Solenoidal lipschitz truncation for parabolic PDEs

Dominic Breit, L. Diening, S. Schwarzacher

Research output: Contribution to journalArticle

31 Citations (Scopus)

Abstract

We consider functions u is an element of L-infinity(L-2) boolean AND L-p(W-1,W- p) with 1 <p <infinity on a time-space domain. Solutions to nonlinear evolutionary PDEs typically belong to these spaces. Many applications require a Lipschitz approximation u(lambda) of u which coincides with u on a large set. For problems arising in fluid mechanics one needs to work with solenoidal (divergencefree) functions. Thus, we construct a Lipschitz approximation, which is also solenoidal. As an application we revise the existence proof for non-stationary generalized Newtonian fluids of Diening, R u. zi. cka and Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010) 1-46. Since div u(lambda) = 0, we are able to work in the pressure free formulation, which heavily simplifies the proof. We also provide a simplified approach to the stationary solenoidal Lipschitz truncation of Breit, Diening and Fuchs, Solenoidal Lipschitz truncation and applications in fluid mechanics.

Original languageEnglish
Pages (from-to)2671-2700
Number of pages30
JournalMathematical Models and Methods in Applied Sciences
Volume23
Issue number14
DOIs
Publication statusPublished - Dec 2013

Keywords

  • Navier-Stokes equations
  • unsteady flows
  • generalized Newtonian fluids
  • existence of weak solutions
  • solenoidal Lipschitz truncation
  • divergence free truncation
  • WEAK SOLUTIONS
  • DEPENDENT VISCOSITY
  • NEWTONIAN FLUIDS
  • EXISTENCE

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