arXiv subjects

Huijiang Zhao

Publications and source records attributed to Huijiang Zhao.

At least 19 recordsLinked to original sources

arXiv2026

Large time behavior of global solutions to one-dimensional non-isentropic compressible Navier-Stokes-Poisson equations with large initial data

In this paper, we are concerned with the construction of global solutions to the Cauchy problem for the one-dimensional non-isentropic compressible Navier-Stokes-Poisson equations, as well as the precise description of their large-time behavior. While results for small initial data are well-established, the case of large initial perturbations remains less satisfactory. Even for constant transport coefficients, only the Nishida-Smoller type solutions have been obtained so far. In this work, we will provide a complete resolution of this problem. The key idea involves revealing favorable hidden identities of the potential force and a new representation for the specific volume, which enables us to derive uniform upper and lower bounds on this quantity. Moreover, this strategy extends to compressible viscous single-ion flow under the Boltzmann relation; notably, this new method for deriving lower and upper bounds on the specific volume also applies to the Navier-Stokes equations. The proof relies on a domain decomposition technique and some new dissipative structures to close the nonlinear energy estimates.

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arXiv2026

Approximation of the relaxed Euler system at the level of the compressible Navier-Stokes equations

We study the global well-posedness of the Cauchy problem in $\mathbb R^d$, $d\geq2$, for the relaxed compressible Euler system with a Maxwell-type constitutive law for the stress tensor. We show that the corresponding compressible Navier-Stokes solutions serve as both relaxation limits and large-time asymptotic profiles of the global relaxed Euler solutions constructed here. Given a Navier-Stokes background satisfying suitable boundedness and decay assumptions, we construct global smooth relaxed Euler solutions for each fixed $α\in(0,1]$ and sufficiently small $\varepsilon>0$. Here, the background need not be small. After subtracting the stress initial layer, the approximation error is $O(\varepsilon^α)$ uniformly in time. Under suitable additional conditions on the initial perturbations, this error decays in $L^2$ faster than the decay of standard Navier-Stokes background, uniformly in $\varepsilon$, identifying the background solution as a large-time asymptotic profile. As an application, we obtain the corresponding small-data results using classical Navier-Stokes theory to verify the background assumptions.

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arXiv2026

Optimal time decay of the Boltzmann equation with cutoff soft potentials in the whole space

Since Strain's work \cite{Strain-2012-KRM} on almost optimal time decay rates for the microscopic part of perturbative classical solutions to the Boltzmann system with soft potentials via the time-velocity splitting method, how to obtain the optimal decay of the microscopic part has remained unsolved. In this paper, we resolve this problem by developing a nonlinear energy estimate method different from the time-velocity splitting approach. The proof relies on a combination of a hierarchy of time-weighted energy estimates, the damping factor $e^{-ν(v)(t-τ)}$, and a dyadic decomposition of velocity space.

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arXiv2026

Global Existence of Strong and Weak Solutions to the 2D Compressible Navier-Stokes-Poisson System in Bounded Domains with Large Data and Vacuum

We study the two-dimensional compressible Navier-Stokes-Poisson system in a general simply connected bounded domain with Navier-slip boundary conditions and a positive, possibly non-flat, doping profile. We establish the global existence of a unique strong solution and of a global weak solution for arbitrarily large initial data allowing vacuum when the shear viscosity is a positive constant and the bulk one proportional to a power of the density with the power bigger than 4/3. The main difficulties arise from the interaction between the Poisson force and the general boundary geometry. To overcome them, we introduce a modified material derivative and derive quantitative $L^p$ estimates for the density by means of a Neumann duality argument together with a harmonic correction generated by the Navier-slip boundary condition. We further obtain a pointwise representation of the effective viscous flux by pulling back the Neumann Green function on the unit disk through a conformal mapping. The resulting commutator estimates, combined with elliptic estimates for the Poisson equation, yield a uniform-in-time upper bound for the density and allow us to close the higher-order a priori estimates. Finally, the global weak solution is constructed through a charge-preserving approximation and compactness argument.

