arXiv subjects

Ling-Yun Shou

Publications and source records attributed to Ling-Yun Shou.

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

Long-time dynamics of partially dissipative hyperbolic systems with non-autonomous coefficients

We study quasilinear symmetrizable partially dissipative hyperbolic systems with non-autonomous relaxation coefficients in $\mathbb{R}^d$ ($d\geq1$). The existence of global strong solutions is established in a critical regularity setting for systems satisfying the so-called Shizuta-Kawashima (SK) and entropy conditions. When the initial data are additionally bounded in a lower-regularity norm, we prove that the corresponding solutions converge to equilibrium at optimal algebraic decay rates. Furthermore, we show that the conservative part of the solution behaves asymptotically as the solution of a non-autonomous parabolic equation. Our results apply to the compressible Euler system with the time-dependent damping coefficient $\frac{K}{(1+t)^α}$ ($α<1$, $K>0$ or $α=1$, $K\gg 1$) in the velocity equation. The natural low/high-frequency splitting of the autonomous theory persists in the non-autonomous setting, but with a frequency-threshold that evolves in time. To handle this moving frequency structure, we introduce a new class of hybrid Besov spaces adapted to time-dependent thresholds and derive hypocoercive estimates in each frequency regime. Our results reveal the qualitative and quantitative effects of general time-dependent relaxation coefficients on dissipation and large-time dynamics.

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arXiv2026

Compressible Euler equations with time-dependent damping in the critical regularity setting: global well-posedness and strong relaxation limit

We investigate the relaxation problem and the diffusion phenomenon for the compressible Euler system with a time-dependent damping coefficient of the form $\tfracμ{(1+t)^λ}$ in $\mathbb{R}^d$ $(d \geq 1)$. We establish uniform regularity estimates with respect to the relaxation parameter $\varepsilon$ and prove the global well-posedness of classical solutions to the Cauchy problem. In addition, we justify the global-in-time strong convergence of the solutions towards those of a general porous medium-type diffusion system, with an explicit rate of convergence, and for ill-prepared initial data. The core of our proof relies on a refined hypocoercivity framework combined with a new time-dependent frequency decomposition, both adapted to handle damping terms with time-dependent coefficients. This enables us to treat the overdamped regime $λ\in (-\infty,0)$ and the underdamped regime $λ\in (0,1)$ for any $μ>0$, and also the borderline critical case $λ=1$ under the improved condition $μ>2\varepsilon^2$.

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arXiv2026

Global relaxation limit for the one-fluid Euler-Poisson system with large smooth data

Whether the multi-dimensional Euler-Poisson system admits global smooth solutions remains a challenging open problem. In this paper, we construct a class of large-data global smooth solutions to the one-fluid Euler-Poisson system in $\mathbb{R}^d$ ($1\leq d\leq 5$) by using the relaxation dissipation mechanism. Precisely, assuming that the initial density is far from vacuum and $\varepsilon E_0$ is sufficiently small, where $E_0$ denotes the initial energy and $\varepsilon$ is the relaxation time, we establish the global well-posedness of smooth solutions to the Cauchy problem. In particular, the size of the initial perturbation may be arbitrarily large, provided that the relaxation time is sufficiently small. Furthermore, we introduce an effective unknown motivated by Darcy's law to derive quantitative error estimates at the rate $\mathcal O(e^{-λt}\varepsilon)$ between the rescaled Euler-Poisson system and the limiting drift-diffusion system for ill-prepared data. The new ingredient lies in developing the maximum principle for the nonlinear drift-diffusion system with nonlocal effect, which leads to the large-data global existence.

