Local existence, uniqueness and smooth dependence for nonsmooth quasilinear parabolic problems
Please always quote using this URN:urn:nbn:de:0296-matheon-6426
- We prove local existence, uniqueness, Hölder regularity in space and time, and smooth dependence in Hölder spaces for a general class of quasilinear parabolic initial boundary value problems with nonsmooth data. As a result the gap between low smoothness of the data, which is typical for many applications, and high smoothness of the solutions, which is necessary for the applicability of differential calculus to abstract formulations of the initial boundary value problems, has been closed. The theory works for any space dimension, and the nonlinearities are allowed to be nonlocal and to have any growth. The main tools are new maximal regularity results [19, 20] in Sobolev–Morrey spaces for linear parabolic initial boundary value problems with nonsmooth data, linearization techniques and the Implicit Function Theorem.
Author: | Jens Griepentrog, Lutz Recke |
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URN: | urn:nbn:de:0296-matheon-6426 |
Referee: | Fredi Tröltzsch |
Document Type: | Preprint, Research Center Matheon |
Language: | English |
Date of first Publication: | 2009/06/08 |
Release Date: | 2009/07/27 |
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Institute: | Humboldt-Universität zu Berlin |
MSC-Classification: | 35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Kxx Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35] / 35K60 Nonlinear initial value problems for linear parabolic equations |
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Kxx Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35] / 35K90 Abstract parabolic equations | |
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Rxx Miscellaneous topics (For equations on manifolds, see 58Jxx; for manifolds of solutions, see 58Bxx; for stochastic PDE, see also 60H15) / 35R05 Partial differential equations with discontinuous coefficients or data | |
Preprint Number: | 658 |