arXiv subjects

Lingda Xu

Publications and source records attributed to Lingda Xu.

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.

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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.

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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.

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arXiv2026

Limiting Pointwise Decay for the compressible isentropic Navier-Stokes equations

We study the long-time pointwise behavior of small localized perturbations of a constant state for the one-dimensional compressible isentropic Navier-Stokes equations. After subtracting the two Burgers diffusion waves, the convergent sum of all higher-order diffusion waves, and the cross-family viscous corrections, we prove a cone-preserving pointwise estimate for the exact physical remainder and, in particular, \[ |R_i(x,t)|\leq C E_N\log(2+t)Ψ_i(x,t), \qquad \sup_{x\in\mathbb R}Ψ_i(x,t)\leq C(1+t)^{-1}. \] Here $E_N$ measures the size of the initial data and $Ψ_i$ is the cone-resolved weight; both are defined precisely in the main theorem below. Thus $\|R_i(t)\|_{L^\infty}\leq C E_N(1+t)^{-1}\log(2+t)$. The key new idea is to apply a familywise Cole--Hopf transformation to the spatial antiderivative of the remainder, which exactly eliminates the critical same-family first-order feedback. We further construct an approximate Green function adapted to the two characteristic families and combine it with Gaussian-mode extraction and a Kawashima-type energy argument. This yields a cone-preserving estimate at the limiting decay rate, up to a logarithmic loss.

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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.

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arXiv2026

Nonlinear stability threshold for 3D compressible Couette flow

We establish the nonlinear stability threshold $O(ν^{3/2})$ for the three-dimensional Couette flow governed by the compressible Navier--Stokes equations. While stability thresholds are well understood in two dimensions for both compressible and incompressible flows, and in three dimensions for incompressible flows, the three-dimensional compressible case remains open due to additional structural features, strong mode interactions, and wave coupling. The proof is based on a refined frequency-space approach. For zero modes, we improve upon two-dimensional methods by clearly separating and precisely estimating the main contributions from diffusion waves, acoustic waves, and the lift-up mechanism, leading to a systematic way to handle their nonlinear coupling. For the non-zero modes, we introduce new multiplier estimates and a decomposition based on the structure of the compressible system, which allows us to track the interaction between dissipation and acoustic effects.

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arXiv2026

Hydrodynamic limit of rarefaction wave for the Vlasov-Maxwell-Landau system with Coulomb potential

In this paper, we investigate the hydrodynamic limit of rarefaction wave for the two-species Vlasov-Maxwell-Landau(VML) system with Coulomb potential. We prove that for any given time interval, the solution of the Vlasov-Maxwell-Landau system with appropriate initial data converges to a rarefaction wave as the Knudsen number $ε$ approaches zero. The main difficulty in the analysis lies in the loss of dissipation in the interaction between the electromagnetic field and the microscopic component, and the weak dissipation induced by the Lorentz force and the scaling with small parameter $ε$. For this, we introduce a velocity weight function and a space-time scaling parameter together with suitable $ε$-dependent energy estimates.

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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.

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arXiv2025

Optimal decay rates to the contact wave for 1-D compressible Navier-Stokes equations

This paper investigates the decay rates of the contact wave in one-dimensional Navier-Stokes equations. We study two cases of perturbations, with and without zero mass condition, i.e., the integration of initial perturbations is zero and non-zero, respectively. For the case without zero mass condition, we obtain the optimal decay rate $(1+t)^{-\frac{1}{2}}$ for the perturbation in $L^\infty$ norm, which provides a positive answer to the conjecture in \cite{HMX}. We applied the anti-derivative method, introducing the diffusion wave to carry the initial excess mass, diagonalizing the integrated system, and estimating the energy of perturbation in the diagonalized system. Precisely, due to the presence of diffusion waves, the decay rates for errors of perturbed system are too poor to get the optimal decay rate. We find the dissipation structural in the diagonalized system, see \cref{ds}. This observation makes us able to fully utilize the fact that the sign of the derivative of the contact wave is invariant and to control the terms with poor decay rates in energy estimates. For the case with zero mass condition, there are also terms with poor decay rates. In this case, note that there is a cancellation in the linearly degenerate field so that the terms with poor decay rates will not appear in the second equation of the diagonalized system. Thanks to this cancellation and a Poincaré type of estimate obtained by a critical inequality introduced by \cite{HLM}, we get the decay rate of $\ln^{\frac{1}{2}} (2+t)$ for $L^2$ norm of anti-derivatives of perturbation and $(1+t)^{-\frac{1}{2}}\ln^{\frac{1}{2}}(2+t)$ for the $L^2$ norm of perturbation itself, the decay rates are optimal, which is consistent with the results obtained by using pointwise estimate in \cite{XZ} for the system with artificial viscosity.

