Weak Asymptotics of Shock Wave Formation Process

有限喷管中稳态一维激波解的唯一性：基于物理参数渐近分析的研究

On Uniqueness of Steady 1-D Shock Solutions in a Finite Nozzle via Asymptotic Analysis for Physical Parameters

本文通过物理参数渐近分析，研究有限喷管中无粘欧拉系统稳态一维激波解的唯一性，发现不同物理假设导致极限激波位置不同，难以确立唯一性准则。

$L^p$ asymptotic behavior of perturbed viscous shock profiles

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…

松弛可压缩 Navier-Stokes 方程的粘性激波与稀疏波复合波的渐近稳定性

Asymptotic stability of composite waves of viscous shock and rarefaction for relaxed compressible Navier-Stokes equations

本文研究了一维松弛可压缩 Navier-Stokes 方程的时间渐近稳定性，证明了在小波强度和初始扰动下粘性激波与稀疏波复合波的渐近非线性稳定性，并利用相对熵等方法证明了松弛参数趋于零时解的全局收敛性。

两组粘性激波复合波的渐近稳定性研究

Asymptotic stability of composite waves of two viscous shocks for relaxed compressible Navier-Stokes equations

本文研究了松弛可压缩纳维-斯托克斯方程中两组粘性激波复合波的渐近稳定性，通过相对熵和能量估计方法证明其非线性稳定性，并展示了松弛参数趋于零时解的全局收敛性。

Exponential asymptotic stability of Riemann shocks of hyperbolic systems of balance laws

For strictly entropic Riemann shock solutions of strictly hyperbolic systems of balance laws, we prove that exponential spectral stability implies large-time asymptotic orbital stability. As a preparation, we also prove similar results for constant solutions of initial value and initial boundary value problems, that seem to be new in this generality. Main key technical ingredients include the design of a nonlinear change of variables providing a hypocoercive Kawashima-type structure with dissipative boundary conditions in the high-frequency regime and the explicit identification of most singul…

粘性激波与粘性接触波复合波的时间渐进行为研究：Navier-Stokes-Fourier方程

Time-asymptotic stability of composite wave of viscous shocks and viscous contact wave for Navier-Stokes-Fourier equations

本文研究了Navier-Stokes-Fourier方程中由两个粘性激波和一个粘性接触间断构成的复合波的时间渐进行为，证明了在小扰动下解的全局存在性和收敛性。

三维松弛可压缩Navier-Stokes方程的平面粘性冲击波的渐近稳定性

Asymptotic stability of planar viscous shock wave to three-dimensional relaxed compressible Navier-Stokes equations

本文针对三维松弛可压缩Navier-Stokes方程（修正Maxwell型模型替代牛顿本构关系），在小冲击波强度和初始扰动下，通过相对熵方法等证明平移平面粘性冲击波的非线性时间渐近稳定性，并验证松弛系统解向经典系统解的收敛性。

退化Oleinik激波与稀疏波的时渐稳定性研究：非凸守恒律与Cattaneo定律的二维超几何方程

Time-asymptotic stability of composite waves of degenerate Oleinik shock and rarefaction for non-convex conservation laws with Cattaneo's law

本文研究非凸守恒律与Cattaneo定律的二维超几何方程组，证明退化Oleinik激波与稀疏波复合波的时渐稳定性。

Korteweg型一维可压缩流体外流问题中粘性激波的时间渐近稳定性

Time-asymptotic stability of viscous shocks for the outflow problem of one-dimensional compressible fluids of Korteweg type

研究等熵Navier-Stokes-Korteweg方程外流问题中粘性-色散激波的时间渐近稳定性，通过a-收缩方法证明在激波振幅和初始扰动足够小时解收敛到粘性-色散激波。

正压Navier-Stokes方程外流问题的粘性激波渐近行为

Asymptotic behavior toward viscous shock for the outflow problem of barotropic Navier-Stokes equations

