Startseite Regularity results for a class of widely degenerate parabolic equations
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Regularity results for a class of widely degenerate parabolic equations

  • Pasquale Ambrosio ORCID logo EMAIL logo und Antonia Passarelli di Napoli ORCID logo
Veröffentlicht/Copyright: 25. August 2023
Advances in Calculus of Variations
Aus der Zeitschrift Advances in Calculus of Variations Band 17 Heft 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.

References

[1] Z. M. Akhmedov, G. I. Barenblatt, V. M. Entov and A. K. Mirzadzhan-Zade, Nonlinear effects in gas filtration, Izv. AN SSSR. Mekhanika Zhidkosti i Gaza 4 (1969), no. 5, 103–109. 10.1007/BF01015960Suche in Google Scholar

[2] P. Ambrosio, Besov regularity for a class of singular or degenerate elliptic equations, J. Math. Anal. Appl. 505 (2022), no. 2, Paper No. 125636. 10.1016/j.jmaa.2021.125636Suche in Google Scholar

[3] V. Bögelein, F. Duzaar, R. Giova and A. Passarelli di Napoli, Higher regularity in congested traffic dynamics, Math. Ann. 385 (2023), no. 3–4, 1823–1878. 10.1007/s00208-022-02375-ySuche in Google Scholar

[4] L. Brasco, Global L ∞ gradient estimates for solutions to a certain degenerate elliptic equation, Nonlinear Anal. 74 (2011), no. 2, 516–531. 10.1016/j.na.2010.09.006Suche in Google Scholar

[5] 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), no. 6, 652–671. 10.1016/j.matpur.2010.03.010Suche in Google Scholar

[6] S.-S. Byun, J. Oh and L. Wang, Global Calderón–Zygmund theory for asymptotically regular nonlinear elliptic and parabolic equations, Int. Math. Res. Not. IMRN2015 (2015), no. 17, 8289–8308. 10.1093/imrn/rnu203Suche in Google Scholar

[7] A. Clop, A. Gentile and A. Passarelli di Napoli, Higher differentiability results for solutions to a class of non-homogeneous elliptic problems under sub-quadratic growth conditions, Bull. Math. Sci. (2023), 10.1142/S166436072350008X. 10.1142/S166436072350008XSuche in Google Scholar

[8] E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer, New York, 1993. 10.1007/978-1-4612-0895-2Suche in Google Scholar

[9] F. Duzaar, G. Mingione and K. Steffen, Parabolic systems with polynomial growth and regularity, Mem. Amer. Math. Soc. 214 (2011), no. 1005, 1–118. 10.1090/S0065-9266-2011-00614-3Suche in Google Scholar

[10] F. Giannetti, A. Passarelli di Napoli and C. Scheven, Higher differentiability of solutions of parabolic systems with discontinuous coefficients, J. Lond. Math. Soc. (2) 94 (2016), no. 1, 1–20. 10.1112/jlms/jdw019Suche in Google Scholar

[11] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, River Edge, 2003. 10.1142/9789812795557Suche in Google Scholar

[12] T. Isernia, BMO regularity for asymptotic parabolic systems with linear growth, Differential Integral Equations 28 (2015), no. 11–12, 1173–1196. 10.57262/die/1439901046Suche in Google Scholar

[13] T. Kuusi and G. Mingione, New perturbation methods for nonlinear parabolic problems, J. Math. Pures Appl. (9) 98 (2012), no. 4, 390–427. 10.1016/j.matpur.2012.02.004Suche in Google Scholar

[14] P. Lindqvist, On the time derivative in a quasilinear equation, Skr. K. Nor. Vidensk. Selsk. (2008), no. 2, 1–7. Suche in Google Scholar

[15] P. Lindqvist, On the time derivative in an obstacle problem, Rev. Mat. Iberoam. 28 (2012), no. 2, 577–590. 10.4171/rmi/685Suche in Google Scholar

[16] P. Lindqvist, The time derivative in a singular parabolic equation, Differential Integral Equations 30 (2017), no. 9–10, 795–808. 10.57262/die/1495850427Suche in Google Scholar

