arXiv subjects

Feimin Huang

Publications and source records attributed to Feimin Huang.

At least 19 recordsLinked to original sources

arXiv2026

Time-Decay Rates toward Rarefaction Waves for the One-Dimensional Isentropic Compressible Navier--Stokes Equations

While the asymptotic stability of rarefaction waves for the compressible Navier--Stokes equations has been well established since the pioneering works of Matsumura--Nishihara \cite{MatsumuraNishihara1986} and Liu--Xin \cite{LiuXin1988}, the classical $L^2$ energy method yields only qualitative convergence in $L^\infty(\R)$ as $t\to\infty$ without any explicit decay rate. In this paper, we establish for the first time a standalone $L^4$ energy estimate for perturbations of a weak rarefaction wave in the one-dimensional isentropic compressible Navier--Stokes system, thereby obtaining the first explicit algebraic time-decay rates for this model. Specifically, for small initial perturbations in $H^2(\R)$ and small wave strength $δ$, we prove that $(v-V,u-U)$ decays in $L^4(\R)$ at the rate \[ \|(v-V,u-U)(t)\|_{L^4}\le C\bigl(\|(v_0-V(0),u_0-U(0))\|_{H^2}+\sqrtδ\bigr)(1+t)^{-1/8}, \] where $(V,U)$ denotes the smooth rarefaction wave, while the relative Riemann invariants $z_\pm$ of the perturbed system decay in $L^\infty(\R)$ at the rate $(1+t)^{-1/6}$. The proof relies on a new local quartic energy estimate for $z_\pm$ and $z_\pm-Q$, where $Q=μ(v)v_x-μ(V)V_x$, as well as a nonlocal half-space wave-interaction functional for corrected characteristic modes and a weighted higher-order derivative energy estimate coupled at the scale of the one-dimensional Nash inequality.

math.AP↗

arXiv2026

Asymptotic Stability and $L^p$ Time-Decay Rates of Rarefaction Waves for Systems of Viscous Conservation Laws

We study the asymptotic stability and quantitative time-decay rates of rarefaction waves for one-dimensional $n\times n$ strictly hyperbolic systems of viscous conservation laws. While the nonlinear stability of rarefaction waves for $2\times 2$ systems was established by Xin \cite{Xin1988,Xin1989}, extending such results to general $n\times n$ systems presents additional difficulties arising from the possible coexistence of positive and negative transverse relative characteristic speeds and nonlinear interactions between distinct characteristic families. Moreover, establishing standalone $L^p$ energy estimates and explicit algebraic decay rates for general viscous systems without $L^1$ localization remains a significant challenge. In this paper, we establish both the global $H^2(\R)$ asymptotic stability and, for the first time, standalone physical-space $L^p$ energy estimates and explicit algebraic $L^p$ time-decay rates for weak rarefaction waves. Specifically, for every integer $p\ge 3$, we prove that the perturbation $U(t)-\bar{U}(t)$ around the smooth Burgers rarefaction profile $\bar{U}$ decays in $L^p(\R)$ at the rate \[ \|U(t)-\bar{U}(t)\|_{L^p}\le C_p\bigl(\|U_0-\bar{U}(0)\|_{H^2}^2+δ^3\bigr)^{1/2}(1+t)^{-\frac{1}{2}(\frac{1}{2}-\frac{1}{p})}, \] where $δ$ denotes the wave strength. Our proof combines a diagonal characteristic weight matrix that yields coercive rarefaction damping in all characteristic modes with directed wave-interaction functionals that exploit the strict separation of characteristic speeds to control nonlinear coupling between distinct characteristic families. These estimates lead to a closed $L^p$ Lyapunov inequality, from which the algebraic time-decay rates follow by the one-dimensional Nash inequality.

math.AP↗

arXiv2026

Decay Rate toward the Rarefaction Wave and the Viscous Contact Wave for Jin--Xin Relaxation Model

