Regularity properties of the Zakharov system on the half line

Abstract

ABSTRACTIn this paper, we study the local and global regularity properties of the Zakharov system on the half line with rough initial data. These properties include local and global wellposedness results, local and global smoothing results, and the behavior of higher order Sobolev norms of the solutions. Smoothing means that the nonlinear part of the solution on the half line is smoother than the initial data. The gain in regularity coincides with the gain that was observed for the periodic Zakharov and the Zakharov on the real line. Uniqueness is proved in the class of smooth solutions. When the boundary value of the Schrödinger part of the solution is zero, uniqueness can be extended to the full range of local solutions. Under the same assumptions on the initial data, we also prove global-in-time existence and uniqueness of energy solutions. For more regular data, we prove that all higher Sobolev norms grow at most polynomially-in-time.