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Hai-Liang Li

Publications and source records attributed to Hai-Liang Li.

At least 19 recordsLinked to original sources
arXiv2026

Global existence and vanishing viscosity limit of the compressible Navier-Stokes-Vlasov-Fokker-Planck system in critical spaces

We study multidimensional compressible fluid-particle systems at critical regularity, in which a carrier fluid and a particle phase with Fokker-Planck diffusion are coupled through a drag force. We prove the existence and uniqueness of strong solutions for the Cauchy problems of the Navier-Stokes-Vlasov-Fokker-Planck and Euler-Vlasov-Fokker-Planck systems near equilibrium in their respective critical Besov spaces. Moreover, we establish regularity estimates for the Navier-Stokes-Vlasov-Fokker-Planck system uniform with respect to the common viscosity parameter $μ=λ=\varepsilon$ and justify the global-in-time vanishing-viscosity limit with the convergence rate $\mathcal O(\varepsilon)$. Finally, under an additional lower-order Besov assumption on the initial data, we obtain optimal time-decay estimates for both systems and derive enhanced decay rates for the relative velocity and the microscopic part of the distribution function.

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arXiv2026

Large-friction and incompressible limits for pressureless Euler-Navier-Stokes flows

We study the global macroscopic limits associated with kinetic-fluid interaction models for sprays. Motivated by the Vlasov-Navier-Stokes system under the monokinetic ansatz, we consider the pressureless Euler-Navier-Stokes (Euler-NS) system in $\mathbb{R}^{d}$ ($d\geq2$) coupled through the singular drag force $\frac{1}τ ρ(u-v)$, where $τ$ is the Stokes relaxation time. For initial data uniformly close to equilibrium in critical Besov spaces, we establish global-in-time regularity estimates of solutions to the Cauchy problem for the Euler-NS system, uniformly with respect to $τ$. These estimates yield the global strong convergence of the Euler-NS system toward a one-velocity two-phase drift-flux (DF) model as $τ\to0$, with an explicit convergence rate of order $\sqrtτ$. A key point in the analysis is the introduction of an effective mixed velocity, which allows us to handle the singular relative-velocity relaxation and obtain global error estimates for ill-prepared data. We also derive large-time asymptotic estimates for the Euler-NS system, uniformly in $τ$, including the improved decay of the relative velocity and the convergence of the non-dissipative density toward an asymptotic profile. Furthermore, after introducing the Mach number $\varepsilon>0$, we justify the incompressible limit of the DF model toward the Transport-Navier-Stokes (TNS) system as $\varepsilon\to0$, and prove the combined large-friction and incompressible limit from the Euler-NS system to the TNS system in the regime $τ=\varepsilon\to0$ in an ill-prepared setting. These results provide a unified and quantitative macroscopic picture connecting the Euler-NS, DF, and TNS systems through the large-friction and incompressible regimes.

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arXiv2024

Global existence and optimal time-decay rates of the compressible Navier-Stokes-Euler system

In this paper, we consider the Cauchy problem of the multi-dimensional compressible Navier-Stokes-Euler system for two-phase flow motion, which consists of the isentropic compressible Navier-Stokes equations and the isothermal compressible Euler equations coupled with each other through a relaxation drag force. We first establish the local existence and uniqueness of the strong solution for general initial data in a critical homogeneous Besov space, and then prove the global existence of the solution if the initial data is a small perturbation of the equilibrium state. Moreover, under the additional condition that the low-frequency part of the initial perturbation also belongs to another Besov space with lower regularity, we obtain the optimal time-decay rates of the global solution toward the equilibrium state. These results imply that the relaxation drag force and the viscosity dissipation affect regularity properties and long time behaviors of solutions for the compressible Navier-Stokes-Euler system.

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arXiv2024

Exponential Stability of the Inhomogeneous Navier-Stokes-Vlasov System in Vacuum

In this paper, we study the asymptotic behaviors of solutions to the inhomogeneous Navier-Stokes-Vlasov system in $\mathbb{R}^{3}\times\mathbb{R}^{3}$, where the initial fluid density is allowed to vanish. We establish the uniform bound of the macroscopic density associated with the distribution function and prove the global existence and uniqueness of strong solutions to the Cauchy problem with vacuum for either small initial energy or large viscosity coefficient. The uniform boundedness and the presence of vacuum enable us to show that as the time evolves, the fluid velocity decays, while the distribution function concentrates towards a Dirac measure in velocity centred at $0$, with an exponential rate. In order to overcome the degeneracy in the momentum equations, we develop an energy argument based on higher order functional inequalities designed for fluid-particle coupled structures.

