arXiv subjects

Fucai Li

Publications and source records attributed to Fucai Li.

At least 19 recordsLinked to original sources

arXiv2026

Enhanced stability and asymptotic limits to the non-isentropic compressible fluid-particle interaction model with thermal effects

In Einstein's seminal work [Ann. Physik, 17 (1905), 549-560], he pointed out that the temperature of a fluid influences the motion of suspended particles dramatically. To describe the effect of the temperature in this physical process more precisely, Boudin et al. [ESAIM Proc., 28 (2009), 195-210] introduced a new fluid-particle interaction model containing of the non-isentropic compressible Euler equations for the fluid and a nonlinear Vlasov-Fokker-Planck type equation for the particles. By adding some viscous and heat conductive terms to the fluid part of this model, Mu and Wang [Calc. Var. Partial Differential Equations, 59 (2020), Paper no. 110] established the global existence of classical solutions near an equilibrium state. In this paper, through establishing the uniform a priori estimates with respect to the viscosity and heat conductivity coefficients and taking the combined zero viscosity and heat conductivity limits, we show that the model introduced by Boudin et al. still admits a global classical solution and enjoys optimal decay rates thereby improving Mu and Wang's results and confirming Einstein's predications. Our work indicates that the presence of particles indeed emanates new dissipation effects on the non-isentropic compressible fluid-particle model via the differences between the macroscopic velocity of the particles and the fluid velocity, and the macroscopic temperature of the particles and the fluid temperature, which is significantly different from the case of pure non-isentropic compressible Euler equations. To achieve these goals, we have developed new ideas and techniques to surmount substantial obstacles caused by the absence of viscosity and heat conductivity, and the nonlinear interactions between the fluid and particles.

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arXiv2026

Global well-posedness and inviscid limit of the compressible Navier-Stokes-Vlasov-Fokker-Planck system with density-dependent friction force

This paper investigates the global dynamics of a three-dimensional fluid-particle interaction system that couples the compressible barotropic Navier-Stokes equations with the Vlasov-Fokker-Planck equation through a density-dependent friction force. The study establishes the global well-posedness, uniform-in-viscosity estimates, the global inviscid limit, and optimal large-time decay rates for classical solutions near equilibrium. First, for initial perturbations in $H^3$ sufficiently close to equilibrium, regularity estimates that are uniform in the viscosity coefficient are derived, and the existence of global classical solutions to the Cauchy problem is obtained. These uniform bounds enable us to rigorously justify the global-in-time inviscid limit as viscosity vanishes, with an explicit convergence rate proportional to the viscosity coefficient. This behavior differs significantly from that of the pure compressible Navier-Stokes system in the absence of particle interactions, emphasizing the stabilizing influence of kinetic coupling. Consequently, we establish for the first time the global existence of classical solutions to the compressible Euler-Vlasov-Fokker-Planck system. Moreover, under an additional mild assumption on the initial data, optimal time decay rates for both the solution and its spatial derivatives are obtained. Notably, the dissipative and microscopic components decay at a rate half an order faster than the macroscopic solution itself, indicating a novel relaxation mechanism induced by fluid-particle interactions. The analysis introduces new energy and dissipation structures for the coupled system, overcoming substantial difficulties arising from fluid-particle interactions.

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arXiv2026

Low Mach number limit and optimal time decay rates of the compressible Navier-Stokes-transport system in critical Besov spaces

In this paper, we investigate the Navier-Stokes-Transport (NST) system in the framework of Besov spaces. This system contains of a compressible Navier-Stokes system for the density and momentum of a fluid, and a transport equation for the potential temperature of the fluid. In stark contrast to the well-known Navier-Stokes-Fourier (NSF) system where the temperature satisfies a parabolic type equation providing dissipative effect for the temperature and the density, the temperature in our NST system enjoys a transport equation which precludes a dissipative mechanism for the density, leading to significant different effects to the whole system. We first establish the global well-posedness of strong solutions to the compressible NST system in critical Besov spaces over $\mathbb{R}^d$ with $d \geq 2$. Furthermore, by introducing the Mach number $\varepsilon > 0$, we rigorously prove the low Mach number limit as $\varepsilon \to 0$, showing that the solutions converge to that of the incompressible inhomogeneous Navier-Stokes system. This singular limit holds globally in time, even for {\it ill-prepared} initial data. To address the challenge posed by the lack of dissipation on the density and temperature, we develop a refined energy analysis and establish optimal time decay rates for strong solutions in $\mathbb{R}^d$ with $d \geq 3$. Notably, the density remains uniformly bounded in time, displaying asymptotic behavior fundamentally distinct from that in the NSF system, where the density possesses a dissipative structure via the momentum and temperature equations and exhibits temporal decay.

