ − ∫_{0}^{t} E_{μ}(t − τ)f^{p}(τ)f_{y}(τ)dτ,  (1.15)

is equivalent to the IVP associated to Eqn.(1.1) with initial datum ϕ ∈ H^{s}. □ Lemma 1.0.21 Let u, v ∈ C([0, T]; H^{s}), s > 2 be two solutions of Eqn.(1.1), with the same initial condition. Then u = v.

Proof: By the hypotheses of the lemma, \frac{d}{dt}u and \frac{d}{dt}v are in L^{2}. Substracting the equations satisfied by u and v and taking the inner product in L^{2} with u − v we obtain,

\frac{1}{2} \frac{d}{dt}∥u − v∥^{2} = (H^{(x)}∂_{xy}^{2}(u − v), u − v) − (u^{p}u_{y} − v^{p}v_{y}, u − v)
= −(u^{p}(u − v)_{y} + (u^{p} − v^{p})v_{y}, u − v)
≤ c_{s,p}(M)∥u − v∥^{2},

where max{∥u(t)∥_{s}, ∥v(t)∥_{s}} ≤ M, ∀t ∈ [0, T]. In the second line we used the skew adjointness of H^{(x)}∂_{xy}^{2} in L^{2}. The result follows now from Gronwall’s inequality. □

Lemma 1.0.22 Let u^{μ} ∈ C([0, T_{s}^{μ}]; H^{s}), s > 2, μ > 0 be the solution of Eqn.(1.1) with initial condition ϕ. Then u^{μ} ∈ C((0, T_{s}^{μ}]; H^{∞}).

Proof: Let k > 0, ϵ ∈ (0, 1) and t ∈ (0, T_{s}^{μ}]. By virtue of (1.9), E_{μ}(t)ϕ ∈ H^{s+k}. For the nonlinear term:

∥∫_{0}^{t}E_{μ}(t − τ)(u^{μ})^{p}(τ)u_{y}^{μ}(τ)dτ∥_{s+ϵ} ≤ \frac{1}{p + 1} ∫_{0}^{t}∥E_{μ}(t − τ)(u^{μ})^{p+1}(τ)∥_{s+1+ϵ}dτ

≤K_{ϵ,p} sup_{τ∈[0,t]} ∥u^{μ}(τ)∥_{s}^{p+1} ∫_{0}^{t}(1 + (2μτ)^{−(ϵ+1)})^{1/2}dτ.

The last improper integral converges and then u(t) ∈ H^{s+ϵ}, ∀t ∈ (0, T_{s}^{μ}], since u solves Eqn.(1.15). Repeating the same procedure as many times as necessary, replacing ϕ by u(t′), for any t′ ∈ (0, T_{s}^{μ}], one gets that u(t) ∈ H^{s+nϵ}, ∀n ∈ ℕ, ∀t ∈ (0, T_{s}^{μ}]. Since there exists n ∈ ℕ such that s + nϵ > k, the result follows. □

Proposition 1.0.23 For the solutions u^{μ}, μ > 0 of Eqn.(1.1) with initial value ϕ ∈ H^{s}, s > 2 the inequality (1.12) holds. Moreover, there exist T_{s} > 0 and a positive function ρ ∈ C([0, T_{s})) independent of μ such that u^{μ}, μ > 0 are defined in [0, T_{s}) (possibly extended) and ∥u^{μ}(t)∥_{s} ≤ ρ^{1/2}(t), ∀t ∈ [0, T_{s}).

Proof: Fix a μ > 0 and t ∈ (0, T_{s}^{μ}). By the preceding lemma, one has that \frac{d}{dt}u^{μ}(t) and u^{μ}(t) belong to H^{∞}. Since J^{s} ∈ B(H^{r}, H^{r−s}), ∀r, s ∈ ℝ then J^{s}u^{μ}(t) also belongs to H^{∞}. Then one may apply the operator J^{s} to Eqn.(1.1) and multiply in L^{2} by J^{s}u^{μ}, obtaining:

\frac{1}{2} \frac{d}{dt}∥J^{s}u^{μ}∥^{2} + (J^{s}H^{(x)}u_{xy}^{μ}, J^{s}u^{μ}) + (J^{s}(u^{μ})^{p}u_{y}^{μ}, J^{s}u^{μ}) = μ(J^{s}Δu^{μ}, J^{s}u^{μ}),

(1.16)

where the functions are evaluated at t. Integrating by parts in the last term and using the commutativity of J^{s} and the derivative operators we can rewrite (1.16) as:

\frac{1}{2} \frac{d}{dt}∥J^{s}u^{μ}∥^{2} + (J^{s}H^{(x)}u_{xy}^{μ}, J^{s}u^{μ}) + ((u^{μ})^{p}J^{s}u_{y}^{μ}, J^{s}u^{μ}) +
+([J^{s}, (u^{μ})^{p}]u_{y}^{μ}, J^{s}u^{μ}) = −μ∥J^{s}∇u^{μ}∥^{2}.

Notice that using the Lemmas 0.2.4 and 0.2.5 and the skew symmetry of J^{s}H^{(x)}∂_{xy}^{2} one has:

(J^{s}H^{(x)}u_{xy}^{μ}, J^{s}u^{μ}) = −(J^{s}H^{(x)}u_{xy}^{μ}, J^{s}u^{μ}) = 0,
−((u^{μ})^{p}J^{s}u_{y}^{μ}, J^{s}u^{μ}) = −\frac{1}{2} ∫ (u^{μ})^{p}(J^{s}u^{μ})_{y}^{2} = \frac{p}{2} ∫ (u^{μ})^{p−1}u_{y}^{μ}(J^{s}u^{μ})^{2}
≤ \frac{p}{2}|u^{μ}|_{∞}^{p−1}|u_{y}^{μ}|_{∞}∥u^{μ}∥_{s}^{2},
|([J^{s}, (u^{μ})^{p}]u_{y}^{μ}, J^{s}u^{μ})| ≤ c(|∇(u^{μ})^{p}|_{∞}∥J^{s−1}u_{y}^{μ}∥ + ∥J^{s}(u^{μ})^{p}∥|u_{y}^{μ}|_{∞})∥u^{μ}∥_{s}
≤ c|u^{μ}|_{∞}^{p−1}|∇u^{μ}|_{∞}∥u^{μ}∥_{s}^{2}.

Thus, we have obtained:

\frac{d}{dt}∥u^{μ}(t)∥_{s}^{2} ≤ c|u^{μ}|_{∞}^{p−1}|∇u^{μ}|_{∞}∥u^{μ}∥_{s}^{2}, t ∈ (0, T_{s}^{μ}).  (1.17)

This and Gronwall’s inequality yield (1.12). Moreover, by the Sobolev’s Embedding Theorem one obtains:

\frac{d}{dt}∥u^{μ}(t)∥_{s}^{2} ≤ c_{s}∥u^{μ}(t)∥_{s}^{p+2}, t ∈ (0, T_{s}^{μ}).  (1.18)

Let now ρ(t), t ∈ [0, T_{s}), be the maximally extended solution of

\frac{dρ}{dt} = C_{s}ρ^{\frac{p+2}{2}}

ρ(0) = ∥ϕ∥_{s}^{2} and fix a value T ∈ (0, T_{s}). Thus ∥u^{μ}(t)∥_{s}^{2} ≤ ρ(t), t ∈ [0, min{T_{s}^{μ}, T }] so all the solutions can be extended to the interval [0, T] and consequently, to all [0, T_{s}). □

Remark 1.0.24 One can obtain the explicit form of ρ:

ρ(t) = \frac{2^{2/p}∥ϕ∥_{s}^{2}}{(2 − C_{s}pt∥ϕ∥_{s}^{p})^{2/p}}  (1.19)

and of T_{s}:

T_{s} = 2(C_{s}p∥ϕ∥_{s}^{p})^{−1}.  (1.20)

For any T ∈ (0, T_{s}), denote

M_{s}(T) = \frac{2^{1/p}∥ϕ∥_{s}}{(2 − C_{s}pT ∥ϕ∥_{s}^{p})^{1/p}} .

As a consequence of Theorem 1.0.23 we obtain:

∥u^{μ}(t)∥_{s} ≤ M_{s}(T), ∀μ > 0, ∀t ∈ [0, T].  (1.21)

The following proposition will be also very useful.

Proposition 1.0.25 Let s > 2, μ_{i} > 0, ϕ_{i} ∈ H^{s}, i ∈ {1, 2}. Denote as u_{1}, u_{2} ∈ C([0, T]; H^{s}) the corresponding solutions of Eqn.(1.1) with μ = μ_{1}, μ = μ_{2} respectively. Let K be such that max{∥u_{1}(t)∥_{s}, ∥u_{2}(t)∥_{s}} ≤ K, ∀t ∈ [0, T]. Then there exists c = c_{s,p}(K, T) such that

∥u_{1}(t) − u_{2}(t)∥^{2} ≤ c(∥ϕ_{1} − ϕ_{2}∥^{2} + |μ_{1} − μ_{2}|), ∀t ∈ [0, T]  (1.22)

Proof: Substracting the differential equations for u_{1} and u_{2} and taking the inner product in L^{2} by u_{1} − u_{2} one gets, for t ∈ (0, T]:

\frac{d}{dt}∥u_{1} − u_{2}∥^{2} = \frac{2}{p + 1}((u_{1})^{p+1} − (u_{2})^{p+1}, (u_{1} − u_{2})_{y}) +
+2μ_{1}(Δ(u_{1} − u_{2}), u_{1} − u_{2}) + 2(μ_{1} − μ_{2})(Δu_{1}, u_{1} − u_{2})
= \frac{1}{(p + 1)}(q_{p+1}(u_{1}, u_{2}), (u_{1} − u_{2})_{y}^{2}) − 2μ_{1}∥∇(u_{1} − u_{2})∥^{2} +
+2(μ_{1} − μ_{2})(Δu_{1}, u_{1} − u_{2})
≤ c(p)K^{p}∥u_{1} − u_{2}∥^{2} + 2|μ_{1} − μ_{2}|K^{2},

where

q_{p+1}(a, b) = a^{p} + a^{p−1}b + ⋯ + ab^{p−1} + b^{p}.  (1.23)

Then the result follows by Gronwall’s inequality. □

Proof of Theorem 1.0.19. For μ > 0, the results follow by virtue of the Lemmas 1.0.20 and 1.0.21 and Proposition 1.0.23.

Suppose now that μ = 0 and let us fix T ∈ (0, T_{s}). By the Proposition 1.0.25 we obtain

∥u^{μ}(t) − u^{ν}(t)∥ ≤ c|μ − ν|^{1/2},

for any t ∈ [0, T]. Then {u^{μ}}_{μ>0} is a Cauchy sequence in C([0, T]; L^{2}). Let us denote u = lim_{μ→0} u^{μ}. This limit also holds in C_{w}([0, T]; H^{s}). In fact, let φ ∈ S, ψ ∈ H^{s} and t ∈ [0, T]. We have,

|(u^{μ}(t) − u^{ν}(t), ψ)_{s}| ≤ |(u^{μ}(t) − u^{ν}(t), ψ − φ)_{s}| + |(u^{μ}(t) − u^{ν}(t), φ)_{s}|
≤ 2M_{s}(T)∥ψ − φ∥_{s} + ∥u^{μ}(t) − u^{ν}(t)∥∥φ∥_{2s}.

So, using the density of S in H^{s} one gets u ∈ C_{w}([0, T]; H^{s}) and the assertion follows.

Now, Lemma 0.2.3 and (1.21) imply that for any r ∈ [0, s), u^{μ} → u in C([0, T]; H^{r}). Then for such values of r, Δu^{μ} → Δu and H^{(x)}u_{xy}^{μ} → H^{(x)}u_{xy} in C_{w}([0, T]; H^{s−2})∩C([0, T]; H^{r−2}) and (u^{μ})^{p}u_{y}^{μ} → u^{p}u_{y} in C_{w}([0, T]; H^{s−1})∩ C([0, T]; H^{r−1}). Taking the limit in H^{r−2} in the equation satisfied by u^{μ},

u^{μ}(t) = ϕ + ∫_{0}^{t}(μΔu^{μ} − H^{(x)}u_{xy}^{μ} − (u^{μ})^{p}u_{y}^{μ})(τ)dτ,  (1.24)

we obtain

u(t) = ϕ + ∫_{0}^{t}(−H^{(x)}u_{xy} − u^{p}u_{y})(τ)dτ.  (1.25)

That is, u is a solution for g2D-BO in H^{r}, r ∈ (2, s). Lemma 1.0.21 implies that there is no solution different from u. To prove that u ∈ C([0, T]; H^{s}), one first shows that it is right continuous at zero. Indeed,

∥ϕ∥_{s} ≤ lim inf_{t↘0} ∥u(t)∥_{s} ≤ lim sup_{t↘0} ∥u(t)∥_{s} ≤ lim sup_{t↘0} ρ^{1/2}(t) = ∥ϕ∥_{s}

⟹

lim_{t↘0} ∥u(t)∥_{s} = ∥ϕ∥_{s} ⇒ lim_{t↘0} u(t) = ϕ in H^{s},

since we know that u(t) ⇀ ϕ in H^{s} as t ↘ 0. The left continuity in T and the continuity at the other points can be proved using the uniqueness of the solutions of Eqn.(10) in C([0, t]; H^{r}), r > 2 and the invariance of this set of solutions by the transformations (t, x, y) ↦ (τ − t, x, −y) and (t, x, y) ↦ (t − τ, x, y), for any τ ∈ ℝ. The theorem follows then also for μ = 0 by taking u^{0} = u, T̃_{s}^{0} = T. The inequality (1.12) can be obtained using the weak lower semicontinuity of the norm and the fact that ∇u^{μ} → ∇u^{0} and u^{μ} → u^{0} in C([0, T]; L^{∞}) by the Sobolev’s Embedding Theorem. □

Remark 1.0.26 When μ = 0 we can solve the equation for negative times. It follows that the interval of existence can be taken to be symmetric with respect to the origin. To see this observe that the solutions of Eqn.(10) are invariant under the transformation (t, x, y) ↦ (−t, x, −y).

Remark 1.0.27 Regarding global existence, observe that since for any μ ≥ 0, T̃_{s}^{μ} can be chosen independently of ϕ in a bounded set, Theorem 1.0.19 holds in any interval [0, T], T ∈ (0, T_{s}^{μ}), where [0, T_{s}^{μ}) is the maximal interval of existence of u^{μ} in H^{s} and T_{s}^{μ} = +∞ or ∥u^{μ}(t)∥_{s} → ∞ as t ↗ T_{s}^{μ}. Moreover, we can prove the following extension principle (see [2], [44]):

Proposition 1.0.28 Under the assumptions of Theorem 1.0.19, if μ > 0 then either T_{s}^{μ} = +∞ or ∥u^{μ}(t)∥_{2} → +∞ as t ↗ T_{s}^{μ}.

Proof: Fix μ > 0, and suppose that M_{1} = sup_{t∈(0,T_{s}^{μ})} ∥u^{μ}(t)∥_{2} < ∞ with T_{s}^{μ} < ∞. Combining Lemma 0.2.3 and (1.17) with Lemma 0.2.6 one obtains that for any ϵ ∈ (0, 1/2] such that 2 + ϵ < s and t ∈ (0, T_{s}^{μ}) we have,

\frac{d}{dt}∥u^{μ}(t)∥_{s}^{2} ≤ \frac{c}{ϵ^{1/2}} ∥u^{μ}∥_{2}^{p−1}∥u^{μ}∥_{2+ϵ}∥u^{μ}∥_{s}^{2}
≤ \frac{c}{ϵ^{1/2}} ∥u^{μ}∥_{2}^{p−θ}∥u^{μ}∥_{s}^{2+θ}
≤ \frac{c(1 + M_{1}^{p})}{ϵ^{1/2}}∥u^{μ}∥_{s}^{2+θ}

where θ = \frac{ϵ}{s−2}. Then ∥u^{μ}(t)∥_{s} ≤ ρ^{1/2}(t), ∀t ∈ [0, min\{T_{ϵ}, T_{s}^{μ}\}) where ρ solves:

\frac{d}{dt}ρ(t) = \frac{c(1 + M_{1}^{p})}{ϵ^{1/2}}ρ^{1+θ/2}
ρ(0) = ∥ϕ∥_{s}^{2}

and [0, T_{ϵ}) is the maximal interval of existence of ρ. It is easy to see that

T_{ϵ} = \frac{2(s − 2)}{c(1 + M_{1}^{p})∥ϕ∥_{s}^{θ}ϵ^{1/2}} .

