Strong solution of the stochastic Burgers equation

This work introduces a pathwise notion of solution for the stochastic Burgers equation, in particular, our approach encompasses the Cole–Hopf solution. The developments are based on regularization arguments from the theory of distributions.     

Stochastic Burgers equation from long range exclusion interactions

We consider one-dimensional exclusion processes with long jumps given by a transition probability of the form $p_n(?)=s(?)+{\gamma }_{n}a(?)$, such that its symmetric part $s(?)$ is irreducible with finite variance and its antisymmetric part is absolutely bounded by $s(?)$. We prove that under diffusive time scaling and strength of asymmetry $\sqrt{n}{\gamma }_n{\to }_{n\to \infty }b\ne 0$, the equilibrium density fluctuations are given by the unique energy solution of the stochastic Burgers equation.     

Multiple-front solutions for the Burgers equation and the coupled Burgers equations

In this work, multiple-front solutions for the Burgers equation and the coupled Burgers equations are examined. The tanh–coth method and the Cole–Hopf transformation are used. The work highlights the power of the proposed schemes and the structures of the obtained multiple-front solutions.     

Burgers equation with random boundary conditions

We prove an existence and uniqueness theorem for stationary solutions of the inviscid Burgers equation on a segment with random boundary conditions. We also prove exponential convergence to the stationary distribution.

     

Symbolic computation of solutions for a forced Burgers equation

In this paper we give exact solutions for a forced Burgers equation. We make use of the generalized Cole-Hopf transformation and the traveling wave method.     

Stationary directed polymers and energy solutions of the Burgers equation

We consider the stationary O’Connell–Yor model of semi-discrete directed polymers in a Brownian environment in the intermediate disorder regime and show convergence of the increments of the log-partition function to the energy solutions of the stochastic Burgers equation.The proof does not rely on the Cole–Hopf transform and avoids the use of spectral gap estimates for the discrete model. The key technical argument is a second-order Boltzmann–Gibbs principle.     

Remarks on global controllability for the Burgers equation with two control forces

In this paper we deal with the viscous Burgers equation. We study the exact controllability properties of this equation with general initial condition when the boundary control is acting at both endpoints of the interval. In a first result, we prove that the global exact null controllability does not hold for small time. In a second one, we prove that the exact controllability result does not hold even for large time.     

随机广义KdV方程的随机精确解

通过一个辅助性方程和埃米尔特变换研究广义随机KdV方程的随机雅克比椭圆函数类波解.此外,还通过椭圆函数在模数取极限m→0和m→1的情况,给出方程的随机类孤子解和随机三角函数波解,所得结果丰富了广义随机KdV方程的精确解.     

Generalized Burgers equation with rough transport noise

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.     

Local exact Lagrangian controllability of the Burgers viscous equation

We study and give the definition of the exact Lagrangian controllability of the viscous Burgers equation and prove a local result. We give similar results for the heat equation in dimension 1.     

The law of the iterated logarithm for the solution of the Burgers equation with random initial data

The law of the iterated logarithm is established for the solution of the one-dimensional Burgers equation in the case where the initial potential is described by a zero-range shot noise.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 812–823, December, 1998.The author wishes to thank Professor A. V. Bulinskii for setting the problem and for his attention to the work on the paper.     

Large time behaviour of solutions of a system of generalized Burgers equation

In this paper we study the asymptotic behaviour of solutions of a system ofN partial differential equations. WhenN = 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.     

分数阶Burgers方程的有限元计算(英文)

建立了一维和二维分数阶Burgers方程的有限元格式.时间分数阶导数使用L1方法离散,空间方向使用有限元方法离散.通过选择合适的基函数,将离散后的方程转化成一个非线性代数方程组,并应用牛顿迭代方法求解.数值实验显示出了方法的有效性.     

Exp-function method for the exact solutions of fifth order KdV equation and modified Burgers equation

In this paper, we implemented the exp-function method for the exact solutions of the fifth order KdV equation and modified Burgers equation. By using this scheme, we found some exact solutions of the above-mentioned equations.     

Numerical similarity reductions of the (1+3)-dimensional Burgers equation

We consider the (1+3)-dimensional Burgers equation u_t = u_xx + u_yy + u_zz + uu_x which has considerable interest in mathematical physics. Lie symmetries are used to reduce it to certain ordinary differential equations. We employ numerical methods to solve a number of these ordinary differential equations.     

Beale-Kato-Majda type condition for Burgers equation

We consider a multidimensional Burgers equation on the torus Td and the whole space Rd. We show that, in case of the torus, there exists a unique global solution in Lebesgue spaces. For a torus we also provide estimates on the large time behaviour of solutions. In the case of Rd we establish the existence of a unique global solution if a Beale-Kato-Majda type condition is satisfied. To prove these results we use the probabilistic arguments which seem to be new.     

On symmetry-preserving difference scheme to a generalized Benjamin equation and third-order Burgers equation

In this paper, an exposition of a method is presented for discretizing a generalized Benjamin equation and third-order Burgers equation while preserving their Lie point symmetries. By using local conservation laws, the potential systems of original equation are obtained, which can be used to construct the invariant difference models and symmetry-preserving difference models of original equation, respectively. Furthermore, this method is very effective and can be applied to discrete high-order nonlinear evolution equations.     

Self-similar solutions of a generalized Burgers equation with nonlinear damping

The nonlinear ordinary differential equation resulting from the self-similar reduction of a generalized Burgers equation with nonlinear damping is studied in some detail. Assuming certain asymptotic conditions at plus infinity or minus infinity, we find a wide variety of solutions—(positive) single hump, monotonic (bounded or unbounded) or solutions with a finite zero. The existence or non-existence of positive bounded solutions with exponential decay to zero at infinity for specific parameter ranges is proved. The analysis relies mainly on the shooting argument.     

A stochastic particle method for the McKean-Vlasov and the Burgers equation

In this paper we introduce and analyze a stochastic particle method for the McKean-Vlasov and the Burgers equation; the construction and error analysis are based upon the theory of the propagation of chaos for interacting particle systems. Our objective is three-fold. First, we consider a McKean-Vlasov equation in with sufficiently smooth kernels, and the PDEs giving the distribution function and the density of the measure , the solution to the McKean-Vlasov equation. The simulation of the stochastic system with particles provides a discrete measure which approximates for each time (where is a discretization step of the time interval ). An integration (resp. smoothing) of this discrete measure provides approximations of the distribution function (resp. density) of . We show that the convergence rate is for the approximation in of the cumulative distribution function at time , and of order for the approximation in of the density at time ( is the underlying probability space, is a smoothing parameter). Our second objective is to show that our particle method can be modified to solve the Burgers equation with a nonmonotonic initial condition, without modifying the convergence rate . This part extends earlier work of ours, where we have limited ourselves to monotonic initial conditions. Finally, we present numerical experiments which confirm our theoretical estimates and illustrate the numerical efficiency of the method when the viscosity coefficient is very small.