On global existence and large-time behaviour of weak solutions to the compressible barotropic Navier--Stokes Equations on ${\mathbb{T}}^2$ with density-dependent bulk viscosity: beyond the Va\uıgant--Kazhikhov regime

Abstract

We are concerned with the compressible barotropic Navier--Stokes equations for a $\gamma$-law gas with density-dependent bulk viscosity coefficient $\lambda =\lambda (\rho )={\rho }^{\beta }$ on the two-dimensional periodic domain ${\mathbb{T}}^2$. The global existence of weak solutions with initial density bounded away from zero and infinity for $\beta >3$, $\gamma >1$ has been established by Va\uıgant--Kazhikhov [Sib. Math. J. 36 (1995), 1283--1316]. When $\gamma =\beta >3$, the large-time behaviour of the weak solutions and, in particular, the absence of formation of vacuum and concentration of density as $t\to \infty$, has been proved by Perepelitsa [\textit{SIAM J. Math. Anal.} 39 (2007/08), 1344--1365]. Huang--Li [J. Math. Pures Appl. 106 (2016), 123--154] extended these results by establishing the global existence of weak solutions and large-time behaviour under the assumptions $\beta >3/2$, $1<\gamma <4\beta -3$, and that the initial density stays away from infinity (but may contain vacuum). Improving upon the works listed above, we prove that in the regime of parameters as in Huang--Li, namely that $\beta >3/2$ and $1<\gamma <4\beta -3$, if the density has no vacuum or concentration at $t=0$, then it stays away from zero and infinity at all later time $t\in ]0,\infty [$. Moreover, assuming $\beta >1$, $\gamma >1$ and a technical condition, we establish the global existence of weak solutions on ${\mathbb{T}}^2$. One of the key ingredients of our proof is a novel application --- motivated by the recent work due to Danchin--Mucha [Comm. Pure Appl. Math. 76 (2023), 3437--3492] --- of Desjardins' logarithmic interpolation inequality.

arXiv commentAn argument on pp.27-28 is flawed in V2: In Equation(68), $\nu$ itself becomes unbounded as ${\hat{\rho }}_T\to \infty$. An additional technical condition is added in V3