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arXiv2026

Global Hilbert expansion for the ionic Vlasov-Poisson-Boltzmann system

We justify the global-in-time validity of Hilbert expansion for the ionic Vlasov-Poisson-Boltzmann system in $\mathbb{R}^3$, a fundamental model describing ion dynamics in dilute collisional plasmas. As the Knudsen number approaches zero, we rigorously derive the compressible Euler-Poisson system governing global smooth irrotational ion flows. The truncated Hilbert expansion exhibits a multi-layered mathematical structure: the expansion coefficients satisfy linear hyperbolic systems, while the remainder equation couples with a nonlinear Poisson equation for the electrostatic potential. This requires refined elliptic estimates addressing the exponential nonlinearities and some new enclosed $L^2\cap W^{1,\infty}$ estimates for the potential-dependent terms.

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arXiv2025

The incompressible inhomogeneous Navier-Stokes-Vlasov-Fokker-Planck equations: global well-posedness and inviscid limit

The global well-posedness and inviscid limit are investigated for the fluid-particle interaction system, described by the Navier-Stokes equations for the inhomogeneous incompressible viscous flows coupled with the Vlasov-Fokker-Planck equation for particles through a density-dependent nonlinear friction force in three-dimensional space. It is challenging to establish the inviscid limit over large time periods for the incompressible Euler equations under the influence of the weak dissipative mechanism generated by the friction force. We first prove the global stability of the equilibrium, in the sense that initial perturbations with appropriate Besov spatial regularity lead to global well-posedness and uniform regularity estimates with respect to the viscosity coefficient for strong solutions of the inhomogeneous Navier-Stokes-Vlasov-Fokker-Planck equations. In particular, we establish the optimal rates of convergence to equilibrium uniformly in Navier-Stokes. Then, we construct global solutions to the inhomogeneous Euler-Fokker-Planck equations via the vanishing viscosity limit. Furthermore, by capturing the dissipation arising from two-phase interactions, we rigorously justify the global-in-time strong convergence of the inviscid limit process, with a convergence rate that is in sharp contrast to that in the pure incompressible fluid case. To achieve this global convergence, novel ideas and new techniques are developed in the analysis and may be applied to other significant problems.

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arXiv2025

Uniform stability and optimal time decay rates of the compressible pressureless Navier-Stokes system in the critical regularity framework

This paper investigates the Cauchy problem for the compressible pressureless Navier-Stokes system in $\mathbb{R}^d$ with $d \geq 2$. Unlike the standard isentropic compressible Navier-Stokes system, the density in the pressureless model lacks a dissipative mechanism, leading to significant coupling effects from nonlinear terms in the momentum equations. We first prove the global well-posedness and uniform stability of strong solutions to the compressible pressureless Navier-Stokes system in the critical Besov space $\dot{B}_{2,1}^{\frac{d}{2}} \times \dot{B}_{2,1}^{\frac{d}{2}-1}$. Then, under the additional assumption that the low-frequency component of the initial density belongs to $\dot{B}_{2,\infty}^{\sigma_0+1}$ and that the initial velocity is sufficiently small in $\dot{B}_{2,\infty}^{\sigma_0}$ with $\sigma_0 \in (-\frac{d}{2}, \frac{d}{2}-1]$, we overcome the challenge of derivative loss caused by nonlinearity and establish optimal decay estimates for $u$ in $\dot{B}_{2,1}^{\sigma}$ with $\sigma \in (\sigma_0, \frac{d}{2}+1]$. In particular, it is shown that the density remains uniformly bounded in time which reveals a new asymptotic behavior in contrast to the isentropic compressible Navier-Stokes system where the density exhibits a dissipative structure and decays over time.