Then one can clearly choose ϵ small enough so that T_{ϵ} > T_{s}^{μ} and an extension argument yields a contradiction to the fact that ∥u_{μ}(t)∥_{s} → ∞ as t ↗ T_{s}^{μ}. □ Corollary 1.0.29 The maximal interval of existence of u^{μ} does not depend on s.

Proof: For the case μ > 0 the result follows immediately from the proposition above. Suppose now that μ = 0, s′ > s > 2 and ϕ ∈ H^{s′}. The inequalities (1.17) and (1.21) imply that for any t ∈ [0, min{T̃_{s}^{0}, T̃_{s′}^{0}}] we have,

\frac{d}{dt}∥u^{0}(t)∥_{s′}^{2} ≤ c_{s,p}(T)∥u^{0}(t)∥_{s′}^{2}.

By Gronwall’s inequality, ∥u^{0}(t)∥_{s′} remains bounded as long as ∥u^{0}(t)∥_{s} does and then, by an extension argument, T_{s′}^{0} ≥ T_{s}^{0}. The result follows since the converse inequality is obvious. □

This corollary allows us to write T^{μ} instead of T_{s}^{μ}.

In order to prove the continuous dependence on the initial data, one may use the Bona-Smith approximation (see [14]). The following lemma (with slight modifications) appears in ([25], Chapter 6).

Lemma 1.0.30 Let ϕ ∈ H^{s}, s ∈ ℝ. For any τ > 0 define

ϕ^{τ} = (exp(−τ(1 + |.|^{2})^{s/2})ϕ̂(.))ˇ .

Then the following assertions hold: i)∥ϕ^{τ} − ϕ∥_{s} → 0, as τ ↘ 0; ii)∥ϕ^{τ} ∥_{s+k} ≤ (\frac{k}{sτ})^{k/s}e^{−k/s}∥ϕ∥_{s}, ∀k > 0; iii)∥ϕ^{τ} − ϕ^{θ}∥ ≤ |τ − θ|∥ϕ∥_{s}, ∀τ ≥ 0, ∀θ ≥ 0; iv)∥ϕ^{τ} ∥ ≤ ∥ϕ^{θ}∥ if and only if τ ≥ θ ≥ 0.

Proof: The lemma follows from the Parseval’s identity, Lebesgue’s Theorem of Dominated Convergence and Lemma 1.0.17. □

Theorem 1.0.31 Let ϕ_{n} → ϕ in H^{s}, s > 2 and let u_{n}^{μ}, u^{μ} be the corresponding solutions of Eqn.(1.1). Then for any T ∈ (0, T_{s}), there exists N ∈ ℕ such that for n ≥ N, u_{n}^{μ} ∈ C([0, T]; H^{s}) and sup_{t∈[0,T]} ∥u_{n}^{μ}(t)−u^{μ}(t)∥_{s} → 0 as n → +∞ uniformly for μ ∈ [0, +∞).

Proof: Let us denote ϕ_{n}^{ϵ} = (ϕ_{n})^{ϵ}, u_{n}^{μ,ϵ}(t) = (u_{n}^{μ}(t))^{ϵ}, t ∈ [0, T_{s,n}), u^{μ,ϵ}(t) = (u^{μ}(t))^{ϵ}, t ∈ [0, T_{s}). It is easy to see, using Lemma 1.0.30, that ∥ϕ_{n}^{ϵ}−ϕ∥_{s} → 0 as ϵ → 0 and n → ∞, then by (1.20) there exist ϵ_{0} > 0 and N ∈ ℕ such that for ϵ ∈ (0, ϵ_{0}) and n ≥ N, u^{ϵ}, u_{n}^{ϵ} and u_{n} are defined in [0, T] for any T ∈ (0, T_{s}). For such values of ϵ and N, for t ∈ [0, T] and μ > 0,

∥u^{μ}(t)−u_{n}^{μ}(t)∥_{s} ≤ ∥u^{μ}(t)−u^{μ,ϵ}(t)∥_{s} +∥u^{μ,ϵ}(t)−u_{n}^{μ,ϵ}(t)∥_{s} +∥u_{n}^{μ,ϵ}(t)−u_{n}^{μ}(t)∥_{s}.

Substracting the differential equations for u^{μ} and u^{μ,ϵ}, taking the inner product in H^{s} and using the skew-adjointness of H^{(x)}∂_{xy}^{2} in H^{s} one gets, for t ∈ (0, T]:

\frac{d}{dt}∥u^{μ} − u^{μ,ϵ}∥_{s}^{2} ≤ −2((u^{μ})^{p}u_{y}^{μ} − (u^{μ,ϵ})^{p}u_{y}^{μ,ϵ}, u^{μ} − u^{μ,ϵ})_{s}
≤ −2((u^{μ})^{p}(u^{μ} − u^{μ,ϵ})_{y}, u^{μ} − u^{μ,ϵ})_{s} +
(((u^{μ})^{p} − (u^{μ,ϵ})^{p})u_{y}^{μ,ϵ}, u^{μ} − u^{μ,ϵ})_{s}.

By Lemma 0.2.4 we have

|((u^{μ})^{p}(u^{μ} − u^{μ,ϵ})_{y}, u^{μ} − u^{μ,ϵ})_{s}| ≤ |([J^{s}, (u^{μ})^{p}](u^{μ} − u^{μ,ϵ})_{y}, J^{s}(u^{μ} − u^{μ,ϵ}))| +
+|((u^{μ})^{p}J^{s}(u^{μ} − u^{μ,ϵ})_{y}, J^{s}(u^{μ} − u^{μ,ϵ}))|
≤ c_{s,p}M_{s}(T)^{p}∥u^{μ} − u^{μ,ϵ}∥_{s}^{2}

and

|(((u^{μ})^{p} − (u^{μ,ϵ})^{p})u_{y}^{μ}, u^{μ} − u^{μ,ϵ})_{s}| ≤ ∥((u^{μ})^{p} − (u^{μ,ϵ})^{p})u_{y}^{μ,ϵ}∥_{s}∥u^{μ} − u^{μ,ϵ}∥_{s}
≤ ∥q_{p}(u^{μ}, u^{μ,ϵ})∥_{s}∥(u^{μ} − u^{μ,ϵ})u_{y}^{μ,ϵ}∥_{s}∥u^{μ} − u^{μ,ϵ}∥_{s}
≤ c_{s,p}M_{s}(T)^{p−1}∥(u^{μ} − u^{μ,ϵ})u_{y}^{μ,ϵ}∥_{s}∥u^{μ} − u^{μ,ϵ}∥_{s}.

By Theorem 1.0.29, Lemma 1.0.30 and again Lemma 0.2.4

∥(u^{μ} − u^{μ,ϵ})u_{y}^{μ,ϵ}∥_{s} ≤ ∥[J^{s}, u^{μ} − u^{μ,ϵ}]u_{y}^{μ,ϵ}∥ + ∥(u^{μ} − u^{μ,ϵ})J^{s}u_{y}^{μ,ϵ}∥
≤ c_{s}(∥u^{μ} − u^{μ,ϵ}∥_{s_{0}}∥u^{μ,ϵ}∥_{s+1} + ∥u^{μ} − u^{μ,ϵ}∥_{s}∥u^{μ,ϵ}∥_{s})

for some s_{0} ∈ (1, s−1). Moreover, Lemmas 0.2.3 and 1.0.30 and Proposition 1.0.25 imply

∥u^{μ} − u^{μ,ϵ}∥_{s_{0}} ≤ c∥u^{μ} − u^{μ,ϵ}∥^{1−s_{0}/s}∥u^{μ} − u^{μ,ϵ}∥_{s}^{s_{0}/s}
≤ c_{s,s_{0},p}(T)∥ϕ − ϕ^{ϵ}∥^{1−s_{0}/s}
≤ c_{s,s_{0}}(T)ϵ^{1−s_{0}/s}

and

∥u^{μ,ϵ}∥_{s+1} ≤ c_{s,p}(T)∥ϕ^{ϵ}∥_{s+1} ≤ c_{s}(∥ϕ∥_{s,p}, T)ϵ^{−1/s}.  (1.26)

Then we get

\frac{d}{dt}∥u^{μ} − u^{μ,ϵ}∥_{s}^{2} ≤ c_{s,s_{0},p}(M, T, ∥ϕ∥_{s})(∥u^{μ} − u^{μ,ϵ}∥_{s}^{2} + ϵ^{1−(s_{0}+1)/s}). (1.27)

Setting γ = 1 − (s_{0} + 1)/s and using the Gronwall’s inequality, we obtain

∥u^{μ}(t) − u^{μ,ϵ}(t)∥_{s}^{2} ≤ c(∥ϕ − ϕ^{ϵ}∥_{s}^{2} + ϵ^{γ}).  (1.28)

Finally,

∥u^{μ}(t)−u^{μ,ϵ}(t)∥_{s}+∥u_{n}^{μ}(t)−u_{n}^{μ,ϵ}(t)∥_{s} ≤ c(∥ϕ−ϕ^{ϵ}∥_{s}+∥ϕ_{n}−ϕ_{n}^{ϵ}∥_{s}+ϵ^{γ/2}). (1.29)

On the other hand, by Lemma 0.2.3 and Proposition 1.0.25 we have,

∥u^{μ,ϵ} − u_{n}^{μ,ϵ}∥_{s}^{2} ≤ c_{s}∥u^{μ,ϵ} − u_{n}^{μ,ϵ}∥_{2s}∥u^{μ,ϵ} − u_{n}^{μ,ϵ}∥
≤ c_{s,p}(T)M_{2s}(T, ∥ϕ^{ϵ}∥_{2s})∥ϕ^{ϵ} − ϕ_{n}^{ϵ}∥
≤ c_{s,p}(T)∥ϕ − ϕ_{n}∥.  (1.30)

Thus, the theorem follows from (1.29) and (1.30). For μ = 0, one may use the weak lower semicontinuity of the norm. □

Remark 1.0.32 Theorem 1.0.31 establishes the continuous dependence in an interval that can be chosen independently of ϕ in a bounded set. Thus, it holds in any interval [0, T], T ∈ (0, T_{s}^{μ}).

Notice that to prove the results of this chapter, we did not use the exact form of the operator H^{(x)}∂_{xy}^{2}, but its skew-adjointness property in H^{s}. Moreover, these results could also have been obtained with a nonlinearity of the form u^{p}u_{x}. After these observations we can state:

Theorem 1.0.33 For any given initial condition ϕ ∈ H^{s}, s > 2, there exists T_{s} = T (∥ϕ∥_{s}) such that the IVP for gZK-BO has a unique solution u in the class:

C([−T_{s}, T_{s}]; H^{s}) ∩ C^{1}([−T_{s}, T_{s}]; H^{s−2})

and the map ϕ ↦ u is continuous from H^{s} to C([−T_{s}, T_{s}]; H^{s}). Moreover, the time of existence does not depend on s in the sense that if ϕ ∈ H^{s′} for some s′ > s, then T_{s} = T_{s′}.

Decay properties

In this chapter we will study the asymptotic behavior of the solutions of Eqn.(1.1), μ ≥ 0 as |(x, y)| → ∞. The procedure that we employ follows the ideas used in [23], [25] and [24] for studying these properties in the case of the BO equation.

2.0.1 Isotropic weights

We begin by studying the solutions of Eqn.(1.2) in the spaces F_{r}^{s}, s ≥ 0, r > 0, defined Section 2. Recall that in (1.5) we set, G_{μ}(t, ξ, η) = e^{t(i|ξ|η−μ(ξ^{2}+η^{2}))}. We will need:

Lemma 2.0.34 For the function G_{μ} we have:

∂_{ξ}G_{μ} = t(isgn(ξ)η − 2μξ)G_{μ}  (2.1)
∂_{ξ}^{2}G_{μ} = [t^{2}(−η^{2} + 4μ^{2}ξ^{2} − 4μi|ξ|η) − 2μt]G_{μ} −  (2.2)

+2itηδ(ξ)e^{−tμη^{2}}

∂_{ξ}^{(2k)}G_{μ} = G_{μ} ∑_{j=0}^{k} t^{2k−j}g_{2k,2(k−j)}(μ, ξ, ηsgn(ξ)) +  (2.3)

+e^{−tμη^{2}} ∑_{j=0}^{k−1} f_{2k,j}(t, μ, η)δ^{(2j)}(ξ)

∂_{ξ}^{(2k−1)}G_{μ} = G_{μ} ∑_{j=0}^{k−1} t^{2k−1−j}g_{2k−1,2(k−j)−1}(μ, ξ, ηsgn(ξ)) + (2.4)

+e^{−tμη^{2}} ∑_{j=1}^{k−1} f_{2k−1,j}(t, μ, η)δ^{(2j−1)}(ξ)

∂_{η}G_{μ} = −t(i|ξ| + 2μη)G_{μ}  (2.5)
∂_{η}^{2}G_{μ} = [t^{2}(−ξ^{2} + 4iμ|ξ|η + 4μ^{2}η^{2}) − 2μt]G_{μ}  (2.6)
∂_{η}^{k}G_{μ} = G_{μ} ∑_{j=0}^{[k/2]} t^{k−j}h_{k,k−2j}(μ, |ξ|, η)  (2.7)

(2.8)

where k ∈ ℕ, h_{j,m} and g_{j,m} are homogeneous polynomials of degree m in the last two variables for any j ∈ ℕ and f_{j,m}(t, μ, η) is a polynomial of degree m in t for μ and η fixed and also a polynomial of the same degree in η for μ and t fixed.

Proof: To get the result observe that G_{μ}(t, ξ, η) = g_{μ}(t, ξ, η)f(t, ξ, η), where f(t, ξ, η) = e^{it|ξ|η} and g_{μ}(t, ξ, η) = e^{−μt(ξ^{2}+η^{2})} belong to L^{∞} and C^{∞} respectively as functions of ξ and η.

The calculations for both variables are similar, and we will describe them only in the case of the derivatives with respect to ξ.

By an induction argument it is easy to show:

f_{ξ}^{(2k)} = (itη)^{2k}f + 2i ∑_{j=0}^{k−1}(−1)^{j}(tη)^{2k−2j−1}δ^{(2j)}

f_{ξ}^{(2k+1)} = (itηsgn(ξ))^{2k+1}f + 2i ∑_{j=1}^{k}(−1)^{j−1}(tη)^{2k−2j+1}δ^{(2j−1)}

for any k ∈ ℤ^{+}. On the other side, \frac{d^{k}}{dξ^{k}} g_{μ} = p_{k}(t, ξ, η)g_{μ}, where p_{k} is a polynomial of degree k in each of the variables t and ξ with its higher order term of the form (−2μtξ)^{k}. Then Leibnitz’s rule (with the derivatives of f taken in a generalized sense) leads to the result. □

To prove the global well-posedness of (1.2)-(1.3) in F_{r}^{s}, it suffices to show the persistence and boundedness of E_{μ}(t) in these spaces.