该研究针对半直线上正压Navier-Stokes方程外流问题，采用欧拉坐标下带平移的a-压缩方法，首次证明了粘性激波的时间渐近稳定性。

三维Navier-Stokes系统外流问题的粘性激波的渐近稳定性

Asymptotic stability of viscous shock for the outflow problem of 3D Navier-Stokes system

本文证明三维可压缩Navier-Stokes方程半空间外流问题中弱粘性激波在小扰动下全局渐近稳定，结合a-收缩法与高阶能量估计，首次给出多维外流问题的激波稳定性结果。

Asymptotic stability of shock profiles and rarefaction waves under periodic perturbations for 1-d convex scalar viscous conservation laws

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…

Instantaneous shock location and one-dimensional nonlinear stability of viscous shock wave

We illustrate in a simple setting the instantaneous shock tracking approach to stability of viscous conservation laws introduced by Howard, Mascia, and Zumbrun. This involves a choice of the definition of instanteous location of a viscous shock-- we show that this choice is time-asymptotically equivalent both to the natural choice of least-squares fit pointed out by Goodman and to a simple phase condition used by Guès, Métivier, Williams, and Zumbrun in other contexts. More generally, we show that it is asymptotically equivalent to any location defined by a localized projection

带空间振荡的平面Navier-Stokes激波的时间渐近稳定性

Time-asymptotic stability of planar Navier-Stokes shocks with spatial oscillations

该研究针对三维可压缩等熵Navier-Stokes方程，证明带空间振荡的平面粘性激波对零质量小扰动的时间渐近稳定性，通过反导数技术与不同方向Poincaré不等式结合消除激波压缩的不利影响。

Uniform asymptotic stability for convection-reaction-diffusion equations in the inviscid limit towards Riemann shocks

The present contribution proves the asymptotic orbital stability of viscous regularizations of stable Riemann shocks of scalar balance laws, uniformly with respect to the viscosity/diffusion parameter $ε$. The uniformity is understood in the sense that all constants involved in the stability statements are uniform and that the corresponding multiscale $ε$-dependent topology reduces to the classical $W^{1,\infty}$-topology when restricted to functions supported away from the shock location. Main difficulties include that uniformity precludes any use of parabolic regularization to close regulari…

Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Non-classical shocks to model oil-trapping

We consider a one-dimensional problem modeling two-phase flow in heterogeneous porous media made of two homogeneous subdomains, with discontinuous capillarity at the interface between them. We suppose that the capillary forces vanish inside the domains, but not on the interface. Under the assumption that the gravity forces and the capillary forces are oriented in opposite directions, we show that the limit, for vanishing diffusion, is not in general the optimal entropy solution of the hyperbolic scalar conservation law as in the first paper of the series \cite{NPCX}. A non-classical shock can …

一维分形Burgers方程的渐近自相似激波形成

Asymptotically self-similar shock formation for 1d fractal Burgers equation

本文针对$0<α<\frac{1}{3}$的一维分形Burgers方程，构造了从光滑一般初始数据出发在有限时间内形成首次激波的唯一解，该奇点为稳定自相似激波剖面的渐近自相似$H^6$扰动，并精确计算了其时空位置与Hölder正则性，是首个显式构造并精确刻画此类奇异解的例子。

Asymptotic stabilization of stationnary shock waves using a boundary feedback law

In this paper we consider scalar conservation laws with a convex flux. Given a stationnary shock, we provide a feedback law acting at one boundary point such that this solution is now asymptotically stable in L 1-norm in the class of entropy solution.

正压 Navier-Stokes 方程不透墙问题与入流问题向粘性激波的渐近行为

Asymptotic behavior toward viscous shock for impermeable wall and inflow problem of barotropic Navier-Stokes equations

本文研究半直线上可压缩正压 Navier-Stokes 方程不透墙与入流问题向传出粘性激波的渐近收敛，利用带平移的 $a$-收缩方法，在小 $H^1$ 初始扰动下证明一致收敛并放宽了既有变量与 $\gamma$ 假设。

Propagation of a Three-dimensional Weak Shock Front Using Kinematical Conservation Laws

