Patrick Bernard
We describe the asymptotic behaviour of solutions of unviscid Burgers equation on the circle with time-periodic forcing. These solutions converge to periodic states, but the period of these limit states may be greater than the period of the forcing. We obtain as a corollary a new result on the flow of the associated Hamiltonian system.
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Polynomial mixing for the white-forced hyperviscous Burgers equation on the whole line
研究全直线上白噪声驱动的超粘性伯格斯方程的遍历性，在相空间多个方向受随机强迫假设下，用耦合方法及准则证明动力学以任意幂次多项式速率吸引到唯一不变概率测度。
We study the non-autonomously forced Burgers equation $$ u_t(x,t) + u(x,t)u_x(x,t) - u_{xx}(x,t) = f(x,t) $$ on the space interval $(0,1)$ with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique $H^1$ bounded trajectory of this equation defined for all $t\in \mathbb{R}$. Moreover we demonstrate that this trajectory attracts all trajectories both in pullback and forward sense. We also prove that for the Dirichlet case this attraction is exponential.
Global unique solution to the perturbation of the Burgers' equation forced by derivatives of space-time white noise
本文研究受时空白噪声分数阶导数驱动的一维Burgers方程，依托Anderson Hamiltonian与二维Navier-Stokes方程的相关方法，证明了其温和解与弱解的全局时间存在性与唯一性。
We study turbulence in the one-dimensional Burgers equation with a white-in-time, Gaussian random force that has a Fourier-space spectrum $\sim 1/k$, where $k$ is the wave number. From very-high-resolution numerical simulations, in the limit of vanishing viscosity, we find evidence for multiscaling of velocity structure functions which cannot be falsified by standard tests. We find a new artifact in which logarithmic corrections can appear disguised as anomalous scaling and conclude that bifractal scaling is likely.
We study generalized variants of the Burgers equation and the KdV equation on the circle. The main goal of the paper is to show that both extensions can be recast as geodesic equations on a suitable diffeomorphism group of the circle and the corresponding Bott-Virasoro group respectively. As a consequence we obtain that the initial value problem for the Burgers equation with an additional quadratic term is well-posed on a scale of Sobolev spaces on the circle.
We consider a non-homogeneous generalised Burgers equation: $$ \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - ν\frac{\partial^2 u}{\partial x^2} = η^ω,\quad t \in \R,\ x \in S^1. $$ Here, νis small and positive, f is strongly convex and satisfies a growth assumption, while η^ω is a space-smooth random "kicked" forcing term. For any solution $u$ of this equation, we consider the quasi-stationary regime, corresponding to t>=2. After taking the ensemble average, we obtain upper estimates as well as time-averaged lower estimates for a class of Sobolev norms of $u$. These esti…
We study topological structure of the $ω$-limit sets of the skew-product semiflow generated by the following scalar reaction-diffusion equation \begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2π\mathbb{Z}, \end{equation*} where $f(t,u,u_x)$ is $C^2$-admissible with time-recurrent structure including almost-periodicity and almost-automorphy. Contrary to the time-periodic cases (for which any $ω$-limit set can be imbedded into a periodically forced circle flow), it is shown that one cannot expect that any $ω$-limit set can be imbedded into an almost-periodically force…
Exponential stability of the flow for a generalised Burgers equation on a circle
本文研究圆环上一维Burgers方程流的稳定性，基于保正性半群理论证明$L^1$范数强收缩性，得出有界外力下解在$H^1$中指数稳定，并证明随机外力下两轨迹之差以概率1趋于零。
We consider the non-homogeneous generalised Burgers equation \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - ν\frac{\partial^2 u}{\partial x^2} = η,\ t \geq 0,\ x \in S^1. Here f is strongly convex and satisfies a growth condition, νis small and positive, while ηis a random forcing term, smooth in space and white in time. For any solution u of this equation we consider the quasi-stationary regime, corresponding to t>=T_1, where T_1 depends only on f and on the distribution of η. We obtain sharp upper and lower bounds for Sobolev norms of $u$ averaged in time and in ensembl…
We describe a probabilistic construction of $H^s$-regular solutions for the spatially periodic forced Burgers equation by using a characterization of this solution through a forward-backward stochastic system.
We propose a method to integrate dissipative PDEs rigorously forward in time with the use of Finite Element Method (FEM). The technique is based on the Galerkin projection on the FEM space and estimates on the residual terms. The proposed approach is illustrated on a periodically forced one-dimensional Burgers equation with Dirichlet conditions. For two particular choices of the forcing we prove the existence of the periodic globally attracting trajectory and give precise bounds on its shape.
We prove that the stochastic Burgers equation on $\mathbf{R}^{d}$, $d<4$, forced by gradient noise that is white in time and smooth in space, admits spacetime-stationary solutions. These solutions are thus the gradients of solutions to the KPZ equation on $\mathbf{R}^{d}$ with stationary gradients. The proof works by proving tightness of the time-averaged laws of the solutions in an appropriate weighted space.
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.
This article deals with stochastic partial differential equations with quadratic nonlinearities perturbed by small additive and multiplicative noise. We present the approximate solution of the original equation via the amplitude equation and give the rigorous error analysis. For illustration, we apply our main theorems to stochastic Burger's equation.