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arXiv2025

Global strong solutions to a compressible fluid-particle interaction model with density-dependent friction force

We investigate the Cauchy problem for a fluid-particle interaction model in $\mathbb{R}^3$. This model consists of the compressible barotropic Navier-Stokes equations and the Vlasov-Fokker-Planck equation coupled together via the density-dependent friction force. Due to the strong coupling caused by the friction force, it is a challenging problem to construct the global existence and optimal decay rates of strong solutions. In this paper, by assuming that the $H^2$-norm of the initial data is sufficiently small, we establish the global well-posedness of strong solutions. Furthermore, if the $L^1$-norm of initial data is bounded, then we achieve the optimal decay rates of strong solutions and their gradients in $L^2$-norm. The proofs rely on developing refined energy estimates and exploiting the frequency decomposition method. In addition, for the periodic domain case, our global strong solutions decay exponentially.

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arXiv2025

Global classical solutions to the ionic Vlasov-Poisson-Boltzmann system in a 3D periodic box

We investigate the global well-posedness of the ionic Vlasov-Poisson-Boltzmann system which models the evolution of dilute collisional ions. This system distinguishes the electronic Vlasov-Poisson-Boltzmann system via an additional exponential nonlinearity in the coupled Poisson-Poincar\'{e} equation, which introduces essential mathematical difficulties. In a three-dimensional periodic box, We establish the existence of a unique global-in-time classical solution with an exponential decay under small initial perturbations of a global Maxwellian that preserve mass, momentum and energy conservation laws. Our approach combines a nonlinear energy method with quantitative nonlinear elliptic estimates and new coercivity inequalities for the linearized collision operator $\mathcal{L}$ in ion dynamics.

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arXiv2025

Global Fujita-Kato solutions of the incompressible inhomogeneous magnetohydrodynamic equations

We investigate the incompressible inhomogeneous magnetohydrodynamic equations in $\mathbb{R}^3$, under the assumptions that the initial density $\rho_0$ is only bounded, and the initial velocity $u_0$ and magnetic field $B_0$ exhibit critical regularities. In particular, the density is allowed to be piecewise constant with jumps. First, we establish the global-in-time well-posedness and large-time behavior of solutions to the Cauchy problem in the case that $\rho_0$ has small variations, and $u_0$ and $B_0$ are sufficiently small in the critical Besov space $\dot{B}^{3/p-1}_{p,1}$ with $1<p<3$. Moreover, the small variation assumption on $\rho_0$ is no longer required in the case $p=2$. Then, we construct a unique global Fujita-Kato solution under the weaker condition that $u_0$ and $B_0$ are small in $\dot{B}^{1/2}_{2,\infty}$ but may be large in $\dot{H}^{1/2}$. Additionally, we show a general uniqueness result with only bounded and nonnegative density, without assuming the $L^1(0,T;L^{\infty})$ regularity of the velocity. Our study systematically addresses the global solvability of the inhomogeneous magnetohydrodynamic equations with rough density in the critical regularity setting.

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arXiv2024

Global well-posedness and optimal decay rates of classical solutions to the compressible Navier-Stokes-Fourier-P$_1$ approximation model in radiation hydrodynamics

In this paper, the compressible Navier-Stokes-Fourier-$P_1$ (NSF-$P_1$) approximation model in radiation hydrodynamics is investigated in the whole space $\mathbb{R}^3$. This model consists of the compressible NSF equations of fluid coupled with the transport equations of the radiation field propagation. Assuming that the initial data are a small perturbation near the equilibrium state, we establish the global well-posedness of classical solutions for this model by performing the Fourier analysis techniques and employing the delicate energy estimates in frequency spaces. Here, we develop a new method to overcome a series of difficulties arising from the linear terms $n_1$ in (3.2)$_2$ and $n_0$ in (3.3)$_3$ related to the radiation intensity. Furthermore, if the $L^1$-norm of the initial data is bounded, we obtain the optimal time decay rates of the classical solution at $L^p$-norm $(2\leq p\leq \infty)$. To the best of our knowledge, this is the first result on the global well-posedness of the NSF-$P_1$ approximation model.