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arXiv2026

Hele-Shaw limit of chemotaxis-Navier-Stokes flows

This paper investigates the connection between the chemotaxis--Navier--Stokes system with porous medium type nonlinear diffusion and the Hele--Shaw problem in $\mathbb{R}^d$ ($d\geq2$). First, we prove the global-in-time existence of weak solutions for the Cauchy problem of the chemotaxis-Navier-Stokes system with the general initial data, uniformly in the diffusion range $m\in [3,\infty)$. Then, we rigorously justify the Hele--Shaw limit for this system as $m\rightarrow\infty$, showing the convergence to a free boundary problem of Hele--Shaw type, where the bacterium (cell) diffusion is governed by the stiff pressure law. Moreover, the complementarity relation characterizing the limiting bacterium (cell) pressure via a degenerate elliptic equation is verified by a novel application of the Hele--Shaw framework.

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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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arXiv2026

Sharp decay characterization for partially dissipative hyperbolic systems of balance laws

The partially dissipative systems that characterize many physical phenomena were first pointed out by Godunov (1961), then investigated by Friedrichs-Lax (1971) who introduced the convex entropy, and later by Shizuta-Kawashima (1984,1985) who initiated a simple sufficient criterion ensuring the global existence of smooth solutions and their large-time asymptotics. There has been remarkable progress in the past several decades, through various different attempts. However, the decay character theory for partially dissipative hyperbolic systems remains largely open, as the Fourier transform of Green's function is generally not explicit in multi-dimensions. In this paper, we provide a positive answer to the open question by means of the general $L^p$ energy method. Precisely, a new {\emph{effective quantity}} $Ψ(t,x)$ motivated by the compressible Euler system with damping is introduced, which enables us to capture leading diffusion profiles of the large-time behavior in the spirit of the Chapman-Enskog expansion. Consequently, we prove that the solutions approach the constant equilibrium state in the $\dot{\!B}^σ_{p,1}$-norm at the rate $t^{-(σ-σ_1)/2}$ as $t\rightarrow\infty$, and the corresponding norm of dissipative components decays at the enhanced rate $t^{-(σ-σ_1+1)/2}$, where the boundedness assumption in the $\dot{B}^{σ_1}_{p,\infty} (-d/p\leq σ_1<d/p-1$)-norm of the low frequencies of conservative components is not only sufficient, but also necessary to achieve those upper bounds of decay estimates. Furthermore, both upper and lower bounds for time-decay estimates are obtained if and only if the low-frequency part of $Ψ_0(x)$ (the initial effective quantity) is bounded in a non-trivial subset of $\dot{B}^{σ_1}_{p,\infty}$.

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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

Relaxation limit and asymptotic stability for the Euler-Navier-Stokes equations

The Euler-Navier-Stokes (E-NS) system arises as a macroscopic description of kinetic-fluid interactions, derived from the local-Maxwellian closure of the Vlasov-Fokker-Planck-Navier-Stokes flow. In this paper, we investigate the singular limit of the system in $\mathbb{R}^d$ ($d\ge2$) when the relaxation parameter $\varepsilon>0$ tends to zero. In contrast to the Euler system with velocity damping, the E-NS model features only a weaker relaxation of the relative velocity, which makes it challenging to analyze its dynamics as $\varepsilon\rightarrow 0$. We develop an energy argument to show global-in-time error estimates between the E-NS system and its limit system, the so-called Kramers-Smoluchowski-Navier-Stokes (KS-NS) system. These error estimates enable us to prove the global existence and uniform-in-$\varepsilon$ regularity of the strong solution to the E-NS system in a hybrid critical Besov space with a sharp frequency threshold of order $\mathcal{O}(\varepsilon^{-1})$ separating the low- and high-frequency regimes. Moreover, the large-time asymptotic stability of the global solution to the E-NS system is established. More precisely, we derive the optimal decay rates of the solution uniformly in $\varepsilon$, and the enhanced decay rates for the difference between the densities of the E-NS system and the KS-NS system.

