arXiv subjects

Kung-Chien Wu

Publications and source records attributed to Kung-Chien Wu.

14 recordsLinked to original sources

arXiv2024

Space-time structure and particle-fluid duality of solutions for Boltzmann equation with hard potentials

We study the quantitative pointwise behavior of solutions to the Boltzmann equation for hard potentials and Maxwellian molecules, which generalize the hard sphere case introduced by Liu-Yu in 2004 (Comm. Pure Appl. Math. 57:1543-1608, 2004). The large time behavior of the solution is dominated by fluid structures, similar to the hard sphere case. However, unlike hard sphere, the spatial decay here depends on the potential power $γ$ and the initial velocity weight. A key challenge in this problem is the loss of velocity weight in linear estimates, which makes standard nonlinear iteration infeasible. To address this, we develop an Enhanced Mixture Lemma, demonstrating that mixing the transport and gain parts of the linearized collision operator can generate arbitrary-order regularity and decay in both space and velocity variables. This allows us to decompose the linearized solution into fluid (arbitrary regularity and velocity decay) and particle (rapid space-time decay, but with loss of velocity decay) parts, making it possible to solve the nonlinear problem through this particle-fluid duality.

math.AP↗

arXiv2024

Hölder regularity of solutions of the steady Boltzmann equation with soft potentials

We consider the Hölder regularity of solutions to the steady Boltzmann equation with in-flow boundary condition in bounded and strictly convex domains $Ω\subset\mathbb{R}^{3}$ for gases with cutoff soft potential $(-3<γ<0)$. We prove that there is a unique solution with a bounded $L^{\infty}$ norm in space and velocity. This solution is Hölder continuous, and it's order depends not only on the regularity of the incoming boundary data, but also on the potential power $γ$. The result for modulated soft potential case $-2<γ<0$ is similar to hard potential case $(0\leqγ<1)$ since we have $C^{1}$ velocity regularity from collision part. However, we observe that for very soft potential case $(-3<γ\leq -2)$, the regularity in velocity obtained by the collision part is lower (Hölder only), but the boundary regularity still can transfer to solution (in both space and velocity) by transport and collision part under the restriction of $γ$.

math.AP↗

arXiv2024

3D hard sphere Boltzmann equation: explicit structure and the transition process from polynomial tail to Gaussian tail

We study the Boltzmann equation with hard sphere in a near-equilibrium setting. The initial data is compactly supported in the space variable and has a polynomial tail in the microscopic velocity. We show that the solution can be decomposed into a particle-like part (polynomial tail) and a fluid-like part (Gaussian tail). The particle-like part decays exponentially in both space and time, while the fluid-like part corresponds to the behavior of the compressible Navier-Stokes equation, which dominates the long time behavior and exhibits rich wave motion. The nonlinear wave interactions in the fluid-like part are precisely characterized and therefore we are able to distinguish the linear and nonlinear wave of the solution. It is notable that although the solution has polynomial tail in the velocity initially, the transition process from the polynomial to the Gaussian tail can be quantitatively revealed due to the collision with the background global Maxwellian.

math.AP↗

arXiv2023

Stability of background perturbation for Boltzmann equation

Consider the Boltzmann equation in the perturbation regime. Since the macroscopic quantities in the background global Maxwellian are obtained through measurements, there are typically some errors involved. This paper investigates the effect of background variations on the solution for a given initial perturbation. Our findings demonstrate that the solution changes continuously with variations in the background and provide a sharp time decay estimate of the associated errors. The proof relies on refined estimates for the linearized solution operator and a proper decomposition of the nonlinear solution.

math.AP↗

arXiv2021

Space-time behavior of the solution to the Boltzmann equation with soft potentials

In this paper, we get the quantitative space-time behavior of the full Boltzmann equation with soft potentials ($-2<γ<0$) in the close to equilibrium setting, under some velocity decay assumption, but without any Sobolev regularity assumption on the initial data. We find that both the large time and spatial behaviors depend on the velocity decay of the initial data and the exponent $γ$. The key step in our strategy is to obtain the $L^{\infty }$ bound of a suitable weighted full Boltzmann equation directly, rather than using Green's function and Duhamel's principle to construct the pointwise structure of the solution as in the paper: T.-P. Liu and S.-H. Yu, The Green function and large time behavier of solutions for the one-dimensional Boltzmann equation, Commun. Pure App. Math.,(2004). This provides a new thinking in the related study.

math.AP↗

arXiv2018

Explicit Structure of the Fokker-Planck Equation with potential

We study the pointwise (in the space and time variables) behavior of the Fokker-Planck Equation with flat confinement. The solution has very clear description in the $xt-$plane, including large time behavior, initial layer and asymptotic behavior. Moreover, the structure of the solution highly depends on the potential function.

