Cauchy Problem of Nonlinear Schrödinger Equation with Initial Data
نویسندگان
چکیده
In this paper, we consider in Rn the Cauchy problem for the nonlinear Schrödinger equation with initial data in the Sobolev space W s,p for p < 2. It is well known that this problem is ill posed. However, we show that after a linear transformation by the linear semigroup the problem becomes locally well posed in W s,p for 2n n+1 < p < 2 and s > n(1 − 1 p ). Moreover, we show that in one space dimension, the problem is locally well posed in Lp for any 1 < p < 2.
منابع مشابه
Existence of Mild Solutions to a Cauchy Problem Presented by Fractional Evolution Equation with an Integral Initial Condition
In this article, we apply two new fixed point theorems to investigate the existence of mild solutions for a nonlocal fractional Cauchy problem with an integral initial condition in Banach spaces.
متن کاملNonuniqueness of Weak Solutions of the Nonlinear Schrödinger Equation
Generalized solutions of the Cauchy problem for the one-dimensional periodic nonlinear Schrödinger equation, with cubic or quadratic nonlinearities, are not unique. For any s < 0 there exist nonzero generalized solutions varying continuously in the Sobolev space H, with identically vanishing initial data.
متن کاملAn Integral Form of the Nonlinear Schrödinger Equation with Variable Coefficients
We discuss an integral form of the Cauchy initial value problem for the nonlinear Schrödinger equation with variable coefficients. Some special and limiting cases are outlined.
متن کاملGlobal Well-posedness and Scattering for Derivative Schrödinger Equation
In this paper we mainly study the Cauchy problem for the derivative nonlinear Schrödinger equation in d-dimension (d ≥ 2). We obtain some global well-posedness results with small initial data. The crucial ingredients are L e , L ∞,2 e type estimates, and inhomogeneous local smoothing estimate (L e estimate). As a by-product, the scattering results with small initial data are also obtained.
متن کاملOn the Cauchy Problem for the Derivative Nonlinear Schrödinger Equation with Periodic Boundary Condition
It is shown that the Cauchy problem associated to the derivative nonlinear Schrödinger equation ∂tu − i∂ xu = λ∂x(|u| u) is locally well-posed for initial data u(0) ∈ H(T), if s ≥ 1 2 and λ is real. The proof is based on an adaption of the gauge transformation to periodic functions and sharp multi-linear estimates for the gauge equivalent equation in Fourier restriction norm spaces. By the use ...
متن کامل