Corrado Mascia, Kevin Zumbrun
Using a combination of Kawashima- and Goodman-type energy estimates, we establish spectral stability of general small-amplitude relaxation shocks of symmetric dissipative systems. This extends previous results obtained by Plaza and Zumbrun by singular perturbation techniques under an additional technical assumption, namely, that the background equation be noncharacteristic with respect to the shock.
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We establish pointwise bounds for the Green function and consequent linearized stability for multidimensional planar relaxation shocks of general relaxation systems whose equilibrium model is scalar, under the necessary assumption of spectral stability. Moreover, we obtain nonlinear $L^{2}$ asymptotic behavior/sharp decay rate of perturbed weak shocks of general simultaneously symmetrizable relaxation systems, under small $L^{1}\cap H^{[d/2]+3}$ perturbations with first moment in the normal direction to the front.
In this paper, we consider the large time asymptotic nonlinear stability of a superposition of shock waves with contact discontinuities for the one dimensional Jin-Xin relaxation system with small initial perturbations, provided that the strengths of waves are small with the same order. The results are obtained by elementary weighted energy estimates based on the underlying wave structure and an estimate on the heat equation.
We establish sharp pointwise Green's function bounds and consequent linearized and nonlinear stability for smooth traveling front solutions, or relaxation shocks, of general hyperbolic relaxation systems of dissipative type, under the necessary assumptions ([G,Z.1,Z.4]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave, and hyperbolic stability of the corresponding ideal shock of the associated equilibrium system. This yields, in particular, nonlinear stability of weak relaxation shocks of the discrete kinetic Jin--Xin and Broadwell models. The techni…
We establish existence with sharp rates of decay and distance from the Chapman--Enskog approximation of small-amplitude shock profiles of a class of semilinear relaxation systems including discrete velocity models obtained from Boltzmann and other kinetic equations. Our method of analysis is based on the macro--micro decomposition introduced by Liu and Yu for the study of Boltzmann profiles, but applied to the stationary rather than the time-evolutionary equations. This yields a simple proof by contraction mapping in weighted $H^s$ spaces.
We establish existence with sharp rates of decay and distance from the Chapman--Enskog approximation of small-amplitude quasilinear relaxation shocks in the general case that the profile ODE may become degenerate. Our method of analysis follows the general approach used by Métivier and Zumbrun in the semilinear case, based on Chapman--Enskog expansion and the macro--micro decomposition of Liu and Yu. In the quasilinear case, however, we find it necessary to apply a parameter-dependent Nash-Moser iteration to close the analysis, whereas, in the semilinear case, a simple contraction-mapping argu…
We study a model of radiating gases that describes the interaction of an inviscid gas with photons. We show the existence of smooth traveling waves called 'shock profiles', when the strength of the shock is small. Moreover, we prove that the regularity of the traveling wave increases when the strength of the shock tends to zero.
We establish nonlinear $H^2\cap L^1 \to H^2$ stability with sharp rates of decay in $L^p$, $p\geq 2$, of general hydraulic shock profiles, with or without subshocks, of the inviscid Saint-Venant equations of shallow water flow, under the assumption of Evans-Lopatinsky stability of the associated eigenvalue problem. We verify this assumption numerically for all profiles, giving in particular the first nonlinear stability results for shock profiles with subshocks of a hyperbolic relaxation system.
This paper is concerned with the time-asymptotic behavior of strong solutions to an initial-boundary value problem of the compressible Navier-Stokes-Korteweg system on the half line $\mathbb{R}^+$. The asymptotic profile of the problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary. Moreover, we prove that if the initial data around the shifted viscous shock profile and the strength of the shifted viscous shock profile are sufficiently small, then the problem has a unique global strong solution, which tends to the shifted viscous shock profile as time …
By reduction to a generalized Sturm Liouville problem, we establish spectral stability of hydraulic shock profiles of the Saint-Venant equations for inclined shallow-water flow, over the full parameter range of their existence, for both smooth-type profiles and discontinuous-type profiles containing subshocks. Together with work of Mascia-Zumbrun and Yang-Zumbrun, this yields linear and nonlinear $H^2\cap L^1 \to H^2$ stability with sharp rates of decay in $L^p$, $p\geq 2$, the first complete stability results for large-amplitude shock profiles of a hyperbolic relaxation system.
