Helge Kristian Jenssen, Gregory Lyng
This paper was withdrawn by the authors due an error in the elimination of the front in the linearized interior equations (28)-(29).
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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.
We evaluate by direct calculation the Lopatinski determinant for ZND detonations in Majda's model for reacting flow, and show that on the nonstable (nonnegative real part) complex half-plane it has a single zero at the origin of multiplicity one, implying stability. Together with results of Zumbrun on the inviscid limit, this recovers the result of RoqueJoffre-Vila that viscous detonations of Majda's model also are stable for sufficiently small viscosity, for any fixed detonation strength, heat release, and rate of reaction.
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…
For strong detonation waves of the inviscid Majda model, spectral stability was established by Jung and Yao for waves with step-type ignition functions, by a proof based largely on explicit knowledge of wave profiles. In the present work, we extend their stability results to strong detonation waves with more general ignition functions where explicit profiles are unknown. Our proof is based on reduction to a generalized Sturm-Liouville problem, similar to that used by Sukhtayev, Yang, and Zumbrun to study spectral stability of hydraulic shock profiles of the Saint-Venant equations.
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。
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.
The vanishing viscosity limit of the two-dimensional (2D) compressible isentropic Navier-Stokes equations is studied in the case that the corresponding 2D inviscid Euler equations admit a planar rarefaction wave solution. It is proved that there exists a family of smooth solutions for the 2D compressible Navier-Stokes equations converging to the planar rarefaction wave solution with arbitrary strength for the 2D Euler equations. A uniform convergence rate is obtained in terms of the viscosity coefficients away from the initial time. In the proof, the hyperbolic wave is crucially introduced to …
The aim of this contribution is to make a connection between two recent results concerning the dynamics of vortices in incompressible planar flows. The first one is an asymptotic expansion, in the vanishing viscosity limit, of the solution of the two-dimensional Navier-Stokes equation with point vortices as initial data. In such a situation, it is known (Gallay, 2011) that the solution behaves to leading order like a linear superposition of Oseen vortices whose centers evolve according to the point vortex system, but higher order corrections can also be computed which describe the deformation …
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…
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…
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…
Small-amplitude finite-depth Stokes waves are transversally unstable
该研究针对三维无粘有限深度重力水波方程，证明了所有小振幅无旋平面周期行波具有谱不稳定性，丰富了Stokes波稳定性理论。
Arnold's variational principle and its application to the stability of planar vortices
该研究围绕二维不可压缩Euler方程稳态解的Arnold稳定性判据相关变分原理展开，明确了对应二次型可用于稳定性分析的条件，给出Oseen涡旋的新几何稳定性证明，并分析了无黏情形及高斯涡旋的黏性影响。
Columnar vortices are stationary solutions of the three-dimensional Euler equations with axial symmetry, where the velocity field only depends on the distance to the axis and has no component in the axial direction. Stability of such flows was first investigated by Lord Kelvin in 1880, but despite a long history the only analytical results available so far provide necessary conditions for instability under either planar or axisymmetric perturbations. The purpose of this paper is to show that columnar vortices are spectrally stable with respect to three-dimensional perturbations with no particu…
Helical symmetry is invariance under a one-dimensional group of rigid motions generated by a simultaneous rotation around a fixed axis and translation along the same axis. The key parameter in helical symmetry is the step or pitch, the magnitude of the translation after rotating one full turn around the symmetry axis. In this article we study the limits of three-dimensional helical viscous and inviscid incompressible flows in an infinite circular pipe, with respectively no-slip and no-penetration boundary conditions, as the step approaches infinity. We show that, as the step becomes large, the…
Extending investigations of Antman & Malek-Madani, Schecter & Shearer, Slemrod, Barker & Lewicka & Zumbrun, and others, we investigate phase-transitional elasticity models of strain-gradient effect. We prove the existence of non-constant planar periodic standing waves in these models with strain-gradient effects by variational methods and phase-plane analysis, for deformations of arbitrary dimension and general, physical, viscosity and strain-gradient terms. Previous investigations considered one-dimensional phenomenological models with artificial viscosity/strain gradient effect, for which th…
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。
This paper is devoted to the well-posedness theory of piston problem for compressible {combustion} Euler flows with physical ignition condition. A significant combustion phenomena called detonation will occur provided the reactant is compressed and ignited by a leading shock. Mathematically, the problem can be formulated as an initial-boundary value problem for hyperbolic balance laws with a large shock front as free boundary. In present paper, we establish the global well-posedness of entropy solutions via wave front tracking scheme within the framework of $BV\cap L^1$ space. The main difficu…
We study the vanishing dissipation limit of the three-dimensional (3D) compressible Navier-Stokes-Fourier equations to the corresponding 3D full Euler equations. Our results are twofold. First, we prove that the 3D compressible Navier-Stokes-Fourier equations admit a family of smooth solutions that converge to the planar rarefaction wave solution of the 3D compressible Euler equations with arbitrary strength. Second, we obtain a uniform convergence rate in terms of the viscosity and heat-conductivity coefficients. For this multi-dimensional problem, we first need to introduce the hyperbolic wa…
Uniqueness and stability of planar shock wave to 2D compressible Euler equations in the vanishing viscosity limit of compressible Navier-Stokes equations
针对二维可压缩欧拉方程平面激波解的非唯一性，通过引入密度依赖黏性的纳维-斯托克斯方程全局弱解框架，建立首个消失黏性极限选择准则，证明近平面初始数据下激波解的唯一性与稳定性。
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。
On existence of Sadovskii vortex patch: A touching pair of symmetric counter-rotating uniform vortex
本文通过求解精确冲量条件和环量上界约束下的能量最大化问题，证明了二维欧拉方程中由一对相切对称反向旋转均匀涡旋组成的Sadovskii涡旋斑的存在性。
Quantitative Hydrodynamic Stability for Couette Flow on Unbounded Domains with Navier Boundary Conditions
本文在三个无界区域上证明了二维 Navier-Stokes 方程 Couette 流的稳定性阈值定理，通过新的低频与中频非线性计算，首次给出无界域上完全非线性剪切流的增强耗散结果，并将结论推广到扰动 β-平面方程。
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