Regularity results for a class of widely degenerate parabolic equations

Pasquale Ambrosio; Antonia Passarelli di Napoli

doi:10.1515/acv-2022-0062

Article

Regularity results for a class of widely degenerate parabolic equations

Published/Copyright: August 25, 2023

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From the journal Advances in Calculus of Variations Volume 17 Issue 3

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.

Keywords: Degenerate parabolic equations; higher differentiability; Sobolev regularity

MSC 2020: 35B45; 35B65; 35D30; 35K10; 35K65

Communicated by Verena Bögelein

Funding source: Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni

Award Identifier / Grant number: CUP_E53C22001930001

Funding source: Università degli Studi di Napoli Federico II

Award Identifier / Grant number: 000022-75-2021-FRA-PASSARELLI

Funding statement: P. Ambrosio and A. Passarelli di Napoli are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). P. Ambrosio has been partially supported through the INdAM–GNAMPA 2023 Project “Risultati di regolarità per PDEs in spazi di funzione non-standard” (CUP_E53C22001930001). A. Passarelli di Napoli has been partially supported through the INdAM–GNAMPA 2023 Project “Su alcuni problemi di regolarità del calcolo delle variazioni con convessità degenere” (CUP_E53C22001930001). A. Passarelli di Napoli has been partially supported by Università degli Studi di Napoli “Federico II” through the Project FRA (000022-75-2021-FRA-PASSARELLI). A. Passarelli di Napoli has been partially supported by the CNMS (CN 00000023) CUP_E63C22000930007 Spoke 10 Logistica Merci.

Acknowledgements

We gratefully acknowledge Lorenzo Brasco for pointing out to us the reference [1]. Moreover, we would like to thank the reviewers for their valuable comments, which helped to improve this work.

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Received: 2022-07-24

Accepted: 2023-06-05

Published Online: 2023-08-25

Published in Print: 2024-07-01

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https://doi.org/10.1515/acv-2022-0062

Keywords for this article

Degenerate parabolic equations; higher differentiability; Sobolev regularity