We show that the 1d viscous Burgers equation considered for complex valued functions develops finite-time singularities from compactly supported smooth data. By means of the Cole-Hopf transformation, the singularities of the solutions are related to zeros of complex-valued solutions of the heat equation. We prove that such zeros are isolated if they are not present in the initial data.
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.
We present a reduced basis offline/online procedure for viscous Burgers initial boundary value problem, enabling efficient approximate computation of the solutions of this equation for parametrized viscosity and initial and boundary value data. This procedure comes with a fast-evaluated rigorous error bound certifying the approximation procedure. Our numerical experiments show significant computational savings, as well as efficiency of the error bound.
We use the Cole--Hopf transformation and the Laplace method for the heat equation to justify the numerical results on enstrophy growth in the viscous Burgers equation on the unit circle. We show that the maximum enstrophy achieved in the time evolution is scaled as $\mathcal{E}^{3/2}$, where $\mathcal{E}$ is the large initial enstrophy, whereas the time needed for reaching the maximal enstrophy is scaled as $\mathcal{E}^{-1/2}$. These bounds are sharp for sufficiently smooth initial conditions.
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.
We consider higher order viscous Burgers' equations with generalized nonlinearity and study the associated initial value problems for given data in the $L^2$-based Sobolev spaces. We introduce appropriate time weighted spaces to derive multilinear estimates and use them in the contraction mapping principle argument to prove local well-posedness for data with Sobolev regularity below $L^2$. We also prove ill-posedness for this type of models and show that the local well-posedness results are sharp in some particular cases viz., when the orders of dissipation $p$, and nonlinearity $k+1$, satisfy…
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 study bounds on the enstrophy growth for solutions of the viscous Burgers equation on the unit circle. Using the variational formulation of Lu and Doering, we prove that the maximizer of the enstrophy's rate of change is sharp in the limit of large enstrophy up to a numerical constant but does not saturate the Poincaré inequality for mean-zero 1-periodic functions. Using the dynamical system methods, we give an asymptotic representation of the maximizer in the limit of large enstrophy as a viscous shock on the background of a linear rarefactive wave. This asymptotic construction is used to …
We prove that the viscous Burgers equation has a globally defined smooth solution in all dimensions provided the initial condition and the forcing term are smooth and bounded together with their derivatives. Such solutions may have infinite energy. The proof does not rely on energy estimates, but on a combination of the maximum principle and quantitative Schauder estimates. We obtain precise bounds on the sup norm of the solution and its derivatives, making it plain that there is no exponential increase in time. In particular, these bounds are time-independent if the forcing term is zero. To g…
A "metastable solution" to a differential equation typically refers to a family of solutions for which nearby initial data converges to the family much faster than evolution along the family. Metastable families have been observed both experimentally and numerically in various contexts, they are believed to be particularly relevant for organizing the dynamics of fluid flows. In this work we propose a candidate metastable family for the Burgers equation with periodic boundary conditions. Our choice of family is motivated by our numerical experiments. We furthermore explain the metastable behavi…
We show that the homogeneous viscous Burgers equation $(\partial_t-ηΔ) u(t,x)+(u\cdot\nabla)u(t,x)=0,\ (t,x)\in{\mathbb{R}}_+\times{\mathbb{R}}^d$ $(d\ge 1, η>0)$ has a globally defined smooth solution if the initial condition $u_0$ is a smooth function growing like $o(|x|)$ at infinity. The proof relies mostly on estimates of the random characteristic flow defined by a Feynman-Kac representation of the solution. Viscosity independent a priori bounds for the solution are derived from these. The regularity of the solution is then proved for fixed $η>0$ using Schauder estimates. The result exten…
Analytic continuation of viscous shock solution for the generalized Burgers equation with polynomial nonlinear source term is investigated. We show that a pertubated wave recovers its analyticity in the space variable in the strip limited by the first pair of singularities of the wave.
