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Linear and nonlinear stability for the $3$D stratified Boussinesq equations with the horizontal viscosity and diffusivity

Mikihiro Fujii, Yang Li

Abstract

In this manuscript, we consider the $3$D Boussinesq equations for stably stratified fluids with the horizontal viscosity and thermal diffusivity and investigate the large time behavior of the solutions. Making use of the anisotropic Littlewood--Paley theory, we obtain their precise $L^1$-$L^p$ decay estimates, which provide us information on both the anisotropic and dispersive structure of the system. More precisely, we reveal that the dispersion from the skew symmetric terms of stratification makes the decay rates of some portions of the solutions faster and furthermore the third component of the velocity field exhibit the enhanced dissipative effect, which provides the additional fast decay rate.

Linear and nonlinear stability for the $3$D stratified Boussinesq equations with the horizontal viscosity and diffusivity

Abstract

In this manuscript, we consider the D Boussinesq equations for stably stratified fluids with the horizontal viscosity and thermal diffusivity and investigate the large time behavior of the solutions. Making use of the anisotropic Littlewood--Paley theory, we obtain their precise - decay estimates, which provide us information on both the anisotropic and dispersive structure of the system. More precisely, we reveal that the dispersion from the skew symmetric terms of stratification makes the decay rates of some portions of the solutions faster and furthermore the third component of the velocity field exhibit the enhanced dissipative effect, which provides the additional fast decay rate.

Paper Structure

This paper contains 13 sections, 18 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Linear stability
  4. Nonlinear stability
  5. Anisotropic Littlewood--Paley theory
  6. Linear stability: Proof of Theorem \ref{['thm:lin']}
  7. Explicit formula for linear solutions
  8. Estimates of semigroups related to the linear solutions
  9. Proof of Theorem \ref{['thm:lin']}
  10. Nonlinear stability: Proof of Theorem \ref{['main-thm']}
  11. Decomposition of nonlinear terms
  12. Decay estimates of Duhamel terms
  13. Proof of Theorem \ref{['main-thm']}

Key Result

Theorem 2.1

Let $0<\varepsilon <1/4$. Then, there exist an absolute positive constant $C$ and a positive constant $C_{\varepsilon}$ depending only on $\varepsilon$ such that the linear solution $(v^{\rm lin},\theta^{\rm lin})$ to eq:B-lin-0, with the initial data $(v_0,\theta_0) \in X^{3,4}(\mathbb{R}^3)$ with for all $\alpha=(\alpha_{\rm h},\alpha_3) \in (\mathbb{N} \cup \{ 0\} )^2 \times (\mathbb{N} \cup \

Theorems & Definitions (36)

  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3: Ji-Yan-Wu-22*Proposition 1.2
  • Theorem 2.4
  • Remark 2.5
  • Lemma 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 26 more