On the well-posedness of 2-D incompressible Navier-Stokes equations with variable viscosity in critical spaces

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• 作者：Huan Xu^a ; ^b ; ^{hxuscut@163.com" class="auth_mail" title="E-mail the corresponding author} ; Yongsheng Li^a ; ^{yshli@scut.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Xiaoping Zhai^a ; ^{pingxiaozhai@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35Q35 ; 76D03 ; 35B45
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 April 2016
• 年：2016
• 卷：260
• 期：8
• 页码：6604-6637
• 全文大小：468 K

文摘

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)$p\in (1,4)$ and $a\in {\dot{B}}_{p,1}^{\frac{2}{p}}({\mathbb{R}}^2)$ that the solution mapping d4561d7a8623c2ec6d89c8b74d35" title="Click to view the MathML source">H_a:F↦∇Π${\mathcal{H}}_a:F\mapsto \mathrm{\nabla }\mathrm{\Pi }$ to the 2-D elliptic equation $\mathrm{div}((1+a)\mathrm{\nabla }\mathrm{\Pi })=\mathrm{div}F$ is bounded on ${\dot{B}}_{p,1}^{\frac{2}{p}-1}({\mathbb{R}}^2)$. More precisely, we prove that The proof of the uniqueness of solution to (1.2) relies on a Lagrangian approach ,  and . When the viscosity coefficient μ(ρ)$\mu (\rho )$ is a positive constant, we prove that (1.2) is globally well-posed.