This paper is concerned with singular shocks for a system of conservation laws modeling incompressible two-phase fluid flow. We prove the existence of viscous profiles using the Geometric Singular Perturbation Theory. Weak convergence and growth rates of the unbounded family of solutions are also obtained.
We carry out the first rigorous numerical proof based on Evans function computations of stability of viscous shock profiles, for the system of isentropic gas dynamics with monatomic equation of state. We treat a selection of shock strengths ranging from the lower stability boundary of Mach number $\approx 1.86 $, below which profiles are known by energy estimates to be stable, to the upper stability boundary of $\approx 1669$, above which profiles are expected to be provable by rigorous asymptotic analysis to be stable. These results open the possibilities of: (i) automatic rigorous verificati…
We investigate existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by Antman and Malek-Madani. We establish that the resulting equations fall into the class of symmetrizable hyperbolic--parabolic systems, hence spectral stability implies linearized and nonlinear stability with sharp rates of decay. The new contributions are treatment of the compressible case, formulation of a rigorous nonlinear stability theory, including verification of stability of small-amplitude Lax shocks, and the systematic incorporation in o…
Pseudodifferential damping estimates and stability of relaxation shocks
本文提出基于频率依赖伪微分阻尼估计的方法，用于研究松弛激波的稳定性，强调高频谱稳定性和转折点问题的重要性。
For the two-dimensional Navier--Stokes equations of isentropic magnetohydrodynamics (MHD) with $γ$-law gas equation of state, $γ\ge 1$, and infinite electrical resistivity, we carry out a global analysis categorizing all possible viscous shock profiles. Precisely, we show that the phase portrait of the traveling-wave ODE generically consists of either two rest points connected by a viscous Lax profile, or else four rest points, two saddles and two nodes. In the latter configuration, which rest points are connected by profiles depends on the ratio of viscosities, and can involve Lax, overcompre…
Long-Time Behavior towards Shock Profiles for the Navier-Stokes-Poisson System
本文研究一维等温NSP系统激波剖面的稳定性，证明初值在$H^2$范数下接近激波剖面时，柯西问题全局解随时间趋于参数化激波剖面构成的光滑流形，采用带平移的$a$-压缩方法且无需零质量条件。
Extending to systems of hyperbolic--parabolic conservation laws results of Howard and Zumbrun for strictly parabolic systems, we show for viscous shock profiles of arbitrary amplitude and type that necessary spectral (Evans function) conditions for linearized stability established by Mascia and Zumbrun are also sufficient for linearized and nonlinear phase-asymptotic stability, yielding detailed pointwise estimates and sharp rates of convergence in $L^p$, $1\le p\le \infty$.
Kinetic shock profiles for the Landau equation
本文研究了Landau方程中弱冲击轮廓的存在，通过Maxwellian渐近状态和Rankine-Hugoniot条件揭示了动能层面激波结构的形成机制。
Existence and uniqueness of viscous-dispersive shock profiles for the Navier-Stokes-Fourier-Korteweg system
该研究针对满足Rankine-Hugoniot与Lax熵条件的小强度激波，证明了一维NSFK系统粘性-弥散激波剖面的存在唯一性，通过几何奇异摄动理论构造剖面并给出逐点估计，为后续黎曼解稳定性分析提供基础。
We establish one-dimensional spectral stability of small amplitude viscous and relaxation shock profiles using Evans function techniques to perform a series of reductions and normal forms to reduce to the case of the scalar Burgers equation. In multidimensions, the canonical behavior is described, rather by a 2x2 viscous conservation law; this case will be treated in a companion paper by similar techniques.
