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### Open Access Journals

KRM

In this work, we show that integral estimates for a linear operator linked with Boltzmann
quadratic operator considered in [1] can also be obtained for the case of higher
singularities. Some estimates proven in this earlier work are improved, as in particular, we do not need any regularity with respect to the first function.

KRM

In order to solve the Boltzmann equation numerically, in the present work, we propose a new model equation to approximate the Boltzmann equation without angular cutoff. Here the approximate equation incorporates Boltzmann collision operator with angular cutoff and the Landau collision operator. As a first step, we prove the well-posedness theory for our approximate equation. Then in the next step we show the error estimate between the solutions to the approximate equation and the original equation. Compared to the standard angular cutoff approximation method, our method results in higher order of accuracy.

DCDS

In two space dimension, we show that the steady state the solution
of fluid/particle system may tend to after a long time is completely
determined by the initial total momentum. Based on this observation,
we prove the global-in-time existence of the classical solutions
for arbitrary initial data to the system that couples the
incompressible Navier-Stokes equations
to the Vlasov-Fokker-Planck equation. By linearized method and Littlewood-Paley analysis, the
exponential rate of the convergence toward steady state is obtained
under some specific assumptions.

KRM

We study the long-time behaviour of a nonlinear Fokker-Planck equation, which models the evolution of rigid polymers in a given flow, after a closure approximation. The aim of this work is twofold: first, we propose a microscopic derivation of the classical Doi closure, at the level of the kinetic equation ; second, under specific assumptions on the parameters and the initial condition, we prove convergence of the solution to the Fokker-Planck equation to a particular periodic solution in the long-time limit.

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