We justify the global-in-time validity of Hilbert expansion for the ionic Vlasov-Poisson-Boltzmann system in 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∩ W^1,∞ estimates for the potential-dependent terms.

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.

In this note, we prove some new -type estimates of strong solutions to the incompressible magnetohydrodynamic equations in by the energy method.

In this note, we prove some new Lp-type estimates of strong solutions to the incompressible magnetohydrodynamic equations in Rn(n≥3) by the Lp energy method.

We investigate the Cauchy problem for a fluid-particle interaction model in 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. (c) 2026 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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 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 the 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 order derivatives of the heat flux, which cannot be derived for the system with Fourier's law by Matsumura and Nishida (1979) [25]. In this sense, our results first reveal the essential differences between the two laws. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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 ℝ^d with d ≥ 2. Furthermore, by introducing the Mach number ε > 0, we rigorously prove the low Mach number limit as ε→ 0, showing that the solutions converge to that of the incompressible inhomogeneous Navier-Stokes system. This singular limit holds globally in time, even for 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 ℝ^d with d ≥ 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.

This paper is concerned with a kinetic-fluid model, which contains a Vlasov-Fokker-Planck equation for particles and an incompressible inhomogeneous Euler equations for a fluid. The two parts in the model twist together via the friction force which reveals the inhibition of the fluid on the motion of particles. Consequentially, the particles provide a damping effect on the motion of the fluid. The global existence and uniqueness classical solutions near the equilibrium are established in a 3-D torus. Moreover, the exponential convergence rate of the solution is given. The proofs are mainly based on the macro-micro decomposition and the classical energy estimates. And there are two main novelties in our arguments: on the one hand, our friction force depends on the fluid density n, which is more physical in modeling but more difficult in computations than the density-independent case. On the other hand, we introduce a new energy method to enclose the a priori estimates. More precisely, we derive the dissipation rate on the gradient of the pressure P instead of the density rho in H-2 norm to enclose the estimates parallel to f parallel to(L2 xi H3x) + parallel to u parallel to(H3). We then apply the equation of rho and Gronwall's inequality to obtain the estimates of parallel to rho parallel to(H3).

In this paper, we study strong solutions and decay estimates for an incompressible Vlasov-magnetohydrodynamic (Vlasov-MHD) model arising in magnetized plasmas. This model consists of the Vlasov equation and the incompressible MHD equations, which interact mutually through the Lorentz force. It can be readily verified that the system has two equilibria: (f̅,u̅,B̅)=(0,0,0) and ( f̃,ũ,B̃)=((2π )^- 3/2e^- |v|^2/2,0,0) . For any prescribed time interval, we establish the existence and uniqueness of strong solutions under appropriately small initial perturbations that depend on the length of the interval. In both the whole-space and periodic settings, we further derive the corresponding decay estimates for the fluid velocity and magnetic field over the interval of existence. The primary difficulty stems from the lack of drag-type or Fokker–Planck dissipation in the kinetic equation, along with the trilinear coupling generated by the Lorentz force. To address this issue, we combine the characteristic method, support propagation for the kinetic component, refined energy estimates, and Fourier analysis.

This paper focuses on the global existence and optimal time-decay rates of strong solutions in the $H^{2}(\mathbb{R}^{3})$ space for the Navier-Stokes-Fourier-$P_{1}$ approximation model in radiation hydrodynamics. This model is composed of a non-isentropic compressible Navier-Stokes system and two coupled radiation transport equations. We establish the global existence of strong solutions under the condition that the initial perturbation is sufficiently small. Meanwhile, if the perturbation is additionally bounded in the $L^1$-norm, the optimal time decay rates of all order derivatives of the global strong solutions can be obtained. In particular, we prove that the decay rates of the time derivative of the solutions are optimal. A novel feature of our analysis is the introduction of a new effective radiation mode, specifically $4\theta - n_0$ and $n_1$, which decays faster than our solutions. This work uncovers a new dissipation mechanism caused by the radiation effects.

We investigate the incompressible inhomogeneous magnetohydrodynamic equations in ℝ^3, under the assumptions that the initial density ρ_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 ρ_0 has small variations, and u_0 and B_0 are sufficiently small in the critical Besov space Ḃ^3/p-1_p,1 with 1<p<3. Moreover, the small variation assumption on ρ_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 Ḃ^1/2_2,∞ but may be large in Ḣ^1/2. Additionally, we show a general uniqueness result with only bounded and nonnegative density, without assuming the L^1(0,T;L^∞) regularity of the velocity. Our study systematically addresses the global solvability of the inhomogeneous magnetohydrodynamic equations with rough density in the critical regularity setting.

In this paper, we investigate the uniform stability and optimal time decay of strong solutions to the incompressible kinetic-magnetohydrodynamic (kinetic-MHD) system in the whole space ℝ^3 . This model consists of a kinetic equation describing the evolution of energetic particles and the incompressible MHD equations governing the dynamics of the fluid and magnetic field, coupled through Lorentz forces. Under the assumption that the initial data are small perturbations near the spatially homogeneous equilibrium (M, 0, 0), we first establish the global existence of strong solutions in the L_v^2(H_x^2)× H_x^2× H_x^2 framework, without imposing any regularity assumption on the velocity derivatives of the kinetic perturbation f. Under an additional smallness assumption in L^1 , by employing the low-high frequency decomposition method and the time-weighted energy method, we address the difficulties arising from the velocity-weighted term and the loss of velocity derivatives in v× B·∇ _v f . As a result, we obtain the optimal time decay rates of all spatial derivatives up to order two, including the highest-order rate (1+t)^-7/4 . Here, a=∫ _ℝ^3√(M)f dv and b=∫ _ℝ^3v√(M)f dv denote the zeroth and first velocity moments of f, respectively. We further identify a magnetic-field-induced dissipation mechanism such that b and ∇ b decay one-half order faster than the corresponding orders of the full solution, while accelerated decay also holds for ∂ _t a and ∂ _t b . Furthermore, a refined difference-energy method yields the uniform stability of strong solutions. In the periodic domain 𝕋^3 , we show that f and B decay exponentially, whereas the fluid velocity u remains uniformly bounded, which differs from kinetic-fluid systems with drag-force coupling. To the best of our knowledge, this is the first result concerning the uniform stability and optimal time decay of strong solutions for this kinetic-MHD model.