This paper studies the stability of weak dispersive shock profiles for a quantum hydrodynamics system in one space dimension with nonlinear viscosity and dispersive (quantum) effects due to a Bohm potential. It is shown that, if the shock amplitude is sufficiently small, then the profiles are spectrally stable. This analytical result is consistent with numerical estimations of the location of the spectrum (Lattanzio, Zhelyazov, Math. Models Methods Appl. Sci. 31, 2021). The proof is based on energy estimates at the spectral level, on the choice of an appropriate weighted energy function for th…
We consider the asymptotic behavior of perturbations of Lax and overcompressive type viscous shock profiles arising in systems of regularized conservation laws with strictly parabolic viscosity, and also in systems of conservation laws with partially parabolic regularizations such as arise in the case of the compressible Navier--Stokes equations and in the equations of magnetohydrodynamics. Under the necessary conditions of spectral and hyperbolic stability, together with transversality of the connecting profile, we establish detailed pointwise estimates on perturbations from a sum of the visc…
We investigate the $L^p $ asymptotic behavior $(1\le p \le \infty)$ of a perturbation of a Lax or overcompressive type shock wave solution to a system of conservation law in one dimension. The system of the equations can be strictly parabolic, or have real viscosity matrix (partially parabolic, e.g., compressible Navier--Stokes equations or equations of Magnetohydrodynamics). We use known pointwise Green function bounds for the linearized equation around the shock to show that the perturbation of such a solution can be decomposed into a part corresponding to shift in shock position or shape, a…
Nonlinear stability of shock profiles to Burgers' equation with critical fast diffusion and singularity
本文提出了首个研究具有临界快扩散和奇性的Burgers方程的框架，通过引入权函数和加权能量方法克服了$u=0$时的强奇性，证明了在远场消失的粘性冲击波剖面的非线性稳定性，并通过数值模拟验证了理论结果。
Oscillating Shock Profiles in Relativistic Fluid Dynamics
研究了具有粘性的相对纯辐射流体模型，发现冲击波的连续轮廓在任何变量下均振荡，与经典流体动力学中状态变量单调性不同。
This paper studies the asymptotic stability of shock profiles and rarefaction waves under space-periodic perturbations for one-dimensional convex scalar viscous conservation laws. For the shock profile, we show that the solution approaches the background shock profile with a constant shift in the $ L^\infty(\mathbb{R}) $ norm at exponential rates. The new phenomena contrasting to the case of localized perturbations is that the constant shift cannot be determined by the initial excessive mass in general, which indicates that the periodic oscillations at infinities make contributions to this shi…
Nonlinear orbital stability of stationary discrete shock profiles for scalar conservation laws
本文证明标量守恒律中谱稳定的驻定Lax离散激波剖面在加权ℓ¹和ℓ∞空间非线性轨道稳定，无需振幅弱假设，适用于含人工粘性的广泛格式族，并给出推广到方程组的思路。
Spectral stability of shock-fronted travelling waves under viscous relaxation
本文严格证明了反应-非线性扩散偏微分方程在粘性松弛下的激波前沿行波解具有谱稳定性，通过几何奇异摄动理论证明对于足够小的参数，完整特征值问题由慢特征值问题控制。
Shock profiles for hydrodynamic models for fluid-particles flows in the flowing regime
本文基于流体-动理学耦合方程，通过 Chapman–Enskog 展开推导粘性修正，研究简化与零温流体动力学系统的激波剖面及其小振幅稳定性。
This work establishes nonlinear orbital asymptotic stability of scalar radiative shock profiles, namely, traveling wave solutions to the simplified model system of radiating gas \cite{Hm}, consisting of a scalar conservation law coupled with an elliptic equation for the radiation flux. The method is based on the derivation of pointwise Green function bounds and description of the linearized solution operator. A new feature in the present analysis is the construction of the resolvent kernel for the case of an eigenvalue system of equations of degenerate type. Nonlinear stability then follows in…
