Publication: On the regularity of the solution map of the incompressible porous media equation
Program
KU-Authors
KU Authors
Co-Authors
Publication Date
Language
Type
Embargo Status
Journal Title
Journal ISSN
Volume Title
Alternative Title
Abstract
In this paper, we consider the incompressible porous media equation on the Sobolev space H-s(R-2), s > 2. We provide a Lagrangian formulation of this equation on the Sobolev-type diffeomorphism group D-s(R-2), s > 2. It turns out that this Lagrangian formulation generates an analytic dynamics. This analyticity in the Lagrangian picture will immediately lead to the result that the particle trajectories of the incompressible porous media flow are analytic curves in R-2. In the Eulerian picture, the situation is drastically different. We prove that for T > 0, the time T solution map of the incompressible porous media equation rho(0) (sic) rho(T), mapping the initial value of the solution to its time T value, is nowhere locally uniformly continuous and hence nowhere locally Lipschitz.
Source
Publisher
Springer Int Publ Ag
Subject
Mathematics
Citation
Has Part
Source
Zeitschrift für Angewandte Mathematik und Physik
Book Series Title
Edition
DOI
10.1007/s00033-023-02084-9