Denis Serre
In several space dimensions, scalar shock waves between two constant states u $\pm$ are not necessarily planar. We describe them in detail. Then we prove their asymptotic stability, assuming that they are uniformly non-characteristic. Our result is conditional for a general flux, while unconditional for the multi-D Burgers equation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We show the vanishing viscosity limit to entropy shocks for the fractal Burgers equation in one space dimension. More precisely, we quantify the rate of convergence of the inviscid limit in $L^2$ for large initial perturbations around the entropy shock on any bounded time interval. This is the first result on the inviscid limit to entropy shock for the fractal Burgers equation with the quantified convergence, for large initial perturbations.
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.
We consider inviscid limits to shocks for viscous scalar conservation laws in one space dimension, with strict convex fluxes. We show that we can obtain sharp estimates in $L^2$, for a class of large perturbations and for any bounded time interval. Those perturbations can be chosen big enough to destroy the viscous layer. This shows that the fast convergence to the shock does not depend on the fine structure of the viscous layers. This is the first application of the relative entropy method developed in [22], [23] to the study of an inviscid limit to a shock.
Extending our earlier work on Lax-type shocks of systems of conservation laws, we establish existence and stability of curved multidimensional shock fronts in the vanishing viscosity limit for general Lax- or undercompressive-type shock waves of nonconservative hyperbolic systems with parabolic regularization. The hyperbolic equations may be of variable multiplicity and the parabolic regularization may be of ``real'', or partially parabolic, type. We prove an existence result for inviscid nonconservative shocks that extends to multidimensional shocks a one-dimensional result of X. Lin proved b…
The uniqueness of entropy solution for the compressible Euler equations is a fundamental and challenging problem. In this paper, the uniqueness of a composite wave of shock and rarefaction of 1-d compressible Euler equations is proved in the inviscid limit of compressible Navier-Stokes equations. Moreover, the relative entropy around the original Riemann solution consisting of shock and rarefaction under the large perturbation is shown to be uniformly bounded by the framework developed in \cite{Kang-Vasseur-2021-Invent}. The proof contains two new ingredients: 1) a cut-off technique and the ex…
Shock formation in 1D conservation laws I: Inviscid structure
本文研究一维双曲守恒律中激波形成的稳定性与结构，证明其在简单波附近稳定，刻画经典发展边界，并用分数次齐次函数展开描述非退化激波形成的精确性质。
The Inviscid Limit of Viscous Burgers through Shock Formation and Development
该研究扩展了Chaturvedi与Graham的分析，证明了粘性伯格斯方程在激波形成后的无粘极限逐点误差界，结合前人结果得到了无粘极限从初始时间到年轻激波发展的定量描述。
We prove the uniqueness and stability of entropy shocks to the isentropic Euler systems among all vanishing viscosity limits of solutions to associated Navier-Stokes systems. To take into account the vanishing viscosity limit, we show a contraction property for any large perturbations of viscous shocks to the Navier-Stokes system. The contraction estimate does not depend on the strength of the viscosity. This provides a good control on the inviscid limit process. We prove that, for any initial value, there exists a vanishing viscosity limit to solutions of the Navier-Stokes system. The converg…
Combining work of Serre and Zumbrun, Benzoni-Gavage, Serre, and Zumbrun, and Texier and Zumbrun, we propose as a mechanism for the onset of cellular instability of viscous shock and detonation waves in a finite-cross-section duct the violation of the refined planar stability condition of Zumbrun--Serre, a viscous correction of the inviscid planar stability condition of Majda. More precisely, we show for a model problem involving flow in a rectangular duct with artificial periodic boundary conditions that transition to multidimensional instability through violation of the refined stability cond…