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arXiv2025

Asymptotic stability of the composite wave of rarefaction wave and contact wave to nonlinear viscoelasticity model with non-convex flux

In this paper, we consider the wave propagations of viscoelastic materials, which has been derived by Taiping-Liu to approximate the viscoelastic dynamic system with fading memory (see [T.P.Liu(1988)\cite{LiuTP}]) by the Chapman-Enskog expansion. By constructing a set of linear diffusion waves coupled with the high-order diffusion waves to achieve cancellations to approximate the viscous contact wave well and explicit expressions, the nonlinear stability of the composite wave is obtained by a continuum argument. It emphasis that, the stress function in our paper is a general non-convex function, which leads to several essential differences from strictly hyperbolic systems such as the Euler system. Our method is completely new and can be applied to more general systems and a new weighted Poincaré type of inequality is established, which is more challenging compared to the convex case and this inequality plays an important role in studying systems with non-convex flux.

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arXiv2025

The stability threshold for 3D MHD equations around Couette with rationally aligned magnetic field

We address a stability threshold problem of the Couette flow $(y,0,0)$ in a uniform magnetic fleld $α(σ,0,1)$ with $σ\in\mathbb{Q}$ for the 3D MHD equations on $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. Previously, the authors in \cite{L20,RZZ25} obtained the threshold $γ=1$ for $σ\in\mathbb{R}\backslash\mathbb{Q}$ satisfying a generic Diophantine condition, where they also proved $γ= 4/3$ for a general $σ\in\mathbb{R}$. In the present paper, we obtain the threshold $γ=1$ in $H^N(N>13/2)$, hence improving the above results when $σ$ is a rational number. The nonlinear inviscid damping for velocity $u^2_{\neq}$ is also established. Moreover, our result shows that the nonzero modes of magnetic field has an amplification of order $ν^{-1/3}$ even on low regularity, which is very different from the case considered in \cite{L20,RZZ25}.

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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.

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arXiv2024

Vanishing viscosity limit to the planar rarefaction wave with vacuum for 3-D full compressible Navier-Stokes equations with temperature-dependent transport coefficients

In this paper, we construct a family of global-in-time solutions of the 3-D full compressible Navier-Stokes (N-S) equations with temperature-dependent transport coefficients (including viscosity and heat-conductivity), and show that at arbitrary times {and arbitrary strength} this family of solutions converges to planar rarefaction waves connected to the vacuum as the viscosity vanishes in the sense of $L^\infty(\R^3)$. We consider the Cauchy problem in $\R^3$ with perturbations of the infinite global norm, particularly, periodic perturbations. To deal with the infinite oscillation, we construct a suitable ansatz carrying this periodic oscillation such that the difference between the solution and the ansatz belongs to some Sobolev space and thus the energy method is feasible. The novelty of this paper is that the viscosity and heat-conductivity are temperature-dependent and degeneracies caused by vacuum. Thus the a priori assumptions and two Gagliardo-Nirenberg type inequalities are essentially used. Next, more careful energy estimates are carried out in this paper, by studying the zero and non-zero modes of the solutions, we obtain not only the convergence rate concerning the viscosity and heat conductivity coefficients but also the exponential time decay rate for the non-zero mode.

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arXiv2024

Nonlinear Stability of Planar Shock Waves for the 3-D Boltzmann Equation

This paper studies the stability and large-time behavior of the three-dimensional (3-D) Boltzmann equation near shock profiles. We prove the nonlinear stability of the composite wave consisting of two shock profiles under general perturbations without the assumption of integral zero of macroscopic quantities. To address the challenge caused by the compressibility of shock profiles, we apply the method of anti-derivative based on macro-micro decomposition. However, the system of anti-derivatives presents certain difficulties. Firstly, general perturbations may generate diffusion waves that evolve and interact with shock profiles, resulting in errors that are not controllable. We therefore introduce a set of coupled diffusion waves to cancel out these poor errors and perform careful estimates on wave interactions. Secondly, we perform diagonalized system estimates to fully exploit the compressibility of shock profiles and control terms that decay slowly. Thirdly, the presence of diffusion waves causes critical terms with decay $(1+t)^{-1}$, and we introduce a Poincaré type of inequality to address these terms. Finally, estimates on anti-derivatives can only control terms along the propagation direction, while for transversal directions, we use the entropy-entropy flux pair as well as the Poincaré inequality to control the lower order terms using diffusion terms. As a result, we obtain nonlinear stability through the energy method, which is the first stability result for the planar shock of the multi-dimensional Boltzmann equation to the best of our knowledge.

