arXiv subjects

Jinkai Ni

Publications and source records attributed to Jinkai Ni.

17 recordsLinked to original sources

arXiv2026

Compressible Navier--Stokes equations with a potential force: global well-posedness and optimal time-decay rates for arbitrarily large $L^2$ initial data

We study the Cauchy problem for the three-dimensional barotropic compressible Navier--Stokes equations with a time-independent potential force near a spatially nonconstant stationary state. The potential is controlled in unweighted homogeneous Besov spaces; in particular, no polynomial spatial-weight condition involving $(1+|x|)^j\nabla^jϕ$ is imposed. For initial data relative to the stationary state that are sufficiently small in $\dot H^{\frac12-δ}\cap\dot H^3$, we establish the existence and uniqueness of a global strong solution in $H^3$, while allowing the initial $L^2$ norm to be arbitrarily large. If the initial data are bounded in $\dot B^s_{2,\infty}$ for $s\in[-\frac32,-1)$, then the solution and its first spatial derivative decay at the optimal rates $(1+t)^{-\frac{k-s}{2}}$ with $k=0$ and $1$, respectively. The analysis relies on refined homogeneous energy estimates and a frequency-localized description for the dissipative and asymptotic structures of the system.

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arXiv2026

Time-periodic solutions of the Vlasov-Poisson-Boltzmann system with a general external force in $\mathbb{R}^3$

In this paper, we study the time-periodic problem for the Vlasov-Poisson-Boltzmann (VPB) system with a given time-periodic external force in the whole space $\mathbb{R}^3$. The force is allowed to be non-potential. Around the global Maxwellian, we prove the global existence of small solutions in a hybrid function space that combines the low-frequency Besov framework for the forced Boltzmann equation with a corresponding control of the self-consistent electric field. The main novelty lies in the treatment of the nonlinear Vlasov force $-\nabla_xϕ\cdot \nabla_vf + \frac{1}{2}(v \cdot \nabla_xϕ)f$ at low frequencies. Rather than treating it as a generic source term, we exploit the Poisson equation and macroscopic balance laws to recover the structural cancellation required for the VPB semi-group estimates, which combined with high-frequency energy estimates and weighted microscopic propagation, yields a closed global well-posedness theory. We further prove the asymptotic stability of small solutions driven by the same force. When the external force is time-periodic, Serrin's method yields a unique time-periodic solution with the same period, together with its stability. As a direct consequence, our result also gives the existence and stability of stationary solutions when the external force is time-independent.

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arXiv2026

Low Mach number limit for the Navier--Stokes--Korteweg equations with a stationary force

In this paper, we investigate the low Mach number limit for the three-dimensional compressible Navier--Stokes--Korteweg equations in the whole space under a small stationary external force. We first construct a family of small stationary solutions uniformly with respect to the Mach number $\epsilon$ and prove that both the stationary density fluctuation and the compressible component of the stationary velocity are of order $\epsilon^2$. For ill-prepared non-stationary perturbations around these stationary solutions, we establish the existence and uniqueness of global strong solution by combining uniform high-order energy estimates with a low-frequency Besov estimate and a Kawashima-type compensating functional. The main difficulty is that Korteweg tensor not only changes the elliptic structure of the stationary problem, but also modifies the dispersive mechanism of the acoustic modes. In Korteweg-symmetric variables, the associated spectral projections are uniformly bounded zero-order Fourier multipliers, while the acoustic-capillary phase is wave-like at low frequencies and Schr\"odinger-like at high frequencies. Since the source terms generated by the stationary coefficients are generally not integrable in time, we decompose the Duhamel source according to its time-integrability and frequency behavior. Dyadic dispersive estimates, high-frequency damping estimates, and maximal regularity for the heat equation yield the global-in-time convergence rate $\epsilon^{\min\{1/r,\,1/2-1/p\}}$ in the mixed Besov norms $L^r(0,\infty;\dot B^s_{p,1})$. As a consequence, Besov embeddings also yield quantitative convergence in the mixed Lebesgue norms $L^r(0,\infty;L^p)$.

