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## Drift diffusion equations with fractional diffusion and the quasigeostrophic equation

Venue: | Ann. Math |

Citations: | 187 - 14 self |

### Citations

179 | An extension problem related to the fractional Laplacian
- Caffarelli, Silvestre
(Show Context)
Citation Context ...) to C∞0 (RN × R+) by: −∆L(θ) = 0 in RN × (0,∞), L(θ)(x, 0) = θ(x) for x ∈ RN (This extension consists simply in convolving θ with the Poisson kernel of the upper half space in one more variable. See =-=[1]-=- for a general discussion). Then the following result holds true: consider θ defined on RN . then: Λθ(x) = ∂ν [Lθ](x), (6) where we denote ∂ν [Lθ] the normal derivative of Lθ on the boundary {(x, 0)|x... |

162 |
Sulla differenziabilit‘a e lanaliticit‘a delle estremali degli integrali multipli regolari
- Giorgi
- 1957
(Show Context)
Citation Context ... gives the desired result. 4 From L 2 to L ∞ : 1 ε ‖v‖2 L∞ (t1,t2;L2N ∫ ∫ t2 (B2)) t1 RN |[∇η][θ]+| 2 dxds, In these two sections (4 and 5) we follow De Giorgi’s ideas in his ”oscillation lemma” (see =-=[7]-=-) to prove Holder continuity: Suppose that θ oscillates in 7Q1 = [−1, 0]×B1 between −2 and 2, but it is negative most of the time. In particular, if ‖θ+‖ L 2 is very small, then we prove that θ+| Q1/... |

135 |
A maximum principle applied to quasi-geostrophic equations
- Córdoba, Córdoba
(Show Context)
Citation Context ...f the harmonic extension of θ to the upper half space. Existence theory is sketching in appendix C. In the case of Quasi-geostrophic equation it can also be seen as a corollary of Cordoba and Cordoba =-=[6]-=-. Those energy inequalities are reminiscent of the notion of entropic solutions for scalar conservation laws. Consider a weak solution of (1) lying in L 2 (H 1/2 ) and for which we can define the equa... |

122 | Global well-posedness for the critical 2D dissipative quasigeostrophic equation
- Kiselev, Nazarov, et al.
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Citation Context ...sted preprint in arXiv, Kiselev, Nazarov, and Volberg present a very elegant proof of the fact that in 2D, solutions with periodic C ∞ data for the quasi-geostrophic equation remain C ∞ for all time (=-=[9]-=-). We conclude our introduction by pointing out that our techniques also can be seen as a parabolic De Giorgi Nash Moser method to treat ”boundary parabolic 3problems” of the type: div(a∇θ) = 0, in Ω... |

94 |
Les Inéquations en Mécanique et en Physique. Dunod
- Duvaut, Lions
- 1972
(Show Context)
Citation Context ...nsion. 1 Introduction Non linear evolution equations with fractional diffusion arise in many contexts: In the quasi-geostrophic flow model (Constantin [3]), in boundary control problems (Duvaut-Lions =-=[8]-=-), in surface flame propagation and in financial mathematics. In this paper, motivated by the quasi-geostrophic model, we study the equation: ∂tθ + v · ∇θ = −Λθ, x ∈ R N , (1) divv = 0, where Λθ = (−∆... |

84 | On the critical dissipative quasi-geostrophic equation. Dedicated to Professors Ciprian Foias and Roger Temam
- Constantin, Cordoba, et al.
(Show Context)
Citation Context ...as been constructed by Resnick in [12]. Constantin and Wu showed in [6] that in the subcritical case any solution with smooth initial value is smooth for all time. Constantin Cordoba 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∞. In both the critical case and supercritical cases, Chae and Lee considered in [3] th... |

82 |
Dynamical problems in nonlinear advective partial differential equations
- Resnick
- 1995
(Show Context)
Citation Context ...e situation is classically decomposed into 3 cases: The subcritical case for β > 1, the critical case for β = 1 and the supercritical case for β < 1. Weak solutions has been constructed by Resnick in =-=[11]-=-. Constantin and Wu showed in [5] that in the subcritical case any solution with smooth initial value is smooth for all time. Constantin Cordoba and Wu showed in [4] that the regularity is conserved f... |

67 |
Behavior of solutions of 2d quasi-geostrophic equations
- Constantin, Wu
- 1999
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Citation Context ...versus transport: The subcritical case for β > 1, the critical case for β = 1 and the supercritical case for β < 1. Weak solutions has been constructed by Resnick in [12]. Constantin and Wu showed in =-=[6]-=- that in the subcritical case any solution with smooth initial value is smooth for all time. Constantin Cordoba and Wu showed in [5] that the regularity is conserved for all time in the critical case ... |