We consider the skew-product semiflow which is generated by a scalar reaction-diffusion equation \begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2π\mathbb{Z}, \end{equation*} where $f$ is uniformly almost periodic in $t$. The structure of the minimal set $M$ is thoroughly investigated under the assumption that the center space $V^c(M)$ associated with $M$ is no more than $2$-dimensional. Such situation naturally occurs while, for instance, $M$ is hyperbolic or uniquely ergodic. It is shown in this paper that $M$ is a $1$-cover of the hull $H(f)$ provided that $M$ is…
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.
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 consider the stochastically forced Burgers equation with an emphasis on spatially rough driving noise. We show that the law of the process at a fixed time $t$, conditioned on no explosions, is absolutely continuous with respect to the stochastic heat equation obtained by removing the nonlinearity from the equation. This establishes a form of ellipticity in this infinite dimensional setting. The results follow from a recasting of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time and not on path-space. The results are proven…
We describe a topological method to study the dynamics of dissipative PDEs on a torus with rapidly oscillating forcing terms. We show that a dissipative PDE, which is invariant with respect to Galilean transformations, with a large average initial velocity can be reduced to a problem with rapidly oscillating forcing terms. We apply the technique to the Burgers equation, and the incompressible 2D Navier-Stokes equations with a time-dependent forcing. We prove that for a large initial average speed the equation admits a bounded eternal solution, which attracts all other solutions forward in time…
Global weak solutions of a Hamiltonian regularised Burgers equation
研究无粘 Burgers 方程的非色散守恒正则化，构造连接其与 Hunter–Saxton 方程的 Galilean 不变模型，并证明两类全局弱解及参数极限。
In this note we discuss the diffusive, vector-valued Burgers equations in a three-dimensional domain with periodic boundary conditions. We prove that given initial data in $H^{1/2}$ these equations admit a unique global solution that becomes classical immediately after the initial time. To prove local existence, we follow as closely as possible an argument giving local existence for the Navier--Stokes equations. The existence of global classical solutions is then a consequence of the maximum principle for the Burgers equations due to Kiselev and Ladyzhenskaya (1957). In several places we encou…
In this survey, we review the results on turbulence for the generalised Burgers equation on the circle: u_t+f'(u)u_x=νu_{xx}+η,\ x \in S^1=\R/\Z, obtained by A.Biryuk and the author in \cite{Bir01,BorK,BorW,BorD}. Here, f is smooth and strongly convex, whereas the constant 0<ν<< 1 corresponds to a viscosity coefficient. We will consider both the case η=0 and the case when ηis a random force which is smooth in x and irregular (kick or white noise) in t. In both cases, sharp bounds for Sobolev norms of u averaged in time and in ensemble of the type C ν^{-δ}, δ>=0, with the same value of δfor upp…
In this paper, we consider the asymptotic dynamics of the skew-product semiflow generated by the following time almost-periodically forced scalar reaction-diffusion equation \begin{equation}\label{eq0} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\, 0<x<L \end{equation} with periodic boundary condition \begin{equation} \label{bdc1} u(t,0)=u(t,L),\quad u_x(t,0)=u_x(t,L), \end{equation} where $f$ is uniformly almost periodic in $t$. In particular, we study the topological structure of the limit sets of the skew-product semiflow. It is proved that any compact minimal invariant set (throughout this paper, we…
Stationary solutions to the stochastic Burgers equation on the line
本文研究实直线上受时空白、空间光滑高斯噪声导数驱动的随机 Burgers 方程，证明每个均值对应唯一极值时空平稳不变测度，并确立衰减扰动初值解的分布收敛性。
The shock discontinuities, generically present in inviscid solutions of the forced Burgers equation, and their bifurcations happening in the course of time (perestroikas) are classified in two and three dimensions -- the one-dimensional case is well known. This classification is a result of selecting among all the perestroikas occurring for minimum functions depending generically on time, the ones permitted by the convexity of the Hamiltonian of the Burgers dynamics. Topological restrictions on the admissible perestroikas of shocks are obtained. The resulting classification can be extended to …
The zero-dispersion limit for the Benjamin--Ono equation on the circle
本文研究了Benjamin-Ono方程在圆上零色散极限的性质，通过显式公式刻画了零色散极限，并给出了正则性性质和最大原理估计。
This is the extended version of a lecture course given at the University of Vienna in the spring term 2005. The main aim of this course was to understand the papers \cite{10} and \cite{11} and to give a complete account of existence and uniqueness of the solutions of the members of higher order of the hierarchies of Burgers' equation and the Korteweg-de Vries equation, including their derivation and all the necessary background, both on the circle and on the real line in the setting of rapidly decreasing functions. These are all geodesic equations of infinite dimensional regular Lie groups, na…
Null coordinates for quasi-periodic $(1+1)$-dimensional wave operators on the circle with applications to reducibility
本文研究圆上拟周期(1+1)维波算子，通过构造零坐标使主象征化为常系数，并将构造问题归结为拉直拟周期向量场，从而为拟周期受迫线性Klein-Gordon方程的可约性提供了不依赖定量Egorov型结果的新证明。
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