Theorem 2.0.35 (a) Let ϕ ∈ F_{1}^{s}, s ≥ 1, t ≥ 0 and μ ≥ 0, then E_{μ}(t) ∈ B(F_{1}^{s}) and it holds:

∥E_{μ}(t)ϕ∥_{s,1} ≤ p_{1}(μ, t)∥ϕ∥_{s,1},  (2.9)

where p_{1} is a polynomial of first degree in t with positive coefficients depending on μ. Moreover, E_{μ}(·)ϕ ∈ C([0, +∞); F_{1}^{s}).

(b) Let ϕ ∈ F_{k}^{s}, s ≥ k ≥ 2, k ∈ ℕ, t ≥ 0, μ ≥ 0 then E_{μ}(t)ϕ ∈ F_{k}^{s} if and only if

∫ x^{l−1}ϕ(x, y)dx = 0, ∀y ∈ ℝ, l = 1, .., k − 1.  (2.10)

This is a necessary and sufficient condition for the existence of solution to (1.2)-(1.3) in F_{k}^{s}. If that condition is satisfied, then E_{μ}(·)ϕ ∈ C([0, +∞); F_{k}^{s}) and an estimate similar to that in (2.9) holds.

Proof: For simplicity’s sake, we write u(t) = u^{μ}(t) = E_{μ}(t)ϕ.

(a) Using (2.1) we have that:

û_{ξ} = (−t(isgn(ξ)η + 2μξ)ϕ̂ + ϕ̂_{ξ})G_{μ},  (2.11)
û_{η} = (−t(i|ξ| + 2μη)ϕ̂ + ϕ̂_{η})G_{μ}.  (2.12)

So if ϕ ∈ F_{1}^{s}, s ≥ 1, then

∥xu∥ = ∥û_{ξ}∥ ≤ t(1 + 2μ)∥ϕ∥_{1} + ∥xϕ∥
∥yu∥ = ∥û_{η}∥ ≤ t(1 + 2μ)∥ϕ∥_{1} + ∥yϕ∥

∥u∥_{s,1} ≤ ∥u∥_{s} + ∥u∥_{0,1}
≤ ∥ϕ∥_{s} + ∥u∥ + ∥xu∥ + ∥yu∥
≤ 4(1 + t(1 + 2μ))∥ϕ∥_{s,1}.

(b) To fix ideas take k = 2, ϕ ∈ F_{2}^{s}, s ≥ 2. Now compute one more derivative:

û_{ξξ} = [ϕ̂(t^{2}(−η^{2} + 4μ^{2}ξ^{2} + 4iμ|ξ|η) − 2tμ) −  (2.13)
-2ϕ̂_{ξ}t(isgn(ξ)η − 2μξ) + ϕ̂_{ξξ}]G_{μ} − 2itδ(ξ)ηϕ̂(0, η)e^{−tμη^{2}}
û_{ηη} = [ϕ̂(t^{2}(4μ^{2}η^{2} − ξ^{2}) − 2tμ) − 2ϕ̂_{η}t(2μη + i|ξ|) + ϕ̂_{ηη}]G_{μ}. (2.14)

Then by Plancherel’s Theorem and Proposition 0.2.10:

∥y^{2}u∥ ≤ c_{2}(μ, t)∥ϕ∥_{2} + c_{1}(μ, t)∥ϕ∥_{2,2} + ∥y^{2}ϕ∥  (2.15)

and x^{2}u ∈ L^{2} if and only if ϕ̂(0, η) = 0, ∀η ≠ 0. Notice that since ϕ ∈ F_{2}^{s}, then ϕ ∈ L^{1}, ϕ̂ is a continuous function and that allows to extend the conclusion to η = 0. In such a case:

∥x^{2}u∥ ≤ d_{2}(μ, t)∥ϕ∥_{2} + d_{1}(μ, t)∥ϕ∥_{2,2} + ∥x^{2}ϕ∥  (2.16)

where c_{1}, d_{1} and c_{2}, d_{2} are polynomials of first and second degree in t, respectively, with positive coefficients depending on μ. Then elementary calculations lead to the boundedness of E_{μ}(t) in F_{2}^{s}.

For any η ∈ ℝ we can write:

ϕ̂(0, η) = \frac{1}{2π} ∫ ϕ(x, y)e^{−iηy}dxdy
= \frac{1}{\sqrt{2π}} ∫ (\frac{1}{\sqrt{2π}} ∫ ϕ(x, y)dx) e^{−iηy}dy
= F^{y} (\frac{1}{\sqrt{2π}} ∫ ϕ(x, ·)dx) (η)
= 0

where F^{y} denotes the Fourier transform with respect to the variable y. So we obtain that

ϕ̂(0, η) = 0, ∀η ∈ ℝ,

is equivalent to

∫ ϕ(x, y)dx = 0, ∀y ∈ ℝ.

For k > 2, one computes ∂_{ξ}^{k}û and ∂_{η}^{k}û with the help of Lemma 2.0.34. The relation (2.10) is equivalent to the vanishing of the coefficients multiplying the delta function and its derivatives in the expression for these quantities. □

The preceding theorem allows us to obtain a unique continuation principle.

Corollary 2.0.36 Let ϕ ∈ S(ℝ^{2}) and ϕ(·, y) ∈ C_{0}^{∞}(ℝ) for any y ∈ ℝ. Take μ ≥ 0 and suppose that there exist positive numbers R and t such that

supp u(t, ·, y) = supp (E_{μ}(t)ϕ)(·, y) ⊂ (−R, R), ∀y ∈ ℝ.

Then ϕ must vanish.

Proof: If the hypotheses hold, then x^{k}u(t) ∈ L^{2}, ∀k ∈ ℤ^{+}, thus ∂_{ξ}^{k}û ∈ L^{2} and then for the singular part in the expression of ∂_{ξ}^{k}û to vanish, the condition (2.10) must be satisfied for any k ∈ ℤ^{+}. But then denoting F^{x} the Fourier transform in the variable x, since for any y ∈ ℝ, x^{k}ϕ(x, y) ∈ L_{x}^{1}(ℝ):

∫ x^{k}ϕ(x, y)dx = F^{x}(x^{k}ϕ(x, y))(0) = i^{k} \frac{d^{k}}{dξ^{k}} F^{x}ϕ(·, y)(0).

Thus, the result follows because the support of ϕ(·, y) is compact and then F^{x}ϕ(·, y) is analytic for any y ∈ ℝ. □

The following result deals with the persistence and gain of regularity of the solutions of (1.2)-(1.3) in some weighted spaces.

Proposition 2.0.37 Let s ≥ 1, λ ≥ 0, γ ∈ [0, 1] and μ > 0. Then for any t > 0, E_{μ}(t) ∈ B(F_{γ}^{s}, F_{γ}^{s+λ}) and for ϕ ∈ F_{γ}^{s}, we have:

∥E_{μ}(t)ϕ∥_{s+λ,γ} ≤ K_{γ}(t; μ, λ)∥ϕ∥_{s,γ}.

Moreover E_{μ}(·)ϕ ∈ C((0, +∞), F_{γ}^{s+λ}).

Proof: By Lemma 1.0.18 it suffices to prove:

∥E_{μ}(t)ϕ∥_{0,γ} ≤ Q_{γ}(t; μ, λ)∥ϕ∥_{0,γ},  (2.17)

for any ϕ ∈ F_{γ}^{s}. This inequality holds for γ = 0, again by Lemma 1.0.18.

Combining the Fourier transform with the relations (2.11)-(2.12) and (1.11) one gets:

∥xu∥ ≤ t(1 + 2μ)∥\sqrt{ξ^{2} + η^{2}}e^{−μt(ξ^{2}+η^{2})}ϕ̂∥ + ∥ϕ̂_{ξ}∥
≤ c(2μ)^{−1/2}(1 + 2μ)t^{1/2}∥ϕ∥ + ∥xϕ∥

and analogously:

∥yu∥ ≤ c(2μ)^{−1/2}(1 + 2μ)t^{1/2}∥ϕ∥ + ∥yϕ∥.  (2.18)

and then (2.17) follows for γ = 1.

Since S ↪ F_{1}^{s} and S is dense in L^{2}((1 + x^{2} + y^{2})dxdy), then E_{μ}(t) can be extended to a bounded operator in L^{2}((1 + x^{2} + y^{2})dxdy) and the result follows for γ ∈ (0, 1) by the Theorem of Stein and Weiss of interpolation of L^{p} spaces with varying measures (see [11]). □

Regarding the nonlinear problem we have the following result:

Theorem 2.0.38 Let ϕ ∈ H^{s}(w^{2}), s > 2 and w be a weight with all its first and second derivatives bounded and such that

|w(x, y)| ≤ c_{λ}e^{λ(x^{2}+y^{2})}, ∀(x, y) ∈ ℝ^{2}, ∀λ ∈ (0, λ^{∗}),  (2.19)

for some λ^{∗} > 0, c_{λ} > 0. Then the solution u^{μ}, μ ≥ 0 of Eqn.(1.1) with initial condition ϕ belongs to C([0, T^{μ}); H^{s}(w^{2})).

For the proof we will need the following lemma:

Lemma 2.0.39 Let w be a weight as in the Theorem 2.0.38. Set w_{λ}(x, y) = w(x, y)e^{−λ(x^{2}+y^{2})}. Then there exist constants c_{j}, j = 1 … 4 independent of λ such that

|∇w_{λ}|_{∞} ≤ c_{1}|∇w|_{∞} + c_{2}  (2.20)
|D^{α}w_{λ}|_{∞} ≤ c_{3}|∇w|_{∞} + |D^{α}w|_{∞} + c_{4}  (2.21)

for any multiindex α = (α_{1}, α_{2}) such that |α| = 2.

Proof: It is easy to obtain that:

w_{λ,x}(x, y) = e^{−λ(x^{2}+y^{2})}(w_{x}(x, y) − 2λxw(x, y)).

Using Lemma 1.0.17 and

|w(x, y) − w(0, 0)| ≤ |∇w|_{∞}\sqrt{x^{2} + y^{2}}, ∀(x, y) ∈ ℝ^{2}

one gets:

|w_{λ,x}(x, y)| ≤ e^{−λ(x^{2}+y^{2})}(|w_{x}(x, y)| + 2λ|x|(w(0, 0) + |∇w|_{∞}\sqrt{x^{2} + y^{2}}))
≤ e^{−λ(x^{2}+y^{2})}(|w_{x}(x, y)| + 2λ|x|w(0, 0) + 2λ|x|\sqrt{x^{2} + y^{2}}|∇w|_{∞})
≤ |w_{x}|_{∞} + 2λw(0, 0)(2λ)^{−1/2}e^{−1/2} + 2λ(λ)^{−1}e^{−1}|∇w|_{∞}
≤ |w_{x}|_{∞} + \sqrt{2λ}e^{−1/2}w(0, 0) + 2e^{−1}|∇w|_{∞}

So (2.20) follows after taking c_{1} = 1 + 2e^{−1} and c_{2} = \sqrt{2λ^{∗}}e^{−1/2}w(0, 0). To prove (2.21) first notice that

w_{λ,xx}(x, y) = e^{−λ(x^{2}+y^{2})}(−4λxw_{x}(x, y) + w_{xx}(x, y) + 2λ(2λx^{2} − 1)w(x, y)),

then:

|w_{λ,xx}(x, y)| ≤ e^{−λ(x^{2}+y^{2})}(4λ|x||w_{x}(x, y)| + |w_{xx}(x, y) + 2λ|2λx^{2} − 1|w(x, y))
≤ e^{−λ(x^{2}+y^{2})}(4λ|x||w_{x}(x, y)| + |w_{xx}(x, y)| +
2λ|2λx^{2} − 1|(w(0, 0) + |∇w(x, y)|\sqrt{x^{2} + y^{2}})
≤ 4λ(2λ)^{−1/2}e^{−1/2}|w_{x}(x, y)| + |w_{xx}(x, y)| +
2λ(2e^{−1} + 1)w(0, 0) + \sqrt{2λ}(e^{−1/2} + (3e^{−1})^{3/2})|∇w|_{∞}
≤ \sqrt{2λ}(3e^{−1/2} + (3e^{−1})^{3/2})|∇w|_{∞} + |w_{xx}|_{∞} + 2λ(2e^{−1} + 1)w(0, 0).

Similar calculations can be carried out for w_{λ,xy} and w_{λ,yy}, so we obtain (2.21) with c_{3} = \sqrt{2λ^{∗}}(3e^{−1/2} + (3e^{−1})^{3/2}), c_{4} = 2λ^{∗}(2e^{−1} + 1)w(0, 0). □

Proof of the Theorem 2.0.38.

In view of Theorem 1.0.19 it suffices to prove that ∥wu(t)∥ remains bounded as long as t ∈ [0, T] for any T ∈ (0, T^{μ}).

Define w_{λ}(x, y) = w(x, y)e^{−λ(x^{2}+y^{2})} and denote u^{μ} by u. Take a T ∈ (0, T^{μ}). By the hypotheses, ∥w_{λ}u(t)∥ < ∞ and ∥w_{λ}u_{t}(t)∥ < ∞; ∀t ∈ (0, T] so we can multiply Eqn.(1.1) by w_{λ} and take the inner product in L^{2} with w_{λ}u to obtain:

\frac{d}{dt}∥w_{λ}u∥^{2} = 2(w_{λ}u, w_{λ}H^{(x)}u_{xy} − w_{λ}u^{p}u_{y} + μw_{λ}Δu).  (2.22)

Notice that we have:

(w_{λ}u, w_{λ}H^{(x)}u_{xy}) = (w_{λ}u, [w_{λ}, H^{(x)}∂_{xy}^{2}]u) + (w_{λ}u, H^{(x)}∂_{xy}^{2}w_{λ}u)
= (w_{λ}u, [w_{λ}, H^{(x)}]∂_{xy}^{2}u) + (w_{λ}u, H^{(x)}[w_{λ}, ∂_{xy}^{2}]u)
(w_{λ}u, w_{λ}Δu) = (w_{λ}u, [w_{λ}, Δ]u) + (w_{λ}u, Δ(w_{λ}u))
= (w_{λ}u, [w_{λ}, Δ]u) − ∥∇w_{λ}u∥^{2}
∥w_{λ}u^{p}u_{y}∥ ≤ |u^{p−1}u_{y}|_{∞}∥w_{λ}u∥ ≤ M_{s}^{p}∥w_{λ}u∥

where M_{s} = M_{s}(T) is as was defined in (1.21). Using the equalities:

[w_{λ}, ∂_{xy}^{2}]u = −(w_{λ,xy}u + w_{λ,x}u_{y} + w_{λ,y}u_{x}),  (2.23)
[w_{λ}, Δ]u = −(Δw_{λ}u + 2∇w_{λ}∇u),  (2.24)

Lemma 2.0.39 and Theorem (0.2.7) one obtains:

\frac{d}{dt}∥w_{λ}u∥^{2} ≤ 2∥w_{λ}u∥((c_{1}|∇w|_{∞} + c_{2})((C + 2)∥u_{y}∥ + 3∥u_{x}∥) +

(3c_{3}|∇w|_{∞}+ |w_{xx}|_{∞}+ |w_{xy}|_{∞}+ |w_{yy}|_{∞}+ 3c_{4})∥u∥ + M_{s}^{p}∥w_{λ}u∥).

From this we get that

\frac{d}{dt}∥w_{λ}u∥^{2} ≤ 2(C_{1}∥u∥_{s}∥w_{λ}u∥ + M_{s}^{p}∥w_{λ}u∥^{2})
≤ C_{1}^{2}∥u∥_{s}^{2} + (1 + M_{s}^{p})∥w_{λ}u∥^{2}
≤ (C_{1}M_{s})^{2} + (1 + M_{s}^{p})∥w_{λ}u∥^{2}

and by Gronwall’s inequality it follows that:

∥w_{λ}u∥^{2} ≤ e^{(1+M_{s}^{p})T} (∥w_{λ}ϕ∥^{2} + T (C_{1}M_{s})^{2}).

