An approach by difference to the porous medium equation with convection
The author discusses the existence and the uniqueness of the solution to the Cauchy problem for the porous medium equation with convection terms: \[ (1)\quad u_t= \Delta \phi(u)+ \sum^N_{i= 1} {\partial\over \partial x_i} F_i(u),\;x\in \mathbb{R}^n,\;t> 0;\qquad (2)\quad u(0, x)= u_0(x),\;x\in \mathbb{R}^n. \] The functions \(\phi\) is assumed to be strictly increasing, locally Lipschitz continuous on \(\mathbb{R}\) and satisfies \(\phi(0)= 0\). The functions \(F_i\) are defined on \(\mathbb{R}\), \(F_i(0)= 0\), and \(|F_i(r)- F_i(s)|/ |\phi(r)- \phi(s)|\) are bounded for \(r\), \(s\) in every bounded subinterval of \(\mathbb{R}\). The author provides a direct method for solving the problem (1)--(2) via a method of difference approximation which converges to a limit of a semigroup \(\{S(t): t\geq 0\}\) of contractions on \(L^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)\) associated with the problem (1), (2).
scientific article; zbMATH DE number 863655