This note studies a four-field hyperbolic PDE model that was recently introduced by Bemfica, Disconzi, and Noronha for the pure radiation fluid with viscosity, and asks whether shock waves admit continuous profiles in this description. The model containing two free parameters mu, nu and being causal whenever one chooses (mu,nu) from a certain range C subset R2, this paper shows that for any choice of (mu,nu) in the interior of C, there is a dichotomy in so far as (i) shocks of sufficiently small amplitude admit profiles and (ii) certain other, thus necessarily non-small, shocks do not. This fi…
Existence and spectral stability analysis of viscous-dispersive shock profiles for isentropic compressible fluids of Korteweg type
本文研究了等熵可压缩流体中粘性-分散型冲击波轮廓的存在性及稳定性，证明了其唯一性及任意幅值特性，并分析了其谱稳定性。
In this paper we study existence and stability of shock profiles for a 1-D compressible Euler system in the context of Quantum Hydrodynamic models. The dispersive term is originated by the quantum effects described through the Bohm potential; moreover we introduce a (linear) viscosity to analyze its interplay with the former while proving existence, monotonicity and stability of travelling waves connecting a Lax shock for the underlying Euler system. The existence of monotone profiles is proved for sufficiently small shocks; while the case of large shocks leads to the (global) existence for an…
Shock profiles for the non-cutoff Boltzmann equation with hard potentials
本研究针对长程分子相互作用的无截断硬势Boltzmann方程，构建了光滑的激波剖面解，证明其可被可压缩Navier-Stokes激波剖面很好逼近，采用能量估计和定量Chapman-Enskog近似完成证明。
This paper shows that in second-order hyperbolic systems of partial differential equations proposed in authors' earlier paper (J. Math. Phys. 59 (2018)) for modelling the relativistic dynamics of barotropic fluids in the presence of viscosity and heat conduction, shock waves of arbitrary strength have smooth, monotone dissipation profiles. The results and arguments extend classical considerations of Weyl (Comm. Pure Appl. Math. 2 (1949)) and Gilbarg (Amer. J. Math. 73 (1951)) to the relativistic setting.
Linear stability of discrete shock profiles for systems of conservation laws
本文证明守恒有限差分格式的谱稳定稳态离散激波剖面具有线性轨道稳定性，借助 Green 函数逐点渐近分析，将结论推广到含人工高阶黏性的广泛格式。
Combining pointwise Green's function bounds obtained in a companion paper [MZ.2] with earlier, spectral stability results obtained in [HuZ], we establish nonlinear orbital stability of small amplitude viscous shock profiles for the class of dissipative symmetric hyperbolic-parabolic systems identified by Kawashima [Kaw], notably including compressible Navier--Stokes equations and the equations of magnetohydrodynamics, obtaining sharp rates of decay in $L^p$ with respect to small $L^1\cap H^3$ perturbations, $2\le p\le \infty$. Our analysis follows the approach introduced in [MZ.1] to treat sta…
Extending previous work with Lattanzio and Mascia on the scalar (in fluid-dynamical variables) Hamer model for a radiative gas, we show nonlinear orbital asymptotic stability of small-amplitude shock profiles of general systems of coupled hyperbolic--eliptic equations of the type modeling a radiative gas, that is, systems of conservation laws coupled with an elliptic equation for the radiation flux, including in particular the standard Euler--Poisson model for a radiating gas. The method is based on the derivation of pointwise Green function bounds and description of the linearized solution op…
Asymptotic stability of composite waves of shock profile and rarefaction for the Navier-Stokes-Poisson system
本文研究了一维等温Navier-Stokes-Poisson系统中激波剖面与稀疏波复合波的渐近稳定性，通过带平移的a-收缩方法和调制相对泛函，证明了在初始数据充分接近Riemann数据时解的收敛性。
