Variation on a theme of caffarelli and vasseur

By Alexander V. Kiselev, Fedor Nazarov. 2009

153 citations recorded by the source corpus.

Recently, using DiGiorgi-type techniques, Caffarelli and Vasseur have shown that a certain class of weak solutions to the drift diffusion equation with initial data in L2 gain H¨older continuity, provided that the BMO norm of the drift velocity is bounded uniformly in time. We show a related result: a uniform bound on the BMO norm of a smooth velocity implies a uniform bound on the Cβ norm of the solution for some β > 0. We apply elementary tools involving the control of H¨older norms by using test functions. In particular, our approach offers a third proof of the global regularity for the critical surface quasigeostrophic (SQG) equation in addition to the two proofs obtained earlier. Bibliography: 6 titles.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/8b34650b2e4512eec9ca08b20ad5410f88dc3868

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

Long Time Dynamics of Forced Critical SQG (2013, background): “In an attempt to find a bridge between these two proofs, in [KN09] a completely different proof of global regularity for critical SQG was obtained.”
Nonlinear maximum principles for dissipative linear nonlocal operators and applications (2011, methodology): “The Lp norms of θ are nonincreasing in time under smooth evolution, and it is known that smooth solutions persist for α ≥ 1 ([3], [15], [22], [23], [24], [29], and many more).”
On the global regularity for the supercritical SQG equation (2014, methodology): “While for the critical case γ = 1 the question of global regularity of (SQGγ) with arbitrarily large datum has been settled [1, 6, 7, 17, 19] (see also [11] for the logarithmically supercritical case), the corresponding result for the supercritical equation γ ∈ (0, 1) remains open.”
On the differentiability of the solution to the Hamilton–Jacobi equation with critical fractional diffusion (2009, background): “1) is interesting by itself and provides a non variational counterpart to the result in [4] and [12].”
Global well-posedness of slightly supercritical active scalar equations (2012, result): “of [20] uses an appropriate set of test functions and estimat es on their evolution.”
On the Loss of Continuity for Super-Critical Drift-Diffusion Equations (2012, result): “Other proofs of this result were given in [25] and [14].”
On the differentiability of the solution to an equation with drift and fractional diffusion (2010, methodology): “It is used for example in [3], [13] and [9] for problems with drift and fractional diffusion.”
Holder estimates for advection fractional-diffusion equations (2010, methodology): “An alternative approach is used in [8] where the Hölder continuity is obtained through the study of the dual equation.”
Finite time singularity for the modified SQG patch equation (2016, methodology): “Global regularity for the critical viscous SQG equation, with β = 1 2 , was proved independently by Caffarelli and Vasseur [3], and by Kiselev, Nazarov, and Volberg [34] (see also the subsequent works [11,12,33] for alternative proofs).”