We consider by a combination of analytical and numerical techniques some basic questions regarding the relations between inviscid and viscous stability and existence of a convex entropy. Specifically, for a system possessing a convex entropy, in particular for the equations of gas dynamics with a convex equation of state, we ask: (i) can inviscid instability occur? (ii) can there occur viscous instability not detected by inviscid theory? (iii) can there occur the ---necessarily viscous--- effect of Hopf bifurcation, or "galloping instability"? and, perhaps most important from a practical point…
Relative Entropy and the Inviscid Limit for Planar Shocks of the Multi-D Burgers' Equation
本文将近期研究的粘性消失极限分析框架应用于多维粘性Burgers方程，证明大粘性激波在特定扰动下的L²稳定性，并研究其向无粘激波的收敛速率。
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…
We study by a combination of analytical and numerical Evans function techniques multi-D viscous and inviscid stability and associated transverse bifurcation of planar slow Lax MHD shocks in a channel with periodic boundary conditions. Notably, this includes the first multi-D numerical Evans function study for viscous MHD. Our results suggest that, rather than a planar shock, a nonplanar traveling wave with the same normal velocity is the typical mode of propagation in the slow Lax mode. Moreover, viscous and inviscid stability transitions appear to agree, answering (for this particular model a…
In this paper, we prove the leading term of time-asymptotics of the moving vacuum boundary for compressible inviscid flows with damping to be that for Barenblatt self-similar solutions to the corresponding porous media equations obtained by simplifying momentum equations via Darcy's law plus the possible shift due to the movement of the center of mass, in the one-dimensional and three-dimensional spherically symmetric motions, respectively. This gives a complete description of the large time asymptotic behavior of solutions to the corresponding vacuum free boundary problems. The results obtain…
Dissipative structures in compressible inviscid fluids
研究可压缩欧拉系统弱解的局部能量平衡，揭示耗散结构的精细性质，比较压缩与不可压缩模型，确认总能量在codimension一奇异结构上的累积是区分两者的特征。
We establish the existence of an axisymmetric weak solution to the steady Euler system with a transonic shock, nonzero vorticity, and nonzero swirl in a three-dimensional cylinder. When prescribing the supersonic solution in the upstream region by axisymmetric functions with variable entropy and variable angular momentum density(=swirl), we construct such a solution by using a Helmholtz decomposition of the velocity field and the method of iteration. An iteration scheme is developed using a delicate decomposition of the Rankine-Hugoniot conditions on the transonic shock via Helmholtz decomposi…
Shock formation in 1D conservation laws II: Vanishing viscosity
本文研究一维双曲守恒律中弱粘性对激波形成的影响，通过匹配渐近展开确定无粘激波形成前强范数的尖锐收敛速率，并识别奇点附近的普适粘性行为，适用于可压缩纳维-斯托克斯方程。
The steady inviscid compressible self-similar flows and the stability analysis
该研究针对球坐标系下定常无粘可压缩自相似流展开，发现多种新型流态结构，揭示激波角与径向速度的单调性新关系，还证明了带非零涡量的光滑跨声速自相似流的存在唯一性及结构稳定性。
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。
Asymptotic stability of the symmetric flow via inviscid damping and enhanced dissipation
该研究针对有限通道无滑移边界下的对称流，建立线性化Navier-Stokes系统的无粘阻尼与增强耗散估计，证明高雷诺数下对称流的渐近稳定性，并给出初始扰动的阈值条件。
The linear stability of shock waves for the nonlinear Schrödinger-Inviscid Burgers system
针对非线性薛定谔方程与无粘Burgers方程耦合系统，通过线性化方法证明了当参考解为Burgers方程满足熵条件的激波时系统的线性稳定性，并基于能量估计和间断系数双曲型方程性质给出证明，数值实验支持理论结果。
We study an Eulerian droplet model which can be seen as the pressureless gas system with a source term, a subsystem of this model and the inviscid Burgers equation with source term. The condition for loss of regularity of a solution to Burgers equation with source term is established. The same condition applies to the Eulerian droplet model and its subsystem. The Riemann problem for the Eulerian droplet model is constructively solved by going through the solution of the Riemann problems for the inviscid Burgers equation with a source term and the subsystem, respectively. Under suitable general…
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。
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.
