General Parabolic Mixed Order Systems in Lp and Applications

• Main Authors: Denk, Robert. (Author), Kaip, Mario. (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic
• Language: English
• Published: Cham : Springer International Publishing : Imprint: Birkh©Þuser, 2013.
• Series: Operator Theory: Advances and Applications, 239
• Subjects: Mathematics.; Operator theory.; Partial differential equations.; Mathematical physics.; Partial Differential Equations.; Mathematical Physics.; Operator Theory.
• Online Access: View fulltext via EzAccess

 In this text, a theory for general linear parabolic partial differential equations is established, which covers equations with inhomogeneous symbol structure as well as mixed order systems. Typical applications include several variants of the Stokes system and free boundary value problems. We show well-posedness in Lp-Lq-Sobolev spaces in time and space for the linear problems (i.e., maximal regularity), which is the key step for the treatment of nonlinear problems. The theory is based on the concept of the Newton polygon and can cover equations that are not accessible by standard methods as, e.g., semigroup theory. Results are obtained in different types of non-integer Lp-Sobolev spaces as Besov spaces, Bessel potential spaces, and TriebelỚ<U+001c>Lizorkin spaces. The latter class appears in a natural way as traces of Lp-Lq-Sobolev spaces. We also present a selection of applications in the whole space and on half-spaces. Among others, we prove well-posedness of the linearizations of the generalized thermoelastic plate equation, the two-phase NavierỚ<U+001c>Stokes equations with BoussinesqỚ<U+001c>Scriven surface, and the Lp-Lq two-phase Stefan problem with GibbsỚ<U+001c>Thomson correction.