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arXiv2025

Large-time asymptotics of periodic two-dimensional Vlasov-Navier-Stokes flows

We study the large-time behavior of finite-energy weak solutions for the Vlasov-Navier-Stokes equations in a two-dimensional torus. We focus first on the homogeneous case where the ambient (incompressible and viscous) fluid carrying the particles has a constant density, and then on the variable-density case. In both cases, large-time convergence to a monokinetic final state is demonstrated. For any finite energy initial data, we exhibit an algebraic convergence rate that deteriorates as the initial particle distribution increases. When the initial particle distribution is suitably small, then the convergence rate becomes exponential, a result consistent with the work of Han-Kwan et al. [17] dedicated to the homogeneous, three-dimensional case, where an additional smallness condition on the velocity was required. In the non-homogeneous case, we establish similar stability results, allowing a piecewise constant fluid density with jumps.

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arXiv2025

High-capillarity limit and smoothing effect of large solutions for a multi-dimensional generic non-conservative compressible two-fluid model

We investigate the global existence and long-time behavior of large solutions, in the high-capillarity regime, for a general multidimensional non-conservative compressible two-fluid model with the capillary pressure relation $f(α^{-}ρ^{-})=P^{+}-P^{-}$. Our main contributions are threefold. First, for sufficiently large capillarity coefficients, we prove the existence and uniqueness of global solutions in critical Besov spaces for large initial perturbations, under the sharp stability condition $-\frac{s_{-}^{2}(1,1)}{α^{-}(1,1)}<f^{\prime}(1)<0$, thereby removing the additional negativity restriction assumed by Evje--Wang--Wen [Arch. Ration. Mech. Anal. 221:1285--1316, 2016]. Second, we give a rigorous justification of the global-in-time convergence to the incompressible Navier-Stokes flows and obtain explicit convergence rates in critical spaces for ill-prepared data. Third, if in addition the initial perturbation lies in a lower-regularity Besov space, we derive optimal decay rates for the solution and for its derivatives of any order, revealing a long-term smoothing effect. To the best of our knowledge, this is the first result on global large-amplitude strong solutions for multidimensional compressible two-fluid flows. Our analysis exploits the interplay between dispersion (two-phase Gross--Pitaevskii structure) and parabolic dissipation, both induced by capillarity effects.

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arXiv2025

Global convergence rates in the relaxation limits for the compressible Euler and Euler-Maxwell systems in Sobolev spaces

We study two relaxation problems in the class of partially dissipative hyperbolic systems: the compressible Euler system and the compressible Euler-Maxwell system. In classical Sobolev spaces, we derive a global convergence rate of $\mathcal{O}(\varepsilon)$ between strong solutions of the relaxed Euler system and the porous medium equation in $\mathbb{R}^d$ ($d\geq1$) for \emph{ill-prepared} initial data. In a well-prepared setting, we derive an enhanced convergence rate of order $\mathcal{O}(\varepsilon^2)$ between the solutions of the relaxed compressible Euler system and their first-order asymptotic approximation. Regarding the relaxed Euler-Maxwell system, we prove the global strong convergence of its solutions to the drift-diffusion model in $\mathbb{R}^3$ in an \emph{ill-prepared} setting. These results are achieved by developing a new asymptotic expansion approach that, combined with stream function techniques, ensures uniform-in-time error estimates.

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arXiv2025

The Boltzmann equation in the homogeneous critical regularity framework

We construct a unique global solution to the Cauchy problem of the 3D Boltzmann equation for initial data around the Maxwellian in the spatially critical homogeneous Besov space $\widetilde{L}^2_ξ(\dot{B}_{2,1}^{1/2}\cap\dot{B}_{2,1}^{3/2})$. In addition, under the condition that the low-frequency part of initial perturbation is bounded in $\widetilde{L}^2_ξ(\dot{B}_{2,\infty}^{σ_{0}})$ with $-3/2\leqσ_{0}<1/2$, it is shown that the solution converges to its equilibrium in large times with the optimal rate of $\mathcal{O}(t^{-(σ-σ_{0})/2})$ in $\widetilde{L}^2_ξ(\dot{B}_{2,1}^σ)$ with some $σ>σ_0$, and the microscopic part decays at an enhanced rate of $\mathcal{O}(t^{-(σ-σ_{0})/2-1/2})$. In contrast to [19], the usual $L^2$ estimates are not necessary in our approach, which provides a new understanding of hypocoercivity theory for the Boltzmann equation allowing to construct the Lyapunov functional with different dissipation rates at low and high frequencies. Furthermore, a time-weighted Lyapunov energy argument can be developed to deduce the optimal time-decay estimates.