Stability of Riemann Shocks for isothermal Euler by Inviscid limits of global-in-time large Navier-Stokes flows
该研究针对一维等温气体动力学系统，建立了无小性条件光滑初值下Navier-Stokes系统强解全局存在性，证明扰动强解的收缩性与一致估计，最终得到Euler系统Riemann激波在消失粘性极限下的稳定性。
We present a streamlined account of recent developments in the stability theory for planar viscous shock waves, with an emphasis on applications to physical models with ``real,'' or partial viscosity. The main result is the establishment of necessary, or ``weak'', and sufficient, or ``strong'', conditions for nonlinear stability analogous to those established by Majda [Ma.1--3] in the inviscid case but (generically) separated by a codimension-one set in parameter space rather than an open set as in the inviscid case. The importance of codimension one is that transition between nonlinear stabil…
Viscous Destabilization for Large Shocks of Conservation Laws
本文研究带位移的a-收缩性质在粘性守恒律系统中对激波振幅的一致性，证明大激波情形下该性质可能失效，揭示了“粘性失稳”现象，并说明a-收缩强于经典非线性稳定性。
Ultrasonic propagation through media with thermal and molecular relaxation can be modeled by third-order in time nonlinear wave-like equations with memory. This paper investigates the asymptotic behavior of a Cauchy problem for such a model, the nonlocal Jordan--Moore--Gibson--Thompson equation, in the so-called critical case, which corresponds to propagation in inviscid fluids. The memory has an exponentially fading character and type I, meaning that involves only the acoustic velocity potential. A major challenge in the global analysis is that the linearized equation's decay estimates are of…
We describe recent analytical and numerical results on stability and behavior of viscous and inviscid detonation waves obtained by dynamical systems/Evans function techniques like those used to study shock and reaction diffusion waves. In the first part, we give a broad description of viscous and inviscid results for 1D perturbations; in the second, we focus on inviscid high-frequency stability in multi-D and associated questions in turning point theory/WKB expansion.
Similarity Solutions of Shock Formation for First-order Strictly Hyperbolic Systems
研究揭示了广义一阶严格双曲方程中激波形成具有自相似性和普遍性，推导出自相似通用解的解析公式。
We consider the Riemann problem composed of two shocks for the 1D Euler system. We show that the Riemann solution with two shocks is stable and unique in the class of weak inviscid limits of solutions to the Navier-Stokes equations with initial data with bounded energy. This work extends to the case of two shocks a previous result of the authors in the case of a single shock. It is based on the method of weighted relative entropy with shifts known as $a$-contraction theory. A major difficulty due to the method is that very little control is available on the shifts. A modification of the constr…
Motivated by radiation hydrodynamics, we analyse a 2x2 system consisting of a one-dimensional viscous conservation law with strictly convex flux -- the viscous Burgers' equation being a paradigmatic example -- coupled with an elliptic equation, named viscous Hamer-type system. In the regime of small viscosity and for large shocks, namely when the profile of the corresponding underlying inviscid model undergoes a discontinuity -- usually called sub-shock -- it is proved the existence of a smooth propagating front, regularising the jump of the corresponding inviscid equation. The proof is based …
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.
