Kung Chien Wu

An Asymptotic Limit of a Navier–Stokes System with Capillary Effects

A combined incompressible and vanishing capillarity limit in the barotropic compressible Navier–Stokes equations for smooth solutions is proved. The equations are considered on the two-dimensional torus with well prepared initial data. The momentum equation contains a rotational term originating from a Coriolis force, a general Korteweg-type tensor modeling capillary effects, and a density-dependent viscosity. The limiting model is the viscous quasi-geostrophic equation for the “rotated” velocity potential. The proof of the singular limit is based on the modulated energy method with a careful choice of the correction terms.

Cauchy Problem and Exponential Stability for the Inhomogeneous Landau Equation

Abstract This work deals with the inhomogeneous Landau equation on the torus in the cases of hard, Maxwellian and moderately soft potentials. We first investigate the linearized equation and we prove exponential decay estimates for the associated semigroup. We then turn to the nonlinear equation and we use the linearized semigroup decay in order to construct solutions in a close-to-equilibrium setting. Finally, we prove an exponential stability for such a solution, with a rate as close as we want to the optimal rate given by the semigroup decay.

Pointwise Behavior of the Linearized Boltzmann Equation on a Torus

We study the pointwise behavior of the linearized Boltzmann equation on a torus for nonsmooth initial perturbations. The result reveals both the fluid and kinetic aspects of this model. The fluid-like waves are constructed as part of the long-wave expansion in the spectrum of the Fourier modes for the space variable, and the time decay rate of the fluid-like waves depends on the size of the domain. We design a Picard-type iteration for constructing the increasingly regular kinetic-like waves, which are carried by the transport equations and have exponential time decay rate. The mixture lemma plays an important role in constructing the kinetic-like waves, and we supply a new proof of this lemma without using the explicit solution of the damped transport equations (compare with Liu and Yus proof [H. W. Kuo, T. P. Liu, and S. E. Noh, Bull. Inst. Math. Acad. Sin. (N.S.), 5 (2010), pp. 1--10; T.-P. Liu and S.-H. Yu, Comm. Pure Appl. Math., 57 (2004), pp. 1543--1608]).

Low Froude number limit of the rotating shallow water and Euler equations

We perform the mathematical derivation of the rotating lake equations (anelastic system) from the classical solution of the rotating shallow water and Euler equations when the Froude number tends to zero.

Nonlinear stability of the Boltzmann equation in a periodic box

We study the nonlinear stability of the Boltzmann equation in the 3-dimensional periodic box with size 1/e of each side, where 0 < e ≪ 1 is the Knudsen number. The initial perturbation is not necessary smooth. The convergence rate is algebraic for small time region and exponential for large time region. Moreover, the algebraic rate is optimal and the exponential rate depends on the size of the domain (Knudsen number).

Quantitative Pointwise Estimate of the Solution of the Linearized Boltzmann Equation

We study the quantitative pointwise behavior of the solutions of the linearized Boltzmann equation for hard potentials, Maxwellian molecules and soft potentials, with Grad’s angular cutoff assumption. More precisely, for solutions inside the finite Mach number region (time like region), we obtain the pointwise fluid structure for hard potentials and Maxwellian molecules, and optimal time decay in the fluid part and sub-exponential time decay in the non-fluid part for soft potentials. For solutions outside the finite Mach number region (space like region), we obtain sub-exponential decay in the space variable. The singular wave estimate, regularization estimate and refined weighted energy estimate play important roles in this paper. Our results extend the classical results of Liu and Yu (Commun Pure Appl Math 57:1543–1608, 2004), (Bull Inst Math Acad Sin 1:1–78, 2006), (Bull Inst Math Acad Sin 6:151–243, 2011) and Lee et al. (Commun Math Phys 269:17–37, 2007) to hard and soft potentials by imposing suitable exponential velocity weight on the initial condition.