Applying now the Monotone Convergence Theorem yields:

∥wu∥^{2} ≤ e^{(1+M_{s}^{p})T} (∥wϕ∥^{2} + T (C_{1}M_{s})^{2}).  (2.25)

Thus, we have proved that u(t) ∈ H^{s}(w^{2}), ∀t ∈ (0, T^{μ}). The continuity can be derived from analogous estimates. □

From the above result about local existence one may derive the continuous dependence in weighted Sobolev spaces.

Corollary 2.0.40 Under the assumptions in the theorem above the continuous dependence of solutions of Eqn.(1.1) also holds in the spaces H^{s}(w^{2}).

Proof: For μ ≥ 0, denote u, v two solutions of Eqn.(1.1) with (possibly different) initial values ϕ, ψ in H^{s}(w^{2}), s > 2 and set z = v − u. By the theorem above, they remain in H^{s}(w^{2}) as long as they do in H^{s}. Let T > 0 be such that both solutions exist for t ∈ [0, T]. For those values of time one gets, after substracting the equations:

z_{t} − H^{(x)}z_{xy} + v^{p}z_{y} + (v^{p} − u^{p})u_{y} = μΔz  (2.26)

Proceeding as in the previous proof we get:

\frac{d}{dt}∥wz∥^{2} = 2(wz, wH^{(x)}z_{xy} − wv^{p}z_{y} − w(v^{p} − u^{p})u_{y} + μwΔz)
≤ 2c∥z∥_{s}∥wz∥ − (wz, wv^{p}z_{y} + wzq_{p}(v, u)u_{y}),  (2.27)

where the constant c depends only on w and on the constant in Theorem (0.2.7) and q is as was defined in (1.23).

(wz, wv^{p}z_{y}) = \frac{1}{2}(w^{2}v^{p}, (z)_{y}^{2}) = −(w_{y}v^{p}z, wz) − \frac{p}{2}(v^{p−1}v_{y}wz, wz)
≤ c′∥v∥_{s}^{p}(∥z∥ + ∥wz∥)∥wz∥ ≤ 2c′∥v∥_{s}^{p}∥wz∥^{2}
(wz, wzq_{p}(v, u)u_{y}) ≤ c′′|q_{p}(v, u)u_{y}|_{∞}∥wz∥^{2} ≤ c′′q_{p}(∥u∥_{s}, ∥v∥_{s})∥u∥_{s}∥wz∥^{2},

where all the constants depends on s and p. Plugging these estimates in (2.27) we obtain:

\frac{d}{dt}∥wz∥^{2} ≤ 2c∥z∥_{s}∥wz∥ + 4c′∥v∥_{s}^{p}∥wz∥^{2} + 2c′′q_{p}(∥u∥_{s}, ∥v∥_{s})∥u∥_{s}∥wz∥^{2}
≤ c∥z∥_{s}^{2} + c(∥v∥_{s}^{p} + q_{p}(∥u∥_{s}, ∥v∥_{s})∥u∥_{s})∥wz∥^{2}.  (2.28)

Suppose now that there exists a sequence {ϕ_{n}} ∈ H^{s}(w^{2}) such that ϕ_{n} → ϕ in H^{s}(w^{2}). Then ϕ_{n} → ϕ in H^{s} and since we have continuous dependence in H^{s}, for any T ∈ (0, T^{μ}), we can find N such that the solutions u_{n} associated to ϕ_{n} are defined in [0, T] for n ≥ N and sup_{t∈[0,T]} ∥u_{n}(t) − u(t)∥_{s} → 0 as n → +∞. It is easy to see that there exists a constant K > 0 such that sup_{t∈[0,T]} ∥u(t)∥_{s} ≤ K and sup_{t∈[0,T]} ∥u_{n}(t)∥_{s} ≤ K, ∀n ≥ N. Then for any ϵ > 0, (2.28) implies that for n large enough:

\frac{d}{dt}∥w(u_{n} − u)∥^{2} ≤ ϵ + 2cK^{p}∥w(u_{n} − u)∥^{2}.  (2.29)

By Gronwall’s inequality:

∥w(u_{n} − u)∥^{2} ≤ e^{2cK^{p}T} ∥w(ϕ_{n} − ϕ)∥^{2} + ϵT e^{2cK^{p}T}  (2.30)

and the result follows. □

Observe that the weights w_{γ}(x, y) = (1 + x^{2} + y^{2})^{γ/2}, γ ∈ [0, 1] satisfy the hypotheses of the preceding two theorems. Then we have local wellposedness for Eqn.(1.1) in the spaces F_{γ}^{s} for γ ∈ [0, 1], s > 2. This result is analogous to that obtained in [24] for BO. Nevertheless, ∇w_{γ} ∈ L^{∞} if and only if γ ≤ 1, then the mentioned theorems do not allow to consider stronger weights. More than that, there is no persistence if γ ≥ 2 and p is odd.

Theorem 2.0.41 Let s > 2 and p odd. Then if u ∈ C([0, T]; F_{2,0}^{s}) is the solution of Eqn.(1.1), with μ ≥ 0, corresponding to an initial condition ϕ ∈ F_{2,0}^{s}, u must vanish identically.

Proof: Suppose that u satisfies the hypotheses of the theorem. It is easy to see that its Fourier transform û must be continuous in the variable ξ.

After multiplying by x^{2} in Eqn.(1.1) we obtain

∂_{t}(x^{2}u) = x^{2}μΔu − x^{2}H^{(x)}u_{xy} − x^{2}u^{p}u_{y}  (2.31)

and since

∥x^{2}u^{p}u_{y}∥ ≤ |u^{p−1}u_{y}|_{∞}∥x^{2}u∥ ≤ c∥u∥_{s}^{p}∥x^{2}u∥,

then x^{2}u^{p}u_{y} ∈ L^{2}.

Similarly it can be proved that this term belongs to C([0, T]; L^{2}). Taking the Fourier transform in Eqn.(2.31) and integrating with respect to t we conclude that for any t ∈ (0, T], the function (ξ, η) ↦ δ(ξ)η ∫_{0}^{t} û(τ, ξ, η)dτ is measurable. Then we must have:

∫_{0}^{t} û(τ, 0, ·)dτ = 0, ∀t ∈ [0, T],

and this in turn implies û(t, 0, .) = 0, ∀t ∈ [0, T]. Using the boundedness of the Fourier transform one gets, for t ∈ [0, T], ξ ∈ ℝ and a.e. η ∈ ℝ:

û(t, ξ, η) = û(0, ξ, η) + ∫_{0}^{t}[H^{(x)}u_{xy} − u^{p}u_{y} + μΔu]ˆ (τ, ξ, η)dτ
= û(0, ξ, η) + ∫_{0}^{t}[i|ξ|ηû − i\frac{η}{p + 1}{ \atop u^{p+1}} − μ(ξ^{2} + η^{2})û](τ, ξ, η)dτ.

Evaluating at ξ = 0, we obtain

∫_{0}^{t} { \atop u^{p+1}}(τ, 0, η)dτ = 0, ∀t ∈ [0, T], η − a.e.

Then it is easy to see that ∫ u^{p+1}(t, x, y)dx = 0, ∀y ∈ ℝ, ∀t ∈ [0, T] and the result follows at once. □

Remark 2.0.42 For the values 1 < γ < 2 we do not know if the persistence holds in the spaces F_{γ}^{s}, s > 2.

We have a unique continuation’s result.

Corollary 2.0.43 Let u ∈ C([0, T]; H^{s}), s > 2, be a solution of Eqn.(1.1), μ ≥ 0 and p odd. If there exists R > 0 such that

supp u(t, ·, y) ⊂ (−R, R), ∀y ∈ ℝ

and for each t ∈ [t_{1}, t_{2}], t_{1} < t_{2}, t_{1}, t_{2} ∈ [0, T], then u must vanish identically in the interval [t_{1}, t_{2}].

Proof: For the proof it suffices to observe that if u satisfies the hypotheses of this corollary, then u(t) must belong to the space F_{2,0}^{s} for t ∈ [t_{1}, t_{2}]. Then apply the Theorem 2.0.41 using u(t_{1}) as initial data. □

2.0.2 Non isotropic weights

The difficulties with the persistence in F_{2}^{s} are closely connected with the appearence of the delta function δ(ξ). One could then use non isotropic weights to analyse the behavior of the solutions considering higher decay rates only in the variable y. Thus, in this section we will investigate the IVP’s associated to Eqns.(1.1) and (1.2) in the spaces F_{1,k}^{s}, k ∈ ℕ, s ≥ 0.

For the linear problem we obtain:

Theorem 2.0.44 Let k ∈ ℕ, s ≥ k, t ≥ 0, μ ≥ 0, and take ϕ ∈ F_{1,k}^{s}. Then E_{μ}(t) ∈ B(F_{1,k}^{s}) and satisfies the estimate:

∥E_{μ}(t)ϕ∥_{F_{1,k}^{s}} ≤ p_{k}(t; μ)∥ϕ∥_{F_{1,k}^{s}},  (2.32)

where p_{k}(t; μ) is a polynomial of k-th degree in t with positive coefficients depending only on μ. Moreover, E_{μ}(.)ϕ ∈ C([0, +∞); F_{1,k}^{s}) .

Proof: Using Proposition 0.2.11, Lemma 2.0.34 and Leibnitz’s rule one gets:

∥y^{k}E_{μ}(t)ϕ∥=∥∂_{η}^{(k)}(G_{μ}(t, ξ, η)ϕ̂)∥  (2.33)
≤∥ ∑_{l=0}^{k} c_{l}∂_{η}^{(l)}G_{μ}(t, ξ, η)∂_{η}^{(k−l)}ϕ̂∥  (2.34)

≤∑_{l=0}^{k} |c_{l}|∥ ∑_{j=0}^{[l/2]} t^{l−j}h_{l,l−2j}(μ, |ξ|, η)∂_{η}^{(k−l)}ϕ̂∥  (2.35)

≤∑_{l=0}^{k} c_{l,μ} ∑_{j=0}^{[l/2]} t^{l−j}∥(ξ^{l−2j} + η^{l−2j})∂_{η}^{(k−l)}ϕ̂∥  (2.36)

≤(∑_{l=0}^{k} d_{l,μ}(1 + t^{l}))∥ϕ∥_{F_{1,k}^{k}}.  (2.37)

The inequality (2.32) follows from the above expression. The continuity of E_{μ}(.)ϕ can be proved making similar calculations. □

The following lemma will be very useful:

Lemma 2.0.45 Let k ∈ ℕ, μ > 0, t > 0 and α multiindex. Then for any ϕ ∈ F_{0,k}^{0},

∥D^{α}E_{μ}(t)ϕ∥_{F_{0,k}^{0}} ≤ c(μ, |α|)(1 + t^{−|α|/2} + t^{(k−|α|)/2})∥ϕ∥_{F_{0,k}^{0}}.  (2.38)

Proof: Using Leibnitz’s rule we obtain:

∥y^{k}D^{α}E_{μ}(t)ϕ ∥ = ∥∂_{η}^{(k)}(ξ^{α_{1}}η^{α_{2}}G_{μ}(t, ξ, η)ϕ̂)∥  (2.39)
≤ ∑_{j=0}^{min {k,α_{2}}} c_{α_{2},j}∥ξ^{α_{1}}η^{α_{2}−j}∂_{η}^{(k−j)}(G_{μ}(t, ξ, η)ϕ̂)∥  (2.40)

≤ ∑_{j=0}^{min {k,α_{2}}} c_{α_{2},j}∥(ξ^{2} + η^{2})^{(|α|−j)/2}∂_{η}^{(k−j)}(G_{μ}(t, ξ, η)ϕ̂)∥. (2.41)

Set Z = ∥(ξ^{2}+η^{2})^{(|α|−j)/2}∂_{η}^{(k−j)}(G_{μ}(t, ξ, η)ϕ̂)∥. Then by Lemmas 1.0.17 and 2.0.34 we get:

Z ≤ ∥(ξ^{2} + η^{2})^{(|α|−j)/2}∑_{m=0}^{k−j} ∑_{l=0}^{[m/2]} c_{m,l,μ}t^{m−l}(ξ^{2} + η^{2})^{\frac{m−2l}{2}} |G_{μ}(t, ξ, η)∂_{η}^{(k−j−m)}ϕ̂|∥

≤ ∑_{m=0}^{k−j} c_{m,μ} ∑_{l=0}^{[m/2]} t^{m−l} sup_{(ξ,η)∈ℝ^{2}}\{(ξ^{2} + η^{2})^{\frac{|α|−j+m−2l}{2}} e^{−μt(ξ^{2}+η^{2})}\}∥∂_{η}^{(k−j−m)}ϕ̂∥

≤ ∑_{m=0}^{k−j} c(m, μ, |α|) ∑_{l=0}^{[m/2]} t^{\frac{m+j−|α|}{2}} ∥∂_{η}^{(k−j−m)}ϕ̂∥
≤ c(μ, |α|)(t^{−|α|/2} + t^{(k−|α|)/2})∥(1 + y^{2k})^{1/2}ϕ∥.

Taking this expression into (2.41), the result follows. □

Theorem 2.0.46 Let k ∈ ℕ, s ≥ 0, λ ≥ 0, μ > 0 and t > 0. Then for any ϕ ∈ F_{1,k}^{s}: a)E_{μ}(t) ∈ B(F_{1,k}^{s}, F_{1,k}^{s+λ}) and satisfies the estimate:

∥E_{μ}(t)ϕ∥_{F_{1,k}^{s+λ}} ≤ K(t; μ, λ)∥ϕ∥_{F_{1,k}^{s}}  (2.42)

where K is a continuous function such that K ∼ O(t^{−λ/2}) as t → 0^{+}. Moreover, E_{μ}(·)ϕ ∈ C((0, +∞); F_{1,k}^{s+λ}). b)For any multiindex α, D^{α}E_{μ}(t) ∈ B(F_{1,k}^{s}) and the following inequality holds:

∥D^{α}E_{μ}(t)ϕ∥_{F_{1,k}^{s}} ≤ Q(t; μ, |α|)∥ϕ∥_{F_{1,k}^{s}},  (2.43)

where Q is continuous, Q ∼ O(t^{−|α|/2}) as t → 0^{+}. It is also true that D^{α}E_{μ}(·)ϕ ∈ C((0, +∞); F_{1,k}^{s}).

Proof: The results are direct consequences of Lemmas 1.0.18 and 2.0.45 and Theorem 2.0.35. □

The next theorem shows the persistence of the solutions of Eqn.(1.1) in the spaces F_{1,k}^{k}.

Theorem 2.0.47 Let u^{μ} ∈ C([0, T^{μ}); H^{k}), k ∈ ℕ, k ≥ 3 be the maximal solution of Eqn.(1.1), μ ≥ 0 with initial data ϕ ∈ F_{1,k}^{k}. Then u^{μ} ∈ C([0, T^{μ}); F_{1,k}^{k}).

Proof: Take μ > 0 and write u^{μ} = u for simplicity. As in the proof of the Theorem 1.0.19, using Proposition 0.2.12 and Theorem 2.0.46 one obtains a local solution to Eqn.(1.1) in the complete metric space

H_{k}(T^{∗}) = {f ∈ C([0, T^{∗}]; F_{1,k}^{k}) : sup_{t∈[0,T^{∗}]} ∥f(t) − E_{μ}(t)ϕ∥_{F_{1,k}^{k}} ≤ ∥ϕ∥_{F_{1,k}^{k}}},

for some T^{∗} > 0. By the uniqueness in H^{k}, this solution must be u and by Theorem 2.0.38 it only remains to prove that it belongs to F_{0,k}^{0} for t ∈ [0, T^{μ}). It suffices to estimate ∥y^{k}u(t)∥, ∀t ∈ [0, T^{∗}].

