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Regularity results for a class of widely degenerate parabolic equations.

Authors :
Ambrosio, Pasquale
Passarelli di Napoli, Antonia
Source :
Advances in Calculus of Variations. Jul2024, Vol. 17 Issue 3, p805-829. 25p.
Publication Year :
2024

Abstract

Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE u t - div ⁡ ( (| D ⁢ u | - ν) + p - 1 ⁢ D ⁢ u | D ⁢ u | ) = f in ⁢ Ω T = Ω × (0 , T) , where Ω is a bounded domain in ℝ n for n ≥ 2 , p ≥ 2 , ν is a positive constant and (⋅) + stands for the positive part. Assuming that the datum f belongs to a suitable Lebesgue–Sobolev parabolic space, we establish the Sobolev spatial regularity of a nonlinear function of the spatial gradient of the weak solutions, which in turn implies the existence of the weak time derivative u t . The main novelty here is that the structure function of the above equation satisfies standard growth and ellipticity conditions only outside a ball with radius ν centered at the origin. We would like to point out that the first result obtained here can be considered, on the one hand, as the parabolic counterpart of an elliptic result established in [L. Brasco, G. Carlier and F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations [corrected version of mr2584740], J. Math. Pures Appl. (9) 93 2010, 6, 652–671], and on the other hand as the extension to a strongly degenerate context of some known results for less degenerate parabolic equations. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
18648258
Volume :
17
Issue :
3
Database :
Academic Search Index
Journal :
Advances in Calculus of Variations
Publication Type :
Academic Journal
Accession number :
178186538
Full Text :
https://doi.org/10.1515/acv-2022-0062