Burgers vortices are explicit stationary solutions of the Navier-Stokes equations which are often used to describe the vortex tubes observed in numerical simulations of three-dimensional turbulence. In this model, the velocity field is a two-dimensional perturbation of a linear straining flow with axial symmetry. The only free parameter is the Reynolds number $Re = Γ/ν$, where $Γ$ is the total circulation of the vortex and $ν$ is the kinematic viscosity. The purpose of this paper is to show that Burgers vortex is asymptotically stable with respect to general three-dimensional perturbations, for all values of the Reynolds number. This definitive result subsumes earlier studies by various authors, which were either restricted to small Reynolds numbers or to two-dimensional perturbations. Our proof relies on the crucial observation that the linearized operator at Burgers vortex has a simple and very specific dependence upon the axial variable. This allows to reduce the full linearized equations to a vectorial two-dimensional problem, which can be treated using an extension of the techniques developped in earlier works. Although Burgers vortices are found to be stable for all Reynolds numbers, the proof indicates that perturbations may undergo an important transient amplification if $Re$ is large, a phenomenon that was indeed observed in numerical simulations.
In this paper we establish rigorously that the family of Burgers vortices of the three-dimensional Navier-Stokes equation is stable for small Reynolds numbers. More precisely, we prove that any solution whose initial condition is a small perturbation of a Burgers vortex will converge toward another Burgers vortex as time goes to infinity, and we give an explicit formula for computing the change in the circulation number (which characterizes the limiting vortex completely.) We also give a rigorous proof of the existence and stability of non-axisymmetric Burgers vortices provided the Reynolds nu…
Burgers vortices are stationary solutions of the three-dimensional Navier-Stokes equations in the presence of a background straining flow. These solutions are given by explicit formulas only when the strain is axisymmetric. In this paper we consider a weakly asymmetric strain and prove in that case that non-axisymmetric vortices exist for all values of the Reynolds number. In the limit of large Reynolds numbers, we recover the asymptotic results of Moffatt, Kida and Ohkitani (1994). We also show that the asymmetric vortices are stable with respect to localized two-dimensional perturbations.
The present paper is devoted to the study of the global solution and nonlinear stability to the coupled complex Ginzburg Landau and Burgers equations for sequential flames
We study the three-dimensional Navier-Stokes equations in the presence of the axisymmetric linear strain, where the strain rate depends on time in a specific manner. It is known that the system admits solutions which blow up in finite time and whose profiles are in a backward self-similar form of the familiar Burgers vortices. In this paper it is shown that the existing stability theory of the Burgers vortex leads to the stability of these blow-up solutions as well. The secondary blow-up is also observed when the strain rate is relatively weak.
The paper is concerned with the steady-state Burgers equation of fractional dissipation on the real line. We first prove the global existence of viscosity weak solutions to the fractal Burgers equation driven by the external force. Then the existence and uniqueness of solution with finite $H^{\fracα{2}}$ energy to the steady-state equation are established by estimating the decay of fractal Burgers' solutions. Furthermore, we show that the unique steady-state solution is nonlinearly stable, which means any viscosity weak solution of fractal Burgers equation, starting close to the steady-state s…
In this paper, we show uniqueness and stability for the piecewise-smooth solutions to the Burgers--Hilbert equation constructed in Bressan and Zhang [Commun. Math. Sci., 15(1):165--184, 2017]. The Burgers--Hilbert equation is $u_t+(\frac{u^2}{2})_x=\mathbf{H}[u]$ where $\mathbf{H}$ is the Hilbert transform, a nonlocal operator. We show stability and uniqueness for solutions amongst a larger class than the uniqueness result in Bressan and Zhang. The solutions we consider are measurable and bounded, satisfy at least one entropy condition, and verify a strong trace condition. We do not have small…
We consider smooth solutions of the Burgers-Hilbert equation that are a small perturbation $δ$ from a global periodic traveling wave with small amplitude $ε$. We use a modified energy method to prove the existence time of smooth solutions on a time scale of $\frac{1}{εδ}$ with $0<δ\llε\ll1$ and on a time scale of $\fracε{δ^2}$ with $0<δ\llε^2\ll1$. Moreover, we show that the traveling wave exists for an amplitude $ε$ in the range $(0,ε^*)$ with $ε^*\sim 0.23$ and fails to exist for $ε>\frac{2}{e}$.