Note that for t ∈ [0, T^{∗}], y^{k}u^{p}u_{y}(t) ∈ L^{2} since

∥y^{k}u^{p}u_{y}(t)∥ ≤ c(k)∥u(t)∥_{k}^{p}∥y^{k}u(t)∥ ≤ c(k)M_{s}^{p}∥y^{k}u(t)∥.

We claim that for |α| = 2, y^{k}D^{α}u(t) ∈ L^{2}, ∀t ∈ (0, T^{∗}]. Indeed, fix α = (0, 2), then using the integral equation (1.15) one obtains:

∥y^{k}u_{y}(t)∥ ≤ ∥y^{k}∂_{y}E_{μ}(t)ϕ∥ + ∫_{0}^{t} ∥y^{k}∂_{y}E_{μ}(t − τ)(u^{p}u_{y})(τ)∥dτ.

By Lemma 2.0.45 it follows that for any t ∈ (0, T^{∗}], y^{k}u_{y}(t) ∈ L^{2}. Since:

∥y^{k}∂_{y}(u^{p}u_{y})(t)∥ ≤ |(u^{p−1}u_{yy})(t)|_{∞}∥y^{k}u(t)∥ + p|(u^{p−1}u_{y})(t)|_{∞}∥y^{k}u_{y}(t)∥,

then one may use again the integral equation (1.15) and get:

∥y^{k}u_{yy}(t)∥ ≤ ∥y^{k}∂_{y}E_{μ}(t)u_{y}(ϵ)∥ + ∫_{ϵ}^{t} ∥y^{k}∂_{y}E_{μ}(t − τ)∂_{y}(u^{p}u_{y})(τ)∥dτ,

for any ϵ ∈ (0, T^{∗}). Using Lemma 2.0.45 again one obtains that y^{k}u_{yy}(t) ∈ L^{2}, ∀t ∈ (ϵ, T^{∗}]. Since ϵ is arbitrary, it holds also for t ∈ (0, T^{∗}]. For the other possible values of α, one can use the same reasoning, obtaining that y^{k}u_{t}(t) ∈ L^{2} ∀t ∈ (0, T^{∗}].

Using Proposition 0.2.11 and integration by parts, one obtains:

(y^{k}u_{xx}, y^{k}u) = −∥∂_{x}(y^{k}u)∥^{2} ≤ 0,
(y^{k}u_{yy}, y^{k}u) = ([y^{k}, ∂_{y}^{2}]u, y^{k}u) − ∥∂_{y}(y^{k}u)∥^{2},
≤ ([y^{k}, ∂_{y}^{2}]u, y^{k}u),
∥[y^{k}, ∂_{y}^{2}]u∥ ≤ k(k − 1)∥y^{k−2}u∥ + k∥y^{k−1}u_{y}∥ ≤ c_{1}(k)∥u∥_{F_{1,k}^{k}}.

Moreover,

|(H^{(x)}(y^{k}u_{xy}), y^{k}u)| = |(H^{(x)}(−ky^{k−1}u_{x} + ∂_{xy}(y^{k}u)), y^{k}u)|
= k|(H^{(x)}(y^{k−1}u_{x}), y^{k}u)|
≤ k∥y^{k−1}u_{x}∥∥y^{k}u∥
≤ c_{2}(k)∥u∥_{F_{1,k}^{k}}∥y^{k}u∥.

These relations yield:

∂_{t}∥y^{k}u∥^{2} ≤ 2(μc_{1}(k)∥u∥_{F_{1,k}^{k}}∥y^{k}u∥ + c_{2}(k)∥u∥_{F_{1,k}^{k}}∥y^{k}u∥ + c(k)M_{k}^{p}∥y^{k}u∥^{2})
≤ d_{1}(μ, k)∥u∥_{k}^{2} + d_{2}(k)∥y^{k}u∥^{2}

where d_{1} is increasing in μ. Gronwall’s inequality implies:

∥y^{k}u(t)∥^{2} ≤ e^{d_{2}(k)T} (∥y^{k}ϕ∥^{2} + T d_{1}(μ, k)M_{k}^{2}).  (2.44)

That allows us to extend the solution u to its interval of existence in H^{k}.

For the case μ = 0, observe that by (2.44) and the Proposition 1.0.23, there exists T > 0 such that

∥y^{k}u^{μ}(t)∥^{2} ≤ K(T, k, μ^{∗}, ∥ϕ∥_{F_{1,k}^{k}}),  (2.45)

for any μ ∈ (0, μ^{∗}), μ^{∗} > 0 arbitrary and for any t ∈ [0, T]. Then for any t ∈ [0, T], there exists a sequence μ_{n} such that u^{μ_{n}}(t) ⇀ v(t) in L^{2}((1 + y^{2k})dxdy), for some v(t) ∈ L^{2}((1+y^{2k})dxdy). But we know that u^{μ} → u^{0} in C([0,T];L^{2}) and then v = u^{0} since L^{2}((1 + y^{2k})dxdy) ↪ L^{2}. The inequality (2.44) follows also for μ = 0 by the weak lower semicontinuity of the norm of L^{2}((1 + y^{2k})dxdy). Finally, an extension argument yields the result. □

Long time behavior of small solutions and nonlinear data scattering

For several nonlinear equations it has been proved that for higher order nonlinearities and small initial data, the solutions behave as the solutions of the associated linear equation (see [30] and [47], for example). In this chapter we are going to discuss results of this nature for g2D-BO.

Consider Eqn.(10) and let us investigate for which values of p one can obtain global wellposedness for small initial data.

In what follows we will write E_{0}(t) = E(t). For any f ∈ S(ℝ^{2}):

E(t)f(x, y) = \frac{1}{2π} ∫ e^{i(|ξ|ηt+xξ+yη)}f̂(ξ, η)dξdη = \frac{1}{2π} I(t) ∗ f(x, y) (3.1)

where I(t)(x, y) = (e^{i|ξ|ηt})ˇ (x, y). We can compute the kernel I(t) explicitly:

Lemma 3.0.48 For any real numbers x, y, t:

I(t)(x, y) = \{ \begin{matrix}\frac{2}{t}cos(\frac{xy}{t})χ_{\{yt≤0\}} & if t ≠ 0 \\ 2πδ(x, y) & if t = 0 \end{matrix}

holds.

Proof: Set f(ξ, η) = e^{iξη}χ_{{ξ≥0}} and note that e^{i|ξ|η} = f(ξ, η) + f(−ξ, η). Straightforward computations show that f satisfies:

f_{ξ} = iηf + δ_{ξ} ⊗ 1_{η}  (3.2)
f_{η} = iξf  (3.3)

in S′. Using the properties of the Fourier transform, one sees that if g = f̌ then we must have:

g_{y} = −ixg − 1_{x} ⊗ δ_{y}  (3.4)
g_{x} = −iyg.  (3.5)

Now observe that if g_{1}, g_{2} ∈ D′ are two solutions of (3.4)-(3.5) then h = g_{1} − g_{2} satisfies:

h_{y} = −ixh
h_{x} = −iyh.

Thus, ∂_{x}(e^{ixy}h) = ∂_{y}(e^{ixy}h) = 0 and h(x, y) = ce^{−ixy} for some c ∈ ℝ. We then have that the solutions of (3.4)-(3.5) are of the form g = g_{0} + ce^{−ixy}, for c ∈ ℝ and g_{0} being any particular solution.

By the symmetry of (3.2)-(3.3) and (3.4)-(3.5) one verifies easily that g_{0}(x, y) = e^{−ixy}χ_{{y≤0}} is a solution of (3.4)-(3.5) and then for some c ∈ ℝ, f̌(x, y) = e^{−ixy}χ_{{y≤0}} + ce^{−ixy}. Then (e^{i|ξ|η})ˇ (x, y) = 2 cos(xy)(χ_{{y≤0}} + c) and

I(t)(x, y) = \frac{2}{t} cos (\frac{xy}{t} ) (χ_{\{yt≤0\}} + c) .

The constant c is determined by the fact that I(t) → 2πδ in S′ as t → 0. In fact, for any t > 0, A ⊂ ℝ and ϕ ∈ S we have:

\frac{2}{t} ∫_{A} dy ∫_{−∞}^{+∞} cos (\frac{xy}{t} )ϕ(x, y)dx = \frac{1}{t} ∫_{A} dy ∫_{−∞}^{+∞}(e^{− \frac{ixy}{t}} + e^{\frac{ixy}{t}})ϕ(x, y)dx
= \frac{\sqrt{2π}}{t} ∫_{A} (F^{x}ϕ (\frac{y}{t} , y) + F^{x}ϕ (−\frac{y}{t} , y)) dy
= \sqrt{2π} ∫_{t^{−1}A}(F^{x}ϕ(y, ty) + F^{x}ϕ(−y, ty))dy.

Then:

lim_{t↘0} < I(t), ϕ > = \sqrt{2π} ∫_{−∞}^{0}(F^{x}(y, 0) + F^{x}(−y, 0))dy +
c\sqrt{2π} ∫_{−∞}^{+∞}(F^{x}(y, 0) + F^{x}(−y, 0))dy
= 2πϕ(0, 0) + 4πcϕ(0, 0).

Then c = 0 and the result is proved. □

The expression just derived allows us to obtain the following L^{p} − L^{q} estimate for the semigroup E(t):

Proposition 3.0.49 For any f ∈ L^{1} ∩ L^{2} holds

|E(t)f|_{\frac{2}{1−θ}} ≤ c|t|^{−θ}|f|_{\frac{2}{1+θ}}  (3.6)

where θ ∈ [0, 1].

Proof: The Young’s inequality and Lemma 3.0.48 yield:

|E(t)f|_{∞} ≤ c|t|^{−1}|f|_{1}

and the result follows by interpolation since E(t) is a unitary operator in L^{2}. □

Proposition 3.0.50 For s > 1 and f ∈ L^{1} ∩ H^{s} holds:

|E(t)f|_{∞} ≤ c(1 + |t|)^{−1}(|f|_{1} + ∥f∥_{s})  (3.7)

Proof: By Proposition (3.0.49) and the Sobolev’s Embedding Theorem:

|t||E(t)f|_{∞} ≤ c|f|_{1}  (3.8)
|E(t)f|_{∞} ≤ c∥E(t)f∥_{s} = c∥f∥_{s}.  (3.9)

Adding this inequalities leads to the desired estimate. □

Now we present the main result of the section.

Theorem 3.0.51 For p ≥ 3 and s > 3 there exist δ > 0 and R = R(δ) > 0 such that if ϕ ∈ L_{1}^{1} ∩ H^{s} with

|ϕ|_{1,1} + ∥ϕ∥_{s} < δ,

then g2D-BO has a unique global strong solution u(t) in the class C_{b}(ℝ, H^{s}) with

sup_{t∈ℝ}(1 + |t|)|u(t)|_{1,∞} ≤ R.  (3.10)

For the proof we will need the following Lemma:

Lemma 3.0.52 Let

J(t) = (1 + t) ∫_{0}^{t} \frac{1}{(1 + t − τ)(1 + τ)^{p−1}} dτ.

Then we have:

i)J(t) = O(1) as t ∼ +∞ for p ≥ 3;

ii)J(t) → +∞ as t → +∞ for p = 1, 2.

Proof: Suppose that t ≠ −1. Making the change of variables τ′ = 1 + τ one gets:

\frac{J(t)}{1 + t} = ∫_{1}^{1+t} \frac{1}{(2 + t − τ′)τ^{′p−1}} dτ′

and expanding the integrand as:

\frac{1}{(2 + t − τ′)τ^{′p−1}} = \frac{(2 + t)^{1−p}}{τ′} + \frac{(2 + t)^{2−p}}{τ^{′2}} + ⋯ + \frac{(2 + t)^{−1}}{τ′^{p−1}} + \frac{(2 + t)^{−1}}{2 + t − τ′}

one can compute the integral explicitly obtaining:

\frac{J(t)}{1 + t} = 2(2 + t)^{1−p} ln(1 + t) + ∑_{j=2}^{p−1} \frac{1}{1 − j} (2 + t)^{j−p} (\frac{1}{(1 + t)^{j−1}} − 1) .

From this expression the result follows easily. □

Proof of the Theorem 3.0.51:

Observe that it suffices to prove that (3.10) holds for some R > 0. Indeed, in such a case and using the hypotheses we have:

∫_{0}^{t} |∇u|_{∞}|u|_{∞}^{p−1}dτ ≤ ∫_{0}^{t} |u|_{1,∞}^{p}dτ ≤ R^{p} ∫_{0}^{t}(1 + |τ|)^{−p}dτ ≤ const. (3.11)

Thus by virtue of (1.12) the solution can be extended to any finite interval.

To prove (3.10) take T ∈ (0, T^{0}). Let K(T) = sup_{t∈[0,T]}(1 + |t|)|u(t)|_{1,∞}. From the integral equation (1.15) we obtain:

|u(t)|_{1,∞} = |E(t)ϕ|_{1,∞} + ∫_{0}^{t} |E(t − τ)u^{p}(τ)u_{y}(τ)|_{1,∞}dτ.  (3.12)

Proposition 3.0.50 yields:

|E(t)ϕ|_{1,∞} ≤ c(1 + t)^{−1}(|ϕ|_{1,1} + ∥ϕ∥_{s}) ≤ cδ(1 + t)^{−1}  (3.13)

For the rest of the terms we get:

|E(t − τ)u^{p}u_{y}|_{∞} ≤ c(1 + t − τ)^{−1}(|u^{p}u_{y}|_{1} + ∥u^{p}u_{y}∥_{s−1})
|u^{p}u_{y}|_{1} ≤ |u|_{∞}^{p−1}|uu_{y}|_{1} ≤ |u|_{∞}^{p−1}∥u∥∥u_{y}∥ ≤ |u|_{∞}^{p−1}∥u∥_{s}^{2}
∥u^{p}u_{y}∥_{s−1} ≤ c∥u_{y}∥_{s−1}∥u^{p}∥_{s−1} ≤ c|u|_{∞}^{p−1}∥u∥_{s}^{2}

where we used the inequalities (3) and (3.7) and the Hölder inequality.

Similarly we obtain:

|E(t − τ)∇(u^{p}u_{y})|_{∞} ≤ c(1 + t − τ)^{−1}(|∇(u^{p}u_{y})|_{1} + ∥∇(u^{p}u_{y})∥_{s−2})
|∇(u^{p}u_{y})|_{1} = |pu^{p−1}∇uu_{y} + u^{p}∇u_{y}|_{1} ≤ c|u|_{∞}^{p−1}∥u∥_{s}^{2}
∥∇(u^{p}u_{y})∥_{s−2} ≤ ∥u^{p}u_{y}∥_{s−1} ≤ c∥u_{y}∥_{s−1}∥u^{p}∥_{s−1} ≤ c|u|_{∞}^{p−1}∥u∥_{s}^{2}.

Thus by Lemma 3.0.52 and the inequality (1.12):

|u(t)|_{1,∞} ≤ cδ(1 + t)^{−1} + c ∫_{0}^{t}(1 + t − τ)^{−1}|u(τ)|_{∞}^{p−1}∥u(τ)∥_{s}^{2}dτ
≤ cδ(1 + t)^{−1} + cδ^{2}K(T)^{p−1} ∫_{0}^{t}(1 + t − τ)^{−1}(1 + τ)^{−(p−1)} ×
× exp (c ∫_{0}^{τ} |u(t′)|_{∞}^{p−1}|∇u(t′)|_{∞}dt′)dτ
(1 + t)|u(t)|_{1,∞} ≤ cδ + cδ^{2}K(T)^{p−1}(1 + t) ∫_{0}^{t}(1 + t − τ)^{−1}(1 + τ)^{−(p−1)} ×
× exp (cK(t)^{p} ∫_{0}^{τ} (1 + t′)^{−p}dt′)dτ
≤ cδ + cδ^{2}K(T)^{p−1}e^{cK(T)^{p}}∫_{0}^{t} (1 + t − τ)^{−1}(1 + τ)^{−(p−1)}dτ
≤ cδ + cδ^{2}K(T)^{p−1}e^{cK(T)^{p}}

where t ∈ [0, T].

