### Citations

2299 | Singular Integrals and Differentiability Properties of Functions - Stein - 1986 |

240 |
A review of mathematical topics in collisional kinetic theory
- Villani
- 2002
(Show Context)
Citation Context ...rites also (1.5) ∂t f + v.∇x f +L( f ) = h. As explained in [6], (1.1) or its version (1.5) is related to the linearized Landau operator, which plays a crucial role in Plasma Physics, see for example =-=[1, 11]-=- and the references therein. Moreover the above assumptions (1.2) and (1.3) are natural in view of Guo’s work [5]. However, note that the model (1.1) does not take into account the kernel which is nat... |

150 |
An introduction to semilinear evolution equations, Translated from the 1990 French original by Yvan Martel and revised by the authors
- Cazenave, Haraux
- 1998
(Show Context)
Citation Context ...s standard in semi group theory, we can get formally more regularity. As explained above, we assume γ ≥ −2 to ensure that we have automatically an L2 bound. Then following formally the arguments from =-=[4]-=-, we see that, using only norms w.r.t. (x, v) variables ‖(−v.∇x f − L( f )) f (t)‖ ≤ 1 t √ 2 ‖ f0‖ By using the main Theorem (without time dependence), we see that we have hypoelliptic results on f (t... |

71 |
The Landau equation in a periodic box
- Guo
(Show Context)
Citation Context ...zed Landau operator, which plays a crucial role in Plasma Physics, see for example [1, 11] and the references therein. Moreover the above assumptions (1.2) and (1.3) are natural in view of Guo’s work =-=[5]-=-. However, note that the model (1.1) does not take into account the kernel which is naturally associated with the true Landau linearized equation. In particular, for applications to the true nonlinear... |

42 | Hypoelliptic regularity in kinetic equations
- Bouchut
(Show Context)
Citation Context ...der kinetic equation, and again the proof therein was using such tools. Recently, we have provided in [2] a very simple proof of the results of [7] by using arguments originally introduced by Bouchut =-=[3]-=- and Perthame [9]. Again herein, we shall give a different proof of this main Theorem by using simple and standard arguments. Moreover, we keep the regularity of the coefficients as low as possible, t... |

24 |
Hypoellipticity for a class of kinetic equations
- Morimoto, Xu
(Show Context)
Citation Context ...Dx| 23 f ‖2 + ‖ < v > γ 3 |v ∧ ∇x | 23 f ‖2 . ‖ f ‖2 + ‖h‖2. The proof in [6] uses pseudo-differential calculus, together with Wick calculus. In the continuation of a previous work of Morimoto and Xu =-=[8]-=-, a similar study was also performed in [7] in order to deduce hypoelliptic results for a fractional order kinetic equation, and again the proof therein was using such tools. Recently, we have provide... |

19 |
A review of Boltzmann equation with singular kernels
- Alexandre
(Show Context)
Citation Context ...rites also (1.5) ∂t f + v.∇x f +L( f ) = h. As explained in [6], (1.1) or its version (1.5) is related to the linearized Landau operator, which plays a crucial role in Plasma Physics, see for example =-=[1, 11]-=- and the references therein. Moreover the above assumptions (1.2) and (1.3) are natural in view of Guo’s work [5]. However, note that the model (1.1) does not take into account the kernel which is nat... |

11 | Hypoelliptic estimates for a linear model of the Boltzmann equation without angular cutoff
- Lerner, Morimoto, et al.
(Show Context)
Citation Context ... . ‖ f ‖2 + ‖h‖2. The proof in [6] uses pseudo-differential calculus, together with Wick calculus. In the continuation of a previous work of Morimoto and Xu [8], a similar study was also performed in =-=[7]-=- in order to deduce hypoelliptic results for a fractional order kinetic equation, and again the proof therein was using such tools. Recently, we have provided in [2] a very simple proof of the results... |

11 |
Higher moments for kinetic equations: the Vlasov-Poisson and Fokker-Planck cases
- Perthame
- 1990
(Show Context)
Citation Context ...ion, and again the proof therein was using such tools. Recently, we have provided in [2] a very simple proof of the results of [7] by using arguments originally introduced by Bouchut [3] and Perthame =-=[9]-=-. Again herein, we shall give a different proof of this main Theorem by using simple and standard arguments. Moreover, we keep the regularity of the coefficients as low as possible, this point being c... |

10 | Anisotropic hypoelliptic estimates for Landau-type operators
- Herau, Pravda-Starov
(Show Context)
Citation Context ...TY R. ALEXANDRE Abstract. We consider the linearized Landau operator for which we provide simple proofs of hypoellipticity, and in particular we recover the recent results of Hérau and Pravda-Starov =-=[6]-=-. Our arguments are elementary and in particular avoids any use of pseudo-differential calculus. 1. Introduction We consider hypoelliptic effects associated with the time version of a kinetic equation... |

5 |
Fractional order kinetic equations and hypoellipticity
- Alexandre
- 2012
(Show Context)
Citation Context ...imilar study was also performed in [7] in order to deduce hypoelliptic results for a fractional order kinetic equation, and again the proof therein was using such tools. Recently, we have provided in =-=[2]-=- a very simple proof of the results of [7] by using arguments originally introduced by Bouchut [3] and Perthame [9]. Again herein, we shall give a different proof of this main Theorem by using simple ... |