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arXiv2025

Strong relaxation limit and uniform time asymptotics of the Jin-Xin model in the $L^{p}$ framework

We investigate the time-asymptotic stability of the Jin-Xin model and its diffusive relaxation limit toward viscous conservation laws in $\mathbb{R}^d$ for $d\geq 1$. First, we establish a priori estimates that are uniform with respect to both the time and the relaxation parameter $\varepsilon>0$, for initial data in hybrid Besov spaces based on $L^{p}$-norms. This uniformity enables us to derive $\mathcal{O}(\varepsilon)$ bounds on the difference between solutions of the viscous conservation law and its associated Jin-Xin approximation, thus justifying the strong convergence of the relaxation process. Furthermore, under an additional condition on the initial data, for instance, that the low frequencies belong to $L^{p/2}(\mathbb{R}^{d})$, we show that the $L^{p}(\mathbb{R}^d)$-norm of the solution to the Jin-Xin model decays at the optimal rate $(1+t)^{-d/{2p}}$, and the $L^{p}(\mathbb{R}^d)$-norm of its difference with the solution of the associated viscous conservation law decays at the enhanced rate $\varepsilon(1+t)^{-d/{2p}-1/2}$.

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arXiv2025

Large-Time Asymptotics for Hyperbolic Systems with Non-Symmetric Relaxation: An Algorithmic Approach

We study the stability of one-dimensional linear hyperbolic systems with non-symmetric relaxation. Introducing a new frequency-dependent Kalman stability condition, we prove an abstract decay result underpinning a form of inhomogeneous hypocoercivity. In contrast with the homogeneous setting, the decay rates depend on how the Kalman condition is fulfilled and, in most cases, a loss of derivative occurs: one must assume an additional regularity assumption on the initial data to ensure the decay. Under structural assumptions, we refine our abstract result by providing an algorithm, of wide applicability, for the construction of Lyapunov functionals. This allows us to systematically establish decay estimates for a given system and uncover algebraic cancellations (beyond the reach of the Kalman-based approach) reducing the loss of derivatives in high frequencies. To demonstrate the applicability of our method, we derive new stability results for the Sugimoto model, which describes the propagation of nonlinear acoustic waves, and for a beam model of Timoshenko type with memory.

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arXiv2025

The non-conservative compressible two-fluid system with common pressure: Global existence and sharp time asymptotics

This paper concerns the global-in-time evolution of a generic compressible two-fluid model in $\mathbb{R}^d$ ($d\geq3$) with the common pressure law. Due to the non-dissipative properties for densities and two different particle paths caused by velocities, the system lacks the usual symmetry structure and is partially dissipative in the sense that the Shizuta-Kawashima condition is violated, which makes it challenging to study its large-time stability. By developing a pure energy method in the framework of Besov spaces, we succeed in constructing a unique global classical solution to the Cauchy problem when the initial data are close to their constant equilibria. Compared to the previous related works, the main novelty lies in that our method is independent of the spectral analysis and does not rely on the $L^1$ smallness of the initial data. Furthermore, if additionally the initial perturbation is bounded in $\dot{B}^{σ_0}_{2,\infty}$ type spaces with lower regularity, the optimal time convergence rates are also obtained. In particular, the asymptotic convergence of the non-dissipative components toward their equilibrium states is first characterized.

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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 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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