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arXiv2023

Nonlinear stability of planar shock wave to 3-D compressible Navier-Stokes equations in half space with Navier Boundary conditions

In this paper, we consider the large time behavior of planar shock wave for 3-D compressible isentropic Navier-Stokes equations (CNS) in half space. Providing the strength of the shock wave and initial perturbations are small, we proved the planar shock wave for 3-D CNS is nonlinearly stable in half space with Navier boundary condition. The main difficulty comes from the compressibility of shock wave, which leads to lower order terms with bad sign, see the third line in \cref{C17}. We apply a decomposition of the solution into zero and non-zero modes: we take the anti-derivative for the zero mode and obtain the space-time estimates for the energy of perturbation itself. Then combining the fact that the Poincaré inequality is available for the non-zero mode, we have successfully controlled the lower order terms with bad sign in \cref{C17}. To overcome the difficulty that comes from the boundary, we introduce the two crucial estimates on boundary \cref{CLem0} and fully utilize the property of Navier boundary conditions, which means that the normal velocity is zero on the boundary and the fluid tangential velocity is proportional to the tangential component of the viscous stress tensor on the boundary. Finally, the nonlinear stability is proved by the weighted energy method.

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arXiv2023

Decay rate to the planar viscous shock wave for multi-dimensional scalar conservation laws

In this paper, we study the time-decay rate toward the planar viscous shock wave for multi-dimensional (m-d) scalar viscous conservation law. We first decompose the perturbation into zero and non-zero mode, and then introduce the anti-derivative of the zero mode. Though an $L^p$ estimate and the area inequality introduced in \cite{DHS2020}, we obtained the decay rate for planar shock wave for n-d scalar viscous conservation law for all $n\geq1$. The initial perturbations we studied are small, i.e., $\|Φ_0\|_{H^2}\bigcap\|Φ_0\|_{L^p}\le \varepsilon$, where $Φ_0$ is the anti-derivative of the zero mode of initial perturbation and $\varepsilon$ is a small constant, see \cref{antiderivative}. It is noted that there is no additional requirement on $Φ_0$, i.e., $Φ_0(x_1)$ only belongs to $H^2(\R)$. Thus, there are essential differences from previous results, in which the initial data is required to belong to some weighted Sobolev space, cf.\cite{Goo1989,KM1985}. Moreover, the exponential decay rate of the non-zero mode is also obtained.

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arXiv2023

Nonlinear asymptotic stability of compressible vortex sheets with viscosity effects

This paper concerns the stabilizing effect of viscosity on the vortex sheets. It is found that although a vortex sheet is not a time-asymptotic attractor for the compressible Navier-Stokes equations, a viscous wave that approximates the vortex sheet on any finite time interval can be constructed explicitly, which is shown to be time-asymptotically stable in the $ L^\infty $-space with small perturbations, regardless of the amplitude of the vortex sheet. The result shows that the viscosity has a strong stabilizing effect on the vortex sheets, which are generally unstable for the ideal compressible Euler equations even for short time [26,8,1]. The proof is based on the $ L^2 $-energy method.In particular, the asymptotic stability of the vortex sheet under small spatially periodic perturbations is proved by studying the dynamics of these spatial oscillations. The first key point in our analysis is to construct an ansatz to cancel these oscillations. Then using the Galilean transformation, we are able to find a shift function of the vortex sheet such that an anti-derivative technique works, which plays an important role in the energy estimates. Moreover, by introducing a new variable and using the intrinsic properties of the vortex sheet, we can achieve the optimal decay rates to the viscous wave.

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arXiv2023

Nonlinear stability of the composite wave of planar rarefaction waves and planar contact waves for viscous conservation laws with non-convex flux under multi-dimensional periodic perturbations

In this paper, we study the nonlinear stability of the composite wave consisting of planar rarefaction and planar contact waves for viscous conservation laws with degenerate flux under multi-dimensional periodic perturbations. To the level of our knowledge, it is the first stability result of the composite wave for conservation laws in several dimensions. Moreover, the perturbations studied in the present paper are periodic, which keep constantly oscillating at infinity. Suitable ansatz is constructed to overcome the difficulty caused by this kind of perturbation and delicate estimates are done on zero and non-zero modes of perturbations. We obtain satisfactory decay rates for zero modes and exponential decay rates for non-zero modes.

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