[17] P. Lindqvist, Notes on the Stationary p-Laplace Equation, Springer Briefs Math., Springer, Cham, 2019. 10.1007/978-3-030-14501-9Suche in Google Scholar

[18] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Gauthier-Villars, Paris, 1969. Suche in Google Scholar

[19] V. Maz’ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Grundlehren Math. Wiss. 342, Springer, Heidelberg, 2011. 10.1007/978-3-642-15564-2Suche in Google Scholar

[20] C. Scheven, Regularity for subquadratic parabolic systems: Higher integrability and dimension estimates, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), no. 6, 1269–1308. 10.1017/S030821050900167XSuche in Google Scholar
Received: 2022-07-24
Accepted: 2023-06-05
Published Online: 2023-08-25
Published in Print: 2024-07-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. A flow approach to the prescribed Gaussian curvature problem in ℍ𝑛+1
  3. A split special Lagrangian calibration associated with frame vorticity
  4. The homogeneous causal action principle on a compact domain in momentum space
  5. The Lp Minkowski problem for q-torsional rigidity
  6. A twist in sharp Sobolev inequalities with lower order remainder terms
  7. Least energy solutions for affine p-Laplace equations involving subcritical and critical nonlinearities
  8. Stochastic homogenisation of free-discontinuity functionals in randomly perforated domains
  9. Existence of minimizers for the SDRI model in 2d: Wetting and dewetting regime with mismatch strain
  10. Generalized minimizing movements for the varifold Canham–Helfrich flow
  11. A characterization of gauge balls in ℍ n by horizontal curvature
  12. Minimizers of 3D anisotropic interaction energies
  13. Regularity results for a class of widely degenerate parabolic equations
  14. On isosupremic vectorial minimisation problems in L ∞ with general nonlinear constraints
  15. Quasiconformal, Lipschitz, and BV mappings in metric spaces
  16. Continuous differentiability of a weak solution to very singular elliptic equations involving anisotropic diffusivity
  17. Optimal transport with nonlinear mobilities: A deterministic particle approximation result
  18. On functions of bounded β-dimensional mean oscillation
  19. Relaxed many-body optimal transport and related asymptotics
  20. Minimizers of nonlocal polyconvex energies in nonlocal hyperelasticity
https://doi.org/10.1515/acv-2022-0062
Schlagwörter für diesen Artikel
Degenerate parabolic equations; higher differentiability; Sobolev regularity

Artikel in diesem Heft

  1. Frontmatter
  2. A flow approach to the prescribed Gaussian curvature problem in ℍ𝑛+1
  3. A split special Lagrangian calibration associated with frame vorticity
  4. The homogeneous causal action principle on a compact domain in momentum space
  5. The Lp Minkowski problem for q-torsional rigidity
  6. A twist in sharp Sobolev inequalities with lower order remainder terms
  7. Least energy solutions for affine p-Laplace equations involving subcritical and critical nonlinearities
  8. Stochastic homogenisation of free-discontinuity functionals in randomly perforated domains
  9. Existence of minimizers for the SDRI model in 2d: Wetting and dewetting regime with mismatch strain
  10. Generalized minimizing movements for the varifold Canham–Helfrich flow
  11. A characterization of gauge balls in ℍ n by horizontal curvature
  12. Minimizers of 3D anisotropic interaction energies
  13. Regularity results for a class of widely degenerate parabolic equations
  14. On isosupremic vectorial minimisation problems in L ∞ with general nonlinear constraints
  15. Quasiconformal, Lipschitz, and BV mappings in metric spaces
  16. Continuous differentiability of a weak solution to very singular elliptic equations involving anisotropic diffusivity
  17. Optimal transport with nonlinear mobilities: A deterministic particle approximation result
  18. On functions of bounded β-dimensional mean oscillation
  19. Relaxed many-body optimal transport and related asymptotics
  20. Minimizers of nonlocal polyconvex energies in nonlocal hyperelasticity
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 20.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/acv-2022-0062/html
Button zum nach oben scrollen