Global well-posedness for the critical 2D dissipative quasi-geostrophic equation

By Alexander V. Kiselev, Fedor Nazarov, Alexander Volberg. 2006

536 citations recorded by the source corpus.

We give an elementary proof of the global well-posedness for the critical 2D dissipative quasi-geostrophic equation. The argument is based on a non-local maximum principle involving appropriate moduli of continuity.

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

View Paper: https://www.semanticscholar.org/paper/b739c4bce220990d2998dc60afaccf5f5bcc03e7

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Citations in the corpus

Classical solutions and higher regularity for nonlinear fractional diffusion equations (2013, background): “give a proof of C regularity of a class of periodic solutions of geostrophic equations in 2D with C data [13].”
Fractional powers of second order partial differential operators: extension problem and regularity theory (2010, methodology): “By using other methods it is shown in [50] that when σ = 1/2 and the initial data is smooth, there exists a unique solution to QG that is also smooth.”
Eulerian dynamics with a commutator forcing III. Fractional diffusion of order 0 (2017, unknown): “[14], in order to verify a Beale–Kato–Majda type criterion ∫ T 0 |ρx(·, t)| 2 ∞ dt < ∞, to guarantee continuation of the solution beyond T .”
On the regularity of a class of generalized quasi-geostrophic equations (2010, methodology): “6 in the sequel, we can improve the modulus of continuity method in [14] to show the global well-posedness.”
Variation on a theme of caffarelli and vasseur (2009, methodology): “In particular, our approach ofiers a third proof of the global regularity for the critical surface quasi-geostrophic (SQG) equation in addition to [ 5 ] and [1].”
Endpoint Sobolev Theory for the Muskat Equation (2020, methodology): “However, Cameron proved in [15] that one can apply themethod introduced byKiselev-Nazarov-Volberg to prove the existence of global in time solutions whenever the product of the maximal and minimal slopes is less than 1.”
Long Time Dynamics of Forced Critical SQG (2013, unknown): “The proof of global regularity in [KNV07] is based on constructing a family of Lipschitz moduli of continuity with the property that if the initial data obeys such a modulus so will the solution of the critical SQG equation for all later times, and so that this family behaves nicely under rescaling.”
On a transport equation with nonlocal drift (2014, methodology): “We note that the method developed in [41] for general linear drift-diﬀusion equations also yields this result (the methods developed in [3,14,15,31] for global regularity of the critically dissipative SQG equation only appear to work in the case of positive initial datum where the L 2 norm of the…”
The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, background): “There is much work in geostrophic flows, like [188, 174, 175].”
Well-posedness for a transport equation with nonlocal velocity (2008, methodology): “For further details, we refer the readers to [9] and [13].”
Finite time singularity for the modified SQG patch equation (2016, methodology): “Global regularity for the critical viscous SQG equation, with β = 12 , was proved independently by Caffarelli and Vasseur [3], and by Kiselev, Nazarov, and Volberg [36].”
Blow up and regularity for fractal Burgers equation (2008, background): “Much of the discussion follows [14]; we reproduce the argument here for the sake of completeness.”
Dissipative quasi-geostrophic equations in critical Sobolev spaces: smoothing effect and global well-posedness (2007, methodology): “Then we can consider �(t1) as a initial data and apply the result of the global existence for smooth initial data in [ 13 ].”
Estimates of the Green Function for the Fractional Laplacian Perturbed by Gradient (2010, methodology): “Gradient perturbations of ∆ were studied by Caffarelli and Vasseur ([16]) and Kiselev, Nazarov, Volberg ([37]).”
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation (2006, background): “In a recently posted preprint in arXiv, Kiselev, Nazarov, and Volberg present a very elegant proof of the fact that in 2D, solutions with periodic C(1) data for the quasi-geostrophic equation remain C(1) for all time ([10]).”
On the global regularity for the supercritical SQG equation (2014, methodology): “ full range γ ∈ (0,1) by means of the methods devised in [17], and ﬁnally in [ 16] for all γ ∈ (0,1) through a modiﬁcation of the modulus of continuity approach that has been successfully employed in [16,18,19]. The simple proof given in this paper is based on the arguments in [6,7], cf. Section 3.5”
Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation (2007, background): “Recently this problem has attracted a significant amount of research ([1],[2],[3],[4],[5],[6],[9], [11],[12],[13],[14],[17],[18],[19],[20], [21],[22],[23]).”