Convergence problem of Schr?dinger equation and wave equation in low regularity spaces

This paper is devoted to investigating the convergence problem of the nonlinear Schrodinger equation and the nonlinear wave equation. Firstly, the almost everywhere pointwise convergence of the one-dimensional nonlinear Schrodinger equation is established in Hs(R)(s >= 14) with the aid of the decomposition of integral form solution to problem. Secondly, the uniform convergence lim t -> 0 sup x is an element of 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 Schrodinger equation and S1(t)f is the solution to the free one-dimensional Schrodinger equation. Finally, the almost everywhere pointwise convergence of the nonlinear wave equation is established in Hs(R3)(s > 1).(c) 2022 Elsevier Inc. All rights reserved.