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Nonlinear interpolation and the flow map for quasilinear equations

Thomas Alazard, Nicolas Burq, Mihaela Ifrim, Daniel Tataru, Claude Zuily

Abstract

We prove an interpolation theorem for nonlinear functionals defined on scales of Banach spaces that generalize Besov spaces. It applies to functionals defined only locally, requiring only some weak Lipschitz conditions, extending those introduced by Lions and Peetre. Our analysis is self-contained and independent of any previous results about interpolation theory. It depends solely on the concepts of Friedrichs' mollifiers, seen through the formalism introduced by Hamilton, combined with the frequency envelopes introduced by Tao and used recently by two of the authors and others to study the Cauchy problem for various quasilinear evolutions in partial differential equations. Inspired by this latter work, our main application states that, for an abstract flow map of a quasilinear problem, both the continuity of the flow as a function of time and the continuity of the data to solution map follow automatically from the estimates that are usually proven when establishing the existence of solutions: propagation of regularity via tame a priori estimates for higher regularities and contraction for weaker norms.

Nonlinear interpolation and the flow map for quasilinear equations

Abstract

We prove an interpolation theorem for nonlinear functionals defined on scales of Banach spaces that generalize Besov spaces. It applies to functionals defined only locally, requiring only some weak Lipschitz conditions, extending those introduced by Lions and Peetre. Our analysis is self-contained and independent of any previous results about interpolation theory. It depends solely on the concepts of Friedrichs' mollifiers, seen through the formalism introduced by Hamilton, combined with the frequency envelopes introduced by Tao and used recently by two of the authors and others to study the Cauchy problem for various quasilinear evolutions in partial differential equations. Inspired by this latter work, our main application states that, for an abstract flow map of a quasilinear problem, both the continuity of the flow as a function of time and the continuity of the data to solution map follow automatically from the estimates that are usually proven when establishing the existence of solutions: propagation of regularity via tame a priori estimates for higher regularities and contraction for weaker norms.

Paper Structure

This paper contains 13 sections, 15 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Generalized Besov spaces
  4. Nonlinear interpolation
  5. Proof of the main result
  6. Reduction to the model case
  7. Reduction to the case $E=F$
  8. Friedrichs's mollifiers
  9. Interpolation
  10. Continuity
  11. Proof of Proposition \ref{['P19']}
  12. Proof of Theorem \ref{['princ']}
  13. Littlewood-Paley decompositions

Key Result

Theorem 1

Consider $\mu \in [2,+\infty]$ and four real numbers $s_0,s,s_1$ and $r$ such that Consider $u_0$ in the Sobolev space $H^s=H^s(\mathbb{R}^d)$ and a (possibly nonlinear) function defined on the ball of radius $r$ around $u_0$: where $B_s(u_0,r) = \{ v_0 \in H^s; \| v_0-u_0\|_{H^s} <r\}$, and satisfying the two following properties: Then we have the three following conclusions:

Theorems & Definitions (40)

  • Theorem 1: Automatic continuity of the flow map
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Definition 7
  • Remark 8
  • Definition 9: Spaces of exponentially decaying sequences
  • Proposition 12
  • ...and 30 more