Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation

By Peter Constantin, Jiahong Wu. 2007

174 citations recorded by the source corpus.

Abstract We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical ( α 1 / 2 ) dissipation ( − Δ ) α : If a Leray–Hopf weak solution is Holder continuous θ ∈ C δ ( R 2 ) with δ > 1 − 2 α on the time interval [ t 0 , t ] , then it is actually a classical solution on ( t 0 , t ] .

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/42d2a02ad33acd91846ce381ef5098fde3e5b88a

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Critical SQG in Bounded Domains (2016, unknown): “, Λ is replaced by Λs with 0 < s ≤ 1 then if θ is bounded in Cα with α > 1 − s, then the solution is smooth ([12]).”
Global well-posedness of slightly supercritical active scalar equations (2012, background): “It was shown by Constantin and Wu [10] that if the solution isC with δ > 1 − α, then it is smooth (see also Silvestre [26] for drift velocity that is not divergence free).”
On the differentiability of the solution to an equation with drift and fractional diffusion (2010, background): “Some applications are: • It implies the result in [5] about a regularity criteria for the supercritical surface quasigeostrophic equation.”
Global regularity for a modified critical dissipative quasi-geostrophic equation (2008, methodology): “Following the method of [6], we essentially show that if a LerayHopf weak solution of (1.”
On the regularity of a class of generalized quasi-geostrophic equations (2010, methodology): “From the direction of weak solutions, Constantin and Wu in [7] showed a regularity criterion for the weak solutions in terms of the uniform (in time) Hölder estimates.”
On the Euler equations of incompressible fluids (2007, background): “Work on the supercritical case along similar lines is in progress ([59], [60]).”
Nonlinear maximum principles for dissipative linear nonlocal operators and applications (2011, methodology): “This recovers the results of [13], without making use of the Besov-space techniques.”