Global solutions to a class of strongly coupled parabolic systems

Wiegner, Michael

doi:10.1007/bf01444644

Cited by 26 publications

(27 citation statements)

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“…Wiegner in [13] provided another example of parabolic system with everywhere regularity. He showed everywhere Hölder continuity of weak solutions of strongly coupled systems…”

Section: Introductionmentioning

confidence: 99%

Everywhere regularity of certain nonlinear diffusion systems

Trokhimtchouk

2009

Calc. Var.

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In this paper I discuss nonlinear parabolic systems that are generalizations of scalar diffusion equations. More precisely, I consider systems of the formwhere (z) is a strictly convex function. I show that when is a function only of the norm of u, then bounded weak solutions of these parabolic systems are everywhere Hölder continuous and thus everywhere smooth. I also show that the method used to prove this result can be easily adopted to simplify the proof of the result due to Wiegner (Math Ann 292(4):711-727, 1992) on everywhere regularity of bounded weak solutions of strongly coupled parabolic systems. Mathematics Subject Classification (2000)

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“…Wiegner in [13] provided another example of parabolic system with everywhere regularity. He showed everywhere Hölder continuity of weak solutions of strongly coupled systems…”

Section: Introductionmentioning

confidence: 99%

Everywhere regularity of certain nonlinear diffusion systems

Trokhimtchouk

2009

Calc. Var.

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“…He employed an alternative analysis on level sets to show the smallness condition for the oscillation of solution so that the regularity result of [14] can be applied. Recently, in [19,20], Ku fner generalized the results on invariant regions in [24] to derive L bounds for solutions to some strongly coupled systems, which also satisfy the structure conditions considered in [30], so that global existence results follow.…”

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confidence: 96%

“…They showed that a bounded solution u is regular on certain subset of 0 T where the oscillation of u is small enough. In [30], Wiegner gave an everywhere regularity result for a strongly coupled system having special structure. He employed an alternative analysis on level sets to show the smallness condition for the oscillation of solution so that the regularity result of [14] can be applied.…”

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confidence: 98%

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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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“…The main idea we are utilizing has been employed earlier in [16] for semilinear systems (see also [10], [17] and [13]), and consists in switching to two new functions of unknowns. For each such function an L ∞ estimate is established in the conventional way, and hence the final conclusion about each component of the vector function solution is inferred.…”

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confidence: 99%