On the critical dissipative quasi-geostrophic equation

By Peter Constantin, D. C'ordoba, Jiahong Wu. 2001

266 citations recorded by the source corpus.

The 2D quasi-geostrophic (QG) equation is a two dimensional model of the 3D incompressible Euler equations. When dissipation is included in the model, then solutions always exist if the dissipation's wave number dependence is super-linear. Below this critical power, the dissipation appears to be insuY- cient. For instance, it is not known if the critical dissipative QG equation has global smooth solutions for arbitrary large initial data. In this paper we prove existence and uniqueness of global classical solutions of the critical dissipative QG equation for ini- tial data that have small L 1 norm. The importance of an L 1 smallness condition is due to the fact thatL 1 is a conserved norm for the non-dissipative QG equation and is non-increasing on all solutions of the dissipative QG, irrespective of size.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d4811639fcd95288a8ef8a60b9bc47412a8fd75f

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Citations in the corpus

Existence and Regularity of Weak Solutions to the Quasi-Geostrophic Equations in the Spaces Lp or $$\dot{H}^{-1/2}$$ (2007, methodology): “The dissipative cases have then been studied by Constantin and Wu in [4] when α> 1/2 and global existence in Sobolev space is studied by Constantin, Cordoba and Wu in [ 2 ] when α = 1/2.”
A New Bernstein’s Inequality and the 2D Dissipative Quasi-Geostrophic Equation (2006, background): “In the critical case, Constantin, Cordoba, and Wu[10] proved the existence and uniqueness of global smooth solution on the spatial periodic domain under the assumption of small L∞ norm.”
On the global regularity for the supercritical SQG equation (2014, unknown): “This program started with [5].”
Refined blowup criteria and nonsymmetric blowup of an aggregation equation (2009, methodology): “Rd jxju(t;x)dxdt: Now use ( 10 ). Taking R ! 1 and applying again Lebesgue ’s Monotone Convergence Theorem immediately yields the result.”
Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation (2007, background): “In Constantin, Córdoba and Wu [6], we proved in the critical case (α = 1 2 ) the global existence and uniqueness of classical solutions corresponding to any initial data with L-norm comparable to or less than the diffusion coefficient κ.”
Finite-Time Singularities of an Aggregation Equation in $${\mathbb {R}^n}$$ with Fractional Dissipation (2009, methodology): “We refer the reader to [10],[9] and [27] where this type of dissipation has been used in the context of the surface quasi-geostrophic equation and other one dimensional models, for a more detailed explanation of the 3 regimes.”
Dissipative Quasi-Geostrophic Equation for Large Initial Data in the Critical Sobolev Space (2006, background): “In the critical case, Constantin, Cordoba and Wu [4] constructed a”
Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations (2007, background): “In a recent work [14], Kiselev, Nazarov and Volberg proved that smooth global solutions persist for any C∞ periodic initial data [7], for the critical QG equation.”
Blow up and regularity for fractal Burgers equation (2008, background): “Many of the results can be extended to a more general class of equations, including the surface quasi-geostrophic equation.”
Critical SQG in Bounded Domains (2016, unknown): “A pleasant aspect of criticality is that solutions with small initial L∞ norm are smooth ([6]).”
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation (2006, background): “Constantin Córdoba and Wu showed in [5] that the regularity is conserved for all time in the critical case provided that the initial value is small in L(1).”
On the regularity of a class of generalized quasi-geostrophic equations (2010, background): “in [3] showed the global well-posedness of the classical solution under the condition that the zero-dimensional L∞ norm of the data is small.”
Global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces (2006, background): “Indeed, global existence and uniqueness for arbitrary initial data are established in various function spaces (see for example [8, 16]).”
Global Well-Posedness in the Super-Critical Dissipative Quasi-Geostrophic Equations (2003, background): “For the critical case(α = 1 ), Constantin, Cordoba, and Wu [ 6 ] proved the existence and uniqueness”
On the Euler equations of incompressible fluids (2007, background): “In the critical case, it was known for some time that if the initial data are small in L∞ they remain small and the solution is regular ([ 44 ], [62]).”