The paper is concerned with the time-periodic (T-periodic) problem of the fractal Burgers equation with a T-periodic force on the real line. Based on the Galerkin approximates and Fourier series (transform) methods, we first prove the existence of T-periodic solution to a linearized version. Then, the existence and uniqueness of T-periodic solution to the nonlinear equation are established by the contraction mapping argument. Furthermore, we show that the unique T-periodic solution is asymptotically stable. This analysis, which is carried out in energy space $ H^{1}(0,T;H^{\fracα{2}}(R))\cap L…
Coherent structures are solutions to reaction-diffusion systems that are time-periodic in an appropriate moving frame and spatially asymptotic at $x=\pm\infty$ to spatially periodic travelling waves. This paper is concerned with sources which are coherent structures for which the group velocities in the far field point away from the core. Sources actively select wave numbers and therefore often organize the overall dynamics in a spatially extended system. Determining their nonlinear stability properties is challenging as localized perturbations may lead to a non-localized response even on the …
Vorticity plays a prominent role in the dynamics of incompressible viscous flows. In two-dimensional freely decaying turbulence, after a short transient period, evolution is essentially driven by interactions of viscous vortices, the archetype of which is the self-similar Lamb-Oseen vortex. In three dimensions, amplification of vorticity due to stretching can counterbalance viscous dissipation and produce stable tubular vortices. This phenomenon is illustrated in a famous model originally proposed by Burgers, where a straight vortex tube is produced by a linear uniaxial strain field. In real f…
We prove $L^2$ stability estimates for entropic shocks among weak, possibly \emph{non-entropic}, solutions of scalar conservation laws $\partial_t u+\partial_x f(u)=0$ with strictly convex flux function $f$. This generalizes previous results by Leger and Vasseur, who proved $L^2$ stability among entropy solutions. Our main result, the estimate \begin{align*} \int_{\mathbb R} |u(t,\cdot)-u_0^{shock}(\cdot -x(t))|^2\,dx\leq \int_{\mathbb R}|u_0-u_0^{shock}|^2 +Cμ_+([0,t]\times\R), \end{align*} for some Lipschitz shift $x(t)$, includes an error term accounting for the positive part of the entropy…
Asymptotic linear stability of columnar vortices driven by Coriolis force
School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；School of Mathematical Sciences, Zhejiang University of Technology（浙江工业大学数学系）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）
The aim of these notes is to present in a comprehensive and relatively self-contained way some recent developments in the mathematical analysis of two-dimensional viscous flows. We consider the incompressible Navier-Stokes equations in the whole plane, and assume that the initial vorticity is a finite measure. This general setting includes vortex patches, vortex sheets, and point vortices. We first prove the existence of a unique global solution, for any value of the viscosity parameter, and we investigate its long-time behavior. We next consider the particular case where the initial flow is a…
We are concerned with the large-time behavior of the radially symmetric solution for multidimensional Burgers equation on the exterior of a ball $\mathbb{B}_{r_0}(0)\subset \mathbb{R}^n$ for $n\geq 3$ and some positive constant $r_0>0$, where the boundary data $v_-$ and the far field state $v_+$ of the initial data are prescribed and correspond to a stationary wave. It is shown in \cite{Hashimoto-Matsumura-JDE-2019} that a sufficient condition to guarantee the existence of such a stationary wave is $v_+<0, v_-\leq |v_+|+μ(n-1)/r_0$. Since the stationary wave is no longer monotonic, its nonline…
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…
The mathematical theory of hydrodynamic stability started in the middle of the 19th century with the study of model examples, such as parallel flows, vortex rings, and surfaces of discontinuity. We focus here on the equally interesting case of columnar vortices, which are axisymmetric stationary flows where the velocity field only depends on the distance to the symmetry axis and has no component in the axial direction. The stability of such flows was first investigated by Kelvin in 1880 for some particular velocity profiles, and the problem benefited from important contributions by Rayleigh in…
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.
We establish the existence, uniqueness, and stability of the stationary solution of the one-dimensional viscous Burgers equation with the Dirichlet boundary conditions on a finite interval. We obtain explicit formulas for solutions and analytically determine the Lyapunov exponents characterizing the asymptotic behavior of arbitrary solutions approaching the stationary one.
Relative Entropy and the Inviscid Limit for Planar Shocks of the Multi-D Burgers' Equation
The University of Kansas（堪萨斯大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Institute of Applied Mathematics, AMSS, CAS（中国科学院数学与系统科学研究院应用数学研究所）；University of Chinese Academy of Sciences（中国科学院大学）
Existence and stability of Sadovskii vortices: from patch to smooth vortices
Osaka Metropolitan University（大阪公立大学）；Ulsan National Institute of Science and Technology（蔚山科学技术院）；Korea Institute for Advanced Study（韩国高等科学研究院）；Universität Basel（巴塞尔大学）
We study the linearized stability of n-vortex solutions of the magnetic Ginzburg-Landau (or Abelian-Higgs) equations. We prove that the fundamental vortices (n=1,-1) are stable for all values of the coupling constant, k, and we prove that the higher-degree vortices (|n| > 1) are stable for k < 1 and unstable for k > 1. This resolves a long-standing conjecture.
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.
In the analysis of the long-time behaviour of two-dimensional incompressible viscous fluids, Oseen vortices play a major role as attractors of any homogeneous solution with integrable initial vorticity. As a first step in the study of the density-dependent case, the present paper establishes the asymptotic stability of Oseen vortices for slighty inhomogeneous fluids with respect to localized regular perturbations.
We study Ginzburg-Landau equations for a complex vector order parameter. We consider the Dirichlet problem in the disk in the plane with a symmetric, degree-one boundary condition, and study its stability, in the sense of the spectrum of the second variation of the energy. We find that the stability of the degree-one equivariant solution depends on both the Ginzburg-Landau parameter as well as the sign of the interaction term in the energy.
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…
We investigate spectral stability of vortex solutions of the Gross-Pitaevskii equation, a mean-field approximation for Bose-Einstein condensates (BEC) in an effectively two-dimensional axisymmetric harmonic trap. We study eigenvalues of the linearization both rigorously and through computation of the Evans function, a sensitive and robust technique whose use we justify mathematically. The absence of unstable eigenvalues is justified a posteriori through use of the Krein signature of purely imaginary eigenvalues, which also can be used to significantly reduce computational effort. In particular…
Both experimental and numerical studies of fluid motion indicate that initially localized regions of vorticity tend to evolve into isolated vortices and that these vortices then serve as organizing centers for the flow. In this paper we prove that in two dimensions localized regions of vorticity do evolve toward a vortex. More precisely we prove that any solution of the two-dimensional Navier-Stokes equation whose initial vorticity distribution is integrable converges to an explicit self-similar solution called ``Oseen's vortex''. This implies that the Oseen vortices are dynamically stable for…
