Renjun Duan
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View article: Global stability of the inhomogeneous sheared Boltzmann equation in torus
Global stability of the inhomogeneous sheared Boltzmann equation in torus Open
Homo-energetic solutions to the spatially homogeneous Boltzmann equation have been extensively studied, but their global stability in the inhomogeneous setting remains challenging due to unbounded energy growth under self-similar scaling a…
View article: Ionic KdV structure in weakly collisional plasmas
Ionic KdV structure in weakly collisional plasmas Open
We consider the one-dimensional ions dynamics in weakly collisional plasmas governed by the Vlasov-Poisson-Landau system under the Boltzmann relation with the small collision frequency $ν>0$. It is observed in physical experiments that the…
View article: Inelastic Boltzmann equation under shear heating
Inelastic Boltzmann equation under shear heating Open
In this paper, we study the spatially homogeneous inelastic Boltzmann equation for the angular cutoff pseudo-Maxwell molecules with an additional term of linear deformation. We establish the existence of non-Maxwellian self-similar profile…
View article: Global dynamics of isothermal rarefied gas flows in an infinite layer
Global dynamics of isothermal rarefied gas flows in an infinite layer Open
Let rarefied gas be confined in an infinite layer with diffusely reflecting boundaries that are isothermal and non-moving. The initial-boundary value problem on the nonlinear Boltzmann equation governing the rarefied gas flow in such setti…
View article: Steady compressible Navier-Stokes-Fourier system with slip boundary conditions arising from kinetic theory
Steady compressible Navier-Stokes-Fourier system with slip boundary conditions arising from kinetic theory Open
This paper studies the boundary value problem on the steady compressible Navier-Stokes-Fourier system in a channel domain $(0,1)\times\mathbb{T}^2$ with a class of generalized slip boundary conditions that were systematically derived from …
View article: The 3D kinetic Couette flow via the Boltzmann equation in the diffusive limit
The 3D kinetic Couette flow via the Boltzmann equation in the diffusive limit Open
In the paper we study the Boltzmann equation in the diffusive limit in a channel domain $\mathbb{T}^2\times (-1,1)$ for the 3D kinetic Couette flow. Our results demonstrate that the first-order approximation of the solutions is governed by…
View article: Compressible fluid limit for smooth solutions to the Landau equation
Compressible fluid limit for smooth solutions to the Landau equation Open
Although the compressible fluid limit of the Boltzmann equation with cutoff has been extensively investigated in Caflisch [Comm. Pure Appl. Math. 33 (1980), 651–666] and Guo, Jang, and Jiang [Comm. Pure Appl. Math. 63 (2010), 337–361], obt…
View article: Polynomial tail solutions of the non-cutoff Boltzmann equation near local Maxwellians
Polynomial tail solutions of the non-cutoff Boltzmann equation near local Maxwellians Open
This paper aims to incorporate the Caflisch's decomposition into the macro-micro decomposition in Boltzmann theory for allowing the microscopic component to exhibit only the polynomial tail in large velocities. In particular, we treat the …
View article: A note on Landau damping of two-species Vlasov-Poisson system
A note on Landau damping of two-species Vlasov-Poisson system Open
In this note we adopt an approach by Grenier, Nguyen and Rodnianski in \cite{GNR} for studying the nonlinear Landau damping of the two-species Vlasov-Poisson system in the phase space $\mathbb{T}^d_x \times \mathbb{R}^d_v$ with the dimensi…
View article: Time-Velocity Decay of Solutions to the Non-Cutoff Boltzmann Equation in the Whole Space
Time-Velocity Decay of Solutions to the Non-Cutoff Boltzmann Equation in the Whole Space Open
In this paper, we consider the perturbed solutions with polynomial tail in\nlarge velocities for the non-cutoff Boltzmann equation near global Maxwellians\nin the whole space. The global in time existence is proved in the weighted\nSobolev…
View article: Boltzmann equation with mixed boundary condition
Boltzmann equation with mixed boundary condition Open
We study the Boltzmann equation in a smooth bounded domain featuring a mixed boundary condition. Specifically, gas particles experience specular reflection in two parallel plates, while diffusive reflection occurs in the remaining portion …
View article: The Boltzmann equation for plane Couette flow
The Boltzmann equation for plane Couette flow Open
In the paper, we study the plane Couette flow of a rarefied gas between two parallel infinite plates at y=\pm L moving relative to each other with opposite velocities (\pm \alpha L,0,0) along the x -direction. Assuming that the stationary …
View article: KdV limit for the Vlasov-Poisson-Landau system
KdV limit for the Vlasov-Poisson-Landau system Open
Two fundamental models in plasma physics are given by the Vlasov-Poisson-Landau system and the compressible Euler-Poisson system which both capture the complex dynamics of plasmas under the self-consistent electric field interactions at th…
View article: Solutions to the non-cutoff Boltzmann equation in the grazing limit
Solutions to the non-cutoff Boltzmann equation in the grazing limit Open
It is known that in the parameter range -2 \leq \gamma <-2s , a spectral gap does not exist for the linearized Boltzmann operator without cutoff, but it does for the linearized Landau operator. This paper is devoted to the understanding of…
