2007, Griepentrog, Jens A.

In this text we introduce new classes of Sobolev-Morrey spaces being adequate for the regularity theory of second order parabolic boundary value problems on Lipschitz domains of space dimension n ≥ 3 with nonsmooth coefficients and mixed boundary conditions. We prove embedding and trace theorems as well as invariance properties of these spaces with respect to localization, Lipschitz transformation, and reflection. In the second part [11] of our presentation we show that the class of second order parabolic systems with diagonal principal part generates isomorphisms between the above mentioned Sobolev-Morrey spaces of solutions and right hand sides.

2014, Griepentrog, Jens A.

An analytical model for multicomponent systems of nonlocally interacting particles is presented. Its derivation is based on the principle of minimization of free energy under the constraint of conservation of particle number and justified by methods established in statistical mechanics. In contrast to the classical Cahn-Hilliard theory with higher order terms, the nonlocal theory leads to an evolution system of second order parabolic equations for the particle densities, weakly coupled by nonlinear and nonlocal drift terms, and state equations which involve both chemical and interaction potential differences. Applying fixed-point arguments and comparison principles we prove the existence of variational solutions in suitable Hilbert spaces for evolution systems. Moreover, using maximal regularity for nonsmooth parabolic boundary value problems in Sobolev-Morrey spaces and comparison principles, we show uniqueness, global regularity and uniform positivity of solutions under minimal assumptions on the regularity of interaction. Applying a refined version of the Lojasiewicz-Simon gradient inequality, this paves the way to the convergence of solutions to equilibrium states. We conclude our considerations with the presentation of simulation results for phase separation processes in ternary systems.

2007, Griepentrog, Jens A.

This text is devoted to maximal regularity results for second order parabolic systems on Lipschitz domains of space dimension n ≥ 3 with diagonal principal part, nonsmooth coefficients, and nonhomogeneous mixed boundary conditions. We show that the corresponding class of initial boundary value problems generates isomorphisms between two scales of Sobolev–Morrey spaces for solutions and right hand sides introduced in the first part [12] of our presentation. The solutions depend smoothly on the data of the problem. Moreover, they are Hölder continuous in time and space up to the boundary for a certain range of Morrey exponents. Due to the complete continuity of embedding and trace maps these results remain true for a broad class of unbounded lower order coefficients.