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- R Alexandre, L Desvillettes, C Villani, And B Wennberg
- 2000

We study Boltzmann's collision operator for long-range interactions, i.e. without Grad's angular cutoff assumption. We establish a functional inequality showing that the entropy dissipa-tion controls smoothness of the distribution function, in a precise sense. Our estimate is optimal, and gives a unified treatment of both the linear and the nonlinear cases.… (More)

- R Alexandre, C Villani
- 2002

We study the Boltzmann equation without Grad's angular cut-oo assumption. We introduce a suitable renormalized formulation , which allows the cross-section to be singular in both the angular and the relative velocity variables. This situation occurs as soon as one is interested in long-range interactions. Together with several new estimates, this enables us… (More)

- R Alexandre, Y Morimoto, S Ukai, C.-J Xu, T Yang
- 2009

The Boltzmann equation without Grad's angular cutoff assumption is believed to have regularizing effect on the solution because of the non-integrable angular singularity of the cross-section. However, even though so far this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially… (More)

- Radjesvarane ALEXANDRE, Mouhamad EL SAFADI
- 2004

We use Littlewood Paley decompositions and related arguments to study some regularity questions in Boltzmann equation. This is done in the framework of homogeneous solutions, with a scattering cross section of Maxwell type, without assuming the usual Grad's cutoff assumption. Although the recent results of Desvillettes and Wennberg include larger… (More)

- R. ALEXANDRE, T. YANG
- 2011

In this paper, we consider the spatially homogeneous Boltzmann equation without angular cutoff. We prove that every L 1 weak solution to the Cauchy problem with finite moments of all order acquires the C ∞ regularity in the velocity variable for the positive time.

For the first time, a proof of the sifting process (SP) and so the empirical mode decomposition (EMD), is given. For doing this, lower and upper envelopes are modeled in a more convenient way that helps us prove the convergence of the SP towards a solution of a partial differential equation (PDE). We also prove that such a PDE has a unique solution, which… (More)

In this work, we start the study of precise functional properties of a linear operator linked with Boltzmann quadratic operator. This is done for singular cross-sections. We show some kernel estimates, which can be used to deduce some functional properties of Boltzmann operator itself.

We consider electromagnetic waves propagating in a periodic medium characterized by two small scales. We perform the corresponding homogenization process, relying on the modelling by Maxwell's partial differential equations.