Homogenization of first order hyperbolic equations and application to miscible flow in porous media
The homogenization of the hyperbolic equation \[ \partial_ tu^{\varepsilon}+a^{\varepsilon}(t,y)\partial_ xu^{\varepsilon}=0,\quad t>0,\quad x\in {\mathbb R},\quad y\in \Omega \subset {\mathbb R}^ N \tag{1} \] with initial data and boundary conditions when \(x\in]0,1[\) is considered. For this purpose the \(L^{\infty}\)-weak*-limit of some holomorphic functions of the type \(\phi^{\varepsilon}_ y(\lambda)=(\lambda - A^{\varepsilon}(y))^{-1}\), \(\lambda\in {\mathbb C}\setminus [m,M]\) with \(0\leq m\leq A^{\varepsilon}(y)\leq M\) for a.e. \(y\), by using the integral representation of Nevanlinna-Pick's holomorphic functions is studied first.
scientific article; zbMATH DE number 4146642
scientific article