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arXiv2024

Global existence and decay rates of strong solutions to the diffusion approximation model in radiation hydrodynamics

In this paper, we study the global well-posedness and optimal time decay rates of strong solutions to the diffusion approximation model in radiation hydrodynamics in $\mathbb{R}^3$. This model consists of the full compressible Navier-Stokes equations and the radiative diffusion equation which describes the influence and interaction between thermal radiation and fluid motion. Supposing that the initial perturbation around the equilibrium is sufficiently small in $H^2$-norm, we obtain the global strong solutions by utilizing method of the frequency decomposition. Moreover, by performing Fourier analysis techniques and using the delicate energy method, we consequently derive the optimal decay rates (including highest-order derivatives) of solutions for this model.

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arXiv2024

Global existence and time decay of strong solutions to a fluid-particle coupled model with energy exchanges

In this paper, we investigate a three-dimensional fluid-particle coupled model. % in whole space $\mathbb{R}^3$. This model combines the full compressible Navier-Stokes equations with the Vlasov-Fokker-Planck equation via the momentum and energy exchanges. We obtain the global existence and optimal time decay rates of strong solutions to the model in whole space $\mathbb{R}^3$ when the initial data are a small perturbation of the given equilibrium in $H^2$. We show that the $L^2$-norms of the solutions and their gradients decay as $(1+t)^{-3/4}$ and $(1+t)^{-5/4}$ respectively. Moreover, we also obtain the decay rates of solutions in $L^p$-norms for $p\in [2,\infty]$, and the optimal time decay rates of the highest-order derivatives of strong solutions which reads as $(1+t)^{-{7}/{4}}$ in $L^2$-norm. % Our decay rates are consistent with those of non-isentropic compressible Navier-Stokes equations. When the model is considered in a periodic domain, besides the global existence results, we show the strong solution decay exponentially. Our proofs rely on the energy method, Fourier analysis techniques, and the method of frequency decomposition. And some new ideas are introduced to achieve the desired convergence rates.

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arXiv2024

Global well-posedness and decay rates of strong solutions to the incompressible Vlasov-MHD system

In this paper, we study the global well-posedness and decay rates of strong solutions to an incompressible Vlasov-MHD model arising in magnetized plasmas. This model is consist of the Vlasov equation and the incompressible magnetohydrodynamic equations which interacts together via the Lorentz forces. It is readily to verify that it has two equilibria $(\bar f,\bar u,\bar B)=(0,0,0)$ and $( \tilde f,\tilde u,\tilde B)=(M,0,0)$, where $M$ is the global maxwellian. For each equilibrium, assuming that the $H^2$ norm of the initial data $(f_0,B_0,U_0)$ is sufficient small and $f_0(x,v)$ has a compact support in the position $x$ and the velocity $v$, we construct the global well-posedness and decay rates of strong solutions near the equilibrium in the whole space $\mathbb{R}^3$. And the solution decays polynomially. The global existence result still holds for the torus $\mathbb{T}^3$ case without the compact support assumption in $x$. In addition, the decay rates are exponential. Lack of dissipation structure in the Vlasov equation and the strong trilinear coupling term $((u-v)\times B)f$ in the model are two main impediments in obtaining our results. To surround these difficulties, we assume that $f_0(x,v)$ has a compact support and utilize the method of characteristics to calculate the size of the supports of $f$. Thus, we overcome the difficulty in estimating the integration $\int_{\mathbb{R}^3} \big((u-v)\times B\big)f\mathrm{d}v$ and obtain the global existence of strong solutions by taking advantage of a refined energy method. Moreover, by making full use of the Fourier techniques, we obtain the optimal time decay rate of the gradient of the solutions. This is the first result on strong solutions to the Vlasov-MHD model containing nonlinear Lorentz forces.

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arXiv2023

Global well-posedness and large-time behavior of the compressible Navier-Stokes equations with hyperbolic heat conduction

The classical Fourier's law, which states that the heat flux is proportional to the temperature gradient, induces the paradox of infinite propagation speed for heat conduction. To accurately simulate the real physical process, the hyperbolic model of heat conduction named Cattaneo's law was proposed, which leads to the finite speed of heat propagation. A natural question is that whether the large-time behavior of the heat flux for compressible flow would be different for these two laws. In this paper, we aim to address this question by studying the global well-posedness and optimal time-decay rates of classical solutions to the compressible Navier-Stokes system with Cattaneo's law. By designing a new method, we obtain the optimal time-decay rates for the highest derivatives of the heat flux, which cannot be derived for the system with Fourier's law by Matsumura and Nishida [Proc. Japan Acad. Ser. A Math. Sci., 55(9):337-342, 1979]. In this sense, our results first reveal the essential differences between the two laws.

