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 Vaĭgant--Kazhikhov regime

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 Vaĭgant–Kazhikhov [$\mathit{\text{Sib.}}\mathit{\text{Math.}}\mathit{\text{J.}}$ 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 $t\to \mathrm{\infty }$, has been proved by Perepelitsa [$\mathit{\text{SIAM}}\mathit{\text{J.}}\mathit{\text{Math.}}\mathit{\text{Anal.}}$ 39 (2007/08), 1344–1365]. Huang–Li [$\mathit{\text{J.}}\mathit{\text{Math.}}\mathit{\text{Pures}}\mathit{\text{Appl.}}$ 106 (2016), 123–154] extended these results by establishing the global existence of weak solutions and large-time behaviour under the assumptions $\beta >3\mathrm{/}2$, $1<\gamma <4\beta -3$, 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 $\beta >3\mathrm{/}2$ 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 $t\in ]0,\mathrm{\infty }[$. Moreover, under the mere assumption that $\beta >1$ and $\gamma >1$, we establish the global existence of weak solutions, thus pushing the global existence theory of the barotropic Navier–Stokes equations on ${\mathbb{T}}^2$ to the most general setting to date. One of the key ingredients of our proof is a novel application – motivated by the recent work due to Danchin–Mucha [$\mathit{\text{Comm.}}\mathit{\text{Pure}}\mathit{\text{Appl.}}\mathit{\text{Math.}}$ 76 (2023), 3437–3492] – of Desjardins’ logarithmic interpolation inequality.