Convergence problem of Schrödinger equation and wave equation in low regularity spaces

Abstract

This paper is devoted to investigating the convergence problem of the nonlinear Schrödinger equation and the nonlinear wave equation. Firstly, the almost everywhere pointwise convergence of the one-dimensional nonlinear Schrödinger equation is established in Hs(R)(s≥14) with the aid of the decomposition of integral form solution to problem. Secondly, the uniform convergencelimt→0⁡supx∈R⁡|u(x,t)−S1(t)f|=0 is established with initial data in Hs(R)(s>0), where u is the solution to the one-dimensional nonlinear Schrödinger equation and S1(t)f is the solution to the free one-dimensional Schrödinger equation. Finally, the almost everywhere pointwise convergence of the nonlinear wave equation is established in Hs(R3)(s>1).