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arXiv2023

On Stability and Instability of Gravity Driven Navier-Stokes-Korteweg Model in Two Dimensions

Bresch-Desjardins-Gisclon-Sart have derived that the capillarity can slow {down} the growth rate of Rayleigh-Taylor (RT) instability in the capillary fluids based on the linearized two-dimensional (2D) Navier-Stokes-Korteweg equations in 2008. Motivated by their linear theory, we further investigate the nonlinear RT problem for the 2D incompressible case in a horizontally periodic slab domain with Navier boundary condition, and rigorously verify that the RT instability can be inhibited by capillarity under our 2D setting. More precisely, if the RT density profile $\barρ$ satisfies an additional stabilizing condition, then there is a threshold $κ_{C}$ of capillarity coefficient, such that if the capillarity coefficient $κ$ is bigger than $κ_{C}$, then the small perturbation solution around the RT equilibrium state is \emph{algebraically} stable in time. In particular, if the RT density profile is linear, then the threshold $κ_{C}$ can be given by the formula $κ_{C}=g /(π^2h^{-2}+L^{-2})\barρ'$, where $2πL$ denotes the length of a periodic cell of the slab domain in the horizontal direction, and $h$ the height of the slab domain. In addition, we also provide a nonlinear instability result for $κ\in[0,κ_{C})$. The instability result presents that the capillarity can not inhibit the RT instability, if its strength is too small.

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arXiv2022

The combined non-equilibrium diffusion and low Mach number limits of the compressible Navier-Stokes-Fourier-P1 approximation radiation model

In this paper, we investigate the combined non-equilibrium diffusion and low Mach number limits of the compressible Navier-Stokes-Fourier-P1 (NSF-P1) model with general initial data, which arises in the radiation hydrodynamics. Compared to the classical compressible Navier-Stokes-Fourier system, the NSF-P1 model has an asymmetric singular structure caused by the radiation field. To handle these singular terms, we introduce an equivalent pressure and an equivalent velocity to balance the order of singularity and establish the uniform estimates of solutions by designating appropriate weighted norms as well as carrying out delicate energy analysis. We conclude that, for partially general initial data and the strong scattering effect, the NSF-P1 model converges to the system of low Mach number heat-conducting viscous flows coupled with a diffusion equation. We also discuss the variations of the limit equations as the scattering intensity changes. Furthermore, when the scattering effect is sufficiently weak, we can obtain the singular limits of the NSF-P1 model with fully general initial data.

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arXiv2022

Global weak solutions to the Vlasov-Poisson-Fokker-Planck-Navier-Stokes system

We consider the compressible Vlasov-Poisson-Fokker-Planck-Navier-Stokes system in a three dimensional bounded domain with nonhomogeneous Dirichlet boundary conditions. The system describes the evolution of charged particles ensemble dispersed in an isentropic fluid. For the adiabatic coefficient $γ>3/2$, we establish the global existence of weak solutions to this system with arbitrary large initial and boundary data.

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arXiv2021

Global strong solutions to the Vlasov-Poisson-Boltzmann system with soft potential in a bounded domain

Boundary effects are crucial for dynamics of dilute charged gases governed by the Vlasov-Poisson-Boltzmann (VPB) system. In this paper, we study the existence and regularity of solutions to the VPB system with soft potential in a bounded convex domain with in-flow boundary condition. We establish the existence of strong solutions in the time interval $[0,T]$ for an arbitrary given $T>0$ when the initial distribution function is near an absolute Maxwellian. Our contribution is based on a new weighted energy estimate in some $W^{1,p}$ space and $L_x^3 L_v^{1+}$ space for soft potential. By using the classical $L^2$--$L^\infty$ method and bootstrap argument, we extend the local solutions from small time scale to large time scale.

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