This paper establishes the time-decay rates toward the rarefaction wave and the viscous contact wave for the one-dimensional Jin--Xin relaxation system under the $L^1$ integrability condition of the initial perturbation. In the case of a strictly convex flux, we employ a decomposition method to derive a logarithmic $L^1$ estimate related to a conjecture proposed by Liu\cite{liu2001lp}. Furthermore, a time-weighted $L^2$ method yields the optimal decay rate toward the rarefaction wave. Regarding the case where the flux is partially linearly degenerate, by employing a similar $L^1$ estimate and time-weighted $L^2$ method, along with a refined estimate of the nonlinear wave interactions, we obtain the corresponding decay rate toward the composite wave. The decay estimates are obtained without smallness assumptions on the initial perturbations or the wave strength.

math.AP↗

arXiv2026

Global classical solutions of the repulsive relativistic Vlasov--Poisson system

Global classical existence for general large data in the three-dimensional repulsive relativistic Vlasov--Poisson system has long been an open problem. We give an affirmative answer: every nonnegative $C^1$ initial datum satisfying \[ f_0(x,p)+|\nabla_{x,p}f_0(x,p)|\lesssim\langle x\rangle^{-a}\langle p\rangle^{-b},\qquad a>1,\quad b>3, \] generates a unique global classical solution. No smallness or symmetry is assumed. The result covers finite and infinite mass and energy, including infinite kinetic energy density. The power thresholds are sharp for general pure-power bounds ensuring finite density and an absolutely convergent Coulomb field. The proof rests on two new mechanisms: local energy recovery for a population selected by energy growth, and a corrected moment--action closure based on nonnegative transverse interactions. The argument also yields global classical solutions of the nonrelativistic equation in the same initial class, allowing infinite mass with polynomial velocity tails of any order $b>3$.

math.AP↗

arXiv2026

Radial pressureless Euler flow: structure and stability of absorption at the origin

We study absorption at the origin in radial pressureless Euler flow in $\mathbb{R}^n$, $n\geq2$. For rotation-invariant nonnegative locally finite initial mass and bounded radial initial velocity vanishing at the origin, we prove global existence and uniqueness in a full-space radial entropy class with a one-sided Lipschitz bound and an initial kinetic-energy trace. Absorption is permanent, including retention of any initial origin atom. For finite positive mass with finite second moment, a Lagrangian projection representation characterizes the absorbed mass and its left limits at positive times through extreme minimizers of a free-position primitive. We derive a radial scalar momentum balance with a nonnegative Stieltjes boundary impulse measure. The minimum value of the same primitive yields a weighted Stieltjes identity equating accumulated time-weighted impulse with the integral of initial radii over absorbed labels, valid also for jumps and singular continuous absorption; the boundary impulse measure and the absorption measure are mutually absolutely continuous on $(0,\infty)$. For such finite-mass data with common total mass, strong $L^2$ convergence of initial quantiles and Lagrangian velocities, together with convergence of initial origin masses, yields convergence of absorbed mass at continuity times of the limiting absorption and local weak convergence of absorption measures on $(0,\infty)$. Finally, even with a fixed smooth initial radial probability density, every probability measure compactly supported in $(0,\infty)$ can be realized as the absorption measure by choosing a bounded radial initial velocity; in particular, purely singular continuous absorption occurs.

math.AP↗

arXiv2026

One-dimensional pressureless flow with a nonlinear velocity law: Bregman variational construction and well-posedness

