Space-time stationary solutions for the Burgers equation

@article{Bakhtin2012SpacetimeSS,
  title={Space-time stationary solutions for the Burgers equation},
  author={Yuri Bakhtin and Eric A. Cator and Konstantin Khanin},
  journal={Journal of the American Mathematical Society},
  year={2012},
  volume={27},
  pages={193-238}
}
We construct space-time stationary solutions of the 1D Burgers equation with random forcing in the absence of periodicity or any other compactness assumptions. More precisely, for the forcing given by a homogeneous Poissonian point field in space-time we prove that there is a unique global solution with any prescribed average ve- locity. These global solutions serve as one-point random attractors for the infinite-dimensional dynamical system associated to solutions to the Cauchy problem. The… 
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52 Citations
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Methods Citations
7
Results Citations
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52 Citations

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References

SHOWING 1-10 OF 36 REFERENCES
Viscosity Limit of Stationary Distributions for the Random Forced Burgers Equation
We prove convergence of stationary distributions for the randomly forced Burgers and Hamilton–Jacobi equations in the limit when viscosity tends to zero. It turns out that for all values of the…
Invariant measures for Burgers equation with stochastic forcing
In this paper we study the following Burgers equation du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t) where f(x,t)=dF/dx(x,t) is a random forcing function, which is periodic in x and white noise…
The Burgers equation with Poisson random forcing
We consider the Burgers equation on the real line with forcing given by Poissonian noise with no periodicity assumption. Under a weak concentration condition on the driving random force, we prove…
AUBRY-MATHER THEORY AND PERIODIC SOLUTIONS OF THE FORCED BURGERS EQUATION
Consider a Hamiltonian system with Hamiltonian of the form H(x;t;p) where H is convex in p and periodic in x ,a ndt and x2 R 1 . It is well-known that its smooth invariant curves correspond to…
Burgers Turbulence and Random Lagrangian Systems
Abstract: We consider a spatially periodic inviscid random forced Burgers equation in arbitrary dimension and the random time-dependent Lagrangian system related to it. We construct a unique…
Random Burgers equation and Lagrangian systems in non-compact domains
In this paper we study stationary distributions for randomly forced Burgers and Hamilton–Jacobi equations in R d in the case when the forcing potentials have a large global maxima and a small global…
A shape theorem and semi-infinite geodesics for the Hammersley model with random weights
In this paper we will prove a shape theorem for the last passage percolation model on a two dimensional $F$-compound Poisson process, called the Hammersley model with random weights. We will also…
Busemann functions and equilibrium measures in last passage percolation models
The interplay between two-dimensional percolation growth models and one-dimensional particle processes has been a fruitful source of interesting mathematical phenomena. In this paper we develop a…
Transversal fluctuations for increasing subsequences on the plane
Abstract. Consider a realization of a Poisson process in ℝ2 with intensity 1 and take a maximal up/right path from the origin to (N, N) consisting of line segments between the points, where maximal…
Random Dynamical Systems
...
...