Compactness of the Green operator of nonlinear diffusion equations: Application to Boussinesq type systems in fluid dynamics (Q1904454)

The authors investigate nonlinear diffusion problems of the type \(v_t- \Delta\beta(v)= g\) in \((0, T)\times \Omega\), \(\beta(v)= 0\) on \((0, T)\times \partial \Omega\), \(v(0, x)= v_0(x)\) in \(\Omega\), where \(\Omega\) is an open regular bounded domain in \(\mathbb{R}^N\) and \(\beta\) is a continuous nondecreasing function such that \(\beta(0)= 0\). First they prove the compactness of the Green type operators \(g\to v\) and \(g\to \beta(v)\) for \(v_0\in L^1(\Omega)\) fixed. Then the results obtained are applied to a generalized Boussinesq type system describing the motion of a fluid inside \(\Omega\), in which a diffusion process take place simultaneously. For piecewise linear or fast diffusion operators, i.e. \(\beta(s)= |s|^{m- 1} s\), \(m\in (0, 1]\), the existence theorem and some regularity results are established.