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arXiv2024

Non-existence of classical solutions to a two-phase flow model with vacuum

In this paper, we study the well-posedness of classical solutions to a two-phase flow model consisting of the pressureless Euler equations coupled with the isentropic compressible Navier-Stokes equations via a drag forcing term. We consider the case that the fluid densities may contain a vacuum, and the viscosities are density-dependent functions. Under suitable assumptions on the initial data, we show that the finite-energy (i.e., in the inhomogeneous Sobolev space) classical solutions to the Cauchy problem of this coupled system do not exist for any small time.

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arXiv2023

Incompressible limit of porous media equation with chemotaxis and growth

We revisit the problem of proving the incompressible limit for the compressible porous media equation with Newtonian drift and growth. The question is motivated by models of living tissues development including chemotaxis. We extend the problem, already treated by the authors and several other contributions, in using a simplified approach, in treating dimensions two or higher, and in incorporating the pressure driven growth term. We also complete the analysis with stronger $L^4$ estimates on the pressure gradient. The major difficulty is to prove the strong convergence of the pressure gradient which is obtained here by a new observation on an algebraic relation involving the pressure gradient for weak limits.

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arXiv2023

Green's function and Pointwise Behavior of the One-Dimensional Vlasov-Maxwell-Boltzmann System

The pointwise space-time behavior of the Green's function of the one-dimensional Vlasov-Maxwell-Boltzmann (VMB) system is studied in this paper. It is shown that the Green's function consists of the macroscopic diffusive waves and Huygens waves with the speed $\pm \sqrt{5/3}$ at low-frequency, the hyperbolic waves with the speed $\pm 1$ at high-frequency, the singular kinetic and leading short waves, and the remaining term decaying exponentially in space and time. Note that these high-frequency hyperbolic waves are completely new and can not be observed for the Boltzmann equation and the Vlasov-Poisson-Boltzmann system. In addition, we establish the pointwise space-time estimate of the global solution to the nonlinear VMB system based on the Green's function. Compared to the Boltzmann equation and the Vlasov-Poisson-Boltzmann system, some new ideas are introduced to overcome the difficulties caused by the coupling effects of the transport of particles and the rotating of electro-magnetic fields, and investigate the new hyperbolic waves and singular leading short waves.

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arXiv2023

On the vacuum free boundary problem of the viscous Saint-Venant system for shallow water in two dimensions

In this paper, we establish the local-in-time well-posedness of classical solutions to the vacuum free boundary problem of the viscous Saint-Venant system for shallow water in two dimensions. The solutions are shown to possess higher-order regularities uniformly up to the vacuum free boundary, although the depth degenerates as a singularity of the distance to the vacuum boundary. Since the momentum equations degenerate in both the dissipation and time evolution, there are difficulties in constructing approximate solutions by the Galerkin's scheme and gaining higher-order regularities uniformly up to the vacuum boundary for the weak solution. To construct the approximate solutions, we introduce some degenerate-singular elliptic operator, whose eigenfunctions form an orthogonal basis of the projection space. Then the high-order regularities on the weak solution are obtained by using some carefully designed higher-order weighted energy functional.

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arXiv2023

Global existence and large time behavior for primitive equations with free boundary

In the present paper, the primitive equations, which can be used to simulate the large scale motion of ocean and atmosphere, are considered in the three-dimensional domain bounded below by a fixed solid boundary and above by a free moving boundary. The global existence and uniqueness of strong solutions are established and the long time convergence to the equilibrium state is showed either at exponential rate for horizontal periodic domain or at algebraic rate for horizontal whole space.

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arXiv2023

Global weak solutions for compressible Navier-Stokes-Vlasov-Fokker-Planck system

The one-dimensional compressible Navier-Stokes-Vlasov-Fokker-Planck system with density-dependent viscosity and drag force coefficients is investigated in the present paper. The existence, uniqueness, and regularity of global weak solution to the initial value problem for general initial data are established in spatial periodic domain. Moreover, the long time behavior of the weak solution is analyzed. It is shown that as the time grows, the distribution function of the particles converges to the global Maxwellian, and both the fluid velocity and the macroscopic velocity of the particles converge to the same speed.

