Global existence and maximal regularity of solutions of gradient systems

Abstract

In this article, we use a Galerkin method to prove a maximal regularity result for the following abstract gradient systemThis abstract result is applied to nonlinear diffusion equations and to nondegenerate quasilinear parabolic equations with nonlocal coefficients.

Existence of solutions for evolution equations in Hilbert spaces with anti-periodic boundary conditions and its applications

“…For the proof of this fact we invite the reader to see [2], Remarks 6 and 7. We note also, from assumption (A4), that we have for every (t, u)…”

“…We check that all assumptions of Theorem 2 are satisfied. The verification of assumption (A7) can be made as in [2], Proof of Corollary 10. As a consequence of Theorem 2, we obtain:…”

Global Existence for Nonlocal Quasilinear Diffusion Systems in Non-Isotropic Non-Divergence Form

“…In particular, we can consider fractional or nonlocal derivatives of any order σ ≤ s ≤ 1. This generalises the classical gradient, and is conceptually similar to the ideas of Boussandel (see [12] and [13]), where he considers the classical gradient weighted by a measure. For general operators A satisfying (2), the theorem applies with θ = 0 and σ < s ≤ 1.…”

Existence and uniqueness of solutions for gradient systems without a compactness embedding condition

“…Problem (3.2) admits a maximal solution u m ∈ W 1,2 loc ([0, T m ); V m ). P r o o f. See [7], Proof of Theorem 4, Part 1. Lemma 3.2.…”

“…g = •, • H , problem (1.1) was considered in the literature; see for example [9], Theorem 6.1. When g depends further on the space variable, problem (1.1) was studied in [7], where the embedding V ֒→ H is supposed to be compact, the energy E is assumed to be H-elliptic (i.e., the functional u → E(u) + 1 2 ω u 2 H is convex and weakly coercive for some constant ω ∈ R) instead of convex and weakly coercive and the metric g satisfied some continuity condition.…”