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Global well-posedness for the logarithmically energy-supercritical Nonlinear Wave Equation with partial symmetry

Aynur Bulut, Benjamin Dodson

Abstract

We establish global well-posedness and scattering results for the logarithmically energy-supercritical nonlinear wave equation, under the assumption that the initial data satisfies a partial symmetry condition. These results generalize and extend work of Tao in the radially symmetric setting. The techniques involved include weighted versions of Morawetz and Strichartz estimates, with weights adapted to the partial symmetry assumptions. In an appendix, we establish a corresponding quantitative result for the energy-critical problem.

Global well-posedness for the logarithmically energy-supercritical Nonlinear Wave Equation with partial symmetry

Abstract

We establish global well-posedness and scattering results for the logarithmically energy-supercritical nonlinear wave equation, under the assumption that the initial data satisfies a partial symmetry condition. These results generalize and extend work of Tao in the radially symmetric setting. The techniques involved include weighted versions of Morawetz and Strichartz estimates, with weights adapted to the partial symmetry assumptions. In an appendix, we establish a corresponding quantitative result for the energy-critical problem.

Paper Structure

This paper contains 9 sections, 9 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['p.1']}
  3. Strichartz estimates for (NLW)
  4. Morawetz estimate and the radial Sobolev embedding
  5. Sketch of the Proof of Theorem \ref{['p.1']}
  6. Weighted Strichartz estimates with symmetry
  7. Proof of Theorem \ref{['t.1']}
  8. Proof of Theorem \ref{['t.2']}
  9. Quantitative Strichartz norm estimates for the energy-critical (NLW)

Key Result

Theorem \oldthetheorem

Suppose that $u:I\times\mathbb{R}^4\rightarrow\mathbb{R}$ is a solution to (NLW) with nonlinearity (3.1) and radially symmetric initial data $(u_0,u_1)\in \tilde{H}^{9/4}(\mathbb{R}^4)\times H^{5/4}(\mathbb{R}^4)$. Then with constant independent of the time interval $I$. In particular, combining this estimate with the usual local theory for (NLW), radial solutions to (NLW) with nonlinearity (3.1)

Theorems & Definitions (16)

  • Theorem \oldthetheorem: Radial log-supercritical NLW on $\mathbb{R}^4$
  • Theorem \oldthetheorem: Axially symmetric log-supercritical NLW on $\mathbb{R}^4$
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • ...and 6 more