Smoothing properties of initial-boundary value problems

Abstract

This thesis discusses the smoothing properties of dispersive partial differential equations. In the first part of the thesis, we consider the Davey–Stewartson system on R 2 and demonstrate that the nonlinear part of the solution flow is smoother than the initial data. As an application of the smoothing result, we address the dissipa tive Davey–Stewartson system and give a simplified proof of the existence of a global attractor for the system. In the next part, we study well- posedness and regularity properties of the biharmonic Schr¨odinger equation on the half-line. More precisely, we prove local existence and uniqueness and show that the data to solution map is continuous. Moreover, we establish global well- posedness and global smoothing for higher regular spaces by showing that the solution grows at most linearly. As regards to the smoothing result, the derivative gain we obtain for the nonlinear part of the so lution is up to full derivative. The last part of the thesis addresses the Hirota–Satsuma system on the torus. The Hirota–Satsuma system is given by two Korteweg- de Vries equations exhibiting different dispersion relations which is due to the coupling coeffi cient a. The main result demonstrates the regularity level of the nonlinear part of the evolution compared to initial data. The gain in regularity depends very much on the arithmetic properties of the coefficient a. Then, we consider the forced and damped Hirota–Satsuma system and establish the analogous smoothing estimates. By the help of the smoothing estimates, we prove the existence and regularity of a global attractor in the energy space.