Qinghua Xiao

The Vlasov-Poisson-Boltzmann system for non-cutoff hard potentials

This paper is concerned with the construction of globally smooth solutions near a given global Maxwellian to the Cauchy problem of the Vlasov-Poisson-Boltzmann system for non-cutoff hard potentials in three space dimensions without the neutral condition imposed on the initial perturbation. Our analysis is based on the time-weighted energy method and some delicate estimates.

Uniform Stability and the Propagation of Regularity for the Relativistic Boltzmann Equation

We present a priori uniform

The vlasov-poisson-boltzmann system near maxwellians for long-range interactions

L^q

L2-stability of the Vlasov-Maxwell-Boltzmann system near global Maxwellians

-stability estimates of classical solutions to the relativistic Boltzmann equation. Our uniform stability analysis does not require any smallness in the amplitudes of solutions, but we need exponentially decaying far-field conditions on density functions in phase space. The uniform stability analysis follows directly from the integrable temporal decay of the collision frequencies along the particle trajectory. Our uniform stability results improve on the previous results [S. Y. Ha, E. Jeong, and R. M. Strain, Comm. Pure Appl. Anal., 12 (2013), pp. 1141--1161] from a uniform

L2-stability of the Landau equation near global Maxwellians

L^1

On the global solvability of the coupled kinetic-fluid system for flocking with large initial data

-stability analysis of the relativistic Boltzmann equation, which relied crucially on the nonlinear functional approach and smallness in the amplitude of the distribution function.

The Vlasov–Poisson–Boltzmann system with angular cutoff for soft potentials

Abstract In this article, we are concerned with the construction of global smooth small-amplitude solutions to the Cauchy problem of the one species Vlasov-Poisson-Boltzmann system near Maxwellians for long-range interactions. Compared with the former result obtained by Duan and Liu in [12] for the two species model, we do not ask the initial perturbation to satisfy the neutral condition and our result covers all physical collision kernels for the full range of intermolecular repulsive potentials.

GLOBAL NONLINEAR STABILITY OF RAREFACTION WAVES FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH TEMPERATURE AND DENSITY DEPENDENT TRANSPORT COEFFICIENTS

We present a L2-stability theory of the Vlasov-Maxwell-Boltzmann system for the two-species collisional plasma. We show that in a perturbative regime of a global Maxwellian, the L2-distance between two strong solutions can be controlled by that between initial data in a Lipschitz manner. Our stability result extends earlier results [Ha, S.-Y. and Xiao, Q.-H., “A revisiting to the L2-stability theory of the Boltzmann equation near global Maxwellians,” (submitted) and Ha, S.-Y., Yang, X.-F., and Yun, S.-B., “L2 stability theory of the Boltzmann equation near a global Maxwellian,” Arch. Ration. Mech. Anal. 197, 657–688 (2010)] on the L2-stability of the Boltzmann equation to the Boltzmann equation coupled with self-consistent external forces. As a direct application of our stability result, we show that classical solutions in Duan et al. [“Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space,” Commun. Pure Appl. Math. 24, 1497–1546 (2011)] and Guo [“The Vlasov-Maxwell-Boltzma...

Remarks on the nonlinear stability of the Kuramoto-Sakaguchi equation

We present an L-2-stability of the kinetic Landau equation for a single species charged plasma with an inverse power-law interaction force in the perturbative regime of global Maxwellians. Our result demonstrates that the L-2-distance between two classical solutions to the Landau equation can be controlled by that between corresponding initial data in a Lipschitz manner. The Coulomb interaction is known to be the singular and marginal case of the theory of the Boltzmann equation where the grazing collisions are the dominant. For some class of classical solutions, we show that our L-2-stability result can provide a uniform L-2-stability

The Vlasov–Poisson–Boltzmann system for the whole range of cutoff soft potentials

We study the dynamics of infinitely many Cucker–Smale (CS) flocking particles under the interplay of random communication and compressible fluids in planar wave case. For the dynamics of an ensemble of flocking particles, we use the kinetic Cucker–Smale–Fokker–Planck (CS–FP) equation with a degenerate diffusion, whereas for the fluid component, we use the compressible Navier–Stokes (NS) equations. These two subsystems are coupled via the drag force. For this coupled model, we present a global existence of classical solutions for arbitrarily large initial data which may contain vacuum.