math-ph↗

arXiv2017

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, 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, 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 largely extend the classical results of Liu-Yu \cite{[LiuYu], [LiuYu2], [LiuYu1]} and Lee-Liu-Yu \cite% {[LeeLiuYu]} to hard and soft potentials by imposing suitable exponential velocity weight on the initial condition.

math.AP↗

arXiv2017

Smoothing effects and decay estimate of the solution of the linearized two species Landau equation

We study the Landau equation for a mixture of two species in the whole space, with initial condition of one species near a vacuum and the other near a Maxwellian equilibrium state. For the linearized level, without any smoothness assumption on the initial data, it is shown that the solution becomes smooth instantaneously in both the space and momentum variables. Moreover, the large time behavior of the solution is also obtained.

math-ph↗

arXiv2017

Solving Linearized Landau Equation Pointwisely

We study the pointwise (in the space and time variables) behavior of the linearized Landau equation for hard and moderately soft potentials. The solution has very clear description in the $(x,t)-$variables, including large time behavior and asymptotic behavior. More precisely, we obtain the pointwise fluid structure inside finite Mach number region, and exponential or sub-exponential decay, depending on interactions between particles, in the space variable outside finite Mach number region. The spectrum analysis, regularization effect and refined weighted energy estimate play important roles in this paper.

math-ph↗

arXiv2015

Cauchy problem and exponential stability for the inhomogeneous Landau equation

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 a exponential stability for such a solution, with a rate as close as we want to the optimal rate given by the semigroup decay.

math.AP↗

arXiv2015

Anelastic Approximation of the Gross-Pitaevskii equation for General Initial Data

We perform a rigorous analysis of the anelastic approximation for the Gross-Pitaevskii equation with $x$-dependent chemical potential. For general initial data and periodic boundary condition, we show that as $\eps\to 0$, equivalently the Planck constant tends to zero, the density $|ψ^{\eps}|^{2}$ converges toward the chemical potential $ρ_{0}(x)$ and the velocity field converges to the anelastic system. When the chemical potential is a constant, the anelastic system will reduce to the incompressible Euler equations. The resonant effects the singular limit process and it can be overcome because of oscillation-cancelation.

math.AP↗

arXiv2013

Global in Time Estimates for the Spatially Homogeneous Landau Equation with Soft Potentials

This paper deals with some global in time a priori estimates of the spatially homogeneous Landau equation for soft potentials $\ga\in[-2,0)$. For the first result, we obtain the estimate of weak solutions in $L^α_{t}L_{v}^{3-\eps}$ for $α=\frac{2(3-\eps)}{3(2-\eps)}$ and $0<\eps<1$, which is an improvement over estimates by Fournier-Guerin [N. Fournier; H. Guerin, Well-posedness of the spatially homogeneous Landau equation for soft potentials. J. Funct. Anal. 25(2009), no. 8, 2542--2560]. Foe the second result, we have the estimate of weak solutions in $L_{t}^{\infty}L^{p}_{v}$, $p>1$, which extends part of results by Fournier-Guerin and Alexandre-Liao-Lin [R. Alexandre, J. Liao, and C. Lin, Some a priori estimates for the homogeneous Landau equation with soft potentials, arXiv:1302.1814]. As an application, we deduce some global well-posedness results for $\ga\in [-2,0)$. Our estimates include the critical case $\ga=-2$, which is the key point in this paper.

math-ph↗

arXiv2013

Pointwise Behavior of the Linearized Boltzmann Equation on Torus

We study the pointwise behavior of the linearized Boltzmann equation on torus for non-smooth initial perturbation. 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 mode for the space variable, 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. Moreover, the mixture lemma plays an important role in constructing the kinetic-like waves, we supply a new proof of this lemma to avoid constructing explicit solution of the damped transport equations

math-ph↗

arXiv2013

Exponential Time Decay Estimates for the Landau Equation on Torus

We study the time decay estimates for the linearized Landau equation on torus when the initial perturbation is not necessarily smooth. Our result reveals the kinetic and fluid aspects of the equation. We design a Picard-type iteration and Mixture lemma for constructing the increasingly regular kinetic like waves, they are carried by transport equations and have exponential time decay rate. The fluid like waves are constructed as part of the long-wave expansion in the spectrum of the Fourier mode for the space variable and the time decay rate depends on the size of the domain. The Mixture lemma plays an important role in this paper, this lemma is parallel to Boltzmann equation but the proof is more challenge.

math-ph↗