Consider the function f_{δ}(x) = cδ + cδ^{2}x^{p−1}e^{cx^{p}} − x. Since f_{δ}(0) = cδ and for any x̄ > 0, f_{δ}(x̄) → −x̄ as δ → 0, then there exists δ_{0} > 0 such that for δ ∈ (0, δ_{0}), f_{δ} has a positive zero. Let R = R(δ) be the first positive zero of this function. The estimates just obtained imply that f_{δ}(K(T)) ≥ 0 and then K(T) ≤ R, since f_{δ} changes its monotonicity only once in ℝ^{+}. By an extension argument, the theorem follows. □

The next natural step in studying the linear behavior of g2D-BO is the search for scattering states (see [47](Chapter 6), [41]). In this context a scattering state can be understood as a solution u(t) for which there exist associated elements ϕ_{±} such that u(t) ∼ E(t)ϕ_{±} as t → ±∞.

It turns out that the global solutions just found are also scattering states.

Theorem 3.0.53 With the assumptions of the above theorem there exist ϕ_{±} ∈ H^{s} such that

∥u(t) − E(t)ϕ_{±}∥_{r} → 0

when t → ±∞, for r ∈ [s − 1, s).

Proof: Assume that u is a scattering state and write:

u(t) = E(t − T)u(T) − ∫_{T}^{t} E(t − τ)u^{p}(τ)u_{y}(τ)dτ.  (3.14)

Applying E(−t) and taking (formally) the limit as T → ∞ one gets the Yang-Feldman equation (see [47]):

E(−t)u(t) = ϕ_{+} − ∫_{+∞}^{t} E(−τ)u^{p}(τ)u_{y}(τ)dτ  (3.15)
ϕ_{+} = E(−t)u(t) − ∫_{t}^{+∞} E(−τ)u^{p}(τ)u_{y}(τ)dτ  (3.16)
= ϕ − ∫_{0}^{+∞} E(−τ)u^{p}(τ)u_{y}(τ)dτ.  (3.17)

The last integral converges in H^{s−1}:

∥ ∫_{0}^{t} E(−τ)u^{p}(τ)u_{y}(τ)dτ∥_{s−1} ≤ ∫_{0}^{t} ∥u^{p}(τ)u_{y}(τ)∥_{s−1}dτ
≤ ∫_{0}^{t} ∥u(τ)∥_{s}∥u^{p}(τ)∥_{s−1}dτ
≤ ∫_{0}^{t} ∥u(τ)∥_{s}^{2}|u(τ)|_{∞}^{p−1}dτ
≤ c ∫_{0}^{t}(1 + τ)^{−(p−1)}dτ ≤ c.

So, we can take (3.17) as a definition for ϕ_{+}. Analogously as above one can show that as t → +∞:

∥ ∫_{t}^{+∞} E(−τ)u^{p}(τ)u_{y}(τ)dτ∥_{s−1} → 0.

Then by (3.15), ∥E(−t)u(t) − ϕ_{+}∥_{s−1} → 0 as t → +∞ and since ∥u(t)∥_{s} is bounded, E(−t)u(t) converges weakly in H^{s} to ϕ_{+}, so in particular ϕ_{+} ∈ H^{s}. The result follows from Lemma 0.2.3. □

Solitary waves

The solitary waves, or waves of permanent form, are known to arise in several nonlinear models as a consequence of the combined effects of the linear and the nonlinear parts.

If we look for travelling wave-type solutions of Eqn.(10) in the form:

u(x, y, t) = v(x − bt, y − ct) b ≥ 0, c ≥ 0 b^{2} + c^{2} > 0,

then the function v must obey

−bv_{x} − cv_{y} − H^{(x)}v_{xy} + v^{p}v_{y} = 0.  (4.1)

Taking the Fourier transform we obtain:

(bξ + cη + |ξ|η) v̂ = \frac{η}{p + 1}v̂^{p+1}.  (4.2)

Suppose now that v ∈ H^{s} ∩ L^{1}, s > 2. Then v̂ must be continuous and bounded, but the function

f (ξ, η) = \frac{η}{bξ + cη + |ξ|η}

is not bounded in zero, so we must have v̂^{p+1}(0, 0) = 0. Thus, any non zero solution in H^{s}, s > 2 of Eqn.(4.1) with p odd, cannot be integrable.

Take b = 0. The equation to analyse in this case is the following:

−cv_{y} − H^{(x)}v_{xy} + v^{p}v_{y} = 0  (4.3)

Assuming that v and Hv_{x} vanish at infinity, we get:

−cv − H^{(x)}v_{x} + \frac{1}{p + 1}v^{p+1} = 0  (4.4)

For any ϕ ∈ H^{1/2} consider the functional:

L(ϕ) = c ∫ ϕ^{2}dxdy + ∫ ϕH^{(x)}ϕ_{x}dxdy − \frac{2}{(p + 1)(p + 2)} ∫ ϕ^{p+2}dxdy. (4.5)

By Theorem 0.2.1-(i), we have

H^{1/2} ↪ L^{p+2}, ∀p ∈ [0, 2],  (4.6)

so, one can guarantee that L is well defined for p ∈ {1, 2} (we are considering p ∈ ℕ). For such values of p, L is also Fréchet differentiable and the solutions of Eqn.(4.4) are the critical points of this functional. The following theorem holds.

Theorem 4.0.54 For any p ∈ {1, 2}, Eqn.(4.4) has no nonvanishing solutions in H^{1/2}.

Proof: Consider first p = 1. It is known (see [8]) that any nonvanishing solution for:

−f(x) − Hf_{x}(x) + f^{2}(x) = 0,

in H^{1/2} must be of the form:

f_{a}(x) = \frac{2}{1 + (x − a)^{2}} , a ∈ ℝ.

On the other hand, if v ∈ H^{1/2} is a solution of Eqn.(4.4), then v(·, y) ∈ H^{1/2}(ℝ) a.e. in y and then

v (x, y) = \frac{4c}{1 + c^{2} (x + g (y))^{2}}  (4.7)

a.e. in y, for some function g. Then the result follows since for any c > 0, the function defined by (4.7) cannot be square integrable.

Now suppose that v ∈ H^{1/2} is a solution for Eqn.(4.4) with p = 2. We shall use a technique presented in [46] for obtaining Pohozaev (see [40]) identities, to show that v must vanish.

Define the mapping

T_{λ}(ϕ)(x, y) = λ^{1/2}ϕ(λx, λy),  (4.8)

so that for any ϕ ∈ H^{r}, r ∈ ℝ, T_{·}(ϕ) ∈ C((0, +∞); H^{r})∩C^{1}((0, +∞); H^{r−1}). For r = 1/2 and λ > 0 it is straightforward to see that

L(T_{λ}(ϕ)) = λ^{−1}c ∫ ϕ^{2}dxdy + ∫ ϕH^{(x)}ϕ_{x}dxdy − \frac{1}{6} ∫ ϕ^{4}dxdy  (4.9)

and then

\frac{d}{dλ}|_{λ=1}L(T_{λ}(ϕ)) = −c ∫ ϕ^{2}dxdy.  (4.10)

Consider a sequence (v_{n})_{n∈ℕ} ⊂ S. For any n ∈ ℕ we have

L′(v_{n}) = 2cv_{n} + 2H^{(x)}v_{n,x} − \frac{2}{3}v_{n}^{3} ∈ L^{2}  (4.11)

and

\frac{d}{dλ}|_{λ=1}T_{λ}(v_{n}) = \frac{1}{2}v_{n} + v_{n,x} + v_{n,y} ∈ H^{1/2}.  (4.12)

Using this and the skew-adjointness of H^{(x)} in L^{2} and integrating by parts we obtain

\frac{d}{dλ}|_{λ=1}L(T_{λ}(v_{n})) = ⟨L′(v_{n}), \frac{d}{dλ}|_{λ=1}T_{λ}(v_{n})⟩_{− \frac{1}{2} , \frac{1}{2}}  (4.13)
= (cv_{n} + H^{(x)}v_{n,x} − \frac{1}{3}v_{n}^{3}, v_{n} + 2v_{n,x} + 2v_{n,y}) (4.14)
= (cv_{n} + H^{(x)}v_{n,x} − \frac{1}{3}v_{n}^{3}, v_{n}),  (4.15)
= ⟨cv_{n} + H^{(x)}v_{n,x} − \frac{1}{3}v_{n}^{3}, v_{n}⟩_{− \frac{1}{2} , \frac{1}{2}} ,  (4.16)

where < ·, · >_{− \frac{1}{2} , \frac{1}{2}} represents the operation of duality between H^{−1/2} and H^{1/2}.

Now, suppose that v_{n} → v in H^{1/2}. Then H^{(x)}v_{n,x} → H^{(x)}v_{x} in H^{−1/2}. By (4.6), H^{1/2} ↪ L^{4} and

∫ |v_{n}^{3} − v^{3}|^{4/3}dxdy = ∫ (|v_{n} − v| · |v_{n}^{2} + v_{n}v + v^{2}|)^{4/3} dxdy
≤ |v_{n} − v|_{4}^{4/3}|v_{n}^{2} + v_{n}v + v^{2}|_{2}^{4/3}
≤ c|v_{n} − v|_{4}^{4/3}(|v_{n}|_{4}^{2} + |v|_{4}^{2})^{4/3}
≤ c|v_{n} − v|_{4}^{4/3},

for n large. By duality we have L^{4/3} ↪ H^{−1/2} and then L′(v_{n}) → L′(v) = 0 in H^{−1/2}. By (4.10) and (4.13)-(4.16), ∫ v_{n}^{2}dxdy → 0 and then v = 0. □

The numerical results presented in [45] indicate the existence of a solitary wave for ZK-BO with the form u(t, x, y) = v(x − ct, y), c > 0. Substituting this expression in gZK-BO and assuming enough regularity and decay at infinity, we obtain that the function v must satisfy:

−cv − D_{x,y}v + \frac{1}{p + 1}v^{p+1} = 0.  (4.17)

That is, v must be a critical point for the functional

N (ϕ) = c ∫ ϕ^{2}dxdy + ∫ ϕD_{x,y}ϕdxdy − \frac{2}{(p + 1)(p + 2)} ∫ ϕ^{p+2}dxdy. (4.18)

Once more, (4.6) assures the finiteness and Fréchet differentiability of N in H^{1/2} for p ∈ {1, 2}. For p = 2, one obtains a non existence result as in Theorem 4.0.54.

Theorem 4.0.55 Eqn.(4.17) has no nonvanishing solutions in H^{1/2} for p = 2.

Proof: Let ϕ ∈ H^{1/2}. For λ > 0 set ϕ_{λ} = T_{λ}(ϕ), where T is as in (4.8). We have

D_{x,y}ϕ_{λ}(x, y) = ((ξ^{2} + η^{2})^{1/2}ϕ̂_{λ})ˇ (x, y)
= ((ξ^{2} + η^{2})^{1/2}λ^{−3/2}ϕ̂(λ^{−1}ξ, λ^{−1}η))ˇ (x, y)
= λ^{−1/2}(((λ^{−1}ξ)^{2} + (λ^{−1}η)^{2})^{1/2}ϕ̂(λ^{−1}ξ, λ^{−1}η))ˇ (x, y)
= λ^{3/2}D_{x,y}ϕ(λx, λy)

and

∫ ϕ_{λ}(x, y)D_{x,y}ϕ_{λ}(x, y)dxdy = λ^{2} ∫ ϕ(λx, λy)D_{x,y}ϕ(λx, λy)dxdy
= ∫ ϕ(x, y)D_{x,y}ϕ(x, y)dxdy.

Then \frac{d}{dλ}|_{λ=1}N (T_{λ}(ϕ)) = −c ∫ ϕ^{2}dxdy and the proof follows as that of Theorem 4.0.54 for p = 2. □

For p = 1, however, the existence of solutions for Eqn.(4.17) is guaranteed by Theorem 0.2.15. Indeed, using this theorem we are able to prove:

Theorem 4.0.56 There exists a nonvanishing solution of Eqn.(4.17) with p = 1 that lies in H^{∞}. Moreover, it can be chosen to be a nonnegative decreasing function of |(x, y)|.

Before proving this theorem, we present the following lemma.

Lemma 4.0.57 J^{1}(1 + D_{x,y})^{−1} ∈ B(L^{p}), p ∈ (1, +∞).

Proof: By virtue of Theorem 0.2.13, it suffices to show that for the function defined by

α(ξ, η) = \frac{\sqrt{1 + ξ^{2} + η^{2}}}{1 + \sqrt{ξ^{2} + η^{2}}} ,

there exists c > 0 such that

(ξ^{2} + η^{2})^{\frac{|β|}{2}} |D^{β}α(ξ, η)| ≤ c, ∀(ξ, η) ∈ ℝ^{2}, |β| ≤ 2.

But this follows immediately using:

α_{ξ}(ξ, η) = \frac{ξ(\sqrt{ξ^{2} + η^{2}} − 1)}{\sqrt{ξ^{2} + η^{2}}\sqrt{1 + ξ^{2} + η^{2}}(1 + \sqrt{ξ^{2} + η^{2}})^{2}} ,

α_{ξξ}(ξ, η) = \frac{η^{6} − η^{2} − 2ξ^{6} − 3ξ^{4}η^{2} + 3ξ^{2}\sqrt{ξ^{2} + η^{2}}(ξ^{2} + η^{2} + 1)}{(ξ^{2} + η^{2})^{3/2}(1 + \sqrt{ξ^{2} + η^{2}})^{3}(1 + ξ^{2} + η^{2})^{3/2}}

and

α_{ξη}(ξ, η) = \frac{ξη(1 − 3ξ^{4} − 6ξ^{2}η^{2} − 3η^{4} + 3\sqrt{ξ^{2} + η^{2}}(ξ^{2} + η^{2} + 1))}{(ξ^{2} + η^{2})^{3/2}(1 + \sqrt{ξ^{2} + η^{2}})^{3}(1 + ξ^{2} + η^{2})^{3/2}},

and the symmetry of α. □

Proof of Theorem 4.0.56: Setting L = D_{x,y}, it is easy to see that the hypotheses of the Theorem 0.2.15 are fulfilled for p = 1, γ = 1/2 and m = M = 1. Then there exists v_{c} ∈ H^{1/2}, v_{c} ≠ 0, that satisfies Eqn.(4.17) with p = 1, c > 0.

Denote K_{t} the convolution kernel of e^{−tD_{x,y}} so that K_{t} = \frac{1}{2π} (e^{−t(ξ^{2}+η^{2})^{1/2}})ˇ . We have

K_{t}(x, y) = \frac{3}{4π} \frac{t}{(t^{2} + x^{2} + y^{2})^{3/2}} ,

which is a positive decreasing function of |(x, y)|, ^{1} so v_{c} can be chosen to be a nonnegative decreasing function of |(x, y)|.

To prove that v_{c} lies indeed in H^{∞}, we will use a bootstrapping argument that combines Theorem 0.2.1 and Lemma 4.0.57. It is easy to see that v_{c}(x, y) = cv_{1}(cx, cy), then we may assume that c = 1. Let us call v = v_{1}.

