Upper Maxwellian bounds for the Boltzmann equation with pseudo-Maxwell molecules
Alexander V. Bobylev Irene M. Gamba
Kinetic & Related Models 2017, 10(3): 573-585 doi: 10.3934/krm.2017023

We consider solutions to the initial value problem for the spatially homogeneous Boltzmann equation for pseudo-Maxwell molecules and show uniform in time propagation of upper Maxwellians bounds if the initial distribution function is bounded by a given Maxwellian. First we prove the corresponding integral estimate and then transform it to the desired local estimate. We remark that propagation of such upper Maxwellian bounds were obtained by Gamba, Panferov and Villani for the case of hard spheres and hard potentials with angular cut-off. That manuscript introduced the main ideas and tools needed to prove such local estimates on the basis of similar integral estimates. The case of pseudo-Maxwell molecules needs, however, a special consideration performed in the present paper.

keywords: Boltzmann kinetic equation pseudo-Maxwell molecules bounded solutions
Gain of integrability for the Boltzmann collisional operator
Ricardo J. Alonso Irene M. Gamba
Kinetic & Related Models 2011, 4(1): 41-51 doi: 10.3934/krm.2011.4.41
In this short note we revisit the gain of integrability property of the gain part of the Boltzmann collision operator. This property implies the $W^{l,r}_k$ regularity propagation for solutions of the associated space homogeneous initial value problem. We present a new method to prove the gain of integrability that simplifies the technicalities of previous approaches by avoiding the argument of gain of regularity estimates for the gain collisional integral. In addition our method calculates explicit constants involved in the estimates.
keywords: Boltzmann equation for hard potentials; Collisional Operator; Regularity in $L^p$ spaces.
A note on the time decay of solutions for the linearized Wigner-Poisson system
Irene M. Gamba Maria Pia Gualdani Christof Sparber
Kinetic & Related Models 2009, 2(1): 181-189 doi: 10.3934/krm.2009.2.181
We consider the one-dimensional Wigner-Poisson system of plasma physics, linearized around a (spatially homogeneous) Lorentzian distribution and prove that the solution of the corresponding linearized problem decays to zero in time. We also give an explicit algebraic decay rate.
keywords: plasma physics. Poisson coupling Wigner transform Landau damping
On the minimization problem of sub-linear convex functionals
Naoufel Ben Abdallah Irene M. Gamba Giuseppe Toscani
Kinetic & Related Models 2011, 4(4): 857-871 doi: 10.3934/krm.2011.4.857
The study of the convergence to equilibrium of solutions to Fokker-Planck type equations with linear diffusion and super-linear drift leads in a natural way to a minimization problem for an energy functional (entropy) which relies on a sub-linear convex function. In many cases, conditions linked both to the non-linearity of the drift and to the space dimension allow the equilibrium to have a singular part. We present here a simple proof of existence and uniqueness of the minimizer in the two physically interesting cases in which there is the constraint of mass, and the constraints of both mass and energy. The proof includes the localization in space of the (eventual) singular part. The major example is related to the Fokker-Planck equation introduced in [6, 7] to describe the evolution of both Bose-Einstein and Fermi-Dirac particles.
keywords: Bose-Einstein distribution sub-linear convex functionals. large time behaviour Fokker-Planck equation

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