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arXiv2026

Time-periodic solutions of the Boltzmann equation with soft potentials in $\mathbb{R}^3$

We establish the existence of time-periodic solutions to the Boltzmann equation for the full range of soft potentials and prove their global stability. The proof relies on refined energy estimates combining Besov and Sobolev regularity with velocity-weighted energy methods. The analysis is complicated by the lack of a spectral gap in the soft potential regime together with the absence of time integrability of the periodic forcing over $\mathbb{R}^+$. This provides a framework for the study of time-periodic behavior for the Boltzmann equation.

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arXiv2025

The non-cutoff Vlasov-Poisson-Boltzmann system with weak collisions

We prove global existence of smooth solutions near Maxwellians for the non-cutoff Vlasov-Poisson-Boltzmann system in the weakly collisional regime. To address the weak dissipation of the non-cutoff linearized Boltzmann operator, we develop a refined velocity-weighted energy framework combined with vector-field techniques to control the transport term, nonlinear collisions, and the self-consistent electric field. This approach yields uniform-in-time bounds, captures enhanced dissipation of the solution, and establishes Landau damping for both the density and electric field, providing the first global-in-time result of this type for the non-cutoff Vlasov-Poisson-Boltzmann system. Our approach is inspired by the recent work of Chaturvedi-Luk-Nguyen ({\it J. Amer. Math. Soc.} {\bf 36} (2023), no. 4, 1103--1189.)

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arXiv2024

Global Smooth Radially Symmetric Solutions to a Multidimensional Radiation Hydrodynamics Model

The motion of a compressible inviscid radiative flow can be described by the radiative Euler equations, which consists of the Euler system coupled with a Poisson equation for the radiative heat flux through the energy equation. Although solutions of the compressible Euler system will generally develop singularity no matter how smooth and small the initial data are, it is believed that the radiation effect does imply some dissipative mechanism, which can guarantee the global regularity of the solutions of the radiative Euler equations at least for small initial data. Such an expectation was rigorously justified for the one-dimensional case, as for the multidimensional case, to the best of our knowledge, no result was available up to now. The main purpose of this paper is to show that the initial-boundary value problem of such a radiative Euler equation in a three-dimensional bounded concentric annular domain does admit a unique global smooth radially symmetric solution provided that the initial data is sufficiently small.

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arXiv2023

Global Existence and Vanishing Dispersion Limit of Strong/Classical Solutions to the One-dimensional Compressible Quantum Navier-Stokes Equations with Large Initial Data

We are concerned with the global existence and vanishing dispersion limit of strong/classical solutions to the Cauchy problem of the one-dimensional barotropic compressible quantum Navier-Stokes equations, which consists of the compressible Navier-Stokes equations with a linearly density-dependent viscosity and a nonlinear third-order differential operator known as the quantum Bohm potential. The pressure $p(ρ)=ρ^γ$ is considered with $γ\geq1$ being a constant. We focus on the case when the viscosity constant $ν$ and the Planck constant $\varepsilon$ are not equal. Under some suitable assumptions on $ν,\varepsilon, γ$, and the initial data, we proved the global existence and large-time behavior of strong and classical solutions away from vacuum to the compressible quantum Navier-Stokes equations with arbitrarily large initial data. This result extends the previous ones on the construction of global strong large-amplitude solutions of the compressible quantum Navier-Stokes equations to the case $ν\neq\varepsilon$. Moreover, the vanishing dispersion limit for the classical solutions of the quantum Navier-Stokes equations is also established with certain convergence rates. The proof is based on a new effective velocity which converts the quantum Navier-Stokes equations into a parabolic system, and some elaborate estimates to derive the uniform-in-time positive lower and upper bounds on the specific volume.

