On a homogenization problem for multi-dimensional parabolic differential operators of hydrodynamics (Q2760707)

The article is devoted to the homogenization problem for the system of parabolic equations which describes the motion of a viscous inhomogeneous incompressible fluid in a bounded domain \(\Omega\subset\mathbb R^d\), \(d > 2\): NEWLINE\[NEWLINE \partial_t\vec v - \text{div}_x(M:\nabla_x \vec v) = \vec f - \nabla_{x} p. NEWLINE\]NEWLINE It is assumed that the tensor viscosity components are rapidly oscillating functions \(M_{\varepsilon}^{ijkl}(\vec x,t)\) where \(\varepsilon\) is a small parameter characterizing oscillations which satisfy the following conditions: NEWLINE\[NEWLINE \begin{gathered} M_{\varepsilon}^{ijkl}(\vec x,t)\;\text{are measurable on } \Omega\times [0,T], \\ |M_{\varepsilon}^{ijkl}(\vec x,t)|\leq\lambda_0 \;\text{a.e. in } \Omega\times [0,T], \\ \int_{\Omega}\sum_{i,j,k,l=1}^{d} M_{\varepsilon}^{ijkl}(\vec x,t)\partial_ju_l(\vec x)\partial_iu_k (\vec x)d\vec x \geq \lambda_1\|\vec u\|^2_{J_0^1(\Omega)}\quad \text{~for almost all } t\in [0,T]. \end{gathered} NEWLINE\]NEWLINE Here \(\lambda_0\), \(\lambda_1\) are positive constants independent of \(\varepsilon\).NEWLINENEWLINENEWLINEThe problem is reduced to studying the limit in the equation NEWLINE\[NEWLINE \partial_t\vec v_{\varepsilon} - \text{div}_x(M_{\varepsilon}:\nabla_x \vec v_{\varepsilon}) = \vec f, \quad\vec v_{\varepsilon} \in W_0, NEWLINE\]NEWLINE for \(f\in L_2(0,T;J^{-1}(\Omega))\), where NEWLINE\[NEWLINEW_0 = \{\vec{\varphi} \mid \vec{\varphi}\in L_2(0,T;J_0^1(\Omega)), \partial_{t}\vec{\varphi}\in L_2(0,T;J^{-1} (\Omega)), \vec{\varphi}|_{t=0} = 0\}.NEWLINE\]NEWLINE The main result reads as follows: Let \(\{\vec v_{\varepsilon}\}_ {\varepsilon > 0} \) be a set of solutions to the equation and let the coefficients \(M_{\varepsilon}^{ijkl}\) satisfy the above-mentioned conditions. Then there exist a subsequence \(\varepsilon_{n}\to 0\) and a collection \(\{M_*^{ijkl}(\vec x,t)\}_ {i,j,k,l = 1,\dots,d}\) such that NEWLINE\[NEWLINE \begin{gathered} \vec v_{\varepsilon_n} \to \vec v_* \quad \text{weakly in } W_0\quad \text{as } n\to +\infty, \\ \partial_t\vec v_* - \text{div}_x(M_*:\nabla_x \vec v_*) = \vec f, \quad\vec v_* \in W_0, \end{gathered} NEWLINE\]NEWLINE where \(M_*\) satisfies the conditions for \(M_{\varepsilon}^{ijkl}\) with a constant \(\tilde{\lambda}_0\) which only depends on \(\lambda_0\), \(\lambda_1\).