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.
In this paper we study the inviscid limit of the Benjamin-Ono-Burgers equation in the energy space $ H^{1/2} (\R) $ or $ H^{1/2}(\T) $. We prove the strong convergence in the energy space of the solution to this equation toward the solution of the Benjamin-Ono equation as the dissipation coefficient converges to $ 0 $.
In this paper, we consider the existence of global weak solutions to a one dimensional fluid-particles interaction model: inviscid Burgers-Vlasov equations with fluid velocity in $L^\infty$ and particles' probability density in $L^1$. Our weak solution is also an entropy solution to inviscid Burgers' equation. The approach is adding ingeniously artificial viscosity to construct approximate solutions satisfying $L^\infty$ compensated compactness framework and weak $L^1$ compactness framework. It is worthy to be pointed out that the bounds of fluid velocity and the kinetic energy of particles' p…
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.
In this paper, we study the Cauchy problem for the Benjamin-Ono-Burgers equation $\partial_t u-ε\partial_x^2 u+\mathcal{H}\partial_x^2u+u u_x=0$, where $\mathcal{H}$ denotes the Hilbert transform. We obtain that it is uniformly locally well-posed for small data in the refined Sobolev space $\widetilde{H}^σ(\mathbb{R})$($σ\geq 0$), whose low-frequency part is scaling critical and high-frequency part is equal to Sobolev space $H^σ$($σ\geq 0$). Furthermore, we also obtain its inviscid limit behavior in $\widetilde{H}^σ(\mathbb{R})$($σ\geq 0$).
Considering the Cauchy problem for the modified Korteweg-de Vries-Burgers equation $u_t+u_{xxx}+ε|\partial_x|^{2α}u=2(u^{3})_x, u(0)=ϕ$, where $0<ε,α\leq 1$ and $u$ is a real-valued function, we show that it is uniformly globally well-posed in $H^s (s\geq1)$ for all $ε\in (0,1]$. Moreover, we prove that for any $s\geq 1$ and $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the MKdV equation if $ε$ tends to 0.
Let $f,g:{\Bbb R}\to{\Bbb R}$ be integrable functions, $f$ nowhere zero, and $ϕ(u)=\int du/f(u)$ be invertible. An exact solution to the generalized nonhomogeneous inviscid Burgers' equation $u_t+g(u).u_x=f(u)$ is given, by quadratures.
Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation \begin{eqnarray*} u_t+u_{xxx}+ε|\partial_x|^{2α}u+(u^2)_x=0, \ u(0)=ϕ, \end{eqnarray*} where $0<ε,α\leq 1$ and $u$ is a real-valued function, we show that it is globally well-posed in $H^s\ (s>s_α)$, and uniformly globally well-posed in $H^s (s>-3/4)$ for all $ε\in (0,1)$. Moreover, we prove that for any $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the KdV equation if $ε$ tends to 0.
In this paper, we explore a nonlocal inviscid Burgers equation. Fixing a parameter $h$, we prove existence and uniqueness of the local solution of the equation $\InviscidBurgersNonlocal{u}$ with periodic initial condition. We also explore the blow up properties of solutions to these kinds of equations with given periodic initial data, and show that there exists solutions that blow up in finite time and solutions that are globally regular. This contrasts with the classical inviscid Burgers equation, for which all non-constant smooth periodic initial data lead to finite time blow up. Finally, we…
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…
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）
In this paper, we provide two-sided estimates for the source solution of $d$-dimensional critical fractal Burgers equation $u_t-Δ^{α/2}+b\cdot \nabla\left(u|u|^q\right)=0$, $q=(α-1)/d$, $α\in(1,2)$, $b\in\mathbb{R}^d$, by the density function of the isotropic $α$-stable process.
In the paper, the large time behavior of solutions of the Cauchy problem for the one dimensional fractal Burgers equation $u_t+(-\partial^2_x)^{α/2} u+uu_x=0$ with $α\in (1,2)$ is studied. It is shown that if the nondecreasing initial datum approaches the constant states $u_\pm$ ($u_-<u_+$) as $x\to \pm\infty$, respectively, then the corresponding solution converges toward the rarefaction wave, {\it i.e.} the unique entropy solution of the Riemann problem for the nonviscous Burgers equation.
