Properties of Solutions to Some Nonlinear Dispersive Models
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.