By (4.6) we have v^{2} ∈ L^{q}, q ∈ [1, 2] and then by Lemma 4.0.57, J^{1}v ∈ L^{q}, q ∈ (1, 2] since

J^{1}v = \frac{1}{2}J^{1}(1 + D_{x,y})^{−1}v^{2}.  (4.19)

In particular v ∈ H^{1}. Let p ∈ (1, 2). By Theorem 0.2.1-(ii), H^{1} ↪ L^{q}, q ∈ [2, +∞) and then

|vv_{x}|_{p} ≤ |v|_{q}∥v_{x}∥, for \frac{1}{q} = \frac{1}{p} − \frac{1}{2}.  (4.20)

1The inverse Fourier transform of e^{−t(ξ^{2}+η^{2})^{1/2}} can be calculated using the radial symmetry of this function and properties of the Bessel functions. In [48] it is obtained by other method.

The relations (4.19) and (4.20) and Lemma 4.0.57 imply that J^{1}v_{x} = J^{1}(1+ D_{x,y})^{−1}vv_{x} ∈ L^{p}. Similarly, it can be shown that J^{1}v_{y} ∈ L^{p} and then v ∈ L_{2}^{p}, since J^{1}v ∈ L^{p}. By Theorem 0.2.1 we have,

\{ \begin{matrix}v ∈ L_{2}^{p} ↪ L^{∞}; \\ v_{x} ∈ L_{1}^{p} ↪ L^{q}, \frac{1}{q} = \frac{1}{p} − \frac{1}{2}; \\ v_{xx} ∈ L^{p} \end{matrix}

and then

|(v_{x})^{2} + vv_{xx}|_{p} ≤ |(v_{x})^{2}|_{p} + |vv_{xx}|_{p} ≤ |v_{x}|_{2p}^{2} + |v|_{∞}|v_{xx}|_{p} < ∞.

Thus, by Lemma 4.0.57 and the equation (4.19) we obtain that

J^{1}v_{xx} = J^{1}(1 + D_{x,y})^{−1}((v_{x})^{2} + vv_{xx}) ∈ L^{p}.

Similarly we get J^{1}v_{yy} ∈ L^{p} and then v ∈ L_{3}^{p} by Proposition 0.2.2.

We proceed now by induction and suppose that v ∈ L_{k}^{p}, k ≥ 3. By Theorem 0.2.1-(iii), we have

\{ \begin{matrix}∂_{x}^{l}v ∈ L_{k−l}^{p} ↪ L^{∞}, 0 ≤ l ≤ k − 2; \\ ∂_{x}^{l}v ∈ L^{p}, 0 ≤ l ≤ k, \end{matrix}

so ∂_{x}^{l}v · ∂_{x}^{k−l}v ∈ L^{p} for 0 ≤ l ≤ k − 1. We obtain as above

∂_{x}^{k}J^{1}v = J^{1}(1 + D_{x,y})^{−1} (∑_{0}^{k−1} \binom{k − 1}{l}∂_{x}^{l}v · ∂_{x}^{k−l}v) ∈ L^{p}.

One may prove in a similar manner that ∂_{y}^{k}J^{1}v ∈ L^{p}, so v ∈ L_{k+1}^{p} by Proposition 0.2.2. Since by Theorem 0.2.1-(i), ⋂_{k≥1} L_{k}^{p} = H^{∞}, the result follows. □

Remarks and conclusions

For both equations g2D-BO and gZK-BO, it holds that if u is a solution with time of existence T (i.e. u ∈ C([−T, T]; H^{s})), then u_{λ} ∈ C([−T_{λ}, T_{λ}]; H^{s}) also is, where u_{λ}(t, x, y) = λ^{1/2p}u(λt, λ^{1/2}x, λ^{1/2}y) and T_{λ} = λ^{−1}T. For s ∈ ℝ we may calculate:

∥D_{x,y}^{s}u_{λ}(0)∥^{2} = ∫ (ξ^{2} + η^{2})^{s}λ^{1/p}λ^{−2}|û(0)(λ^{−1/2}ξ, λ^{−1/2}η)|^{2}dξdη
= λ^{s+1/p−1} ∫ (ξ^{2} + η^{2})^{s}|û(0)(ξ, η)|dξdη
= λ^{s+1/p−1}∥D_{x,y}^{s}u(0)∥^{2}.

Thus, s_{p} = 1 − \frac{1}{p} is a critical index in the sense that if s < s_{p}, then ∥D_{x,y}^{s}u_{λ}(0)∥ → ∞ as λ ↘ 0, but T_{λ} → +∞. However, the time of existence is supposed to be inversely proportional to the norm in H^{s} of the solutions, so we do not expect to have well-posedness in H^{s}, s < s_{p}. Observe that s_{1} = 0, which is far from s = 2. In order to obtain well-posedness for s ≤ 2, one should use methods that exploit the dispersive structure of the equations. This has been done by several authors (see [15], [31], [33]) implementing a Picard iterative scheme in appropriate spaces using the integral formulation of the equations. Molinet, Saut and Tzvetkov proved in [37] and [38] that such a procedure cannot be used for BO and KP-I, respectively. Following closely their ideas, we obtain an analogous result for 2D-BO.

Theorem 5.0.58 Let s ∈ ℝ, T > 0 and let X_{T} be a space continuously embedded in C([−T, T]; H^{s}). Then there is no c > 0 such that,

∥E(t)ϕ∥_{X_{T}} ≤ c∥ϕ∥_{s}, ∀ϕ ∈ H^{s}  (5.1)

and

∥ ∫_{0}^{t} E(t − τ)[u(τ)u_{y}(τ)]dτ∥_{X_{T}} ≤ c∥u∥_{X_{T}}^{2} , ∀u ∈ X_{T} .  (5.2)

Since the proof of this theorem is rather technical and very similar to those in [37] and [38], it will be presented in the Appendix.

The lack of estimates of the form (5.1)-(5.2) obstruct strongly the obtention of solutions of 2D-BO as fixed points for the nonlinear mapping (1.13) and leaves open the question about which methods could be used to improve the well-posedness results in H^{s}. On the other hand, it is possible that an approach similar to that of [32] can be used to treat g2D-BO, p ≥ 2 and gZK-BO, probably yielding also better results than Theorems 3.0.51 and 3.0.53. This may be a subject for further work.

Another consequence of Theorem 5.0.58 is that the application datasolution, that is continuous by Theorem 1.0.31, cannot be C^{2} at zero.

Corollary 5.0.59 Let T > 0 and V a bounded neighbourhood of zero in H^{s}, s > 2 be such that the mapping F : V ↦ C([−T, T]; H^{s}), given by F_{t}(ϕ) = u(t), where u is the solution of 2D-BO with initial value ϕ, is well defined (Theorem 1.0.19 ensures the existence of such a T). Then F is not C^{2} at the origin.

Proof: Let us suppose that F is C^{2} at the origin. For φ ∈ H^{s}, s > 2 and 0 < δ ≪ 1 denote u^{δ}(t) = F_{t}(δφ). Using the integral equation (1.15) we obtain,

∥u^{δ}(t) − δE(t)φ∥_{s−1} = \frac{1}{2}∥ ∫_{0}^{t} E(t − τ)∂_{y}(u^{δ}(τ))^{2}dτ∥_{s−1}
≤ c_{T} max_{{|t|≤T }} ∥u^{δ}(t)∥_{s}^{2}.

Thus, multiplying by δ^{−1} and taking the limit as δ → 0 one gets

\frac{du^{δ}}{dδ} |_{δ=0}(t) = E(t)φ,

since u^{0} = 0. Now we compute the second derivative at zero. Observe that we have

u^{δ}(t) = δE(t)φ − \frac{δ^{2}}{2} ∫_{0}^{t} E(t − τ)∂_{y}(E(t)φ)^{2}dτ +

+\frac{1}{2} ∫_{0}^{t} E(t − τ)∂_{y}(δ^{2}(E(τ)φ)^{2} − (u^{δ}(τ))^{2})dτ.

So, using

∥∫_{0}^{t} E(t − τ)∂_{y}((u^{δ}(τ))^{2}−δ^{2}(E(τ)φ)^{2})dτ∥_{s−1}≤ c_{T} max_{{|t|≤T }}∥(u^{δ}(τ))^{2}−δ^{2}(E(τ)φ)^{2}∥_{s}
≤ c_{T} max_{{|t|≤T }} ∥u^{δ}(t) − δE(t)φ)∥_{s}∥u^{δ}(t) + δE(t)φ)∥_{s},

we obtain that in the topology of H^{s−1},

\frac{d^{2}u^{δ}}{dδ^{2}} |_{δ=0}(t) = −\frac{1}{2} ∫_{0}^{t} E(t − τ)∂_{y}(E(t)φ)^{2}dτ.

By the assumptions made we have that

F′′_{t}(0)(φ, φ) = −\frac{1}{2} ∫_{0}^{t} E(t − τ)∂_{y}(E(t)φ)^{2}dτ

where F′′(0) ∈ Bil(H^{s}, C([−T, T]; H^{s})) is the second Fréchet derivative of F evaluated at zero, but this contradicts (5.5) in the Appendix and then the result follows. □

Although we have not succeeded in proving a result similar to Theorem 5.0.58 for ZK-BO, we can follow the ideas used by Biagioni and Linares in [12], to prove that the flow map cannot be uniformly continuous in L^{2}, in case it is well-defined. In [12] it is shown that the flow map for gBO is not uniformly continuous in H^{s}(ℝ), s < s_{1}, p = 1 and in Ḣ^{s_{p}}(ℝ), p > 1, where s_{p} = 1/2 − 1/p is the critical index for gBO. We have,

Theorem 5.0.60 Suppose that ZK-BO defines a local flow in L^{2}. Then it cannot be uniformly continuous.

Proof: It suffices to find two sequences of solutions of ZK-BO, u_{n}, w_{n} ∈ C(ℝ; L^{2}) and m > 0, so that lim_{n→∞} ∥u_{n}(0) − w_{n}(0)∥ = 0 and for any t ≠ 0 and n large, ∥u_{n}(t) − w_{n}(t)∥ ≥ m.

With the notations of Theorem 4.0.56, u_{n}(0) = v_{n}, w_{n}(0) = v_{n+1}. Then u_{n}(t, x, y) = v_{n}(x − nt, y) = nv(n(x − nt), ny) and w_{n}(t, x, y) = v_{n+1}(x − (n + 1)t, y) = (n + 1)v((n + 1)(x − (n + 1)t), (n + 1)y), where v solves the Eqn.(4.17) with c = 1. Performing suitable changes of variables we obtain,

∥u_{n}(0) − w_{n}(0)∥^{2} = ∫ |nv(nx, ny) − (n + 1)v((n + 1)x, (n + 1)y)|^{2}dxdy
= \frac{n^{2}}{(n + 1)^{2}} ∫ |v(\frac{n}{n + 1}x, \frac{n}{n + 1}y) − v(x, y)|^{2}dxdy.

Using that v ∈ L^{2} and the Lebesgue’s Theorem of Dominated Convergence we obtain that as n → ∞,

∫ |v(\frac{n}{n + 1}x, \frac{n}{n + 1}y) − v(x, y)|^{2}dxdy → 0,  (5.3)

and then ∥u_{n}(0) − w_{n}(0)∥ → 0. Now let us fix t > 0, for t < 0 we can proceed similarly. Again with the help of some simple changes of variables we get

∥u_{n}(t) − w_{n}(t)∥^{2}=∫ |nv(nx, ny) − (n + 1)v((n + 1)(x − t), (n + 1)y)|^{2}dxdy
=∫ |\frac{n}{n + 1}v(\frac{n}{n + 1}x, \frac{n}{n + 1}y) − v(x−(n + 1)t, y)|^{2}dxdy
=2∥v∥^{2}− \frac{2n}{n + 1} ∫ v(\frac{n}{n + 1}x, \frac{n}{n + 1}y)v(x−(n + 1)t, y)dxdy.

By the relation (5.3) and the Cauchy-Schwartz inequality, it suffices to estimate ∫ v(x, y)v(x − (n + 1)t, y)dxdy. Writing this integral as:

∫_{|(x,y)|≤R}v(x, y)v(x − (n + 1)t, y)dxdy + ∫_{|(x,y)|≥R}v(x, y)v(x − (n + 1)t, y)dxdy

(5.4)

and choosing R > 0 so that ∫_{|(x,y)|≥R} v^{(}x, y)dxdy < ϵ/∥v∥, for ϵ > 0 arbitrary, we obtain that the second term above is bounded by ϵ, while the first term goes to zero as n → 0, since |v(x, y)| → 0 as |(x, y)| → ∞. We have that as n → ∞, ∫ v(x, y)v(x − (n + 1)t, y)dxdy → 0 and ∥u_{n}(t) − w_{n}(t)∥ → 2∥v∥ and the theorem follows. □

The local results obtained here for g2D-BO in F_{r}^{s}(ℝ^{2}), s > 2, r ∈ {0, 1} follow from a theorem proved by Abdelouhab in [1]. We have also obtain well-posedness in spaces with nonisotropic weights and an ill-posedness result. That illustrates once more ^{1} the relation between the smoothness of the symbol of the dispersive operator and the decay rate of the solutions. Such matters remain to be studied for the solutions of gZK-BO as well as the obtention of sharp unique continuation principles.

One could attempt to perform a stability analysis for the solitary waves of ZK-BO, but this appears to be very difficult. Indeed, to use the classical Liapunov method, one would expect that under some assumptions on the operator D_{x,y} + c − \frac{1}{2}v_{c}^{2}, where v_{c} denotes the solution of Eqn.(4.17), the orbital stability (instability) of v_{c} results from the condition \frac{d}{dc} ∫ v_{c}^{2}dxdy < 0 (> 0). However, in this case \frac{d}{dc} ∫ v_{c}^{2}dxdy = 0. On the other hand, to apply the method of concentration-compactness one should rely on the fact that

1As has been already done in [20], [24], [29] and [51].

I_{q} = inf{H_{2}(φ) : φ ∈ H^{1/2}, ∥φ∥ = q} < ∞

for any q > 0, but observe that if we set φ_{c}(x, y) = cφ(cx, cy), then ∥φ_{c}∥ = ∥φ∥ and H_{2}(φ_{c}) = cH_{2}(φ). Thus I_{q} cannot be finite. Eqn.(4.17) is then a critical case for which more elaborated methods must be used.

Finally, we would like to remark that although the results obtained for 2D-BO and ZK-BO are somewhat analogous to those that exist for BO, we think that 2D-BO is not an appropriate model from the physical point of view, since the degeneration of its symbol makes it very sensitive against small perturbations or in other words, its solutions do not depend continuously on the symbol. Indeed, if we replace H^{(x)}∂_{xy} in 2D-BO by H^{(x)}∂_{xy} + ϵH^{(y)}∂_{yy}, for example, with 0 < ϵ ≪ 1, we obtain an equation (2D-BO_{ϵ}) which differs in many properties from 2D-BO. By Theorem 0.2.15, we obtain inmediately, for example, that 2D-BO_{ϵ} possesses solitary waves of the form u(t, x, y) = v(x, y − ct), v ∈ H^{1/2}. However, by Theorem 4.0.54 it does not hold for 2D-BO. As it was remarked by Benjamin in [9], this kind of robustness against small perturbations is expected in equations that model real phenomena.

5.1 Appendix

Proof of the Theorem 5.0.58:

By the assumptions of the theorem it suffices to prove that for some ϕ ∈ H^{s},

∥ ∫_{0}^{t} E(t − τ)[E(τ)ϕ · E(τ)ϕ_{y}]dτ∥_{s} ≤ c∥ϕ∥_{s}^{2}, t ∈ [−T, T]  (5.5)

does not hold for any c > 0.

We will define such a ϕ through its Fourier transform as:

ϕ̂(ξ, η) = α^{−3/4}χ_{Q_{1}} + α^{−3/4}N^{−s}χ_{Q_{2}},  (5.6)

where Q_{1} = [α/2, α] × [\sqrt{α}/2, \sqrt{α}], Q_{2} = [α/2, 3α/2] × [N, N + \sqrt{α}]. It will be assumed that α ≪ 1 and N ≫ 1 so that all the calculations below work. Their exact range of values will be precisely set at the end of the proof.

