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Regularity of solution maps of the generalized surface quasi-geostrophic equations

Gerard Misiołek, Xuan-Truong Vu, Tsuyoshi Yoneda

Abstract

We study regularity properties of the data-to-solution maps of the family of generalized surface quasi-geostrophic equations which includes both the 2D incompressible Euler and the standard surface quasi-geostrophic equations. We prove that the Lagrangian solution maps, interpreted as Riemannian exponential maps on the group of exact Sobolev class diffeomorphisms, are real analytic and, consequently, the Cauchy problems are locally well-posed in the sense of Hadamard. On the other hand, we also show that the corresponding Eulerian solution maps are nowhere locally uniformly continuous on bounded subsets in the Sobolev topology and fail to be continuous in the standard (large-) Hölder topologies. These results sharpen earlier theorems and further highlight the striking dichotomy between regularity properties of the solution maps in the Lagrangian and Eulerian formulations.

Regularity of solution maps of the generalized surface quasi-geostrophic equations

Abstract

We study regularity properties of the data-to-solution maps of the family of generalized surface quasi-geostrophic equations which includes both the 2D incompressible Euler and the standard surface quasi-geostrophic equations. We prove that the Lagrangian solution maps, interpreted as Riemannian exponential maps on the group of exact Sobolev class diffeomorphisms, are real analytic and, consequently, the Cauchy problems are locally well-posed in the sense of Hadamard. On the other hand, we also show that the corresponding Eulerian solution maps are nowhere locally uniformly continuous on bounded subsets in the Sobolev topology and fail to be continuous in the standard (large-) Hölder topologies. These results sharpen earlier theorems and further highlight the striking dichotomy between regularity properties of the solution maps in the Lagrangian and Eulerian formulations.
Paper Structure (9 sections, 16 theorems, 122 equations)

This paper contains 9 sections, 16 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Technical preliminaries and Lagrangian reformulation
  3. Local well-posedness
  4. Proof of Theorem \ref{['analytic']}
  5. Proof of Corollary \ref{['lwp']}
  6. Non-uniform dependence of the solution map for $0\le \beta\le 1$.
  7. Non-continuity of the solution map for $C^\alpha$
  8. Analyticity in real Banach spaces
  9. Inequalities for fractional Sobolev functions

Key Result

Theorem 1

For $s > 2$ and any $0\le \beta\le 1$ the (Lagrangian) solution map $u_0\mapsto \exp_e^{_{\beta}} u_0$ from $T_{e}\mathcal{D}_{ex}^s$ to $\mathcal{D}_{ex}^s$ is analytic.

Theorems & Definitions (30)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Lemma 6
  • proof
  • Corollary 7
  • Lemma 8
  • proof
  • ...and 20 more