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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

Modified compensating functions for the incompressible Euler--Vlasov--Fokker--Planck system: Global classical solutions and pointwise-in-space decay

We consider the Cauchy problem for the incompressible Euler-Vlasov-Fokker-Planck (Euler-VFP) system in the whole space $\mathbb R^3$ near the global Maxwellian equilibrium. The Fokker-Planck operator and the particle-fluid drag dissipate the relative momentum but do not separately control the common particle-fluid momentum; in Fourier variables, this degeneracy occurs in the transverse momentum components. To recover the missing coercivity, we augment the classical four-moment compensator with a finite-rank skew-adjoint correction constructed from second-order Hermite modes. Combined with the cancellation between the kinetic and fluid drag terms and the incompressibility constraint, the resulting compensated Fourier energy yields a unique global classical solution for sufficiently small initial data $(u_0,f_0)\in H^N\times L_v^2(H^N)$, with $N\geq 4$. The high-order energy argument involves only spatial derivatives of the kinetic perturbation and requires no mixed $x$-$v$ derivative estimates. We further construct a positive-order Lyapunov functional and establish the decay rate $(1+t)^{-1/2}$ for all positive-order spatial derivatives in the $L^2$-norm and for the corresponding pointwise-in-space norms, without any additional $L^1$ integrability or low-frequency assumption on the initial data. Although no uniform algebraic decay rate is asserted for the zero-order energy of $(u,f)$, the directly dissipative variables $u-J(f)$ and $\{\mathbf I-\mathbf P_0\}f$ decay in the $L^2$-norm at the same rate, where $ J(f)=\int_{\mathbb R^3}v\sqrt M f\,{\rm d}v$ denotes the particle momentum and $\mathbf P_0$ is the orthogonal projection onto $\operatorname{span}\{\sqrt M,v_1\sqrt M,v_2\sqrt M,v_3\sqrt M\}$. To the best of our knowledge, these positive-order and zero-order decay estimates have not previously been established for the incompressible Euler-VFP system.

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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

High Mach number limit of the compressible Navier--Stokes equations in critical Besov spaces

We investigate the high Mach number limit for the scaled compressible Navier--Stokes system in the critical Besov framework. In the scaled momentum equation, the pressure force is represented by the term $\varepsilon^2\nabla a^\varepsilon$, where $\varepsilon$ is the inverse Mach number; as $\varepsilon\to0$, the formal limiting system is the compressible pressureless Navier--Stokes system. The analysis is complicated by the absence of density dissipation in the limiting model and by the highest-order coupling created by the viscous terms. For $d\geq2$, we prove the global well-posedness of the scaled system for small initial data and obtain estimates that are uniform with respect to $\varepsilon$. A crucial ingredient is a parameter-dependent lower-order estimate for $\varepsilon a^\varepsilon$, which compensates for the purely transport nature of the density equation and allows the uniform bounds to be closed. Based on these estimates, we justify the high Mach number limit and recover a global strong solution to the pressureless Navier--Stokes system. For $d\geq3$, we further derive quantitative error estimates between the scaled solutions and the pressureless limiting solution. More precisely, on each fixed finite time interval, if the initial discrepancy is of order $\mathcal{O}(\varepsilon)$, then the corresponding lower-order critical Besov error satisfies the same rate, which yields a quantitative justification of the pressureless limit.

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arXiv2026

Three-dimensional time-periodic problem on the Boltzmann equation with external force

The time-periodic problem on the Boltzmann equation with a given time-periodic external force in the three-dimensional whole space has remained open since it was first studied in [15] for only spatial dimensions not less than five. The goal of this paper is to give an affirmative answer to this problem provided that the external force is sufficiently small in the function space $\mathcal{C}(\mathbb{R};\dot{B}^{-3/2}_{2,\infty}\cap\dot{H}^N)$ with $N\geq 4$. The proof is based on Serrin's method through studying the global-in-time stability of the Cauchy problem with time-periodic external forces. As a direct consequence, the result also yields the existence and stability of stationary solutions to the physically realistic three-dimensional Boltzmann equation when the external force is time-independent.

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arXiv2026

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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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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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 $\dot{B}_{2,\infty}^{σ_0+1}$ and that the initial velocity is sufficiently small in $\dot{B}_{2,\infty}^{σ_0}$ with $σ_0 \in (-\frac{d}{2}, \frac{d}{2}-1]$, we overcome the challenge of derivative loss caused by nonlinearity and establish optimal decay estimates for $u$ in $\dot{B}_{2,1}^σ$ with $σ\in (σ_0, \frac{d}{2}+1]$. 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 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 $ρ_0$ 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 $\dot{B}^{3/p-1}_{p,1}$ 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 $\dot{B}^{1/2}_{2,\infty}$ 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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