We establish global existence, uniqueness and quantitative stability for one-dimensional pressureless flow governed by \[ \partial_tρ+\partial_x\bigl(ρf(u)\bigr)=0, \qquad \partial_t(ρu)+\partial_x\bigl(ρu f(u)\bigr)=0, \] where $ρ$ is the mass density, possibly measure-valued, $u$ is the physical velocity, and $f\in C^1(\mathbb R)$, with $f'>0$, determines the transport velocity $f(u)$. The initial mass has finite positive mass and finite second moment, allowing atoms and unbounded support, and the initial physical velocity is bounded. The difficulty at a cluster collision is that mass and momentum conservation determine a mass-weighted average of physical velocity, whereas transport is governed by the nonlinear law $f$ applied to that average. We construct global entropy solutions by a Bregman variational time discretization whose cost is chosen to reproduce the transport velocity determined by this mass-weighted average, coupled with conditional-expectation updates of physical velocity. For every entropy solution satisfying the Oleinik condition on the transport velocity and initial weak continuity of the kinetic energy measure, we establish a Lagrangian representation and prove uniqueness within this class by comparing flows with possibly different cluster intervals. For two such solutions with the same velocity law and common mass, we obtain a $W_\infty$ estimate for the normalized mass measures in terms of the initial position and physical-velocity discrepancies in mass coordinates, whenever the initial $W_\infty$ distance is finite. The Lagrangian evolution has the semigroup property.

math.AP↗

arXiv2026

Stability of composite waves consisting of a non-classical Liu shock and a rarefaction wave for a non-convex viscoelastic system

We investigate the time-asymptotic stability of composite waves consisting of a non-classical Liu shock and a rarefaction wave for non-convex viscoelastic system. By choosing delicate time-dependent shift functions to the viscous shock and suitable weight functions for the relative entropy, we succeed in proving the time-asymptotic stability of the composite of a viscous Liu shock and a rarefaction wave from two different characteristic families for the one-dimensional viscoelastic system with a cubic non-convex stress-strain relation by $a$-contraction method. Here the non-classical Liu shock can be either degenerate or non-degenerate, which can be seen as the system version of the scalar Oleinik shock. Note that both the shift and the weight functions are chosen quite differently in the two cases of nearly degenerate and fully non-degenerate. The paper represents the first time-asymptotic stability result for the composite waves consisting of both non-classical Liu shock and rarefaction wave from two different families for the non-convex physical system.

math.AP↗

arXiv2026

Global Dynamic Patterns of Entropy Solutions for One-dimensional Pressureless Euler System

In this paper, we are concerned with the fine properties of entropy solutions of the Cauchy problem for the one-dimensional pressureless Euler system, wherein the initial density $ρ_0$ is a locally finite Radon measure and the initial velocity $u_0\in L^\infty_{ρ_0}$. We employ the solution formula introduced by [F.M. Huang and Z. Wang, Comm. Math. Phys. 222(1) (2001), 117--146.] for this Cauchy problem to analyze the entropy solutions and obtain various new fine properties of entropy solutions; these can be summarized in four aspects: (i) Characteristics and initial waves for the Cauchy problem; (ii) Fine local structures of entropy solutions; (iii) Divides and global structures of entropy solutions; (iv) Invariants and asymptotic behaviors of entropy solutions including the asymptotic profile and the corresponding decay rates. Through these results (i)-(iv), we establish the global dynamic patterns of entropy solutions of the Cauchy problem for the $1$-D pressureless Euler system with general initial data $ρ_0$ being locally finite Radon measures and $u_0\in L^\infty_{ρ_0}$.

math.AP↗

arXiv2026

Nonlinear stability and optimal decay rate of the planar entropy wave for Landau equation

This paper investigates the nonlinear asymptotic stability and optimal decay rates of entropy waves for the Landau equation with physically realistic Coulomb interactions under general perturbations. We consider the infinite channel domain $\mathbb{R} \times \mathbb{T}^2$ in three dimensions, which possesses both one-dimensional and high-dimensional characteristics, thereby posing two primary analytical challenges: (i) for the one-dimensional Landau equation with Coulomb potentials, the absence of a spectral gap in the linearized operator has obstructed the derivation of wave pattern stability results with explicit time decay rates; (ii) in the study of contact discontinuities, the multidimensional case fundamentally differs from the one-dimensional setting due to lack of a key structural condition. We develop effective analytical approaches to treat those difficulties. To overcome the weak dissipation caused by the spectral gap deficiency, we implement a time-velocity interpolation technique to enhance dissipation and simultaneously construct coupled diffusion waves to compensate for the loss of time decay. To address the missing structural condition in higher dimensions, a novel transformation is introduced to recover the two-sided structural condition within the perturbation system. By developing a derivative-level transformation and a refined energy framework, we restore the necessary structural condition for derivatives, establish the optimal decay of the solution, and prove the stretched exponential decay of its non-zero modes. In contrast to previous methods that rely on artificial viscosity or the Navier--Stokes approximation, our approach directly leverages the intrinsic physical dissipation of the equation and its coupling with the microscopic kinetic component, ensuring broader applicability.

