Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum

Xin Zhong

doi:10.3934/dcds.2019292

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2019, Volume 39, Issue 11: 6713-6745. Doi: 10.3934/dcds.2019292

This issue Previous Article Electro-magneto-static study of the nonlinear Schrödinger equation coupled with Bopp-Podolsky electrodynamics in the Proca setting Next Article The jumping problem for nonlocal singular problems

Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum

Xin Zhong^,

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

^* Corresponding author: Xin Zhong
^* Corresponding author: Xin Zhong

Received: April 2019

Published: August 2019

The author is supported by Fundamental Research Funds for the Central Universities (No. XDJK2019B031) and Chongqing Research Program of Basic Research and Frontier Technology (No. cstc2018jcyjAX0049)

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Abstract

This paper concerns the Cauchy problem of the two-dimensional density-dependent Boussinesq equations on the whole space $ \mathbb{R}^{2} $ with zero density at infinity. We prove that there exists a unique global strong solution provided the initial density and the initial temperature decay not too slow at infinity. In particular, the initial data can be arbitrarily large and the initial density may contain vacuum states and even have compact support. Moreover, there is no need to require any Cho-Choe-Kim type compatibility conditions. Our proof relies on the delicate weighted estimates and a lemma due to Coifman-Lions-Meyer-Semmes [J. Math. Pures Appl., 72 (1993), pp. 247-286].

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Mathematics Subject Classification: Primary: 35Q35; Secondary: 76D03.

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