On Besov Regularity of Temperatures

Abstract

We prove space-time parabolic Besov regularity in terms of integrability of Besov norms in the space variable for solutions of the heat equation on cylindrical regions based on Lipschitz domains.

Regularity in Sobolev and Besov Spaces for Parabolic Problems on Domains of Polyhedral Type

“…The restriction m ≥ 2 in Theorem 7 comes from the fact that we require s 2 = m > d 2 = 3 2 in (4.23). This assumption can probably be weakened, since we expect the solution to satisfy u ∈ L 2 ([0, T ], H s (D)) for all s < 3 2 , see also Remark 11 and the explanations given there.…”

“…To the best of our knowledge, not so many results in this direction are available so far. For parabolic equations, first results for the special case of the heat equation have been reported in [2][3][4], but for a slightly different scale of Besov spaces.…”

Parabolic Besov Regularity for the Heat Equation

“…For our first result, we shall use the parabolic analog of Theorem 3.1 in [8], essentially contained in Corollary 5.2 of [2]. Our second result is a corollary of the first one if we use Theorem 1.1 in [3].…”

“…For the proof of Theorem 2, we shall make use of the next result which is contained in Theorem 1.1 in [3] and remarks following it.…”

Besov regularity of parabolic and hyperbolic PDEs

“…The restriction m ≥ 2 in Theorem 4.10 comes from the fact that we require m > d p = 3 2 in (4.26). This assumption can probably be weakened, since we expect the solution to satisfy u ∈ L 2 ((0, T ), W s 2 (K)) for all s < 3 2 , see also Remark 4.7 and the explanations given there. Moreover, the restriction a ≥ − 1 2 in Theorem 4.10 comes from Corollary 2.5 that we applied.…”