# The Burgers equation with Poisson random forcing

@article{Bakhtin2013TheBE, title={The Burgers equation with Poisson random forcing}, author={Yuri Bakhtin}, journal={Annals of Probability}, year={2013}, volume={41}, pages={2961-2989} }

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 existence and uniqueness of a global solution in a certain class. We describe its basin of attraction that can also be viewed as the main ergodic component for the model. We establish existence and uniqueness of global minimizers associated to the variational principle underlying the dynamics. We also…

## Figures from this paper

## 19 Citations

Stochastic fractional conservation laws

- Mathematics
- 2022

. In this paper, we consider the Cauchy problem for the nonlinear fractional conservation laws driven by a multiplicative noise. In particular, we are concerned with the well-posedness theory and the…

Space-time stationary solutions for the Burgers equation

- Mathematics
- 2012

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…

Inviscid Burgers equation with random kick forcing in noncompact setting

- Mathematics
- 2014

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…

Homogenization of a stochastically forced Hamilton-Jacobi equation

- MathematicsAnnales de l'Institut Henri Poincaré C, Analyse non linéaire
- 2021

Ergodicity and Hopf–Lax–Oleinik formula for fluid flows evolving around a black hole under a random forcing

- MathematicsStochastics and Partial Differential Equations: Analysis and Computations
- 2018

We study the ergodicity properties of weak solutions to a relativistic generalization of Burgers equation posed on a curved background and, specifically, a Schwarzschild black hole. We investigate…

Zero Temperature Limit for Directed Polymers and Inviscid Limit for Stationary Solutions of Stochastic Burgers Equation

- MathematicsJournal of Statistical Physics
- 2018

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…

Invariant measure of scalar first-order conservation laws with stochastic forcing

- Mathematics
- 2013

Under an hypothesis of non-degeneracy of the flux, we study the long-time behaviour of periodic scalar first-order conservation laws with stochastic forcing in any space dimension. For sub-cubic…

Ergodicity of spherically symmetric fluid flows outside of a Schwarzschild black hole with random boundary forcing

- Mathematics
- 2017

We consider the Burgers equation posed on the outer communication region of a Schwarzschild black hole spacetime. Assuming spherical symmetry for the fluid flow under consideration, we study the…

Thermodynamic Limit for Directed Polymers and Stationary Solutions of the Burgers Equation

- MathematicsCommunications on Pure and Applied Mathematics
- 2018

The first goal of this paper is to prove multiple asymptotic results for a time‐discrete and space‐continuous polymer model of a random walk in a random potential. These results include: existence of…

Long‐Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws

- Mathematics
- 2014

We study the long‐time behavior and regularity of the pathwise entropy solutions to stochastic scalar conservation laws with random‐in‐time spatially homogeneous fluxes and periodic initial data. We…

## References

SHOWING 1-10 OF 15 REFERENCES

Burgers Turbulence and Random Lagrangian Systems

- Mathematics
- 2003

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…

Stationary measures for a randomly forced burgers equation

- Mathematics
- 2005

We study a randomly forced Burgers equation and its corresponding Hamilton‐Jacobi equation on the line. The forcing is of the form of a randomly modulated and shifted potential. We prove the…

Viscosity Limit of Stationary Distributions for the Random Forced Burgers Equation

- Mathematics
- 2005

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

- Mathematics
- 2000

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…

Random Burgers equation and Lagrangian systems in non-compact domains

- Mathematics
- 2003

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 MICROSCOPIC MODEL FOR THE BURGERS EQUATION AND LONGEST INCREASING SUBSEQUENCES

- Mathematics
- 1996

We introduce an interacting random process related to Ulam's problem, or finding the limit of the normalized longest increasing subsequence of a random permutation. The process describes the…

Burgers equation with random boundary conditions

- Mathematics
- 2007

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…

Hammersley's interacting particle process and longest increasing subsequences

- Mathematics
- 1995

SummaryIn a famous paper [8] Hammersley investigated the lengthLn of the longest increasing subsequence of a randomn-permutation. Implicit in that paper is a certain one-dimensional continuous-space…