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…
The paper is a comprehensive study of the existence, uniqueness, blow up and regularity properties of solutions of the Burgers equation with fractional dissipation. We prove existence of the finite time blow up for the power of Laplacian $α< 1/2,$ and global existence as well as analyticity of solution for $α\geq 1/2.$ We also prove the existence of solutions with very rough initial data $u_0 \in L^p,$ $1 < p < \infty.$ Many of the results can be extended to a more general class of equations, including the surface quasi-geostrophic equation.
In this paper, we provide two-sided estimates and uniform asymptotics for the solution of $d$-dimensional critical fractal Burgers equation $u_t-Δ^{α/2}u+b\cdot \nabla\left(u|u|^q\right)=0$, $α\in(1,2)$, $b\in\mathbb R^d$ for $q = (α-1)/d$ and $u_0 \in L^1(\mathbb R^d)$. We consider also $q > (α-1)/d$ under additional condition $u_0 \in L^\infty(\mathbb R^d)$. In both cases we assume $u_0\geq0$, which implies that the solution is non-negative. The estimates are given in the terms of the function $P_t u_0$, where $P_t$ is the stable semigroup operator.
The notion of Kruzhkov entropy solution was extended by the first author in 2007 to conservation laws with a fractional laplacian diffusion term; this notion led to well-posedness for the Cauchy problem in the $L^\infty$-framework. In the present paper, we further motivate the introduction of entropy solutions, showing that in the case of fractional diffusion of order strictly less than one, uniqueness of a weak solution may fail.
In this paper, we constuct the multi-point blowup solutions of self-similar type for the inviscid Burgers equation. The shape and blowup dynamics are precisely described. Moreover, the solutions we construct are stable under small perturbations on initial data restricted in a compact set.
We consider an initial value problem for a quadratically nonlinear inviscid Burgers-Hilbert equation that models the motion of vorticity discontinuities. We use a modified energy method to prove the existence of small, smooth solutions over cubically nonlinear time-scales.
We develop ergodic theory of the inviscid Burgers equation with random kick forcing in noncompact setting. The results are parallel to those in our recent work on the Burgers equation with Poissonian forcing. However, the analysis based on the study of one-sided minimizers of the relevant action is different. In contrast with previous work, finite time coalescence of the minimizers does not hold, and hyperbolicity (exponential convergence of minimizers in reverse time) is not known. In order to establish a One Force --- One Solution principle on each ergodic component, we use an extremely soft…
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 vanishing viscosity limit of viscous Burgers-Vlasov equations for one dimensional kinetic model about interactions between a viscous fluid and dispersed particles by using compensated compactness technique and the evolution of level sets arguments. The limit we obtained is exactly a finite-energy weak solution to the inviscid equations.
The aim of this paper is to show how a weakly dispersive perturbation of the inviscid Burgers equation improve (enlarge) the space of resolution of the local Cauchy problem. More generally we will review several problems arising from weak dispersive perturbations of nonlinear hyperbolic equations or systems.
In this paper we study the asymptotic behaviour of solutions of a system of $N$ partial differential equations. When $N = 1$ the equation reduces to the Burgers equation and was studied by Hopf. We consider both the inviscid and viscous case and show a new feature in the asymptotic behaviour.
The equivalence of the Choice Axiom and Tychonov compactness was proved by Kelley in 1950. Tychonov compactness is required to prove existence of the minimiser of an action functional under standard hypotheses. Solutions to the inviscid Burgers equation may be constructed by considering the minimising trajectories of associated action functionals. This construction has been used in the literature to construct an invariant measure for the stochastic inviscid Burgers equation. This article compares results that may be obtained with and without using this representation and draws conclusions abou…
We consider a space-continuous and time-discrete polymer model for positive temperature and the associated zero temperature model of last passage percolation type. In our previous work, we constructed and studied infinite-volume polymer measures and one-sided infinite minimizers for the associated variational principle, and used these objects for the study of global stationary solutions of the Burgers equation with positive or zero viscosity and random kick forcing, on the entire real line. In this paper, we prove that in the zero temperature limit, the infinite-volume polymer measures concent…
We consider an initial value problem for a quadratically nonlinear inviscid Burgers-Hilbert equation that models the motion of vorticity discontinuities. We use a normal form transformation, which is implemented by means of a near-identity coordinate change of the independent spatial variable, to prove the existence of small, smooth solutions over cubically nonlinear time-scales. For vorticity discontinuities, this result means that there is a cubically nonlinear time-scale before the onset of filamentation.