math.AP↗

arXiv2026

Nonlinear stability of compressible vortex sheets in three-dimensional elastodynamics

We investigate the nonlinear stability of compressible vortex sheet solutions for three-dimensional (3D) isentropic elastic flows. Building upon previous results on the weakly linear stability of elastic vortex sheets [19], we perform a detailed study of the roots of the Lopatinskii determinant and identify a geometric stability condition associated with the deformation gradient. We employ an upper triangularization technique that isolates the outgoing modes into a closed system, where they appear only at the leading order. This enables us to derive energy estimates despite derivative loss. The major novelty of our approach includes the following two key aspects: (1) For the 3D compressible Euler vortex sheets, the front symbol exhibits degenerate ellipticity in certain frequency directions, which makes it challenging to ensure the front's regularity using standard energy estimates. Our analysis reveals that the non-parallel structure of the deformation gradient tensor plays a crucial role in recovering ellipticity in the front symbol, thereby enhancing the regularity of the free interface. (2) Another significant challenge in 3D arises from the strong degeneracy caused by the collision of repeated roots and poles. Unlike in 2D, where such interactions are absent, we encounter a co-dimension one set in frequency space where a double root coincides with a double pole. To resolve this, we refine Coulombel's diagonalization framework [21] and construct a suitable transformation that reduces the degeneracy order of the Lopatinskii matrix, enabling the use of localized Garding-type estimates to control the characteristic components. Finally, we employ a Nash-Moser iteration scheme to establish the local existence and nonlinear stability of vortex sheets under small initial perturbations, showing stability within a subsonic regime.

math.AP↗

arXiv2026

Well-Posedness and Asymptotic Decay of Solutions to the Three-Dimensional Euler Equations with Damping

The global well-posedness of the multi-dimensional compressible Euler equations with damping remains a longstanding open problem. This problem has been partially resolved in the isentropic regime ({\it i.e.}, the adiabatic exponent $γ>1$) for small smooth initial data (see \cite{WY, STW}). In this paper, we establish the global well-posedness and asymptotic decay of smooth solutions of the Cauchy problem of the three-dimensional compressible Euler equations with damping for the isentropic regime $γ>1$ and the isothermal regime $γ=1$, allowing for partially large initial data. More precisely, the $L^2$-norm of the initial data is allowed to be large, while the third-order Sobolev norm of the initial data is assumed to be small. For the isentropic case, we develop a new analytical framework in which all required {\it a priori} estimates of solution $(ρ,u)$ can be derived under the condition that $\int_0^T \big( \|\nablaρ\|_{L^\infty} + \|\nabla u\|_{L^\infty} \big) \, \mathrm{d}t$ remains sufficiently small. Moreover, we obtain the optimal algebraic decay rates of global solutions. Furthermore, we study the isothermal limit of solutions of the isentropic regime as $γ\to 1$, and establish the global well-posedness and asymptotic decay of solutions to the isothermal Euler equations with damping.

