On 385-01385-01385-01regularity of the gradient of solutions of degenerate parabolic systems

Wiegner, Michael

doi:10.1007/bf01790549

Cited by 59 publications

(40 citation statements)

References 6 publications

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“…The brilliant idea to use a certain intrinsic geometry which reflects the inhomogeneity of the parabolic system was invented by DiBenedetto & Friedman [12,13] who proved Hölder continuity of the spatial gradient of the solution in the interior of the domain. For the C 1,α -estimate we also refer to Wiegner [29]. In this setting everywhere regularity cannot be expected.…”

Section: Introductionmentioning

confidence: 99%

Global gradient bounds for the parabolicp-Laplacian system

Bögelein

2015

Proc. London Math. Soc.

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ABSTRACT. A by now classical result due to DiBenedetto states that the spatial gradient of solutions to the parabolic p-Laplacian system is locally Hölder continuous in the interior. However, the boundary regularity is not yet well understood. In this paper we prove a boundary L ∞ -estimate for the spatial gradient Du of solutions to the parabolic p-Laplacian systemfor p ≥ 2, together with a quantitative estimate. In particular, this implies the global Lipschitz regularity of solutions. The result continues to hold for the so called asymptotically regular parabolic systems.

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Section: Introductionmentioning

confidence: 99%

Global gradient bounds for the parabolicp-Laplacian system

Bögelein

2015

Proc. London Math. Soc.

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“…case of problem 1.5 , existence and uniqueness of weak solutions under w x general initial data were proved in 24 . For the Holder continuity of w x solutions and their spatial gradients we refer to 22,23,52,21,42 .…”

Section: Introductionmentioning

confidence: 99%

The Number of Peaks of Nonnegative Solutions to Some Nonlinear Degenerate Parabolic Equations

Sakaguchi

1996

Journal of Mathematical Analysis and Applications

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“…The latter implies the Ho lder continuity. The proof is somewhat complicated but the ideas can be extended to some systems (see [29,30]). …”

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“…The idea of using logarithmic function was originally employed by Moser and other authors ( [29,30]) in the aforementioned works to obtain auxiliary results but did not play a direct role to derive regularity results. In contrast, our proof makes use of these functions to get a simpler and straightforward proof compared to those of the methods mentioned above.…”

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confidence: 99%

Hölder Regularity for Certain Strongly Coupled Parabolic Systems

1999

Journal of Differential Equations

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We present a shorter proof to show Ho lder continuity of bounded solutions to a general class of quasilinear parabolic equations. The proof will be extended to obtain regularity results for bounded solutions to certain strongly coupled (or cross-diffusion) quasilinear parabolic systems. Academic PressIn this paper we study the Ho lder continuity of bounded solutions to a class of certain strongly coupled quasilinear parabolic systems of the formis a vector valued function defined in 0 T and div, D denotes the divergence and spatial derivative opeartors. A i , f i are accordingly vector value functions. To the author's knowledge there are only few works on Ho lder regularity of solutions to systems of the type (0.1). In contrast to the case of scalar equations or reaction diffusion systems with the coupling occurs only in the reaction terms, counterexamples (see [26]) indicate that one cannot expect bounded solutions to general strongly coupled systems to be regular everywhere. Also, concerning the problem of global existence of solutions, a priori L bounds are not enough to conclude that the solutions exist on the infinite time interval. The works of Amann [1,3,2] show that, in important cases, it suffices to find a priori L bounds to guarantee global existence provided that we can also prove uniform Ho lder continuity in space and time ([2, Theorem 4.1]).Partial regularity results were obtained by Giaquinta and Struwe in [14] for a fairly general class of systems. Everywhere regularity results for bounded solutions were proven only in few situations assuming additional structure conditions on the system (0.1). Among these are diagonal systems (see Article ID jdeq.1998.3510, available online at http:ÂÂwww.idealibrary.com on 313

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On 385-01385-01385-01regularity of the gradient of solutions of degenerate parabolic systems

Cited by 59 publications

References 6 publications

Global gradient bounds for the parabolicp-Laplacian system

Global gradient bounds for the parabolicp-Laplacian system

The Number of Peaks of Nonnegative Solutions to Some Nonlinear Degenerate Parabolic Equations

Hölder Regularity for Certain Strongly Coupled Parabolic Systems

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