A semi-explicit formula of solution to the boundary layer system for thermal layer derived from the compressible Navier-Stokes equations with the non-slip boundary condition when the viscosity coefficients vanish is given, in particular in three space dimension. In contrast to the inviscid Prandtl system studied by [7] in two space dimension, the main difficulty comes from the coupling of the velocity field and the temperature field through a degenerate parabolic equation. The convergence of these boundary layer equations to the inviscid Prandtl system is justified when the initial temperature…
Well-posedness, magnetic helicity conservation, inviscid limit and asymptotic stability for the generalized Navier-Stokes-Maxwell equations with the Hall effect
本文研究带霍尔效应的广义Navier-Stokes-Maxwell方程的适定性、磁螺旋度守恒、无粘极限与渐近稳定性，证明了粘性小初值全局适定、无粘大初值局部适定等结果，明确了其与霍尔磁流体系统的渐近联系及磁螺旋度守恒的条件差异。
We initiate the study of stability of solutions of the 2D inviscid incompressible porous medium equation (IPM). We begin by classifying all stationary solutions of the inviscid IPM under mild conditions. We then prove some linear stability results. We then study solutions of the IPM equation which are sufficiently regular perturbations of linearly stable steady states. We prove that sufficiently regular perturbations which are also small must be globally regular and strongly converge to a steady state. The mechanism behind the stability is \emph{stratification} as opposed to previous stability…
We study the existence of different vortex-wave systems for inviscid gSQG flow, where the total circulation are produced by point vortices and vortices with compact support. To overcome several difficulties caused by the singular formulation and infinite kinetic energy, we introduce a modified reduction method. Several asymptotic properties of the system are also given.
Weak-BV Stability and Inviscid Limits for Barotropic Flows with Outflow Boundary Conditions
本文证明一维可压缩正压欧拉方程的小BV解可作为外流边界问题的正压纳维-斯托克斯系统的无粘极限，并采用物理有效的边界条件，稳定性结果涵盖一般波模式与反射激波等复杂结构。
Refined wave breaking for the generalized Fornberg-Whitham equation
本文研究了广义Fornberg-Whitham方程的波破碎现象，通过构造破裂解，揭示了有限时间破裂的机制，并展示了其在自相似变量下的收敛性及正则性性质。
We prove nonlinear asymptotic stability of a large class of monotonic shear flows among solutions of the 2D Euler equations in the channel $\mathbb{T}\times[0,1]$. More precisely, we consider shear flows $(b(y),0)$ given by a function $b$ which is Gevrey smooth, strictly increasing, and linear outside a compact subset of the interval $(0,1)$ (to avoid boundary contributions which are incompatible with inviscid damping). We also assume that the associated linearized operator satisfies a suitable spectral condition, which is needed to prove linear inviscid damping. Under these assumptions, we sh…
This note studies classical magnetohydrodynamic shock waves in an inviscid fluidic plasma that is assumed to be a perfect conductor of heat as well as of electricity. For this mathematically prototypical material, it identifies a critical manifold in parameter space, across which slow classical MHD shock waves undergo emergence of a complex conjugate pair of unstable transverse modes. In the reflectionally symmetric case of parallel shocks, this emergence happens at the spectral value 0, and the critical manifold possesses a simple explicit algebraic representation. Results of refined numerica…
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…
In their classical work Caflisch and Sammartino proved the inviscid limit of the incompressible Navier-Stokes equations for well-prepared data with analytic regularity in the half-space. Their proof is based on the detailed construction of Prandtl's boundary layer asymptotic expansions. In this paper, we give a direct proof of the inviscid limit for general analytic data without having to construct Prandtl's boundary layer correctors. Our analysis makes use of the boundary vorticity formulation and the abstract Cauchy-Kovalevskaya theorem on analytic boundary layer function spaces that capture…
The primitive equations (PEs) model large scale dynamics of the oceans and the atmosphere. While it is by now well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces, and that there are solutions to the inviscid PEs (also called the hydrostatic Euler equations) that develop singularities in finite time, the qualitative description of the blowup still remains undiscovered. In this paper, we provide a full description of two blowup mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and …
