Viscosity Limit of Stationary Distributions for the Random Forced Burgers Equation

@article{Gomes2005ViscosityLO,
  title={Viscosity Limit of Stationary Distributions for the Random Forced Burgers Equation},
  author={Diogo A. Gomes and Renato Iturriaga and Konstantin Khanin and Pablo Padilla},
  journal={Moscow Mathematical Journal},
  year={2005},
  volume={5},
  pages={613-631}
}
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 viscosity ν there exists a unique (up to an additive constant) global stationary solution to the randomly forced Hamilton–Jacobi equation. The main result follows from the convergence of these solutions in a limit when ν tends to zero without changing its sign. The two limiting solutions (for different… 
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References

SHOWING 1-10 OF 34 REFERENCES
Instantons in the Burgers equation.
  • GurarieMigdal
  • Mathematics
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 1996
The instanton solution for the forced Burgers equation is found. This solution describes the exponential tail of the probability distribution function of velocity differences in the region where
Two results concerning asymptotic behavior of solutions of the Burgers equation with force
We consider the Burgers equation with an external force. For the case of the force periodic in space and time we prove the existence of a solution periodic in space and time which is the limit of a
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
Scaling and intermittency in Burgers turbulence.
  • BouchaudMézardParisi
  • Mathematics
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 1995
TLDR
An ansatz is proposed for the velocity field in the large-Reynolds-number limit of the forced Burgers equation in N dimensions, which should become exact in the limit N\ensuremath{\rightarrow}\ensure Math{\infty}.
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
Controlled Markov processes and viscosity solutions
This book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. The authors approach stochastic control problems
Kicked Burgers turbulence
Burgers turbulence subject to a force f(x, t) = [sum ]jfj(x)δ(t − tj), where tj are 'kicking times' and the 'impulses' fj(x) have arbitrary space dependence, combines features of the purely decaying
Velocity-difference probability density functions for Burgers turbulence
In this paper the Polyakov equation [Phys. Rev. E {bold 52}, 6183 (1995)] for the velocity-difference probability density functions, with the random Gaussian external force, with the correlation
Steady-state Burgers turbulence with large-scale forcing
Steady-state Burgers turbulence supported by white-in-time random forcing at low wave numbers is studied analytically and by computer simulation. The peak of the probability distribution function
...
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