Spectral stability of shock profiles for hyperbolically regularized systems of conservation laws
本文证明在自然假设且激波振幅足够小时，双曲正则化守恒律系统的异宿行波激波剖面谱稳定，Evans 函数仅在原点有一个单零点。
Spectral stability of shock profiles for the Navier-Stokes-Poisson system
研究Navier-Stokes-Poisson系统中小振幅冲击轮廓的谱稳定性，通过扩展的Evans函数框架证明零特征值的简并性。
Existence and stability of shock profiles for a one-dimensional full compressible Navier--Stokes--Poisson equations
本文证明一维完全可压缩Navier--Stokes--Poisson方程小振幅激波剖面存在唯一且渐近稳定，通过中心流形定理与积分系统能量估计实现。
Motivated by the study of conditional stability of traveling waves, we give an elementary $H^2$ center stable manifold construction for quasilinear parabolic PDE, sidestepping apparently delicate regularity issues by the combination of a carefully chosen implicit fixed-point scheme and "damping-type" $H^s$ energy estimates of a type familiar from the study of hyperbolic--parabolic and relaxation systems. An important feature of these methods is that they generalize to situations such as the hyperbolic--parabolic or relaxation case for which parabolic-type smoothing estimates are unavailable. A…
Non-linear stability of shock profiles in dissipative hyperbolic-hyperbolic systems
本文首次证明谱稳定假设下二阶耗散双曲-双曲系统光滑激波剖面的非线性稳定性，适用于维数≥2的小振幅剖面，涵盖相对论气体模型和Jin-Xin松弛格式推广，创新点在于仿微分型非线性阻尼估计。
Using a simplified pointwise iteration scheme, we establish nonlinear phase-asymptotic orbital stability of large-amplitude Lax, undercompressive, overcompressive, and mixed under--overcompressive type shock profiles of strictly parabolic systems of conservation laws with respect to initial perturbations $|u_0(x)|\le E_0 (1+|x|)^{-3/2}$ in $C^{0+α}$, $E_0$ sufficiently small, under the necessary conditions of spectral and hyperbolic stability together with transversality of the connecting profile. This completes the program initiated by Zumbrun and Howard in \cite{ZH}, extending to the general…
By a combination of asymptotic ODE estimates and numerical Evans function calculations, we establish stability of viscous shock solutions of the isentropic compressible Navier--Stokes equations with $γ$-law pressure (i) in the limit as Mach number $M$ goes to infinity, for any $γ\ge 1$ (proved analytically), and (ii) for $M\ge 2,500$, $γ\in [1,2.5]$ (demonstrated numerically). This builds on and completes earlier studies by Matsumura--Nishihara and Barker--Humpherys--Rudd--Zumbrun establishing stability for low and intermediate Mach numbers, respectively, indicating unconditional stability, in…
Stability of viscous shock profile for convective porous-media flow with degenerate viscosity
本文研究退化粘性对流多孔介质方程的粘性激波大时间行为，通过利用扰动符号并分解积分区域，证明激波在小扰动下的非线性稳定性并得到 $L^\infty$ 衰减率。
Shock profiles for the Navier--Stokes--Fourier--Poisson system under the Boltzmann relation
本文证明 Boltzmann 关系下 Navier--Stokes--Fourier--Poisson 系统存在唯一且轨道稳定的 Lax 型小振幅粘性激波剖面，采用中心流形约简和带平移的 $a$-收缩方法。
It was shown by Schecter (2004, J. Differential Equations, 205, 185-210), using the methods of Geometric Singular Perturbation Theory, that the Dafermos regularization $u_t+f(u)_x= εtu_{xx}$ for the Keyfitz-Kranzer system admits an unbounded family of solutions. Inspired by that work, in this paper we provide a more intuitive approach which leads to a stronger result. In addition to the existence of viscous profiles, we also prove the weak convergence and show that the maximum of the solution is of order $ε^{-2}$. This asymptotic behavior is distinct from that obtained in the author's recent w…
