On the long-time behavior of compressible fluid flows excited by random forcing (Q6569712)

A compressible and viscous fluid occupies a bounded domain \(Q\subset \mathbb{ R}^{d}\), \(d=2,3\), with Lipschitz continuous boundary \(\partial Q\). The authors consider the compressible Navier-Stokes system driven by a stochastic forcing term: \(d\varrho +\operatorname{div}_{x}(\varrho u)dt=0\), \( d(\varrho u)+\operatorname{div}_{x}(\varrho u\otimes u)dt+\nabla _{x}p(\varrho )dt=\operatorname{div}_{x} \mathbb{S}(\nabla _{x}u)dt+\varrho g(\varrho ,u)dt+\varrho F(\varrho ,u)dW\), \(\mathbb{S}(\nabla _{x}u)=\mu (\nabla _{x}u+\nabla _{x}^{t}u-\frac{2}{d} \operatorname{div}_{x}u\mathbb{I})+\lambda \operatorname{div}_{x}u\mathbb{I}\), where \(\varrho \) denotes the mass density, \(u\) the bulk velocity, \(g\) a deterministic force, and \(W\) a stochastic force driven by a Wiener process. The no-slip boundary condition \(u\mid _{\partial Q}=0\) is imposed. The authors consider the physically relevant hard sphere pressure-density equation of state: there is a limit density \(\overline{\varrho }>0\) such that \(p\in C^{1}[0,\overline{ \varrho })\), \(p(0)=0\), \(p^{\prime }(\varrho )>0\) for any \(0<\varrho < \overline{\varrho }\), \(p^{\prime }(\varrho )\geq a\varrho ^{\gamma -1}\), \(a>0 \), \(\lim_{\varrho \rightarrow \overline{\varrho }-}(\overline{\varrho } -\varrho )^{\beta }p(\varrho )=\overline{p}>0\), for some exponents \(\gamma >1 \), \(\beta >3\). They define the notions of dissipative martingale solution, of entire solution, and of stationary (entire) solution \(((\Omega ,\mathfrak{ F},(\mathfrak{F}_{t})_{t\geq -T},\mathbb{P}),\varrho ,u,W)\) to the above problem on the time interval \([-T,\infty )\). The main result proves that, under appropriate hypotheses on the pressure \(p\), the deterministic driving force \(g\), and the noise diffusion coefficients \(\mathbb{F}\), if \(((\Omega , \mathfrak{F},(\mathfrak{F}_{t})_{t\geq -T},\mathbb{P}),\varrho ,u,W)\) is a dissipative martingale solution to the above problem, such that \(\mathbb{E}[ \mathcal{E}(0)^{4}]<\infty \), \(\mathbb{P}[\overline{\varrho }-\frac{1}{ \left\vert Q\right\vert }\int_{Q}\varrho (0,\cdot )dx>\delta ]=1\), \(\mathbb{P }[\frac{\varrho u(0)}{\varrho (0)}\in W_{0}^{1,2}(Q;\mathbb{R}^{d})]=1\), for some deterministic constant \(\delta >0\), there is a sequence \( T_{n}\rightarrow \infty \) and a stationary solution \(((\widetilde{\Omega }, \widetilde{\mathfrak{F}},(\widetilde{\mathfrak{F}}_{t})_{t\geq -T}, \widetilde{\mathbb{P}}),\widetilde{\varrho },\widetilde{u},\widetilde{W})\) such that \(\frac{1}{T_{n}}\int_{0}^{T_{n}}\mathcal{L}_{\mathcal{T}}[\mathcal{ S}_{t}[\varrho ,u,W]]dt\rightarrow \mathcal{L}_{\mathcal{T}}[\mathcal{S}_{t}[ \widetilde{\varrho },\widetilde{u},\widetilde{W}]\) narrowly as \(n\rightarrow \infty \), where \(\mathcal{L}_{\mathcal{T}}\) is the law of the solution, \( \mathcal{E}\) is càdlàg in \([0,\infty )\), thus progressively measurable, and \(\mathcal{E}(t)=\int_{Q}E(\varrho ,\varrho u)(t,\cdot )dx\), a.a. in \((-T,\infty )\) \(\mathbb{P}\)-a.s, \(E\) being the total energy defined as: \(E(\varrho ,m)=\frac{1}{2}\frac{\left\vert m\right\vert ^{2}}{\varrho } +P(\varrho )\), for \(\varrho >0\), \(E(\varrho ,m)=0\), if \(\varrho =0\), \(m=0\), \( E(\varrho ,m)=\infty \), otherwise, the pressure potential \(P\) being defined through \(P^{\prime }(\varrho )\varrho -P(\varrho )=p(\varrho )\), \(P(0)=0\). For the proof, the authors prove an exponential bound for \(\mathbb{E}[ \mathcal{E}(t)^{m}]\), assuming appropriate hypotheses on the initial data. They also prove an asymptotic compactness result for a sequence of dissipative martingale solutions. The paper ends with a study concerning the ergodic structure of the system under consideration. In an appendix, the authors prove the existence of a dissipative martingale solution to the initial value problem.