In this work, we examine the solution properties of the Burgers' equation with stochastic transport. First, we prove results on the formation of shocks in the stochastic equation and then obtain a stochastic Rankine-Hugoniot condition that the shocks satisfy. Next, we establish the local existence and uniqueness of smooth solutions in the inviscid case and construct a blow-up criterion. Finally, in the viscous case, we prove global existence and uniqueness of smooth solutions.
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…
This paper proves the existence of unstable shocks of the Burgers-Hilbert equation conjectured in arXiv:2006.05568. More precisely, we construct smooth initial data with finite $H^9$-norm such that the solution in self-similar coordinates is asymptotic to the first unstable solution to the self-similar inviscid Burgers equation. The blowup profile is a cusp with Hölder 1/5 continuity with explicit blowup time and location. Unlike the previously established stable shocks, the initial data cannot be taken in an open set; instead, we control the two unstable directions by Newton's iteration.
We study properties of solutions of the initial value problem for the nonlinear and nonlocal equation u_t+(-\partial^2_x)^{α/2} u+uu_x=0 with alpha in (0,1], supplemented with an initial datum approaching the constant states u+/u- (u_-smaller than u_+) as x goes to +/-infty, respectively. It was shown by Karch, Miao & Xu (SIAM J. Math. Anal. 39 (2008), 1536--1549) that, for alpha in (1,2), the large time asymptotics of solutions is described by rarefaction waves. The goal of this paper is to show that the asymptotic profile of solutions changes for alpha \leq 1. If alpha=1, there exists a self…
We give a substantially simplified proof of near-optimal estimate on the Kuramoto-Sivashinsky equation from [F. Otto, "Optimal bounds on the Kuramoto-Sivashinsky equation", JFA 2009], at the same time slightly improving the result. The result in the above cited paper relied on two ingredients: a regularity estimate for capillary Burgers and an a novel priori estimate for the inhomogeneous inviscid Burgers equation, which works out that in many ways the conservative transport nonlinearity acts as a coercive term. It is the proof of the second ingredient that we substantially simplify by proving…
A class of differentiable solutions is proved for the isentropic Euler equations in two and three space dimensions. The solutions are explicitly given in terms of solutions to inviscid Burgers equations, and several directions of propagation. The relative orientation of the directions is critical. Within the directional constraints, the Burgers solutions are arbitrary. The several velocities add, and the pressures combine nonlinearly. These solutions cannot exist beyond the time when shocks develop in any of the Burgers solutions.
