Integral estimates for a linear singular operator linked with Boltzmann operators part II: High singularities $1\le\nu<2$
Radjesvarane Alexandre Lingbing He
Kinetic & Related Models 2008, 1(4): 491-513 doi: 10.3934/krm.2008.1.491
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.
keywords: Boltzmann operator non cut off kernels harmonic analysis
High order approximation for the Boltzmann equation without angular cutoff
Lingbing He Yulong Zhou
Kinetic & Related Models 2018, 11(3): 547-596 doi: 10.3934/krm.2018024

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.

keywords: Homogeneous Boltzmann equation full-range interactions hard potentials high order approximation global solutions
On the global smooth solution to 2-D fluid/particle system
Lingbing He
Discrete & Continuous Dynamical Systems - A 2010, 27(1): 237-263 doi: 10.3934/dcds.2010.27.237
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.
keywords: global existence Vlasov-Fokker-Planck Navier-Stokes long time behavior.
Periodic long-time behaviour for an approximate model of nematic polymers
Lingbing He Claude Le Bris Tony Lelièvre
Kinetic & Related Models 2012, 5(2): 357-382 doi: 10.3934/krm.2012.5.357
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.
keywords: Entropy method nematic polymer. Doi closure periodic solution liquid crystal

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