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In this paper we deal with the extension of the Fast Kinetic Scheme (FKS) [J. Comput. Phys., Vol. 255, 2013, pp 680-698] originally constructed for solving the BGK equation, to the more challenging case of the Boltzmann equation. The scheme…
We propose a particle method for numerically solving the Landau equation, inspired by the score-based transport modeling (SBTM) method for the Fokker-Planck equation. This method can preserve some important physical properties of the Landau…
In this paper, we consider the Boltzmann equation for a polyatomic gas. We establish that the mild solution to the Boltzmann equation on the torus is globally well-posed, provided the initial data that satisfy bounded velocity-weighted…
We study the convergence from the multi-species Boltzmann equations to the non-isothermal Maxwell-Stefan system. The global-in-time well-posedness of the Maxwell-Stefan system is first established. The solution is utilized as the fluid…
The transport and continuum equations exhibit a number of conservation laws. For example, scalar multiplication is conserved by the transport equation, while positivity of probabilities is conserved by the continuum equation. Certain…
We construct stable manifolds for a class of degenerate evolution equations including the steady Boltzmann equation, establishing in the process exponential decay of associated kinetic shock and boundary layers to their limiting equilibrium…
We present a general methodology for constructing lattice Boltzmann models of hydrodynamics with certain desired features of statistical physics and kinetic theory. We show how a methodology of linear programming theory, known as…
We present a method for bounding, and in some cases computing, the spectral gap for systems of many particles evolving under the influence of a random collision mechanism. In particular, the method yields the exact spectral gap in a model…
We consider the spatially inhomogeneous Boltzmann equation for inelastic hard-spheres, with constant restitution coefficient $\alpha\in(0,1)$, under the thermalization induced by a host medium with a fixed Maxwellian distribution and any…
Solving the stationary nonlinear Fokker-Planck equations is important in applications and examples include the Poisson-Boltzmann equation and the two layer neural networks. Making use of the connection between the interacting particle…
Stochastic-periodic homogenization is studied for the Maxwell equations with nonlinear and periodic electric conductivity. It is shown by the stochastic-two-scale convergence method that the sequence of solutions of a class of highly…
We develop an efficient algorithm for a spatially inhomogeneous matrix-valued quantum Boltzmann equation derived from the Hubbard model. The distribution functions are $2 \times 2$ matrix-valued to accommodate the spin degree of freedom,…
We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative and…
A simplified relativistic kinetic theory for gases with internal degrees of freedom, based on a BGK-type collision term, is considered. First the Boltzmann equation is rewritten in tetrad form and then thermal coefficients are determined to…
Solution of Helmholtz equation with impedance boundary condition on finite interval is equivalently reformulated as steady state of initial boundary value problem for first order hyperbolic system of partial differential equations.…
This work proves the global stability of the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse-power intermolecular potentials, $r^{-(p-1)}$ with $p>2$, for initial…
We develop a nonlinear multigrid method to solve the steady state of microflow, which is modeled by the high order moment system derived recently for the steady-state Boltzmann equation with ES-BGK collision term. The solver adopts a…
We propose an entropic Fourier method for the numerical discretization of the Boltzmann collision operator. The method, which is obtained by modifying a Fourier Galerkin method to match the form of the discrete velocity method, can be…
Any numerical method fails to provide us with acceptable results if not equipped with appropriate boundary conditions. Catering to more realistic applications, in the present article we have extended the work done on the one plus one…
The Koopman operator approach provides a powerful linear description of nonlinear dynamical systems in terms of the evolution of observables. While the operator is typically infinite-dimensional, it is crucial to develop finite-dimensional…