A relativistic generalization of the inviscid Burgers equation was proposed by LeFloch, Makhlof, and Okutmustur and then investigated on a Schwarzschild background. Here, we extend their analysis to a Friedmann-Lemaitre-Robertson-Walker (FLRW) background. This problem is more challenging due to the existence of non-trivial spatially homogeneous solutions. First, we derive the relativistic Burgers model of interest and determine its spatially homogeneous solutions. Second, we design a numerical scheme based on the finite volume methodology, which is well-balanced in the sense that spatially hom…
We prove finite time blowup of the Burgers-Hilbert equation. We construct smooth initial data with finite $H^5$-norm such that the $L^\infty$-norm of the spacial derivative of the solution blows up. The blowup is an asymptotic self-similar shock at one single point with an explicitly computable blowup profile. The blowup profile is a cusp with Hölder $1/3$ continuity. The blowup time and location are described in terms of explicit ODEs. Our proof uses a transformation to modulated self-similar variables, the quantitative properties of the stable self-similar solution to the inviscid Burgers eq…
We study a formulation of Burgers equation on the Sierpinski gasket, which is the prototype of a p.c.f. self-similar fractal. One possibility is to implement Burgers equation as a semilinear heat equation associated with the Laplacian for scalar functions, just as on the unit interval. Here we propose a second, different formulation which follows from the Cole-Hopf transform and is associated with the Laplacian for vector fields. The difference between these two equations can be understood in terms of different vertex conditions for Laplacians on metric graphs. For the second formulation we sh…
We develop a Lagrangian approach to conservation-law anomalies in weak solutions of inviscid Burgers equation, motivated by previous work on the Kraichnan model of turbulent scalar advection. We show that the entropy solutions of Burgers possess Markov stochastic processes of (generalized) Lagrangian trajectories backward in time for which the Burgers velocity is a backward martingale. This property is shown to guarantee dissipativity of conservation-law anomalies for general convex functions of the velocity. The backward stochastic Burgers flows with these properties are not unique, however. …
We introduce a special stochastic perturbation of the flow of diffuse matter as a curve in the group of diffeomorphisms of flat n-dimensional torus such that the perturbed system yields a solution of Burgers equation in the tangent space at unit of the diffeomorphism group. The same perturbation of the flow of perfect incompressible fluid yields a solution of Reynolds type equation but under some special external force on the diffeomorphism group it transforms into a solution of Navier-Stokes equation without external force.
We show that, for first-order systems of conservation laws with a strictly convex entropy,in particular for the very simple so-called "inviscid" Burgers equation,it is possible to address the Cauchy problem by a suitable convex minimizationproblem, quite similar to some problems arising in optimal transport or variational mean-field game theory.In the general case, we show that smooth, shock-free, solutions can be recoveredon some sufficiently small interval of time. In the special situation of the Burgers equation, we furthershow that every "entropy solution" (in the sense of Kruzhkov)includi…
Persistence problems in weighted spaces have been studied for different dispersive models involving non-local operators. Generally, these models do not propagate polynomial weights of arbitrary magnitude, and the maximum decay rate is associated with the dispersive part of the equation. Altogether, this analysis is complemented by unique continuation principles that determine optimal spatial decay. This work is intended to establish the above questions for a weakly dispersive perturbation of the inviscid Burgers equation. More precisely, we consider the fractional Korteweg-de Vries equation, w…
We consider a class of dispersive and dissipative perturbations of the inviscid Burgers equation, which includes the fractional KdV equation of order $α$, and the fractal Burgers equation of order $β$, where $α, β\in [0,1)$, and the Whitham equation. For all $α, β\in [0,1)$, we construct solutions whose gradient blows up at a point, and whose amplitude stays bounded, which therefore display a "shock-like" singularity. We moreover provide an asymptotic description of the blow-up. To our knowledge, this constitutes the first proof of gradient blow-up for the fKdV equation in the range $α\in [2/3…
Relativistic Burgers equations on curved spacetimes. Derivation and finite volume approximation
Laboratoire Jacques–Louis Lions & Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6)（雅克·路易·利翁斯实验室与法国国家科学研究中心，皮埃尔和玛丽·居里大学（巴黎第六大学））；Department of Mathematics, Middle East Technical University（中东技术大学数学系）
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 …
In this paper, we consider the long wavelength limit for the quantum Euler-Poisson equation. Under the Gardner-Morikawa transform, we derive the quantum Korteweg-de Vries (KdV) equation by a singular perturbation method. We show that the KdV dynamics can be seen at time interval of order $O(ε^{-3/2})$. When the nondimensional quantum parameter $H=2$, it reduces to the inviscid Burgers equation.
We consider the "convection-diffussion" equation $u_t=J*u-u-uu_x,$ where $J$ is a probability density. We supplement this equation with step-like initial conditions and prove a convergence of corresponding solution towards a rarefaction wave, i.e. a unique entropy solution of the Riemann problem for the nonviscous Burgers equation. Methods and tools used in this paper are inspired by those used in [Karch, Miao and Xu, SIAM J. Math. Anal. {\bf 39} (2008), no. 5, 1536--1549.], where the fractal Burgers equation was studied.