math.AP↗

arXiv2025

Global Martingale Entropy Solutions to the Stochastic Isentropic Euler Equations

We establish the existence and compactness of global martingale entropy solutions with finite relative-energy for the stochastically forced system of isentropic Euler equations governed by a general pressure law. To achieve these, a stochastic compensated compactness framework in $L^p$ is developed to overcome the difficulty that the uniform $L^{\infty}$ bound for the stochastic approximate solutions is unavailable, owing to the stochastic forcing term. The convergence of the vanishing viscosity method is established by employing the stochastic compactness framework, along with careful uniform estimates of the stochastic approximate solutions, to obtain the existence of global martingale entropy solutions with finite relative-energy. In particular, in the polytropic pressure case for all adiabatic exponents, we prove that the global solutions satisfy the local mechanical energy inequality when the initial data are only required to have finite relative-energy (while the higher moment estimates for entropy are not required here, as needed in the earlier work). Higher-order relative energy estimates for approximate solutions are also derived to establish the entropy inequality for more convex entropy pairs and to then prove the compactness of solutions to the stochastic isentropic Euler system. The stochastic compensated compactness framework and the uniform estimate techniques for approximate solutions developed in this paper should be useful in the study of other similar problems.

math.AP↗

arXiv2025

Asymptotic stability of planar entropy wave for 3-d Navier-Stokes equations in Eulerian coordinates

We investigate the large-time asymptotic behavior toward the planar entropy wave for the three-dimensional Navier-Stokes equations in Eulerian coordinates, considering two types of initial perturbations -- with and without the assumption that the integral of the initial perturbation is zero. Generic perturbations generate diffusion waves, and structural conditions fail for multi-dimensional Navier-Stokes equations in Eulerian coordinates. These two aspects have posed significant challenges and left the problem unresolved for years. On one hand, since \cite{LX}, the study of the entropy wave has been based on the left-right structural conditions. Without these structural conditions, the decay rates of lower-order terms become too slow to close the {\it a priori} assumption. On the other hand, the presence of diffusion waves yields problematic error terms in the perturbation system. In this work, we introduce a new transformation to ensure that both left-right structural conditions hold for the perturbation system. Additionally, using the fact that the derivative of the entropy wave maintains a fixed sign, we employ well-designed weighted energy estimates to control the slowly decaying terms. This enables us to establish asymptotic stability and derive the optimal decay rate. Furthermore, we address the case of initial perturbations with the zero mass condition and obtain the optimal decay rate by additionally developing a Poincaré type inequality and a key cancellation.

math.AP↗

arXiv2025

Hilbert expansion of the Boltzmann equation on a 2-dimensional disk with specular boundary condition

In the present paper, we concern the hydrodynamic limit of Boltzmann equation with specular reflection boundary condition in a two-dimensional disk to the compressible Euler equations. Due to the non-zero curvature and non-zero tangential velocity of compressible Euler solution on the boundary, new difficulties arise in the construction of Knudsen boundary layer. By employing the geometric correction, and an innovative and refined $L^2-L^\infty$ method, we establish the existence and space-decay for a truncated Knudsen boundary layer. Then, by the Hilbert expansion of multi-scales, we successfully justify the hydrodynamic limit of Boltzmann equation with specular reflection boundary condition to the compressible Euler equations in the two-dimensional disk.

math.AP↗

arXiv2024

Nonlinear asymptotic stability of non-self-similar rarefaction wave for two-dimensional viscous Burgers equation

We investigate the large time behavior of solutions to the two-dimensional viscous Burgers equation $u_t+uu_x+uu_y=Δu$, toward a non-self-similar rarefaction wave of inviscid Burgers equation with two initial constant states, seperated by a curve $y=φ(x)$, and prove that the above 2D non-self-similar rarefaction wave is time-asymptotically stable. Furthermore, we also get the decay rate. Both the rarefaction wave strength and the initial perturbation can be large.

math.AP↗

arXiv2024

The Isometric Immersion of Negatively Curved Surfaces with Finite Total Curvature