Inviscid incompressible limit for capillary fluids with density dependent viscosity
本文在相对能量框架下证明了具有密度依赖粘性的Navier-Stokes-Korteweg系统弱解在低马赫数且粘性消失时收敛到不可压缩Euler系统强解，并利用受Korteweg张量影响的声学系统色散估计完成证明。
Well-posedness, magnetic helicity conservation, inviscid limit and asymptotic stability for the generalized Navier-Stokes-Maxwell equations
本文研究广义 Navier-Stokes-Maxwell 方程的适定性、磁螺度守恒、无粘极限及渐近稳定性，在不同分数阶拉普拉斯参数下建立了全局或局部适定性，并揭示了 NSM 与 MHD 在磁场收敛性和衰减速率上的差异。
We prove asymptotic stability of shear flows close to the planar Couette flow in the 2D inviscid Euler equations on $\Torus \times \Real$. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L^2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically driven to small scales by a linear evolution and weakly converges as $t \rightarrow \pm\infty$. The strong convergence of the velocity field is sometimes referred to as inviscid damping…
Almost global existence and nonlinear asymptotic stability for bubble dynamics in inviscid compressible liquid
本文研究无界等熵可压缩无粘液体中气泡的非线性动力学，通过耦合广义加权时空估计与特征线分析克服拟线性导致的导数损失，证明了几乎全局存在性和非线性渐近稳定性。
The stability of solutions under periodic perturbations for both inviscid and viscous conservation laws is an interesting and important problem. In this paper, a large-amplitude viscous shock under space-periodic perturbation for the isentropic Navier-Stokes equations is considered. It is shown that if the initial perturbation around the shock is suitably small and satisfies a zero-mass type condition (2.17), then the solution of the N-S equations tends to the viscous shock with a shift, which is partially determined by the periodic oscillations. In other words, the viscous shock is nonlinearl…
Uniform Inviscid Damping and Inviscid Limit of the 2D Navier-Stokes equation with Navier Boundary Conditions
本文研究了带 Navier 边界条件的二维 Navier-Stokes 方程在 Couette 流附近的渐近稳定性，首次在非线性层面结合了无粘阻尼、增强耗散和长时间无粘极限，并引入了以相空间技术为核心的新方法。
Extending results of Humpherys-Lyng-Zumbrun in the one-dimensional case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the multidimensional stability of planar Navier--Stokes shocks across the full range of shock amplitudes, including the infinite-amplitude limit, for monatomic or diatomic ideal gas equations of state and viscosity and heat conduction coefficients $μ$, $μ+η$, and $ν=κ/c_v$ in the physical ratios predicted by statistical mechanics, with Mach number $M>1.035$. Our results indicate unconditional stability within the paramete…
On the Cauchy problem for acoustic waves in hereditary fluids: decay properties and inviscid limits
本文研究遗传流体中MGT方程的衰减性质与无粘极限，通过新初始条件推导最优L2估计，并利用多尺度分析证明无粘极限，应用于非线性Jordan-MGT方程。
We construct an asymptotic (in a weak sense) solution corresponding to the shock wave formation in a special situation.
Enhanced Dissipation, Taylor Dispersion, and Inviscid Damping of Couette flow in the Boussinesq system on the Plane
本文在平面全无界区域上证明了具有大 Richardson 数的稳定分层 Couette 流在二维全耗散 Boussinesq 系统中的定量渐近稳定性，首次给出显式增强耗散、Taylor 色散和无粘阻尼估计。
In this paper we study large time behaviors toward shock waves and rarefaction waves under periodic perturbations for 1-D convex scalar conservation laws. The asymptotic stabilities and decay rates of shock waves and rarefaction waves under periodic perturbations are proved.
Three-dimensional structural stability of shock waves in elastodynamics
本文研究了弹性动力学中激波的三维结构稳定性，证明了平面激波至少弱稳定并给出均匀稳定的充要条件，指出弹性力对均匀稳定性起稳定作用。
Multidimensional stability and transverse bifurcation of hydraulic shocks and roll waves in open channel flow
结合分析与数值方法研究倾斜浅水流方程平面水力激波与滚动波解的多维稳定性及横向分岔，获得完整稳定性图谱并揭示新型横向分岔与亚稳态现象。
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.
We prove the large-time asymptotic orbital stability of strictly entropic Riemann shock solutions of first order scalar hyperbolic balance laws, under piecewise regular perturbations provided that the source term is dissipative about endstates of the shock. Moreover the convergence towards a shifted reference state is exponential with a rate predicted by the linearized equations about constant endstates.
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 …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。