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Global results for a Cauchy problem related to biharmonic wave maps

Tobias Schmid

Abstract

We prove global existence of a derivative bi-harmonic wave equation with a non-generic quadratic nonlinearity and small initial data in the scaling critical space $\dot{B}^{2,1}_{\frac{d}{2}}(\mathbb{R}^d) \times \dot{B}^{2,1}_{\frac{d}{2}-2}(\mathbb{R}^d)$ for $ d \geq 3 $. Since the solution persists higher regularity of the initial data, we obtain a small data global regularity result for the biharmonic wave maps equation for a certain class of target manifolds including the sphere.

Global results for a Cauchy problem related to biharmonic wave maps

Abstract

We prove global existence of a derivative bi-harmonic wave equation with a non-generic quadratic nonlinearity and small initial data in the scaling critical space for . Since the solution persists higher regularity of the initial data, we obtain a small data global regularity result for the biharmonic wave maps equation for a certain class of target manifolds including the sphere.

Paper Structure

This paper contains 12 sections, 16 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. The Cauchy problem and outline
  3. The main results
  4. Linear estimates and function spaces
  5. Notation
  6. Linear estimates
  7. Function spaces
  8. Embeddings, linear estimates and continuous operator
  9. Bilinear estimates
  10. Higher regularity
  11. Proof of the main theorem
  12. Local Smoothing & lateral Strichartz inequalities

Key Result

Theorem 1.1

Theorems & Definitions (39)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Lemma 2.2: Strichartz
  • proof
  • Corollary 2.3
  • proof
  • Remark 2.4
  • ...and 29 more