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Compactness of the Green operator of nonlinear diffusion equations: Application to Boussinesq type systems in fluid dynamics. (English) Zbl 0841.35048
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.
Reviewer: O.Titow (Berlin)

##### MSC:
 35K57 Reaction-diffusion equations 35Q35 PDEs in connection with fluid mechanics
##### Keywords:
fast diffusion operators; existence theorem; regularity
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