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Low-Regularity Solutions of the Nonlinear Schrödinger Equation on the Spatial Quarter-Plane

Dionyssios Mantzavinos, Türker Ozsarı

Abstract

The Hadamard well-posedness of the nonlinear Schrödinger equation with power nonlinearity formulated on the spatial quarter-plane is established in a low-regularity setting with Sobolev initial data and Dirichlet boundary data in appropriate Bourgain-type spaces. As both of the spatial variables are restricted to the half-line, a different approach is needed than the one previously used for the well-posedness of other initial-boundary value problems. In particular, now the solution of the forced linear initial-boundary problem is estimated \textit{directly}, both in Sobolev spaces and in Strichartz-type spaces, i.e. without a linear decomposition that would require estimates for the associated homogeneous and nonhomogeneous initial value problems. In the process of deriving the linear estimates, the function spaces for the boundary data are identified as the intersections of certain modified Bourgain-type spaces that involve spatial half-line Fourier transforms instead of the usual whole-line Fourier transform found in the definition of the standard Bourgain space associated with the one-dimensional initial value problem. The fact that the quarter-plane has a corner at the origin poses an additional challenge, as it requires one to expand the validity of certain Sobolev extension results to the case of a domain with a non-smooth (Lipschitz) and non-compact boundary.

Low-Regularity Solutions of the Nonlinear Schrödinger Equation on the Spatial Quarter-Plane

Abstract

The Hadamard well-posedness of the nonlinear Schrödinger equation with power nonlinearity formulated on the spatial quarter-plane is established in a low-regularity setting with Sobolev initial data and Dirichlet boundary data in appropriate Bourgain-type spaces. As both of the spatial variables are restricted to the half-line, a different approach is needed than the one previously used for the well-posedness of other initial-boundary value problems. In particular, now the solution of the forced linear initial-boundary problem is estimated \textit{directly}, both in Sobolev spaces and in Strichartz-type spaces, i.e. without a linear decomposition that would require estimates for the associated homogeneous and nonhomogeneous initial value problems. In the process of deriving the linear estimates, the function spaces for the boundary data are identified as the intersections of certain modified Bourgain-type spaces that involve spatial half-line Fourier transforms instead of the usual whole-line Fourier transform found in the definition of the standard Bourgain space associated with the one-dimensional initial value problem. The fact that the quarter-plane has a corner at the origin poses an additional challenge, as it requires one to expand the validity of certain Sobolev extension results to the case of a domain with a non-smooth (Lipschitz) and non-compact boundary.
Paper Structure (5 sections, 19 theorems, 221 equations, 1 figure)

This paper contains 5 sections, 19 theorems, 221 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Linear estimates on the spatial quarter-plane
  3. Nonlinear analysis: proof of Theorem \ref{['lwp-t']}
  4. Solution of the forced linear Schrödinger equation on the spatial quarter-plane
  5. Zero extension operator for Sobolev spaces on the quarter-plane

Key Result

Theorem 1.1

Let $0\leq s < \frac{1}{2}$, $2 \leq \alpha \leq \frac{3-s}{1-s}$ and $(q, p)$ be the admissible pair Then, the initial-boundary value problem qnls-ibvp for the NLS equation on the spatial quarter-plane has a unique solution that admits the estimate where $c_{s, \alpha}>0$ is a constant that only depends on $s$ and $\alpha$, and the lifespan $T = T(u_0, g_0, h_0, s, \alpha) > 0$ satisfies wit

Figures (1)

  • Figure 2.1: The region $D_j$ and its positively oriented boundary $\partial D_j$, $j=1,2$.

Theorems & Definitions (29)

  • Theorem 1.1: Hadamard well-posedness for NLS on the spatial quarter-plane
  • Theorem 2.1: Sobolev estimate
  • proof
  • Theorem 2.2: Strichartz estimates
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.1: Chain rule for fractional derivatives
  • Lemma 3.2: Fractional Gagliardo-Nirenberg inequality
  • ...and 19 more