Global existence and maximal regularity of solutions of gradient systems
The author applies the Galerkin method in order to prove a maximal regularity result for the following abstract gradient system \[ \begin{cases} u'(t)+\nabla_g E(u(t))=f(t)\quad \text{for a.a.}\;t\in[0,T],\\ u(0)=u_0. \end{cases} \] Here, \(E: V\to\mathbb R\) is a continuously differentiable function over a real Banach space \(V\) densely and compactly embedded into a real Hilbert space \(H,\) \(\nabla_g E\) stands for the gradient of \(E\) with respect to some metric \(g,\) \(f\in L^2(0,T;H)\) and \(u_0\in V.\) The abstract result is applied to nonlinear diffusion equations and to nondegenerate quasilinear parabolic equations with nonlocal coefficients.
scientific article; zbMATH DE number 5838454