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Global existence and uniqueness of strong solutions to the 2D nonhomogeneous primitive equations with density-dependent viscosity

Quansen Jiu, Lin Ma, Fengchao Wang

Abstract

This paper is concerned with an initial-boundary value problem of the two-dimensional inhomogeneous primitive equations with density-dependent viscosity. The global well-posedness of strong solutions is established, provided the initial horizontal velocity is suitably small, that is, $\|\nabla u_{0}\|_{L^{2}}\leq η_{0}$ for suitably small $η_{0}>0$. The initial data may contain vacuum. The proof is based on the local well-posedness and the blow-up criterion proved in \cite{0}, which states that if $T^{*}$ is the maximal existence time of the local strong solutions $(ρ,u,w,P)$ and $T^{*}<\infty$, then \begin{equation*} \sup_{0\leq t<T^{*}}(\left\|\nabla ρ(t)\right\|_{L^{\infty}}+\left\|\nabla^{2}ρ(t)\right\|_{L^{2}}+\left\|\nabla u(t)\right\|_{L^{2}})=\infty. \end{equation*} To complete the proof, it is required to make an estimate on a key term $\|\nabla u_{t}\|_{L_{t}^{1}L_Ω^{2}}$. We prove that it is bounded and could be as small as desired under certain smallness conditions, by making use of the regularity result of hydrostatic Stokes equations and some careful time weighted estimates.

Global existence and uniqueness of strong solutions to the 2D nonhomogeneous primitive equations with density-dependent viscosity

Abstract

This paper is concerned with an initial-boundary value problem of the two-dimensional inhomogeneous primitive equations with density-dependent viscosity. The global well-posedness of strong solutions is established, provided the initial horizontal velocity is suitably small, that is, for suitably small . The initial data may contain vacuum. The proof is based on the local well-posedness and the blow-up criterion proved in \cite{0}, which states that if is the maximal existence time of the local strong solutions and , then \begin{equation*} \sup_{0\leq t<T^{*}}(\left\|\nabla ρ(t)\right\|_{L^{\infty}}+\left\|\nabla^{2}ρ(t)\right\|_{L^{2}}+\left\|\nabla u(t)\right\|_{L^{2}})=\infty. \end{equation*} To complete the proof, it is required to make an estimate on a key term . We prove that it is bounded and could be as small as desired under certain smallness conditions, by making use of the regularity result of hydrostatic Stokes equations and some careful time weighted estimates.
Paper Structure (5 sections, 15 theorems, 136 equations)

This paper contains 5 sections, 15 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['the-2']}
  4. A priori estimates
  5. Proof of Theorem \ref{['the-2']}

Key Result

Theorem 1.1

Suppose that the initial data $(\rho_0,u_0)$ satisfies the regularity conditions and the compatibility condition for some $(P_{0},V_{1})\in V\times L^{2}$. Then there exists a small time $T_{*}\in(0,T)$, such that the initial boundary value problem $(equ11)$-$(144)$ has a unique strong solution $(\rho, u, w ,P)$ satisfying where $1\leqslant q<\infty$. Furthermore, if $T^{*}$ is the maximal exis

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 6 more