Besov regularity of parabolic and hyperbolic PDEs

Abstract

This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations on nonsmooth domains. In particular, we study the smoothness in the specific scale B r τ,τ , 1 τ = r d + 1 p of Besov spaces. The regularity in these spaces determines the approximation order that can be achieved by adaptive and other nonlinear approximation schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms.

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

“…3.3,3.4]. In [22] we extended and generalized these results, which we now wish to transfer to our setting of polyhedral type domains D ⊂ R 3 .…”

“…They will form the basis for obtaining regularity results in Besov spaces later on via suitable embeddings. The results in Sobolev and Kondratiev spaces for Problems However, the extension of the regularity results for Problem 1 to polyhedral type domains follows from very similar arguments as in [22], which is why we merely state the results in Sects. 4.1 and 4.2 and give references for the proofs wherever necessary.…”

“…3.3] we are now able to show the following regularity result in Kondratiev spaces. The proof follows along the same lines as [22,Thm. 3.6].…”

“…However, as a drawback, these results are hard to apply to nonlinear equations since the right-hand sides are not taken from a Banach or quasi-Banach space. The proof of the following theorem is similar to [22,Thm. 3.9] adapted to our setting.…”

“…Throughout the paper we use the same notation as in [22], which for the convenience of the reader is recalled in Appendix 1.…”