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Some results on homogenization of convection-diffusion equations. (English) Zbl 0742.35007

The authors consider some models of degenerate convection-diffusion equations with oscillating coefficients. They prove that the homogenization process produces nonlocal and memory effects when the diffusion is longitudinal.
Let \(\Omega_ x\) be an open subset of \(\mathbb{R}^ n\) with smooth boundary \(\partial\Omega_ x\), and \(\Omega_ y\) any open subset of \(\mathbb{R}^ m\); set \(\Omega=\Omega_ x\times\Omega_ y\). Let \(C\) be a vector field in \((C^ 1(\overline\Omega))^ n\). Consider the homogenization equation \[ \partial_ tv^ \varepsilon+q^ \varepsilon(y)\hbox{div}_ x(Cv^ \varepsilon)-\hbox{div}_ x(\sigma^ \varepsilon(x) \hbox{grad} v^ \varepsilon)=0,\quad \hbox{in } \mathbb{R}_ +\times\Omega, \]
\[ v^ \varepsilon|_{\partial\Omega_ x}=0\quad\hbox{ in }\mathbb{R}_ +\times\Omega_ y\quad\hbox { and }\quad v^ \varepsilon|_{t=0}=v_ 0\quad\hbox { in } \Omega. \] Assume that \(\sigma^ \varepsilon(x)\in(L^ \infty(\Omega_ x))^{n\times n}\) is a positive definite tensor which converges to \(\sigma(x)\) in the sense of homogenization for \(\varepsilon\to 0\), and \(q^ \varepsilon(y)\) in \(L^ \infty(\Omega_ y)\) converges to \(q(y)\) in \(L^ \infty(\Omega_ y)\)- weak star for \(\varepsilon\to 0\) and \(q_ -\leq q(y)\leq q_ +\). Then there exists a nonnegative parametrized measure \(d\omega_ y\) with support in \(I=[q_ -,q_ +]\) such that \(v^ \varepsilon\) of the above equation converges weakly in \(L^ 2(\mathbb{R}_ +\times\Omega_ y;H^ 1_ 0(\Omega_ x))\) to the solution \(v\) of the system \[ \partial_ tv+q(y)\hbox{div}_ x(Cv)-\hbox{div}_ x(\sigma(x)\hbox{grad} v)=\int_ I\hbox{div}_ x(CV)d\omega_ y(\mu), \]
\[ \partial_ tV+\mu\hbox{div}_ x(CV)-\hbox{div}_ x(\sigma(x)\hbox{grad}_ xV)=\hbox{div}_ x(CV),\quad (t,x,\mu)\in\mathbb{R}_ +\times\Omega\times I. \]
\[ v\mid_{\partial\Omega_ x}=0\quad\hbox{ in }\mathbb{R}_ +\times\Omega_ y,\qquad V\mid_{\partial\Omega_ x}=0\quad\hbox{ in }\mathbb{R}_ +\times\Omega_ y\times I, \]
\[ v\mid_{t=0}=v_ 0\quad\hbox{ in } \Omega\quad\hbox { and }\quad V\mid_{t=0}=0\quad\hbox{ in } \Omega\times I. \]

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K20 Initial-boundary value problems for second-order parabolic equations
35K65 Degenerate parabolic equations
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