In this paper, we study the smooth isometric immersion of a complete, simply connected surface with a negative Gauss curvature into the three-dimensional Euclidean space. A fundamental and longstanding problem is to find a sufficient condition for a complete negatively curved surface to be isometrically embedded in R^3 [67]. It can be described as an initial and/or boundary value problem for a hyperbolic system of nonlinear partial differential equations derived from the Gauss-Codazzi equations. The mathematical theory associated with this system is largely incomplete. The global smooth isometric immersion has been proven in the literature when the Gauss curvature decays rapidly and monotonically. However, when the Gauss curvature oscillates or decays slowly, the problem becomes much more challenging and little is known. In our paper, we find a sufficient condition, consisting of a finite total Gauss curvature and appropriate oscillations of the Gauss curvature. Under this condition we prove the global existence of a smooth solution to the Gauss-Codazzi system, achieving a global smooth isometric immersion of the surface into R^3. Furthermore, we show that the finite total Gauss curvature is necessary for the existence of a solution in a special case of the Gauss-Codazzi system. New techniques are developed to overcome the difficulties posed by the slow decay and oscillations of the Gauss curvature. By observing that certain combinations of the Riemann invariants decay faster than others, we reformulate the Gauss-Codazzi equations as a symmetric hyperbolic system and uncover a crucial structure of partial dampings. These partial dampings, along with the finite total curvature and appropriate oscillations of the Gauss curvature, enable us to obtain a global smooth solution through delicate analysis, and consequently establish a global smooth isometric immersion of such surfaces.

math.DG↗

arXiv2024

Nonlinear stability threshold for compressible Couette flow

This paper concerns the Couette flow for 2-D compressible Navier-Stokes equations (N-S) in an infinitely long flat torus $\Torus\times\R$. Compared to the incompressible flow, the compressible Couette flow has a stronger lift-up effect and weaker dissipation. To the best of our knowledge, there has been no work on the nonlinear stability in the cases of high Reynolds number until now and only linear stability was known in \cite{ADM2021,ZZZ2022}.In this paper, we study the nonlinear stability of 2-D compressible Couette flow in Sobolev space at high Reynolds numbers. Moreover, we also show the enhanced dissipation phenomenon and stability threshold for the compressible Couette flow. First, We decompose the perturbation into zero and non-zero modes and obtain two systems for these components, respectively. Different from \cite{ADM2021,ZZZ2022}, we use the anti-derivative technique to study the zero-mode system. We introduce a kind of diffusion wave to remove the excessive mass of the zero-modes and construct coupled diffusion waves along characteristics to improve the resulting time decay rates of error terms and derive a new integrated system \cref{anti}. Secondly, we observe a cancellation with the new system \cref{anti} so that the lift-up effect is weakened. Thirdly, the large time behavior of the zero-modes is obtained by the weighted energy method and a weighted inequality on the heat kernel \cite{HLM2010}.In addition, with the help of the Fourier multipliers method, we can show the enhanced dissipation phenomenon for the non-zero modes by commutator estimates to avoid loss of derivatives. Finally, we complete the higher-order derivative estimates to close the a priori assumptions by the energy method and show the stability threshold.

math.AP↗

arXiv2024

Time-asymptotic stability of composite waves of degenerate Oleinik shock and rarefaction for non-convex conservation laws

We are concerned with the large-time behavior of the solution to one-dimensional (1D) cubic non-convex scalar viscous conservation laws. Due to the inflection point of the cubic non-convex flux, the solution to the corresponding inviscid Riemann problem can be the composite wave of a degenerate Oleinik shock and a rarefaction wave and these two nonlinear waves are always attached together. We give a first proof of the time-asymptotic stability of this composite wave, up to a time-dependent shift to the viscous Oleinik shock, for the viscous equation. The Oleinik shock wave strength can be arbitrarily large. The main difficulty is due to the incompatibility of the time-asymptotic stability proof framework of individual viscous shock by the so-called anti-derivative method and the direct $L^2$-energy method to rarefaction wave. Here we develop a new type of $a$-contraction method with suitable weight function and the time-dependent shift to the viscous shock, which is motivated by [9,12]. Another difficulty comes from that the Oleinik shock and rarefaction wave are always attached together and their wave interactions are very subtle. Therefore, the same time-dependent shift needs to be equipped to both Oleinik shock and rarefaction wave such that the wave interactions can be treated in our stability proof. Time-asymptotically, this shift function grows strictly sub-linear with respect to the time and then the shifted rarefaction wave is equivalent to the original self-similar rarefaction wave.

math.AP↗