View article: Global quasineutral Euler limit for the Vlasov-Poisson-Landau system with rarefaction waves
Global quasineutral Euler limit for the Vlasov-Poisson-Landau system with rarefaction waves Open
In the paper, we consider the Cauchy problem on the spatially one-dimensional Vlasov-Poisson-Landau system modelling the motion of ions under a generalized Boltzmann relation. Let the Knudsen number and the Debye length be given as $\varep…
View article: Global mild solution with polynomial tail for the Boltzmann equation in the whole space
Global mild solution with polynomial tail for the Boltzmann equation in the whole space Open
We are concerned with the Cauchy problem on the Boltzmann equation in the whole space. The goal is to construct global-in-time bounded mild solutions near Maxwellians with the perturbation admitting a polynomial tail in large velocities. T…
View article: Uniform shear flow via the Boltzmann equation with hard potentials
Uniform shear flow via the Boltzmann equation with hard potentials Open
The motion of rarefied gases for uniform shear flow at the kinetic level is governed by the spatially homogeneous Boltzmann equation with a deformation force. In the paper we study the corresponding Cauchy problem with initial data of fini…
View article: Global bounded solutions to the Boltzmann equation for a polyatomic gas
Global bounded solutions to the Boltzmann equation for a polyatomic gas Open
In this paper we consider the Boltzmann equation modelling the motion of a polyatomic gas where the integration collision operator in comparison with the classical one involves an additional internal energy variable $I\in\mathbb{R}_+$ and …
View article: An $L^1_k\cap L^p_k$ approach for the non-cutoff Boltzmann equation in $\mathbb{R}^3$
An $L^1_k\cap L^p_k$ approach for the non-cutoff Boltzmann equation in $\mathbb{R}^3$ Open
In the paper, we develop an $L^1_k\cap L^p_k$ approach to construct global solutions to the Cauchy problem on the non-cutoff Boltzmann equation near equilibrium in $\mathbb{R}^3$. In particular, only smallness of $\|\mathcal{F}_x{f}_0\|_{L…
View article: Compressible Euler-Maxwell limit for global smooth solutions to the Vlasov-Maxwell-Boltzmann system
Compressible Euler-Maxwell limit for global smooth solutions to the Vlasov-Maxwell-Boltzmann system Open
Two fundamental models in plasma physics are given by the Vlasov-Maxwell-Boltzmann system and the compressible Euler-Maxwell system which both capture the complex dynamics of plasmas under the self-consistent electromagnetic interactions a…
View article: Compressible fluid limit for smooth solutions to the Landau equation
Compressible fluid limit for smooth solutions to the Landau equation Open
Although the compressible fluid limit of the Boltzmann equation with cutoff has been well investigated in [6] and [13], it still remains largely open to obtain analogous results in case of the angular non-cutoff or even in the grazing limi…
View article: Heat transfer problem for the Boltzmann equation in a channel with diffusive boundary condition
Heat transfer problem for the Boltzmann equation in a channel with diffusive boundary condition Open
In this paper, we study the 1D steady Boltzmann flow in a channel. The walls of the channel are assumed to have vanishing velocity and given temperatures $θ_0$ and $θ_1$. This problem was studied by Esposito et al [13,14] where they showed…
View article: On Smooth Solutions to the Thermostated Boltzmann Equation with Deformation
On Smooth Solutions to the Thermostated Boltzmann Equation with Deformation Open
This paper concerns a kinetic model of the thermostated Boltzmann equation with a linear deformation force described by a constant matrix. The collision kernel under consideration includes both the Maxwell molecule and general hard potenti…
View article: On smooth solutions to the thermostated Boltzmann equation with deformation
On smooth solutions to the thermostated Boltzmann equation with deformation Open
This paper concerns a kinetic model of the thermostated Boltzmann equation with a linear deformation force described by a constant matrix. The collision kernel under consideration includes both the Maxwell molecule and general hard potenti…
View article: Low Regularity Solutions for the Vlasov-Poisson-Landau/Boltzmann System
Low Regularity Solutions for the Vlasov-Poisson-Landau/Boltzmann System Open
In the paper, we are concerned with the nonlinear Cauchy problem on the Vlasov-Poisson-Landau/Boltzmann system around global Maxwellians in torus or finite channel. The main goal is to establish the global existence and large time behavior…
View article: The Boltzmann equation for plane Couette flow
The Boltzmann equation for plane Couette flow Open
In the paper, we study the plane Couette flow of a rarefied gas between two parallel infinite plates at $y=\pm L$ moving relative to each other with opposite velocities $(\pm αL,0,0)$ along the $x$-direction. Assuming that the stationary s…
View article: Compressible Navier-Stokes approximation for the Boltzmann equation in bounded domains
Compressible Navier-Stokes approximation for the Boltzmann equation in bounded domains Open
It is well known that the full compressible Navier-Stokes equations can be deduced via the Chapman-Enskog expansion from the Boltzmann equation as the first-order correction to the Euler equations with viscosity and heat-conductivity coeff…