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Unique existence of solutions to the inviscid SQG equation in a critical space

Tsukasa Iwabuchi

Abstract

We study the Cauchy problem for the surface quasi-geostrophic (SQG) equations in a two-dimensional bounded domain with the homogeneous Dirichlet boundary condition. We establish the unique existence of strong solutions in the critical Besov space $\dot B^2_{2,1}$, which is embedded in $C^1$. The proof is based on spectral localization using dyadic decomposition associated with the Dirichlet Laplacian. We obtain the solution by establishing uniform estimates for a sequence of solutions to the equation with a regularized nonlinear term.

Unique existence of solutions to the inviscid SQG equation in a critical space

Abstract

We study the Cauchy problem for the surface quasi-geostrophic (SQG) equations in a two-dimensional bounded domain with the homogeneous Dirichlet boundary condition. We establish the unique existence of strong solutions in the critical Besov space , which is embedded in . The proof is based on spectral localization using dyadic decomposition associated with the Dirichlet Laplacian. We obtain the solution by establishing uniform estimates for a sequence of solutions to the equation with a regularized nonlinear term.

Paper Structure

This paper contains 6 sections, 9 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Besov spaces
  4. Independence from boundary conditions for low regularity
  5. Bilinear Estimates
  6. Proof of Theorem \ref{['thm:1']}

Key Result

Theorem 1.1

Let $\theta_0 \in \dot B^2_{2,1}(A_D)$. Then, there exists a time $T > 0$ and a solution $\theta \in C([0,T], \dot B^2_{2,1}(A_D)) \cap C^1 ([0,T], \dot B^1_{2,1}(A_D))$ to the Cauchy problem QG1--QG2. Furthermore, the solution is unique in the class $C([0,T], W^{1,\infty}(\Omega)) \cap C^1 ([0,T],

Theorems & Definitions (9)

  • Theorem 1.1
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Lemma 3.1
  • Proposition 3.2