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Strong convergence of bounded sequences of solutions of porous medium equations. (English) Zbl 1007.35109
The paper deals with the differential inclusion \((\beta (u))_t \ni \Delta u + g(x,t)\) on \(\Omega \times (0,T)\), where \(\beta \) is a maximal monotone graph. The equation is completed with boundary and initial conditions \(u=\varphi \). For sufficiently smooth data the \(L^2\)-gradient estimate of the solution is derived; the estimate is on the set where the solution is small. The estimate allows to show that approximating smooth solutions converge strongly to the weak solution. The theorem is illustrated by an example.
35R70 PDEs with multivalued right-hand sides
76S05 Flows in porous media; filtration; seepage
35Q35 PDEs in connection with fluid mechanics