Unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrödinger equation

Abstract

We study the unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrödinger equation. We show the uniqueness of solution in C([0, T]; H^s) for the critical case or L^\infty(0, T; H^s) for the subcritical case under certain assumptions on spatial dimensions and power of nonlinearity. We do not assume the solution belongs to any auxiliary spaces associated with the Strichartz estimate. For that purpose, we also prove the estimate of product between functions and distributions and the continuity of mapping: u \to |u| on the homogeneous Sobolev or Besove space.