The paper is devoted to studying the 1D viscous Burgers equation controlled by an external force. It is assumed that the initial state is essentially bounded, with no decay condition at infinity, and the control is a trigonometric polynomial of low degree with respect to the space variable. We construct explicitly a control space of dimension 11 that enables one to steer the system to any neighbourhood of a given final state in local topologies. The proof of this result is based on an adaptation of the Agrachev-Sarychev approach to the case of an unbounded domain.
We investigate the time periodic solutions to the viscous Burgers equation $u_t -μu_{xx} + uu_x = f$ for irregular forcing terms. We prove that the corresponding Burgers operator is a diffeomorphism between appropriate function spaces.
This paper is devoted to study the controllability of a one-dimensional fluid-particle interaction model where the fluid follows the viscous Burgers equation and the point mass obeys Newton's second law. We prove the null controllability for the velocity of the fluid and the particle and an approximate controllability for the position of the particle with a control variable acting only on the particle. One of the novelties of our work is the fact that we achieve this controllability result in a uniform time for all initial data and without any smallness assumptions on the initial data.
We define a notion of a viscous shock solution of the stochastic Burgers equation that connects "top" and "bottom" spatially stationary solutions of the same equation. Such shocks generally travel in space, but we show that they admit time-invariant measures when viewed in their own reference frames. Under such a measure, the viscous shock is a deterministic function of the bottom and top solutions and the shock location. However, the measure of the bottom and top solutions must be tilted to account for the change of reference frame. We also show a convergence result to these stationary shock …
Proceeding from the method of stochastic perturbation of a Langevin system associated with the non-viscous Burgers equation we construct a solution to the Riemann problem for the non-interacting particles and sticky particles systems. We analyze the difference in the behavior of discontinuous solutions for these two models and relations between them.
In this work, we are interested in the small time global null controllability for the viscous Burgers' equation y_t - y_xx + y y_x = u(t) on the line segment [0,1]. The second-hand side is a scalar control playing a role similar to that of a pressure. We set y(t,1) = 0 and restrict ourselves to using only two controls (namely the interior one u(t) and the boundary one y(t,0)). In this setting, we show that small time global null controllability still holds by taking advantage of both hyperbolic and parabolic behaviors of our system. We use the Cole-Hopf transform and Fourier series to derive p…
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…
We present here a version of the existence and uniqueness result of time periodic solutions to the viscous Burgers equation with irregular forcing terms (with Sobolev regularity -1 in space). The key result here is an a priori estimate which is simpler than the previously treated case of forcing terms with regularity -1/2 in time.
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.
In this article, global stabilization results for the two dimensional (2D) viscous Burgers' equation, that is, convergence of unsteady solution to its constant steady state solution with any initial data, are established using a nonlinear Neumann boundary feedback control law. Then, applying $C^0$-conforming finite element method in spatial direction, optimal error estimates in $L^\infty(L^2)$ and in $L^\infty(H^1)$- norms for the state variable and convergence result for the boundary feedback control law are derived. All the results preserve exponential stabilization property. Finally, severa…
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.
It is well known that the solutions to the non-viscous Burgers equation develop a gradient catastrophe at a critical time provided the initial data have a negative derivative in certain points. We consider this equation assuming that the particle paths in the medium are governed by a random process with a variance which depends in a polynomial way on the velocity. Given an initial distribution of the particles which is uniform in space and with the initial velocity linearly depending on the position we show both analytically and numerically that there exists a threshold effect: if the power in…
We consider the large time behavior of strong solutions to a kind of stochastic Burgers equation, where the position x is perturbed by a Brownian noise. It is well known that both the rarefaction wave and viscous shock wave are time-asymptotically stable for deterministic Burgers equation since the pioneer work of A. Ilin and O. Oleinik [21] in 1964. However, the stability of these wave patterns under stochastic perturbation is not known until now. In this paper, we give a definite answer to the stability problem of the rarefaction and viscous shock waves for the 1-d stochastic Burgers equatio…
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）
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…
Crossover asymptotics and a sharp confinement rate for the viscous Burgers equation
Instituto de Matemática e Computação, Universidade Federal de Itajubá（伊塔朱巴联邦大学数学与计算研究所）；Friedrich–Alexander-Universität Erlangen–Nürnberg（埃尔朗根-纽伦堡弗里德里希·亚历山大大学）；University of Deusto（德乌斯托大学）；Universidad Autónoma de Madrid（马德里自治大学）
We provide a new analytical approach to operator splitting for equations of the type $u_t=Au+u u_x$ where $A$ is a linear differential operator such that the equation is well-posed. Particular examples include the viscous Burgers' equation, the Korteweg-de Vries (KdV) equation, the Benney-Lin equation, and the Kawahara equation. We show that the Strang splitting method converges with the expected rate if the initial data are sufficiently regular. In particular, for the KdV equation we obtain second-order convergence in $H^r$ for initial data in $H^{r+5}$ with arbitrary $r\ge 1$.
