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Smoothing and Strichartz estimates for degenerate Schrödinger-type equations

Serena Federico, Michael Ruzhansky

Abstract

In this paper we focus on the validity of some fundamental estimates for time-degenerate Schrödinger-type operators. On one hand we derive global homogeneous smoothing estimates for operators of any order by means of suitable comparison principles (that we shall obtain here). On the other hand, we prove weighted Strichartz-type estimates for time-degenerate Scrhödinger operators and apply them to the local well-posedness of the semilinear Cauchy problem. Most of our results apply to nondegenerate operators as well, recovering, in these cases, the well-known standard results.

Smoothing and Strichartz estimates for degenerate Schrödinger-type equations

Abstract

In this paper we focus on the validity of some fundamental estimates for time-degenerate Schrödinger-type operators. On one hand we derive global homogeneous smoothing estimates for operators of any order by means of suitable comparison principles (that we shall obtain here). On the other hand, we prove weighted Strichartz-type estimates for time-degenerate Scrhödinger operators and apply them to the local well-posedness of the semilinear Cauchy problem. Most of our results apply to nondegenerate operators as well, recovering, in these cases, the well-known standard results.

Paper Structure

This paper contains 5 sections, 19 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Comparison principles
  3. Application of comparison principles to smoothing estimates
  4. Weighted Strichartz estimate
  5. Local well-posedness of the semilinear Cauchy problem

Key Result

Lemma 2.1

Let $n,p\geq 1$, and let $b\in C^1(\mathbb{R})$ be such that $b(0)=0$. Let also $a\in C^1(\mathbb{R}^n)$ and $\sigma\in C^0(\mathbb{R}^n)$.Then we have that If $b$ is strictly monotone then, for all $u_0\in L^{2}_x(\mathbb{R}^n)$, where equality holds if $b$ also satisfies $\lim_{t\rightarrow \infty}b(t)=\infty$. If $b'$ is such that $\sharp\{t\in\mathbb{R}; b'(t)=0\}=k\geq 1$, then, for all $u_0

Theorems & Definitions (26)

  • Lemma 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Corollary 2.5
  • Remark 2.6
  • Theorem 2.7
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • ...and 16 more