General Parabolic Mixed Order Systems in Lp and Applications

  • Robert Denk
  • Mario Kaip

Part of the Operator Theory: Advances and Applications book series (OT, volume 239)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Robert Denk, Mario Kaip
    Pages 1-10
  3. Robert Denk, Mario Kaip
    Pages 11-67
  4. Robert Denk, Mario Kaip
    Pages 69-141
  5. Robert Denk, Mario Kaip
    Pages 143-185
  6. Robert Denk, Mario Kaip
    Pages 187-228
  7. Back Matter
    Pages 229-250

About this book


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 Triebel–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 Navier–Stokes equations with Boussinesq–Scriven surface, and the Lp-Lq two-phase Stefan problem with Gibbs–Thomson correction.


Newton-polygon Triebel-Lizorkin spaces free boundary problems mixed order systems parabolic differential equations

Authors and affiliations

  • Robert Denk
    • 1
  • Mario Kaip
    • 2
  1. 1.FB Mathematik und StatistikUniversität KonstanzKonstanzGermany
  2. 2.FB Mathematik und StatistikUniversität KonstanzKonstanzGermany

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