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arXiv2023

Global Hilbert expansion for some non-relativistic kinetic equations

The Vlasov-Maxwell-Landau (VML) system and the Vlasov-Maxwell-Boltzmann (VMB) system are fundamental models in dilute collisional plasmas. In this paper, we are concerned with the hydrodynamic limits of both the VML and the non-cutoff VMB systems in the entire space. Our primary objective is to rigorously prove that, within the framework of Hilbert expansion, the unique classical solution of the VML or non-cutoff VMB system converges globally over time to the smooth global solution of the Euler-Maxwell system as the Knudsen number approaches zero. The core of our analysis hinges on deriving novel interplay energy estimates for the solutions of these two systems, concerning both a local Maxwellian and a global Maxwellian, respectively. Our findings address a problem in the hydrodynamic limit for Landau-type equations and non-cutoff Boltzmann-type equations with a magnetic field. Furthermore, the approach developed in this paper can be seamlessly extended to assess the validity of the Hilbert expansion for other types of kinetic equations.

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arXiv2022

Hilbert expansion for kinetic equations with non-relativistic Coulomb collision

In this paper, we study the hydrodynamic limits of both the Landau equation and the Vlasov-Maxwell-Landau system in the whole space. Our main purpose is two-fold: the first one is to give a rigorous derivation of the compressible Euler equations from the Landau equation via the Hilbert expansion; while the second one is to prove, still in the setting of Hilbert expansion, that the unique classical solution of the Vlasov-Maxwell-Landau system converges, which is shown to be globally in time, to the resulting global smooth solution of the Euler-Maxwell system, as the Knudsen number goes to zero. The main ingredient of our analysis is to derive some novel interplay energy estimates on the solutions of the Landau equation and the Vlasov-Maxwell-Landau system which are small perturbations of both a local Maxwellian and a global Maxwellian, respectively. Our result solves an open problem in the hydrodynamic limit for the Landau-type equations with Coulomb potential and the approach developed in this paper can seamlessly be used to deal with the problem on the validity of the Hilbert expansion for other types of kinetic equations.

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arXiv2022

Global Regularity for A Radiation Hydrodynamics Model with Viscosity and Thermal Conductivity

In this paper, we study the global wellposedness of a radiation hydrodynamics model with viscosity and thermal conductivity. It is now well-understood that, unlike the compressible Euler equations whose smooth solutions must blow up in finite time no matter how small and how smooth the initial data is, the dissipative structure of such a radiation hydrodynamics model can indeed guarantee that its one-dimensional Cauchy problem admits a unique global smooth solution provided that the initial data is sufficiently small, while for large initial data, even if the heat conductivity is taken into account but the viscosity effect is ignored, shock type singularities must appear in finite time for smooth solutions of the Cauchy problem of one-dimensional radiation hydrodynamics model with thermal conductivity and zero viscosity. Thus a natural question is, if effects of both the viscosity and the thermal conductivity are considered, does the one-dimensional radiation hydrodynamics model with viscosity and thermal conductivity exist a unique global large solution? We give an affirmative answer to this problem and show in this paper that the initial-boundary value problem to the radiation hydrodynamics model in an one-dimensional periodic box T = R/Z with viscosity and thermal conductivity does exist a unique global smooth solution for any large initial data. The main ingredient in our analysis is to introduce some delicate estimates, especially an improved estimate on the absolute temperature and a pointwise estimate between the absolute temperature, the specific volume, and the first-order spatial derivative of the macro radiation flux, to deduce the desired positive lower and upper bounds on the density and the absolute temperature.

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arXiv2021

On the Vlasov-Poisson-Boltzmann limit of the Vlasov-Maxwell-Boltzmann system

For the whole range of cutoff intermolecular interactions, we give a rigorous mathematical justification of the limit from the Vlasov-Maxwell-Boltzmann system to the Vlasov-Poisson-Boltzmann system as the light speed tends to infinity.Such a limit is shown to hold global-in-time in the perturbative framework.The key point in our analysis is to deduce certain a priori estimates which are independent of the light speed and the main difficulty is due to the degeneracy of the dissipative effect of the electromagnetic field for large light speed.

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arXiv2020

Radially Symmetric Stationary Wave for Two-dimensional Burgers Equation

We are concerned with the radially symmetric stationary wave for the exterior problem of two-dimensional Burgers equation. A sufficient and necessary condition to guarantee the existence of such a stationary wave is given and it is also shown that such a stationary wave satisfies nice decay estimates and is time-asymptotically nonlinear stable under radially symmetric perturbation.