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.

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-Poincaré 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 ℒ in ion dynamics.

In this paper we prove vanishing viscosity limits of a coupled chemotaxis-fluid model in a bounded domain Ω ⊂ ℝ3. The proof is based on the Banach’s fixed point theorem and the Lp-energy method. In addition, the L∞-estimates and gradient estimates of the heat equations also play a crucial role.

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 R3. 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 H2-norm, we obtain the global strong solutions by utilizing the method of frequency decomposition. Moreover, by performing Fourier analysis techniques and using the delicate energy method, we consequently derive the optimal decay rates (including highest-order spatial derivatives) of solutions for this model.

We consider a kinetic-magnetohydrodynamic model arising in magnetized plasmas in the whole space R-3. This model contains of the Vlasov-Fokker-Planck equation for energetic particles and the incompressible magnetohydrodynamic equations for the fluid and the magnetic field, and they twist together via the Lorentz forces. Assuming that the initial perturbation parallel to(u(0), B-0)(0))parallel to N-H +parallel to f(0)parallel to(Hx,vN) (N >= 4) is small enough, we prove the existence of smooth solutions to the Cauchy problem by employing the delicate energy analysis. Furthermore, if parallel to(u(0), B-0)parallel to(L1) + parallel to f(0)parallel to(Z1) is bounded, we obtain the optimal time decay rates of solutions and their gradients through constructing refined functional and utilizing the Fourier techniques. This is the first result on classical solutions to kinetic-magnetohydrodynamic model involving nonlinear Lorentz forces.

In this paper, we investigate the non-equilibrium diffusion limit of the compressible Navier-Stokes-Fourier-P1 (NSF-P1) approximation model at low Mach number, which arises in radiation hydrodynamics, with general initial data and a parameter $\delta \in [0,2]$ describing the intensity of scatting effect. In previous literature, only $\delta =2$ and well-prepared initial data case to the NSF-P1 model was considered. Here we prove that, for partial general initial data and $\delta =2$, this model converges to the system of low Mach number heat-conducting viscous flows coupled with a diffusion equation as the parameter $\epsilon \rightarrow 0$. Compared to the classical NSF system, the NSF-P1 model has additional new singular structures caused by the radiation pressure. To handle these structures, we construct 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 and carrying out delicate energy analysis. We then obtain the convergence of the pressure and velocity from the local energy decay of the equivalent pressure and equivalent velocity. We also briefly discuss the variations of the limit equations as the scattering intensity changes, i.e., $\delta \in (0,2)$. We find that, with the weakening of scattering intensity, the ``diffusion property" of radiation intensity gradually weakens. Furthermore, when the scattering effect is sufficiently weak ($\delta =0$), we can obtain the singular limits of the NSF-P1 model with general initial data. To our best knowledge, this is the first result on the influence of scattering intensity in the non-equilibrium diffusion limit of the NSF-P1 model.

In this paper we consider the two-species Vlasov-Poisson system with a radiation damping term D^[3](t) in the whole space ℝ^3 , which was introduced by Bauer [Kinet. Relat. Models 11 (2018), 25–42] to approximate the relativistic Vlasov-Maxwell system, a fundamental model of dynamics of collisionless plasma. We obtain the global existence of solutions and optimal pointwise decay estimates of the charge densities and the electrostatic potential to this system for small initial data without any compact support assumptions. To prove our results, we mainly use the modified vector field method and a bootstrap method. There are two main novelties in our arguments: we introduce new modified functions of modified vector fields to control the troublesome terms involving D^[3](t) since it leads to loss an order derivative, and we raise a new bootstrap assumption and carry out new bootstrap arguments.

In this paper, we investigate a fluid-particle model defined in the whole space R-3 and the periodic domain T-3. This model consists of the incompressible inhomogeneous Navier-Stokes (NS) equations and the Vlasov-Fokker-Planck (VFP) equation which couple together via the friction force. Under the assumption that the initial data satisfy & Vert;rho(0)& Vert;H-3+& Vert;u(0)& Vert;H-3 boolean AND L-1+& Vert;f(0)& Vert;L-v(2)(H-3 boolean AND L-1) is sufficiently small, we establish the global existence and optimal time decay rates of classical solutions to the system in R-3. Specifically, we derive optimal decay rates of (u, f) and their gradients in the L-2-norm. Compared with the incompressible homogeneous NS-VFP system, our velocity equation contains an additional factor of 1/1+rho, which leads to a loss of derivatives. We address this issue by employing a refined energy method. To control & Vert;rho & Vert;H-3, we derive precise algebraic decay estimates for & Vert;del u & Vert;(2)(H) through Fourier analysis techniques and semigroup theory, while carefully handling the loss of derivatives. By using the time-weighted energy method, we also prove the propagation effects of rho, u and f which is the first result in fluid-particle systems. The short-time decay rate of del kf is t(-k/2), which is k order faster than t(-3k/2) for the general Fokker-Planck equation. This is a new phenomenon due to the coupling effects. Moreover, in the T-3 case, we observe exponential decay for (u, f), whereas the H-3-norm of the density rho remains bounded. This behavior represents a novel phenomenon compared to the compressible NS-VFP system, where rho typically exhibits temporal decay.