We study the dissipation mechanism of a stochastic particle system for the Burgers equation. The velocity field of the viscous Burgers and Navier-Stokes equations can be expressed as an expected value of a stochastic process based on noisy particle trajectories [Constantin and Iyer Comm. Pure Appl. Math. 3 (2008) 330-345]. In this paper we study a particle system for the viscous Burgers equations using a Monte-Carlo version of the above; we consider N copies of the above stochastic flow, each driven by independent Wiener processes, and replace the expected value with $\frac{1}{N}$ times the su…
We introduce a new technique for studying well posedness and energy estimates for evolution equations with a rough transport term. The technique is based on finding suitable space-time weight functions for the equations at hand. As an example we study the well posedness of the generalized viscous Burgers equation perturbed by a rough path transport noise.
In this paper, we consider the back and forth nudging algorithm that has been introduced for data assimilation purposes. It consists of iteratively and alternately solving forward and backward in time the model equation, with a feedback term to the observations. We consider the case of 1-dimensional transport equations, either viscous or inviscid, linear or not (Bürgers' equation). Our aim is to prove some theoretical results on the convergence, and convergence properties, of this algorithm. We show that for non viscous equations (both linear transport and Burgers), the convergence of the algo…
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.
We consider the motion of a point mass in a one-dimensional viscous compressible barotropic fluid. The fluid--point mass system is governed by the barotropic compressible Navier--Stokes equations and Newton's equation of motion. Our main result concerns the long-time behavior of the fluid and the point mass, and it gives pointwise convergence estimates of the volume ratio and the velocity of the fluid to their equilibrium values. As a corollary, it is shown that the velocity $V(t)$ of the point mass satisfies a decay estimate $|V(t)|=O(t^{-3/2})$ --- a faster decay compared to $t^{-1/2}$ known…
In this paper, we investigate and prove the nonlinear stability of viscous shock wave solutions of a scalar viscous conservation law, using the methods developed for general systems of conservation laws by Howard, Mascia, Zumbrun and others, based on instantaneous tracking of the location of the perturbed viscous shock wave. In some sense, this paper extends the treatment in a previous expository work of Zumbrun ["Instantaneous shock location ..."] on Burgers equation to the general case, giving an exposition of these methods in the simplest setting of scalar equations. In particular we give b…
We are concerned with a viscous Burgers equation forced by a perturbation of white noise type. We study the corresponding transition semigroup in a space of continuous functions weighted by a proper potential, and we show that the infinitesimal generator is the closure (with respect to a suitable topology) of the Kolmogorov operator associated to the stochastic equation. In the last part of the paper we use this result to solve the corresponding Fokker-Planck equation.
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 a nonlocal viscous Burgers equation and study the well-posedness and asymptotic behaviour of its solutions. We prove that under the smallness assumption on the initial data the solutions behave as the self similar profiles of the Burgers equation with Dirac mass as the initial datum. The first term in the asymptotic expansion of the solutions is obtained by rescaling the solutions and proving the compactness of their trajectories.
