Preprint C25/2004
Properties of Solutions to Some Nonlinear Dispersive Models

Mahendra Panthee

**Keywords: **
nonlinear dispersive equations | Korteweg-de Vries equation | Well-posedness.

We study the local and global well-posedness issues of the initial
value problem (IVP) associated to the coupled system of Korteweg-de
Vries equations. Using the bilinear estimates established by Kenig,
Ponce and Vega in the Fourier transform restriction space we prove a
local result for given data in the Sobolev spaces of indices greater
than -3/4. We prove that this local result is optimal by showing that
the map data-solution is not twice differentiable at the
origin. Further, under certain restrictions on the coefficients, we
extend the local solution to a global one when the data is in the
Sobolev spaces of indices greater than -3/10. We also consider the IVP
associated to a coupled system of modified Korteweg-de Vries
equations. We further refine the low-high frequency technique
introduced by Bourgain and simplified by Fonseca, Linares and Ponce to
develop an iteration process below the energy space and prove a global
well-posedness result for data in the Sobolev spaces with indices
greater than 4/9. Finally we consider a bi-dimensional generalization
of the Korteweg-de Vries equation, called Zakharov-Kuznetsov equation,
and prove that if a sufficiently smooth solution to the associated IVP
is supported in a non-trivial time interval then it vanishes
identically.