Statistical solutions for the Navier-Stokes-Fourier system (Q6571441)

The authors consider the Navier-Stokes-Fourier system that describes the time evolution of the mass density \(\varrho =\varrho (t,x)\), the (absolute) temperature \(\vartheta =\vartheta (t,x)\), and the velocity \(u=u(t,x)\) of a general Newtonian compressible heat conducting fluid through the mass conservation (equation of continuity): \(\partial _{t}\varrho +\operatorname{div}_{x}(\varrho u)=0\), the momentum balance (Newton's second law): \( \partial _{t}(\varrho u)+\operatorname{div}_{x}(\varrho u\otimes u)+\nabla _{x}p(\varrho ,\vartheta )=\operatorname{div}_{x}\mathbb{S}(\mathbb{D}_{x}u)+g\), and the internal energy balance (first law of thermodynamics): \(\partial _{t}(\varrho e(\varrho ,\vartheta ))+\operatorname{div}_{x}(\varrho e(\varrho ,\vartheta )u)+\operatorname{div}_{x}q=\mathbb{S}( \mathbb{D}_{x}u):\mathbb{D}_{x}u-p(\varrho ,\vartheta )\operatorname{div}_{x}u+\varrho Q\), completed with the constitutive relations: equations of state (Boyle-Mariotte law): \(p(\varrho ,\vartheta )=\varrho \vartheta \), \( e(\varrho ,\vartheta )=c_{v}\vartheta \), Newton's rheological law: \(\mathbb{S }(\mathbb{D}_{x}u)=\mu (\nabla _{x}u+\nabla _{x}u^{t}-\frac{2}{3} \operatorname{div}_{x}uI+\eta \operatorname{div}_{x}uI)\), \(\mu >0\), \(\eta \geq 0\), and Fourier's law: \( q=-\kappa \nabla _{x}\vartheta \), \(\kappa >0\). The initial conditions \( \varrho (0,\cdot )=\varrho _{0}\), \(u(0,\cdot )=u_{0}\), \(\vartheta (0,\cdot )=\vartheta _{0}\) are imposed. The authors recall the local existence of a unique regular solution to this problem, as proved by \textit{Y. Cho} and \textit{H. Kim} in [J. Differ. Equations 228, No. 2, 377--411 (2006; Zbl 1139.35384)], under regularity results on the data. The life span (maximal existence time) \(T_{\max}\) of the local solution is defined as \( 0<T_{\max}=\sup\{T>0\mid \) a regular solution \((\varrho ,\vartheta ,u)\) exists in \([0,T]\}\). Assuming that \(c_{v}>1\), \(\mu >0\), \(\eta \geq 0\), and \(\kappa >0\), the authors then quote from \N[``On Nash's conjecture for models of viscous, compressible, and heat conducting fluids'', Preprint No. IM 2022 6 IM ASCR Prague] by \textit{E. Feireisl}, \textit{H. Wen}, and \textit{C. Zhu}, \Nuniform bounds on a local strong solution \((\varrho ,\vartheta ,u)\) to the above problem and the partial derivatives with respect to time. They prove a stability result with respect to the data of the problem: For any sequence \( D_{n}\) of data that converges to admissible data \(D\) in some appropriate topology, \(\liminf_{n\rightarrow \infty }T_{\max}[D_{n}]\geq T_{\max}[D]\), and \( (\varrho ,\vartheta ,u)[D_{n}]\rightarrow (\varrho ,\vartheta ,u)[D]\) weakly-(\(\ast \)) in \(X_{T}\), for any \(0<T<T_{\max}[D]\). They recall local existence results from \textit{A. Valli} [Ann. Mat. Pura Appl., IV. Ser. 130, 197--213 (1982; Zbl 0599.76081)] and [Ann. Mat. Pura Appl., IV. Ser. 132, 399--400 (1982; Zbl 0599.76082)] for regular initial data \(\varrho _{0}\in W^{1,q}(\mathbb{T}^{3})\), \(\varrho _{0}\geq 0\), \(\vartheta _{0}\in W^{2,2}(\mathbb{T}^{3})\), \(\vartheta _{0}>0\), \(u_{0}\in W^{2,2}(\mathbb{T} ^{3};\mathbb{R}^{3})\), and from \textit{D. Serre} [Contemp. Math. 526, 339--358 (2010; Zbl 1223.35230)], under the regularity hypotheses \(\varrho _{0}\in W^{k,2}( \mathbb{T}^{3})\), \(\varrho _{0}>0\), \(\vartheta _{0}\in W^{k,2}(\mathbb{T} ^{3})\), \(\vartheta _{0}>0\), \(u_{0}\in W^{k,2}(\mathbb{T}^{3};\mathbb{R}^{3})\). They prove a local existence result assuming that \(\varrho _{0}\in W^{1,q}(\mathbb{T}^{3})\), \(\varrho _{0}>0\), \(\vartheta _{0}\in W^{2,2}( \mathbb{T}^{3})\), \(\vartheta _{0}>0\), \(u_{0}\in W^{2,2}(\mathbb{T}^{3}; \mathbb{R}^{3})\). In the three cases, regularity results are also imposed on \(g\), \(Q\), and their derivatives with respect to time, and the authors prove a regularity result for a solution to the above problem. The paper ends with the introduction of a suitable topology on the phase space \(X\) and the corresponding notion of statistical solution via the push forward measure argument. Let \(\mathbf{U}=(\varrho ,\vartheta ,u)\) the solution to the Navier-Stokes system, \(\mathbf{U}_{0}=(\varrho _{0},\vartheta _{0},u_{0})\) the initial data, and \((f,p)\in F\times P\) the driving force and the parameter set, respectively. The authors extend the strong solution to the above problem as: \(\mathbf{U}[\mathbf{U}_{0};f;p](t,\cdot )\) = the strong solution to the Navier-Stokes-Fourier system, if \(t<T_{\max}[\mathbf{U} _{0};f;p]\), \(U_{\infty }=(0,0,0)\), if \(t\geq T_{\max}\) and \(\mathbf{U}_{0}\in X^{+}=\{(\varrho ,\vartheta ,u)\in X\mid \) \(\left\Vert \varrho ^{-1}\right\Vert _{C(\mathbb{T}^{3})}<\infty \), \(\left\Vert \vartheta ^{-1}\right\Vert _{C(\mathbb{T}^{3})}<\infty \}\), \(U_{\infty }=(0,0,0)\), if \( \mathbf{U}_{0}=\mathbf{U}_{\infty }\). The family of Markov operators \( \mathcal{M}_{t}:\mathfrak{P}[D_{X,F,P}]\rightarrow \mathfrak{P}[X_{\infty }^{+}]\) defined as \(\mathcal{M}_{t}(\mathcal{V})=\int_{X^{+}\times F\times P}\delta \mathbf{U}[D](t,\cdot )d\mathcal{V}(D)\), \(t\geq 0\), for any \( \mathcal{V}\) supported by admissible data, is called a statistical solution to the Navier-Stokes-Fourier system. In an autonomous case, the authors prove a semigroup property for this statistical solution. In the last part of their paper, the authors consider the Monte Carlo-type approximation of statistical solutions with random initial data leaving the forcing and parameters \((f,p)\) deterministic.