We assert that for any s ∈ ℝ there exists m(s) > 0 independent of α and N such that for α and N^{−1} sufficiently small, we have ∥ϕ∥_{s} ≤ m(s). Obviously, it suffices to prove it for s > 0. In such a case,

∥ϕ∥_{s}^{2} ≤ c ∫ (1 + ξ^{2s} + η^{2s})|ϕ̂(ξ, η)|^{2}dξdη
≤ c (α^{−3/2} ∫_{α/2}^{α} dξ ∫_{\sqrt{α}/2}^{√\overline{α}} dη + cα^{−3/2}N^{−2s} ∫_{α/2}^{3α/2} dξ ∫_{N}^{N+\sqrt{α}} η^{2s}dη)
≤ c_{s}(1 + N^{−2s}(N + \sqrt{α})^{2s})
≤ m(s).

Let p(ξ, η) = |ξ|η and I = ∫_{0}^{t} E(t − τ)[E(τ)ϕ · E(τ)ϕ_{y}]dτ, then

I =c∫ e^{i(xξ+yη)} (∫_{0}^{t} E(t − τ)[E(τ)ϕ · E(τ)ϕ_{y}]dτ) (ξ, η)dξdη
=c∫ e^{i(xξ+yη)}( ∫_{0}^{t} e^{i(t−τ)p(ξ,η)}η( ∫ e^{iτ{p(ξ−ξ_{1},η−η_{1})+p(ξ_{1},η_{1})}}

ϕ̂(ξ − ξ_{1}, η − η_{1})ϕ̂(ξ_{1}, η_{1})dξ_{1}dη_{1})dτ)dξdη

=c∫∫e^{i(xξ+yη)+itp(ξ,η)}ηϕ̂(ξ − ξ_{1}, η − η_{1})ϕ̂(ξ_{1}, η_{1})(∫_{0}^{t}e^{iτχ(ξ,ξ_{1},η,η_{1})}dτ)dξ_{1}dη_{1}dξdη
=c∫∫e^{i(xξ+yη)+itp(ξ,η)}ηϕ̂(ξ − ξ_{1}, η − η_{1})ϕ̂(ξ_{1}, η_{1})\frac{1 − e^{itχ(ξ,ξ_{1},η,η_{1})}}{χ(ξ, ξ_{1}, η, η_{1})} dξ_{1}dη_{1}dξdη,

where we have set

χ(ξ, ξ_{1}, η, η_{1}) = p(ξ_{1}, η_{1}) + p(ξ − ξ_{1}, η − η_{1}) − p(ξ, η),

and the calculations have been made formally. Taking ϕ as in (5.6), we have that (E(t − τ)[E(τ)ϕ · E(τ)ϕ_{y}])ˆ ∈ L^{1}, ∀t ∈ [−T, T] , so we can use the inversion formula in the first line and we can also use Fubini’s theorem to justify the third line. Moreover,

∫_{0}^{t} E(t − τ)[E(τ)ϕ · E(τ)ϕ_{y}]dτ = f_{1} + f_{2} + f_{3},

where

f_{1} = cα^{−3/2}F^{−1} (e^{itp(ξ,η)}η ∫_{A_{ξ,η}^{1,1}} \frac{1 − e^{itχ(ξ,ξ_{1},η,η_{1})}}{χ(ξ, ξ_{1}, η, η_{1})} dξ_{1}dη_{1}) ,

f_{2} = cα^{−3/2}N^{−2s}F^{−1} (e^{itp(ξ,η)}η ∫_{A_{ξ,η}^{2,2}} \frac{1 − e^{itχ(ξ,ξ_{1},η,η_{1})}}{χ(ξ, ξ_{1}, η, η_{1})} dξ_{1}dη_{1}) ,

f_{3} = cα^{−3/2}N^{−s}F^{−1} (e^{itp(ξ,η)}η ∫_{A_{ξ,η}^{1,2}∪A_{ξ,η}^{2,1}} \frac{1 − e^{itχ(ξ,ξ_{1},η,η_{1})}}{χ(ξ, ξ_{1}, η, η_{1})} dξ_{1}dη_{1}) ,

for

A_{ξ,η}^{i,j} = {(ξ_{1}, η_{1}) : (ξ_{1}, η_{1}) ∈ Q_{i}, (ξ − ξ_{1}, η − η_{1}) ∈ Q_{j}, i, j ∈ {1, 2}}.

Observe that

supp F(f_{1}) ⊂ [α, 2α] × [\sqrt{α}, 2\sqrt{α}], supp F(f_{2}) ⊂ [α, 3α] × [2N, 2N + 2\sqrt{α}],

and

supp F(f_{3}) ⊂ [α, 5α/2] × [N + \sqrt{α}/2, N + 2\sqrt{α}].

Thus, they are disjoint sets for α and N^{−1} sufficiently small and we get the estimate

∥ ∫_{0}^{t} E(t − τ)[E(τ)ϕ · E(τ)ϕ_{y}]dτ∥_{s} ≥ ∥f_{3}∥_{s}.  (5.7)

Note that for (ξ_{1}, η_{1}) ∈ A_{ξ,η}^{1,2} ∪ A_{ξ,η}^{2,1},

|χ(ξ, ξ_{1}, η, η_{1})| = ξ_{1}(η − η_{1}) + η_{1}(ξ − ξ_{1}) ≤ cNα.  (5.8)

Suppose now that α and N are such that αN = N^{−ϵ}, 0 < ϵ ≪ 1. Observe that for any t ∈ [−T, T], t ≠ 0 we have,

\frac{|1 − e^{itχ}|}{|χ|} = 2\frac{sin(|tχ|)}{|χ|} = \frac{|tχ| + O(|tχ|^{2})}{|χ|} = |t| + O(N^{−ϵ}),

and then \frac{|1−e^{itχ}|}{|χ|} ≥ \frac{|t|}{2} for N ≫ 1. We get,

∥f_{3}∥_{s}^{2} ≥ c ∫ η^{2s}|f̂_{3}|^{2}(ξ, η)dξdη
≥ cα^{−3}N^{−2s} ∫ η^{2s+2}| ∫_{A_{ξ,η}^{1,2}∪A_{ξ,η}^{2,1}} \frac{1 − e^{itχ(ξ,ξ_{1},η,η_{1})}}{χ(ξ, ξ_{1}, η, η_{1})} dξ_{1}dη_{1}|^{2}dξdη

Using the mean value integral theorem we obtain that

| ∫_{A_{ξ,η}^{1,2}∪A_{ξ,η}^{2,1}} \frac{1 − e^{itχ(ξ,ξ_{1},η,η_{1})}}{χ(ξ, ξ_{1}, η, η_{1})} dξ_{1}dη_{1}| ≥ \frac{|t|}{2}meas(A_{ξ,η}^{1,2} ∪ A_{ξ,η}^{2,1}).

Moreover, it is not difficult to see that A_{ξ,η}^{1,2} ⊃ [α/2, α] × [\sqrt{α}/2, \sqrt{α}], for (ξ, η) ∈ [3α/2, 2α] × [N + \sqrt{α}, N + 3\sqrt{α}/2]. Then we have

≥ cα^{−3}N^{−2s} ∫_{3α/2}^{2α} dξ ∫_{N+\sqrt{α}}^{N+3\sqrt{α}/2} η^{2s+2}α^{3}dη
≥ cα^{3/2}N^{2}.

So, we obtain that ∥f_{3}∥_{s} ≥ cα^{3/4}N = cN^{\frac{1−3ϵ}{4}} , but this contradicts the fact that ∥f_{3}∥_{s} is bounded independently of N. □

Bibliography

[1] L. Abdelouhab, Nonlocal dispersive equations in weighted Sobolev spaces, Differential Integral Equations 5 (1992), no. 2, 307-338.

[2] L. Abdelouhab, J. L. Bona, M. Felland, and J.-C. Saut, Nonlocal models for nonlinear, dispersive waves, Physica D, 40, (1989), 360-392.

[3] M. J. Ablowitz and P.A. Clarkson, Nonlinear Evolution equations and Inverse Scattering, London Math. Soc. Lect. Note Series 149, Cambridge University Press, 1991.

[4] M. J. Ablowitz, A.S. Fokas and M.D. Kruskal, Note on the solution to a class of nonlinear singular integro-differential equations, Physics Letters A, 215-218, 1987.

[5] M. J. Ablowitz, A. S. Fokas and P. M. Santini, On the limit from the intermediate long wave equation to the Benjamin-Ono equation, Jr. Math. Phys., 25, 892-899, 1984.

[6] M. J. Ablowitz, H. Segur, Long internal waves in fluids of great depth, Stud. Appl. Math. 62, 249-262, 1980.

[7] L. A. Abramyan, Yu.A. Stepanyants and V. I. Shrira, Multidimensional solitons in shear flows,Sov. Phys. Dokl. 37(12), 575-578, 1992.

[8] C. J. Amick, J. F. Toland, Uniqueness of Benjamin’s Solitary-Wave Solutions of the Benjamin-Ono equation, IMA Journal of Appl. Math., 46, 21-28, 1991.

[9] T. B. Benjamin, Lectures on nonlinear wave motion. Nonlinear wave motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, N.Y., 1972), p. 3-47. Lectures in Appl. Math., Vol. 15, Amer. Math. Soc., Providence, R.I., 1974.

[10] T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 26, 559-592, 1989.

[11] J. Bergh and J. Löfström, Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.

[12] H. A. Biagioni and F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc. 353 (2001), no. 9, 3649-3659

[13] H. Brézis, Análisis Funcional. Teoria y aplicaciones., Alianza Editorial, S. A., Madrid, 1984.

[14] J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A 278 (1975), no. 1287, 555-601.

[15] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, GAFA. 3 (1993), no. 2, 107-156, 209-262.

[16] J. Bourgain, On the compactness of the support of solutions of dispersive equations, Internat. Math. Res. Notices 1997, no. 9, 437-447.

[17] A. P. Calderón, Commutators of Singular Integral Operators, Proc. Nat. Acad. Sc., USA 53 (1965), 1092-1099.

[18] R. R. Coifman, A. M. McIntosh and Y. Meyer, L’intégrale de Cauchy sur les curves lipschitziannes, Ann. of Math. (2) 116 (1982), no. 2, 361-387.

[19] A.I. D’yachenko and E.A. Kuznetsov, Two dimensional wave collapse in the boundary layer, Physica D 87, 301-313, 1995.

[20] J. Ginibre and Y. Tsutsumi, Uniqueness of solutions for the generalized Korteweg-de Vries equation, SIAM J. Math. Anal. 20 (1989), no. 6, 1388-1425.

[21] J. Ginibre and G. Velo, Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation, J. Differential Equations 93 (1991), no. 1, 150-212.

[22] N. Hayashi; P. I. Naumkin, Large time asymptotics of solutions to the generalized Benjamin-Ono equation, Trans. Amer. Math. Soc. 351 (1999), no. 1, 109-130.

[23] R. J. Iorio, Jr., On the Cauchy problem for the BO equation, Comm. Part. Diff. Eqn. 11 (1986), no. 10, 1031-1081.

[24] R. J. Iorio, Jr., The Benjamin-Ono Equation in Weighted Sobolev Spaces, J. Math. Anal. Appl. 157 (1991), no. 2, 577–590.

[25] R. J. Iorio, Jr. and V. M. Iorio, Fourier Analysis and Partial Differential Equations, Cambridge University Press, 2001.

[26] R. J. Iorio, Jr., Unique Continuation Principles for the Benjamin-Ono Equation, Preprint IMPA (2001).

[27] R. I. Joseph, Solitary waves in a finite depth fluid, Jr. Phys. A: Math. Gen., 10, L225-L227, 1977.

[28] B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl. 15 (1970), 539-549.

[29] T. Kato, On the Cauchy problem for the (Generalized) KortewegdeVries Equation, Stud. Appl. Math., Adv. Suppl. Stud. 8, 93-128, 1983.

[30] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier Stokes equations, Comm. Pure Appl. Math. 41, 93-128, 1988.

[31] C. E. Kenig, G. Ponce, and L. Vega Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), no. 4, 527-620.

[32] C. E. Kenig, G. Ponce, and L. Vega, On the generalized Benjamin-Ono equation, Trans. Amer. Math. Soc. 342 (1994), no. 1, 155-172.

[33] C. E. Kenig, G. Ponce, and L. Vega A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), no. 2, 573-603.

[34] D. J. Korteweg and G. de Vries, On the change of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. 39 (1895), no. 5, 422-443.

[35] T. Kubota, D. R. S. Ko and L. D. Dobbs, Weakly nonlinear long internal gravity waves in stratified fluids of finite depth, AIAA J. Hydronautics, 12, 157-165, 1978.

[36] P. D. Lax, A Hamiltonian approach to the KdV and other equations. Nonlinear evolution equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977), pp. 207-224, Publ. Math. Res. Center Univ. Wisconsin, 40, Academic Press, New York-London, 1978.

[37] L. Molinet, J.-C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono equation and related equations, preprint Université de Paris-Sud, 2001.

[38] L. Molinet, J.-C. Saut and N. Tzvetkov, Well-posedness and illposedness results for the Kadomtsev-Petviashvili-I equation, preprint Université de Paris-Sud, 2001.

[39] H. Ono, Algebraic solitary waves in stratified fluids, Jr. Phys. Soc. Japan 39, 1082-1091, 1975.

[40] S. I. Pokhozhaev, On the eigenfunctions of the equation Δu+λf(u) = 0, (Russian) Dokl. Akad. Nauk SSSR 165 1965 36-39.

[41] G. Ponce and L. Vega, Nonlinear small data scattering for the generalized Korteweg-de Vries equation, J. Funct. Anal. 90 (1990), no. 2, 445-457.

[42] M. Reed and B. Simon, Methods of modern mathematical physics. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.

[43] L. G. Redekopp, Similarity solutions of some two-space dimensional nonlinear wave evolution equations, Stud. Appl. Math. 63, 185-207, 1980.

[44] J.-C. Saut, Sur quelques généralisations de l’équation de Korteweg and de Vries. J. Math. Pures Appl. (9) 58 (1979), no. 1, 21-61.

[45] V. I. Shrira, On the subsurface waves in the oceanic upper mixed layer, Dokl. Akad. Nauk SSSR 308 (1989), no. 3, 732-736.

[46] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), no. 2, 149-162.

[47] W. A. Strauss, Nonlinear wave equations. CBMS Regional Conference Series in Mathematics, 73. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society,Providence, RI, 1989.

[48] R. S. Strichartz, A guide to distribution theory and Fourier transforms. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1994.

[49] M. Tajiri, Similarity reductions of the one and two dimensional Benjamin-Ono equations, J. Phys. Soc. Japan, vol 56, nro 4, 1327-1333, 1987.

[50] H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978. 528 pp.

[51] M. Tsutsumi, Weighted Sobolev spaces and rapidly decreasing solutions of some nonlinear dispersive wave equations, J. Differential Equations 42 (1981), no. 2, 260-281.

[52] Weinstein, Michael I., Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. Partial Differential Equations 12 (1987), no. 10, 1133-1173.

[53] Z. Yong, Strongly oblique interactions between internal solitary waves with the same mode, Appl. Math and Mech.(English ed.), 18, No.10, 957-962, 1997.

[54] B. Y. Zhang, Unique continuation for the Korteweg-de Vries equation, SIAM J. Math. Anal. 23 (1992), no. 1, 55-71.

[55] V. E. Zakharov and E. A. Kuznetsov, Three dimensional solitons, Sov. Phys. JETP. 39, No.2, 285-286, 1974.