We study the 2D isentropic Euler equations with the ideal gas law. We exhibit a set of smooth initial data that give rise to shock formation at a single point near the planar symmetry. These solutions are associated with non-zero vorticity at the shock and have uniform-in-time 1/3-Hölder bound. Moreover, these point shocks are of self-similar type and share the same profile, which is a solution to the 2D self-similar Burgers equation. Our proof, following the 3D shock formation result of Buckmaster, Shkoller and Vicol, is based on the stable 2D self-similar Burgers profile and the modulation m…
Shock profiles for the cutoff Boltzmann equation of a binary gas mixture
该研究证明了-3<γ≤1范围内带角截断势的一维二元气体混合物玻尔兹曼方程小振幅行波激波剖面的存在，扩展了经典构造，采用多种方法克服软势相关问题，得到激波剖面的渐近行为。
Convergence to shock profiles for Burgers equation with singular fast-diffusion and boundary effect
本文研究了Burgers方程在奇异快速扩散和边界效应下的激波剖面收敛性，通过新的加权能量估计技术克服奇异性与边界层影响。
Shock solutions for the one-dimensional information geometric regularization of compressible flow
研究可压缩流一维信息几何正则化中激波解的相关问题，通过行波假设将方程简化，证明了跨声速压缩IGR激波剖面的存在性、唯一性和平移模下正则性，分析了正则化消失极限，表明激波宽度缩放规律及IGR剖面收敛情况。
We examine the existence of shock profiles for a hyperbolic-elliptic system arising in radiation hydrodynamics. The algebraic-differential system for the wave profile is reduced to a standard two-dimensional form that is analyzed in details showing the existence of heteroclinic connection between the two singular points of the system for any distance between the corresponding asymptotic states of the original model. Depending on the location of these asymptotic states, the profile can be either continuous or possesses at most one point of discontinuity. Moreover, a sharp threshold relative to …
We are concerned with the long-time behavior of the growth-fragmentation equation. We prove fine estimates on the principal eigenfunctions of the growth-fragmentation operator, giving their first-order behavior close to 0 and $+\infty$. Using these estimates we prove a spectral gap result by following the technique in [Caceres, Canizo, Mischler 2011, JMPA], which implies that solutions decay to the equilibrium exponentially fast. The growth and fragmentation coefficients we consider are quite general, essentially only assumed to behave asymptotically like power laws.
Asymptotic stability of shock profiles and rarefaction waves to the Navier-Stokes-Poisson system under space-periodic perturbations
本文证明了一维可压缩 Navier-Stokes-Poisson 系统在空间周期扰动下，粘性激波剖面和稀疏波分别具有渐近稳定性，其中激波情形需满足零质量条件，证明依赖二次拟设和能量方法。
Nonlinear orbital stability of stationary shock profiles for the Lax-Wendroff scheme
本文研究了双曲守恒律Lax-Wendroff格式驻定激波剖面的稳定性，通过详细分析线性化格式Green函数的高度振荡行为，利用激活函数刻画特征值1的贡献，在多项式加权空间中建立衰减估计，最终在缺乏一致$\ell$ 1估计的色散情形下证明了谱稳定驻定激波剖面的非线性轨道稳定性。
Existence and stability of nonmonotone hydraulic shocks for the Saint Venant equations of inclined thin-film flow
本研究拓展已有工作，严格分类倾斜薄膜流Saint Venant方程非单调水力激波的存在性与对流稳定性，并通过数值实验验证不稳定区域内黎曼数据的两种渐近动力学模式。
Shock Structure for Hyperbolic Balance Laws with Evolutionary Constraints
研究带演化约束的双曲平衡律行波激波剖面，证明Boillat-Ruggeri速度界限在齐次及满行秩约束下不变，秩亏时修正受Lyapunov方程控制，并给出退化模型与数值验证。
Orbital stability of undercompressive viscous shock waves under $L^1\cap H^4$ perturbation
本文利用作者引入的新纵向估计，简化并扩展了 Mascia 和 Zumbrun 关于粘性激波的基本 $L^1\cap H^3$ 稳定性结果，特别是将 Lax 激波的结果扩展到了欠压缩情况。
A compressible viscous-dispersive Euler system in one space dimension in the context of quantum hydrodynamics is considered. The dispersive term is due to quantum effects described through the Bohm potential and the viscosity term is of linear type. It is shown that small-amplitude viscous-dispersive shock profiles for the system under consideration are spectrally stable, proving in this fashion a previous numerical observation by Lattanzio et al. (Phys. D 402, 2020, p. 132222; Appl. Math. Comput. 385, 2020, p. 125450). The proof is based on spectral energy estimates which profit from the mono…
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