Radjesvarane Alexandre
Shanghai Jiao Tong University
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Publication
Featured researches published by Radjesvarane Alexandre.
Journal of Functional Analysis | 2012
Radjesvarane Alexandre; Yoshinori Morimoto; Seiji Ukai; Chao-Jiang Xu; Tong Yang
It is known that the singularity in the non-cutoff cross-section of the Boltzmann equation leads to the gain of regularity and a possible gain of weight in the velocity variable. By defining and analyzing a non-isotropic norm which precisely captures the dissipation in the linearized collision operator, we first give a new and precise coercivity estimate for the non-cutoff Boltzmann equation for general physical cross-sections. Then the Cauchy problem for the Boltzmann equation is considered in the framework of small perturbation of an equilibrium state. In this part, for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator, we derive some new functional estimates on the nonlinear collision operator. Together with the coercivity estimates, we prove the global existence of classical solutions for the Boltzmann equation in weighted Sobolev spaces.
Journal of the American Mathematical Society | 2014
Radjesvarane Alexandre; Ya-Guang Wang; Chao-Jiang Xu; Tong Yang
We develop a new approach to study the well-posedness theory of the Prandtl equation in Sobolev spaces by using a direct energy method under a monotonicity condition on the tangential velocity field instead of using the Crocco transformation. Precisely, we firstly investigate the linearized Prandtl equation in some weighted Sobolev spaces when the tangential velocity of the background state is monotonic in the normal variable. Then to cope with the loss of regularity of the perturbation with respect to the background state due to the degeneracy of the equation, we apply the Nash-Moser-Hormander iteration to obtain a well-posedness theory of classical solutions to the nonlinear Prandtl equation when the initial data is a small perturbation of a monotonic shear flow.
Archive for Rational Mechanics and Analysis | 2010
Radjesvarane Alexandre; Yoshinori Morimoto; Seiji Ukai; Chao-Jiang Xu; Tong Yang
The Boltzmann equation without Grad’s angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non-integrable angular singularity of the cross-section. However, even though this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo-differential operators, we prove the regularizing effect in all (time, space and velocity) variables on the solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial data has this mild regularity and a Maxwellian type decay in the velocity variable, there exists a unique local solution with the same regularity, so that this solution acquires the C∞ regularity for any positive time.
Archive for Rational Mechanics and Analysis | 2011
Radjesvarane Alexandre; Yoshinori Morimoto; Seiji Ukai; Chao-Jiang Xu; Tong Yang
This is a continuation of our series of works for the inhomogeneous Boltzmann equation. We study qualitative properties of classical solutions; the full regularization in all variables, uniqueness, non-negativity and convergence rate to the equilibrium, to be precise. Together with the results of Parts I and II about the well-posedness of the Cauchy problem around the Maxwellian, we conclude this series with a satisfactory mathematical theory for the Boltzmann equation without angular cutoff.
Analysis and Applications | 2012
Radjesvarane Alexandre
We give simple proofs of hypoelliptic estimates for some models of kinetic equations with a fractional order diffusion part. The proofs are based on energy estimates together with the previous ideas of Bouchut and Perthame.
Communications in Mathematical Physics | 2011
Radjesvarane Alexandre; Yoshinori Morimoto; Seiji Ukai; Chao-Jiang Xu; Tong Yang
Journal of Functional Analysis | 2008
Radjesvarane Alexandre; Yoshinori Morimoto; Seiji Ukai; Chao-Jiang Xu; Tong Yang
Analysis and Applications | 2011
Radjesvarane Alexandre; Yoshinori Morimoto; Seiji Ukai; Chao-Jiang Xu; Tong Yang
Communications in Mathematical Sciences | 2009
Salma Bougacha; Jean-Luc Akian; Radjesvarane Alexandre
Comptes Rendus Mathematique | 2007
Radjesvarane Alexandre; Yoshinori Morimoto; Seiji Ukai; Chao-Jiang Xu; Tong Yang