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Weak solutions and maximal regularity for abstract evolution inequalities. (English) Zbl 0858.35073

Summary: We study the regularity and the approximation of the solution of a parabolic evolution inequality \[ u'(t)+A(t)u(t)+ \partial\phi(u(t))\ni f(t), \qquad u(0)=u_0\tag{\(*\)} \] in the framework of a Hilbert triple \(\{V,H,V'\}\); here \(A(t)\), \(t\in]0,T[\), is a family of linear continuous and coercive operators from \(V\) to \(V'\) and \(\phi\) is a proper convex l.s.c. function defined in \(V\) with values in \(]-\infty,\infty]\). We give a weak formulation of \((*)\) which allows \(f\in H^{-1/2+\delta}(0,T;H)\), for some \(\delta>0\) and we prove that the weak solution belongs to \(H^{1/2-\varepsilon} (0,T;H)\), \(\forall \varepsilon>0\), besides the standard \(L^2(0,T;V)\cap L^\infty(0,T;H)\).
Under suitable regularity hypotheses on the data \(f\) (in particular \(f\in H^{1/2+\delta}(0,T;H)\) for some \(\delta>0\) is sufficient) and \(u_0\), the strong solution is in \(H^{3/2-\varepsilon} (0,T;H)\), \(\forall \varepsilon>0\). In this case we prove an optimal error estimate for the backward Euler discretization of \((*)\) in \(L^2(0,T;V)\cap L^\infty(0,T;H)\) and in \(H^1(0,T;H)\). We apply these results to the porous medium and the Stefan problem.

MSC:

35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
35D10 Regularity of generalized solutions of PDE (MSC2000)
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