49 |
Global well-posedness in the super-critical dissipative quasi-geostrophic equations
- Chae, Lee
(Show Context)
Citation Context ... [4] that the regularity is conserved for all time in the critical case provided that the initial value is small in L ∞ . In both the critical case and supercritical cases, Chae and Lee considered in =-=[2]-=- the well-posedness of solutions with initial conditions small in Besov spaces (see also Wu [15]). Notice that our case corresponds to the critical case and global regularity in C 1,β , β < 1 is showe... |

40 | Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces
- Wu
(Show Context)
Citation Context ...al value is small in L ∞ . In both the critical case and supercritical cases, Chae and Lee considered in [2] the well-posedness of solutions with initial conditions small in Besov spaces (see also Wu =-=[15]-=-). Notice that our case corresponds to the critical case and global regularity in C 1,β , β < 1 is showed for any initial value in the energy space without hypothesis of smallness. This ensures that t... |

35 | A new proof of partial regularity of solutions to Navier-Stokes equations
- VASSEUR
- 2007
(Show Context)
Citation Context ...v(a∇θ) = 0, in Ω × [0, T] [f(θ)]t = θν on ∂Ω × [0, T], that arise in boundary control (see Duvaut Lions [8]). Note also that similar results to Theorem 1 can be obtained even for systems (See Vasseur =-=[14]-=- and Mellet, Vasseur [10] for applications of the method in fluid mechanics). 2 L ∞ bounds This section is devoted to the proof of Theorem 1. We use the level set energy inequality for: λ = Ck = M(1 −... |

31 |
Behaior of solutions of 2D quasi-geostrophic equations
- Constantin, Wu
- 1999
(Show Context)
Citation Context ...sed into 3 cases: The subcritical case for β > 1, the critical case for β = 1 and the supercritical case for β < 1. Weak solutions has been constructed by Resnick in [11]. Constantin and Wu showed in =-=[5]-=- that in the subcritical case any solution with smooth initial value is smooth for all time. Constantin Cordoba and Wu showed in [4] that the regularity is conserved for all time in the critical case ... |

24 | Asymptotic behavior to dissipative quasi-geostrophic flows
- Schonbek, Schonbek
(Show Context)
Citation Context ...allness. This ensures that the solutions are classical. Let us also cite a result of maximum principle due to Cordoba and Cordoba [6], results of behavior in large time due to Schonbeck and Schonbeck =-=[13]-=-, [12], and a criteria for blow-up in Chae [1]. Remark 2: In a recently posted preprint in arXiv, Kiselev, Nazarov, and Volberg present a very elegant proof of the fact that in 2D, solutions with peri... |

17 |
On the regularity conditions for the dissipative quasi-geostrophic equations
- Chae
(Show Context)
Citation Context ...assical. Let us also cite a result of maximum principle due to Cordoba and Cordoba [6], results of behavior in large time due to Schonbeck and Schonbeck [13], [12], and a criteria for blow-up in Chae =-=[1]-=-. Remark 2: 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 ∞ data for the quasi-geostrophic equati... |

17 |
Moments and lower bounds in the far-field of solutions to quasigeostrophic flows, Discrete Contin
- Schonbek, Schonbek
(Show Context)
Citation Context ...s. This ensures that the solutions are classical. Let us also cite a result of maximum principle due to Cordoba and Cordoba [6], results of behavior in large time due to Schonbeck and Schonbeck [13], =-=[12]-=-, and a criteria for blow-up in Chae [1]. Remark 2: 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... |

8 |
Lp estimates for quantities advected by a compressible flow
- MELLET, VASSEUR
- 2009
(Show Context)
Citation Context ... [f(θ)]t = θν on ∂Ω × [0, T], that arise in boundary control (see Duvaut Lions [8]). Note also that similar results to Theorem 1 can be obtained even for systems (See Vasseur [14] and Mellet, Vasseur =-=[10]-=- for applications of the method in fluid mechanics). 2 L ∞ bounds This section is devoted to the proof of Theorem 1. We use the level set energy inequality for: λ = Ck = M(1 − 2 −k ), where M will be ... |

7 |
equations, Navier-Stokes equations and turbulence. In Mathematical foundation of turbulent viscous flows
- Euler
- 2006
(Show Context)
Citation Context ... (−∆) 1/2 , are locally smooth for any space dimension. 1 Introduction Non linear evolution equations with fractional diffusion arise in many contexts: In the quasi-geostrophic flow model (Constantin =-=[3]-=-), in boundary control problems (Duvaut-Lions [8]), in surface flame propagation and in financial mathematics. In this paper, motivated by the quasi-geostrophic model, we study the equation: ∂tθ + v ·... |