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 $γ$-law gas with density-dependent bulk viscosity coefficient $λ=λ(ρ)=ρ^β$ 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 $β>3$, $γ>1$ has been established by Va\uıgant--Kazhikhov [Sib. Math. J. 36 (1995), 1283--1316]. When $γ=β>3$, 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 $β>3/2$, $1< γ<4β-3$, 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 $β>3/2$ and $1< γ<4β-3$, 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 $β>1$, $γ>1$ 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.