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arXiv2019

Asymptotics of Radially Symmetric Solutions for the Exterior Problem of Multidimensional Burgers Equation

We are concerned with the large-time behavior of the radially symmetric solution for multidimensional Burgers equation on the exterior of a ball $\mathbb{B}_{r_0}(0)\subset \mathbb{R}^n$ for $n\geq 3$ and some positive constant $r_0>0$, where the boundary data $v_-$ and the far field state $v_+$ of the initial data are prescribed and correspond to a stationary wave. It is shown in \cite{Hashimoto-Matsumura-JDE-2019} that a sufficient condition to guarantee the existence of such a stationary wave is $v_+<0, v_-\leq |v_+|+μ(n-1)/r_0$. Since the stationary wave is no longer monotonic, its nonlinear stability is justified only recently in \cite{Hashimoto-Matsumura-JDE-2019} for the case when $v_\pm<0, v_-\leq v_++μ(n-1)/r_0$. The main purpose of this paper is to verify the time asymptotically nonlinear stability of such a stationary wave for the whole range of $v_\pm$ satisfying $v_+<0, v_-\leq |v_+|+μ(n-1)/r_0$. Furthermore, we also derive the temporal convergence rate, both algebraically and exponentially. Our stability analysis is based on a space weighted energy method with a suitable chosen weight function, while for the temporal decay rates, in addition to such a space weighted energy method, we also use the space-time weighted energy method employed in \cite{Kawashima-Matsumura-CMP-1985} and \cite{Yin-Zhao-KRM-2009}.

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arXiv2018

Global Spherical Symmetric Flows for a Viscous Radiative and Reactive Gas in an Exterior Domain with Large Initial Data

In this paper, we study the global existence, uniqueness and large-time behavior of spherically symmetric solution of a viscous radiative and reactive gas in an unbounded domain exterior to the unit sphere in $\mathbb{R}^{n}$ for $n\geq 2$. The key point in the analysis is to deduce certain uniform estimates on the solutions, especially on the uniform positive lower and upper bounds on the specific volume and the temperature.

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arXiv2017

Global Existence and Large-time Behavior of Solutions to the Cauchy Problem of One-dimensional Viscous Radiative and Reactive Gas

Although there are many results on the global solvability and the precise description of the large time behaviors of solutions to the initial-boundary value problems of the one-dimensional viscous radiative and reactive gas in bounded domain with two typical types of boundary conditions, no result is available up to now for the corresponding problems in unbounded domain. This paper focuses on the Cauchy problem of such a system with prescribed large initial data and the main purpose is to construct its global smooth non-vacuum solutions around a non-vacuum constant equilibrium state and to study the time-asymptotically nonlinear stability of such an equilibrium state. The key point in the analysis is to deduce the uniform positive lower and upper bounds on the specific volume and the temperature.

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arXiv2017

Convergence to the Self-similar Solutions to the Homogeneous Boltzmann Equation

The Boltzmann H-theorem implies that the solution to the Boltzmann equation tends to an equilibrium, that is, a Maxwellian when time tends to infinity. This has been proved in varies settings when the initial energy is finite. However, when the initial energy is infinite, the time asymptotic state is no longer described by a Maxwellian, but a self-similar solution obtained by Bobylev-Cercignani. The purpose of this paper is to rigorously justify this for the spatially homogeneous problem with Maxwellian molecule type cross section without angular cutoff.

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arXiv2016

One-Dimensional Compressible Heat-Conducting Gas with Temperature-Dependent Viscosity

We consider the one-dimensional compressible Navier--Stokes system for a viscous and heat-conducting ideal polytropic gas when the viscosity $μ$ and the heat conductivity $κ$ depend on the specific volume $v$ and the temperature $θ$ and are both proportional to $h(v)θ^α$ for certain non-degenerate smooth function $h$. We prove the existence and uniqueness of a global-in-time non-vacuum solution to its Cauchy problem under certain assumptions on the parameter $α$ and initial data, which imply that the initial data can be large if $|α|$ is sufficiently small. Our result appears to be the first global existence result for general adiabatic exponent and large initial data when the viscosity coefficient depends on both the density and the temperature.

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