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.


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