On the Well-posedness of 2-D Incompressible Navier-Stokes Equations with Variable Viscosity in Critical Spaces

2015/10/28 by Huan Xu, Yongsheng Li, Xu, Huan +3
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions #math.AP

paper · pdf · doi:10.48550/arxiv.1510.08196

arxiv created 2015/10/28 · openalex publication_date 2015/10/28 · arxiv updated 2015/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we first prove the local well-posedness of the 2-D incompressible Navier-Stokes equations with variable viscosity in critical Besov spaces with negative regularity indices, without smallness assumption on the variation of the density. The key is to prove for p∈(1,4) and a∈B_p,1^\frac2p(ℝ²) that the solution mapping H_a:F↦∇Π to the 2-D elliptic equation div((1+a)∇Π)=div F is bounded on B_p,1^\frac2p-1(ℝ²). More precisely, we prove that ‖∇Π‖_B_p,1^\frac2p-1≤ C(1+‖a‖_B_p,1^\frac2p)²‖F‖_B_p,1^\frac2p-1. The proof of the uniqueness of solution to (1.2) relies on a Lagrangian approach [15]-[17]. When the viscosity coefficient μ(ρ) is a positive constant, we prove that (1.2) is globally well-posed.

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