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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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... . ‖ 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...

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...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...

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...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...

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...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 ...