In this paper we present a mathematical theory and a numerical method to study the propagation of a three-dimensional (3-D) weak shock front into a polytropic gas in a uniform state and at rest, though the method can be extended to shocks moving into nonuniform flows. The theory is based on the use of 3-D kinematical conservation laws (KCL), which govern the evolution of a surface in general and a shock front in particular. The 3-D KCL, derived purely on geometrical considerations, form an under-determined system of conservation laws. In the present paper the 3-D KCL system is closed by using …

向边界层、稀疏波和激波的叠加收敛的流入问题的Navier-Stokes方程的1D情况

Convergence to Superposition of Boundary Layer, Rarefaction and Shock for the Inflow Problem of the 1D Navier--Stokes Equations

本文研究了亚音速边界值下1D准静Navier-Stokes方程流入问题的渐进行为稳定性，证明了边界层、稀疏波和激波叠加解的渐近收敛性。

Pointwise Asymptotic Behavior of Perturbed Viscous Shock Profiles

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…

Ion-acoustic shock in a collisional plasma

The paper is concerned with the propagation of ion-acoustic shock waves in a collision dominated plasma. We firstly establish the existence and uniqueness of a small-amplitude smooth travelling wave, then justify its approximation to the shock profile of the KdV-Burgers equations in a suitable asymptotic regime where dissipation in terms of viscosity coefficient is much stronger than dispersion by the Debye length, and prove in the end the large time asymptotic stability of travelling waves under suitably small smooth perturbations.

Soliton asymptotics for KdV shock waves via classical inverse scattering

We show how the inverse scattering transform can be used as a convenient tool to derive the long-time asymptotics of shock waves for the Korteweg-de Vries (KdV) equation in the soliton region. In particular, we improve the results previously obtained via the nonlinear steepest decent approach both with respect to the decay of the initial conditions as well as the region where they are valid.

Navier-Stokes-Poisson系统激波剖面与稀疏波复合波的渐近稳定性

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。

Soliton asymptotics for the KdV shock problem of low regularity

We revisit the asymptotic analysis of the KdV shock problem in the soliton region. Our approach is based on the analysis of the associated Riemann-Hilbert problem and we extend the domain of validity of the asymptotic formulas while at the same time requiring less decay and smoothness for the initial data.

Asymptotics of KdV shock waves via the Riemann-Hilbert approach

This paper discusses some general aspects and techniques associated with the long-time asymptotics of steplike solutions of the Korteweg--de Vries (KdV) equation via vector Riemann--Hilbert problems. We also elaborate on an ill-posedness of the matrix Riemann--Hilbert problem for the KdV case in the class of matrices with square integrable singularities. Furthermore, we refine the asymptotics for the shock wave in the Whitham zone derived previously and rigorously justify it for a more general class of initial data. In particular, we clarify the influence of resonances and of the discrete spec…

Burgers-Hilbert 方程的激波相互作用

Shock interactions for the Burgers-Hilbert Equation

研究 Burgers-Hilbert 方程中两激波相互作用点附近解的渐近结构，通过 $H^2$ 正则部分与 |x|ln|x| 型校正项刻画解，并构造一般初值的单激波分段光滑解。

Spectral stability of weak dispersive shock profiles for quantum hydrodynamics with nonlinear viscosity

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…

Delta shock wave formation in the case of triangular hyperbolic system of conservation laws

We describe $δ$ shock wave arising from continuous initial data in the case of triangular conservation law system arising from "generalized pressureless gas dynamics model". We use the weak asymptotic method.

非凸粘弹性系统中由非经典Liu激波与稀疏波组成的复合波的稳定性

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

本文研究非凸粘弹性系统中由非经典Liu激波与稀疏波组成的复合波的时间渐近稳定性，采用a-收缩方法完成证明，为该类复合波的稳定性提供了首个相关结果。

Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary

A free boundary problem for the one-dimensional compressible Navier-Stokes equations is investigated. The asymptotic stability of the viscous shock wave is established under some smallness conditions. The proof is given by an elementary energy estimate.