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arXiv2022

Incompressible limits of Patlak-Keller-Segel model and its stationary state

We complete previous results about the incompressible limit of both the $n$-dimensional $(n\geq3)$ compressible Patlak-Keller-Segel (PKS) model and its stationary state. As in previous works, in this limit, we derive the weak form of a geometric free boundary problem of Hele-Shaw type, also called congested flow. In particular, we are able to take into account the unsaturated zone, and establish the complementarity relation which describes the limit pressure by a degenerate elliptic equation. Not only our analysis uses a completely different framework than previous approaches, but we also establish a novel uniform $L^3$ estimate of the pressure gradient, regularity à la Aronson-Bénilan, and a uniform $L^1$ estimate for the time derivative of the pressure. Furthermore, for the Hele-Shaw problem, we prove the uniqueness of solutions, meaning that the incompressible limit of the PKS model is unique. In addition, we establish the corresponding incompressible limit of the stationary state for the PKS model with a given mass, where, different from the case of PKS model, we obtain the uniform bound of pressure and the uniformly bounded support of density.

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arXiv2022

Well-posedness of classical solutions to the vacuum free boundary problem of the viscous Saint-Venant system for shallow waters

We establish the local-in-time well-posedness of classical solutions to the vacuum free boundary problem of the viscous Saint-Venant system for shallow waters derived rigorously from incompressible Navier-Stokes system with a moving free surface by Gerbeau-Perthame. Our solutions (the height and velocity) are smooth (the solutions satisfy the equations point-wisely) all the way to the moving boundary, although the height degenerates as a singularity of the distance to the vacuum boundary. The proof is built on some new higher-order weighted energy functional and weighted estimates associated to the degeneracy near the moving vacuum boundary.

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arXiv2021

Global well-posedness of one-dimensional compressible Navier-Stokes-Vlasov system

A fluid-particle model is investigated in the present paper, which consists of the compressible Navier-Stokes equations coupled with the Vlasov equation though a nonlinear drag force. We consider the initial value problem for the one-dimensional compressible Navier-Stokes-Vlasov system and establish the global existence and uniqueness of the weak solution for general initial data in either spatial periodic domain or spatial real line, which is shown to be a classical solution for regular initial data.

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arXiv2021

Existence and Nonlinear Stability of Steady-States to Outflow Problem for the Full Two-Phase Flow

The outflow problem for the viscous full two-phase flow model in a half line is investigated in the present paper. The existence, uniqueness and nonlinear stability of the steady-state are shown respectively corresponding to the supersonic, sonic or subsonic state at far field. This is different from the outflow problem for the isentropic Navier-Stokes equations, where there is no steady-state for the subsonic state. Furthermore, we obtain either exponential time decay rates for the supersonic state or algebraic time decay rates for supersonic and sonic states in weighted Sobolev spaces.

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arXiv2020

Diffusion Limit of the Vlasov-Poisson-Boltzmann System

In the present paper, we study the diffusion limit of the classical solution to the unipolar Vlasov-Poisson-Boltzmann (VPB) system with initial data near a global Maxwellian. We prove the convergence and establish the convergence rate of the global strong solution to the unipolar VPB system towards the solution to an incompressible Navier-Stokes-Poisson-Fourier system based on the spectral analysis with precise estimation on the initial layer.

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arXiv2018

Formation of Singularities of Spherically Symmetric Solutions to the 3D Compressible Euler Equations and Euler-Poisson Equations

By introducing a new averaged quantity with a fast decay weight to perform Sideris's argument (Commun Math Phys, 1985) developed for the Euler Equations, we extend the formation of singularities of classical solution to the 3D Euler Equations established in Sideris (1985) and Makino et al. (Jpn J Appl Math, 1986) for the initial data with compactly supported disturbances to the spherically symmetric solution with general initial data in Sobolev space. Moreover, we also prove the formation of singularities of the spherically symmetric solutions to the 3D Euler-Poisson Equations, but remove the compact support assumptions on the initial data in Makino and Perthame (Jpn J Appl Math, 1990) and Perthame (Jpn J Appl Math, 1990). Our proof also simplifies that of Lei et al. (Math Res Lett, 2013) for the Euler Equations and is undifferentiated in dimensions.

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