Well-posedness theory for degenerate parabolic equations on Riemannian manifolds

Abstract

We consider the degenerate parabolic equationThe fact that the notion of divergence appearing in the equation depends on the metric g requires revisiting the standard entropy admissibility concept. We derive it under an additional geometry compatibility condition and, as a corollary, we introduce the kinetic formulation of the equation on the manifold. Using this concept, we prove well-posedness of the corresponding Cauchy problem.2010 Mathematics Subject Classification. 35K65, 42B37, 76S99.

Connection between a degenerate particle flow model and a free boundary problem

“…follows immediately from (14). Therefore Leray-Schauder fix point theorem yields the existence of a fix point ρ = ρ (ε,δ) for T (•, 1), that is, a solution ρ (ε,δ) ∈ L 2 (0, T; H m (Ω)) to (10), (11).…”

“…to be solved for ρ (ε,δ) ∈ L 2 (0, T; H m (Ω)). We reformulate (10) (11) as a fix point problem for the mapping…”

“…Uniform estimates and compactness argument. Now we aim at taking the limit ε → 0 in (10), (11). First notice that testing (10) against ρ (ε,δ) ∈ L 2 (0, T; H m (Ω)) and exploiting the nonnegativity of f δ lead easily to the bounds…”

“…For more results on entropy solution we refer to [1,5,6,16]. The well-posedness theory of weak entropy solution to strongly degenerate parabolic equation on Riemannian manifold is established in [10]. For general theory of weak solution of degenerate parabolic equation we refer to [15,17], and furthermore to [12,Section 5] for a short review of the development of the mathematical theory in this direction.…”

Well-posedness theory for stochastically forced conservation laws on Riemannian manifolds

“…For scalar conservation laws defined on manifolds, the development of a theory of well-posedness and numerical approximations (of Kružkov-DiPerna solutions) was initiated by LeFloch and co-authors [1,2,6,7,8,44,45,46] (see also Panov [52,53]). The subject has been extended in several directions by different authors, including Giesselmann [30], Dziuk, Kröner, and Müller [24], Lengeler and Müller [47], Giesselmann and Müller [31], and Kröner, Müller, and Strehlau [40], and Graf, Kunzinger, and Mitrovic [32].…”