Relative entropy and contraction for extremal shocks of Conservation Laws up to a shift

We consider systems of conservation laws endowed with a convex entropy. We show the contraction, up to a translation, to extremal entropic shocks, for a pseudo-distance based on the notion of relative entropy. The contraction holds for bounded entropic weak solutions having an additional trace property. The pseudo-distance depends only on the fixed extremal entropic shocks in play. In particular, it can be chosen uniformly for any entropic weak solutions which are compared to a fixed shock. The boundedness of the solutions controls the strength of the shift needed to get the contraction. Howev…

时间周期边界对标量守恒律激波的影响

Nonlinear effects of incoming time-periodic boundary conditions on shock waves for scalar conservation laws

针对时间周期边界条件下半直线上标量守恒律黎曼解长时间行为不明的问题，本文严格证明无粘与粘性情形下解的渐近状态由平移背景激波和边界诱导周期解共同主导，效应表现为影响激波动力学的传播边界波。

三维正压Navier-Stokes方程的平面粘性激波与稀疏波复合波的稳定性

Stability of composite Wave of Planar Viscous Shock and Rarefaction for 3D Barotropic Navier-Stokes Equations

针对三维正压Navier-Stokes方程，证明了平面粘性激波与稀疏波复合波在一般小扰动下的非线性时间渐近稳定性，无需零质量条件，通过a-压缩方法得到全局强解及渐近收敛结果。

Stability of noncharacteristic boundary layers in the standing shock limit

We investigate one- and multi-dimensional stability of noncharacteristic boundary layers in the limit approaching a standing planar shock wave $\bar U(x_1)$, $x_1>0$, obtaining necessary conditions of (i) weak stability of the limiting shock, (ii) weak stability of the constant layer $u\equiv U_-:=\lim_{z\to -\infty} \bar U(z)$, and (iii) nonnegativity of a modified Lopatinski determinant similar to that of the inviscid shock case. For Lax 1-shocks, we obtain equally simple sufficient conditions; for $p$-shocks, $p>1$, the situation appears to be more complicated. Using these results, we deter…

Spectral stability of weak relaxation shock profiles

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.

Non-existence of strong regular reflections in self-similar potential flow

We consider shock reflection which has a well-known local non-uniqueness: the reflected shock can be either of two choices, called weak and strong. We consider cases where existence of a global solution with weak reflected shock has been proven, for compressible potential flow. If there was a global strong-shock solution as well, then potential flow would be ill-posed. However, we prove non-existence of strong-shock analogues in a natural class of candidates.

Convergence rate toward shock wave under periodic perturbation for generalized Korteweg-de Vries-Burgers equation

In this paper, a viscous shock wave under space-periodic perturbation of generalized Korteweg-de Vries-Burgers equation is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover the exponential time decay rate toward the viscous shock wave is also obtained for some certain perturbations.

Navier-Stokes-Poisson 系统在空间周期扰动下激波剖面和稀疏波的渐近稳定性

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。

Landau方程的动能冲击轮廓

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。

可压缩磁流体动力学流动的大振幅粘性激波的非线性稳定性

Nonlinear stability of large amplitude viscous shock wave for compressible magnetohydrodynamics flows

本文针对可压缩磁流体动力学系统柯西问题，在无需假设粘性激波振幅与速度的情况下，证明了任意大振幅粘性激波在小初始扰动下的时间渐近稳定性，且适用于一般体积粘度，为该领域的稳定性研究提供了关键进展。

弱解中超音速激波中断经典发展的拟线性波动方程

A quasilinear wave with a supersonic shock in a weak solution interrupting the classical development

本文研究一维拟线性波动方程柯西问题，首次构造了激波形成解的可证明唯一最大全局双曲发展，并证明同一初值下存在全局弱熵解，且经典解与弱解在激波后出现分歧。

Asymptotic stability of viscous shocks in the modular Burgers equation

Dynamics of viscous shocks is considered in the modular Burgers equation, where the time evolution becomes complicated due to singularities produced by the modular nonlinearity. We prove that the viscous shocks are asymptotically stable under odd and general perturbations. For the odd perturbations, the proof relies on the reduction of the modular Burgers equation to a linear diffusion equation on a half-line. For the general perturbations, the proof is developed by converting the time-evolution problem to a system of linear equations coupled with a nonlinear equation for the interface positio…

Weak perturbations of shock waves

The Cauchy problem is considered for the perturbed strictly hyperbolic 2x2 system of quasilinear equations. The unperturbed problem has a persistent solution with two discontinuity lines (shock waves). Both an asymptotics of shock waves position in the plane (x,t) and an asymptotics of the perturbed problem solution are discussed.

