Local Existence and Uniqueness of Strong Solutions to the Two Dimensional Nonhomogeneous Incompressible Primitive Equati
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
LOCAL EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS TO THE TWO DIMENSIONAL NONHOMOGENEOUS INCOMPRESSIBLE PRIMITIVE EQUATIONS∗
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Quansen JIU (
School of Mathematical Sciences, Capital Normal University, Beijing 100048, China E-mail : [email protected]
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Fengchao WANG (
†
Advanced Institute of Natural Sciences, Beijing Normal University at Zhuhai, Zhuhai 519087, China School of Mathematical Sciences, Beijing Normal University and Key Laboratory of Mathematics and Complex Systems, Ministry of Education Beijing 100875, China E-mail : [email protected] Abstract In this article, we study the initial boundary value problem of the two-dimensional nonhomogeneous incompressible primitive equations and obtain the local existence and uniqueness of strong solutions. The initial vacuum is allowed. Key words
local existence and uniqueness; strong solutions; nonhomogeneous incompressible primitive equations
2010 MR Subject Classification
1
35Q35
Introduction
In this article, we consider the two-dimensional nonhomogeneous primitive equations, which can be written as ρt + u∂x ρ + w∂z ρ = 0, ρ(u + u∂ u + w∂ u) + ∂ P − △u = 0, t x z x (1.1) ∂x u + ∂z w = 0, ∂z P = 0. ∗ Received
February 15, 2019; revised May 9, 2020. Q.S. Jiu was partially supported by the National Natural Science Foundation of China (11671273 and 11931010), key research project of the Academy for Multidisciplinary Studies of CNU and Beijing Natural Science Foundation (1192001). † Corresponding author: Fengchao WANG.
No.5
Q.S. Jiu & F.C. Wang: NONHOMOGENEOUS INCOMPRESSIBLE PRIMITIVE EQUATIONS
1317
Here (x, z) ∈ Ω = M × (0, 1), with M = (0, L1 ). For simplicity, we suppose that L1 = 1. The boundary conditions are that ρ, u are periodic in the direction x, (1.2)
w|z=0 = w|z=1 = 0, u|z=0 = u|z=1 = 0, and the initial conditions are ρ|t=0 = ρ0 , ρu|t=0 = ρ0 u0 .
(1.3)
The primitive equations for atmosphere dynamics are one of the fundamental models in geophysical flows (see [22]). Since the rigorous arguments by Lions, Temam and Wang in [20], mathematical theories regarding the incompressible primitive equations have been widely developed (see [1, 5–11, 18, 23, 26] and references therein). In recent years, there have appeared some mathematical analyses on compressible atmosphere models (see [12–14, 17]). Very recently, Jiu, Li and Wang [16] proved the uniqueness of the weak solutions obtained by Gatapov and Kazhikhov [14], and the global existence of weak solutions for three-dimensional compressible primitive equations was proved by Wang, Dou and Jiu [25]. However, up until now, there has been little study on the nonhomogeneous incompressible primitive equations. In this article, we will prove the local existence and uniqueness of strong solutions to the twodimensional nonhomogeneous primitive equations (1.1)–(1.3) by a semi-Galerkin method as in [3] where the compressible Navier-Stokes equations in three-dimensional is consi
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