Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations

By Peter Constantin, Jiahong Wu. 2007

119 citations recorded by the source corpus.

Abstract We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical ( α 1 / 2 ) dissipation ( − Δ ) α . This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical ( α = 1 / 2 ) QG equation [L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, arXiv: math.AP/0608447, 2006]. Their approach successively increases the regularity levels of Leray–Hopf weak solutions: from L 2 to L ∞ , from L ∞ to Holder ( C δ , δ > 0 ), and from Holder to classical solutions. In the supercritical case, Leray–Hopf weak solutions can still be shown to be L ∞ , but it does not appear that their approach can be easily extended to establish the Holder continuity of L ∞ solutions. In order for their approach to work, we require the velocity to be in the Holder space C 1 − 2 α . Higher regularity starting from C δ with δ > 1 − 2 α can be established through Besov space techniques and will be presented elsewhere [P. Constantin, J. Wu, Regularity of Holder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincare Anal. Non Lineaire, in press].

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/87b84d63acb61a5905ad2ef711a65c7816d309e5

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Citations in the corpus

Nonlinear maximum principles for dissipative linear nonlocal operators and applications (2011, background): “In the super-critical case α < 1, with large initial data, only eventual regularity [17, 30] and conditional regularity [13, 14] have been established.”
Regularity theory for parabolic nonlinear integral operators (2011, unknown): “It was followed by several works on the same subject in the super-critical case (see for instance [11]).”
On the Euler equations of incompressible fluids (2007, background): “Work on the supercritical case along similar lines is in progress ([59], [60]).”
Holder estimates for advection fractional-diffusion equations (2010, result): “It is interesting to compare our result in the case s < 1/2 with the result of [7].”
Fractional powers of second order partial differential operators: extension problem and regularity theory (2010, methodology): “The technique was also applied for instance in [28] and [65] for the supercritical case σ < 1/2.”
Partial Regularity of Suitable Weak Solutions to the Fractional Navier–Stokes Equations (2014, background): “For more related works, we refer readers to [3,4].”