标量粘性守恒律大激波的$L^2$收缩与渐近稳定性

$L^2$-contraction and asymptotic stability of large shock for scalar viscous conservation laws

该研究针对带多项式通量的标量粘性守恒律，采用带时变位移与权函数的$a$收缩方法，在$H^1$框架下证明了严格凸通量下任意大粘性激波的$L^2$收缩与渐近稳定性，并给出$L^1$初始扰动下的$L^2$衰减速率。

Stability of a superposition of shock waves with contact discontinuities for the Jin-Xin relaxation system

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.

Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non BV perturbations

We develop a theory based on relative entropy to show the uniqueness and L^2 stability (up to a translation) of extremal entropic Rankine-Hugoniot discontinuities for systems of conservation laws (typically 1-shocks, n-shocks, 1-contact discontinuities and n-contact discontinuities of large amplitude) among bounded entropic weak solutions having an additional trace property. The existence of a convex entropy is needed. No BV estimate is needed on the weak solutions considered. The theory holds without smallness condition. The assumptions are quite general. For instance, strict hyperbolicity is…

The inviscid limit of viscous Burgers at nondegenerate shock formation

We study the vanishing viscosity limit of the one-dimensional Burgers equation near nondegenerate shock formation. We develop a matched asymptotic expansion that describes small-viscosity solutions to arbitrary order up to the moment the first shock forms. The inner part of this expansion has a novel structure based on a fractional spacetime Taylor series for the inviscid solution. We obtain sharp vanishing viscosity rates in a variety of norms, including $L^\infty$. Comparable prior results break down in the vicinity of shock formation. We partially fill this gap.

粘性近似守恒律的局部渐进行为

Local Asymptotic Patterns for Viscous Approximations of Conservation Laws

研究粘性消失近似在激波附近的行为，通过局部坐标缩放获得极限解，揭示近似方程的永恒解特性。

Weak periodic solutions and numerical case studies of the Fornberg-Whitham equation

Spatially periodic solutions of the Fornberg-Whitham equation are studied to illustrate the mechanism of wave breaking and the formation of shocks for a large class of initial data. We show that these solutions can be considered to be weak solutions satisfying the entropy condition. By numerical experiments, we show that the breaking waves become shock-wave type in the time evolution.

Periodic perturbations of a composite wave of two viscous shocks for 1-D full compressible Navier-Stokes equations

This paper is concerned with the asymptotic stability of a composite wave of two viscous shocks under spatially periodic perturbations for the 1-D full compressible Navier-Stokes equations. It is proved that as time increases, the solution approaches the background composite wave with a shift for each shock, where the shifts can be uniquely determined if both the periodic perturbations and strengths of two shocks are small. The key of the proof is to construct a suitable ansatz such that the anti-derivative method works.

一般$n\ imes n$多维粘性系统复合弱平面波的时间渐近稳定性

Time-asymptotic stability of composite weak planar waves for a general $n\times n$ multi-D viscous system

该研究证明一般$n\ imes n$多维粘性系统中弱平面粘性激波与稀疏波或激波叠加的时间渐近稳定性，推广了Kang-Vasseur-Wang的工作，采用带平移的$a$-压缩方法并发展高阶项分类控制技术。

非凸守恒律中退化Oleinik激波与稀疏波复合波的时间渐近稳定性

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

本文首次证明了一维三次非凸粘性守恒律中退化Oleinik激波与稀疏波复合波的时间渐近稳定性，通过发展带权函数和依赖时间位移的新型a-收缩方法，克服了反导数方法与L²能量方法不兼容及波相互作用微妙的困难。