Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation

By Luis Caffarelli, Alexis Vasseur. 2006

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Motivated by the critical dissipative quasi-geostrophic equation, we prove that drift-diffusion equations with L 2 initial data and minimal assumptions on the drift are locally Holder continuous. As an application we show that solutions of the quasi-geostrophic equation with initial L 2 data and critical diffusion (-Δ) 1/2 are locally smooth for any space dimension.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/184c4e28b95e313f821da64cb2c02461bc4f5fed

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, background): “A number of ideas come from Caffarelli-Vasseur [69] and [66] with difficult modifications due to the degenerate nonlinearity.”
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.”
Hitchhiker's guide to the fractional Sobolev spaces (2011, background): “equations, are now experiencing impressive applications in different subjects, such as, among others, the thin obstacle problem [87, 69], optimization [37], finance [26], phase transitions [2, 14, 88, 40, 45], stratified materials [83, 23, 24], anomalous diffusion [68, 98, 65], crystal dislocation [”
Nonlinear Diffusion with Fractional Laplacian Operators (2012, methodology): “This was done for instance by Caffarelli and Vasseur [29] by using the Caffarelli-Silvestre extension [25].”
Nonuniqueness of Weak Solutions to the SQG Equation (2016, background): “In fact, the argument in [CV10] (see [CZV16]) shows that L 8 t L 2 x X L 2 t 9 H γ{2 x suitable weak solutions (weak solutions which obey the local energy inequality) are bounded L8pT2q for positive time.”
Endpoint Sobolev Theory for the Muskat Equation (2020, methodology): “In the critical case α = 1, the global in time well-posedness has been proved by Kiselev-Nazarov-Volberg [42], Caffarelli-Vasseur [14] and Constantin-Vicol [30] (see also [41,48,52,57]).”
Fractional Gagliardo–Nirenberg and Hardy inequalities under Lorentz norms (2012, background): “They play a crucial role in the study of many important PDEs [1–11], among them the quasi-geostrophic equation in geophysical fluid dynamics [12], which is crucial to the understanding of atmosphere and ocean, as well as being an enlightening model of 3D Euler equations and Navier–Stokes equations.”
Global well-posedness of slightly supercritical active scalar equations (2012, methodology): “9) On the other hand, when x ∈ [1, 2], it is convenient to work with (4.”
On a transport equation with nonlocal drift (2014, methodology): “ caseγ = 1, with general initial datum was ﬁrst shown in [ 23]. We note that the method developed in [41] for general linear drift-diffusion equations also yield this result (the methods developed in [31, 3, 15, 14] for global regularity of the critically dissipative SQG equation only appear to work”
Lipschitz regularity of solutions for mixed integro-differential equations (2011, methodology): “In between we have the critical value β = 1, when the drift and the diffusion are of the same order. ha l-0 06 08 84 8, v er si on 2 - 5 Ja n 20 12 LIPSCHITZ REGULARITY FOR MIXED PIDES 21 In the critical case, it was shown by Caffarelli and Vasseur [6] by using De Giorgi’s approach that the solution”