Global well-posedness and exponential decay to the Cauchy problem of nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum in ℝ²

2021/02/25 by Xin Zhong, Zhong, Xin
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2102.12940

openalex publication_date 2021/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study global well-posedness of strong solutions for the nonhomogeneous Navier-Stokes equations with density-dependent viscosity and initial density allowing vanish in ℝ². Applying a logarithmic interpolation inequality and delicate energy estimates, we show the global existence of a unique strong solution provided that ‖∇μ(ρ₀)‖_L^q is suitably small, which improves the previous result of Huang and Wang [SIAM J. Math. Anal. 46, 1771--1788 (2014)] to the whole space case. Moreover, we also derive exponential decay rates of the solution. In particular, there is no need to require additional initial compatibility condition despite the presence of vacuum.

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