Global strong solution of compressible flow with spherically symmetric data and density-dependent viscosities

In this article, the Cauchy problem of a compressible Navier-Stokes system with density-dependent viscosities when the initial data are spherically symmetric is considered. Firstly, we construct the classical solution for the system in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mi>a</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mi>r</m:mi> <m:mo>:</m:mo> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:mi>r</m:mi> <m:mo>≤</m:mo> <m:mi>a</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:math> {B}ₐ₍ₜ₎=\{r:0≤ r≤ a(t)\} with the stress-free boundary, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>a</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> a(t) is the “particle path,” which arising from <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>a</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>></m:mo> <m:mn>0</m:mn> </m:math> a(0)={a}₀> 0 . Next, the strong solution for the system with the stress-free condition on the boundary <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>a</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> a(t) in the exterior domain <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>\</m:mo> <m:msub> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mi>a</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msub> </m:math> {{R}}³ {B}ₐ₍ₜ₎ is proved by the entropy estimate and the subtle analysis on the upper and lower bounds of the density. Finally, by continuously splicing the two parts of the solution via the stress-free boundary, we establish the global strong solution of the Cauchy problem. In particular, our analysis gives a positive example that it does not exhibit vacuum states provided that no vacuum states are present initially for multi-dimensional compressible flow with density-dependent viscosities, which is consistent with Hoff and Smoller [Hoff and Smoller, Non-formation of vacuum states for compressible Navier-Stokes equations, Comm. Math. Phys. 216 (2001), no. 2, 255–276], where the viscosity coefficients are assumed to be positive constants for one-dimensional case.