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)
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)
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)
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)
The empirical mode decomposition is a powerful tool for signal processing. Because of its original algorithmic, recent works have contributed to its theoretical framework. Following these works, some mathematical contributions on its comprehension and formalism are provided. In this paper, the so called <i>local mean</i> is computed in such a way that it… (More)
In this paper, we provide some theoretical contributions on the study of the 2D empirical mode decomposition (EMD). For doing this, we model the 2D sifting process (SP) in a suitable way, which helps us prove its convergence. Indeed, we prove that the 2D SP converges towards the solution of a fourth order partial differential equation (PDE). We show that 2D… (More)
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 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.
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.