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

In this paper, we show that the stationary solution u(t, ω) of the differentiable random dynamical system U: ℝ+ × L2[0, 1] × Ω → L2[0, 1] generated by the stochastic Burgers' equation with large viscosity, denoted by ν, driven by a Brownian motion in L2[0, 1], is given by: u(t, ω) = U(t, Y(ω), ω) = Y(θ(t, ω)), where Y(ω) can be represented by the following integral equation: [Formula: see text] Here θ is the group of P-preserving ergodic transformations on the canonical probability space [Formula: see text] such that θ(t, ω)(s) = W(t + s) - W(t), where W is the L2[0, 1]-valued Brownian motion on the probability space [Formula: see text], Tν is the linear operator semigroup on L2[0, 1] generated by νΔ.

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Zero temperature limit for (1 + 1) directed polymers with correlated random potential

Journal of Statistical Mechanics Theory and Experiment ◽

10.1088/1742-5468/aa4e5e ◽

2016 ◽

Vol 2016 (12) ◽

pp. 123304 ◽

Cited By ~ 2

Author(s):

Victor Dotsenko

Keyword(s):

Random Potential ◽

Temperature Limit ◽

Directed Polymers ◽

Zero Temperature

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Stationary Solutions to the Stochastic Burgers Equation on the Line

Communications in Mathematical Physics ◽

10.1007/s00220-021-04025-x ◽

2021 ◽

Vol 382 (2) ◽

pp. 875-949

Author(s):

Alexander Dunlap ◽

Cole Graham ◽

Lenya Ryzhik

Keyword(s):

Burgers Equation ◽

Stationary Solutions ◽

Stochastic Burgers Equation

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Stochastic turbulence for Burgers equation driven by cylindrical Lévy process

Stochastics and Dynamics ◽

10.1142/s0219493722400044 ◽

2021 ◽

Author(s):

Shenglan Yuan ◽

Dirk Blömker ◽

Jinqiao Duan

Keyword(s):

Kinematic Viscosity ◽

Burgers Equation ◽

Inviscid Limit ◽

Regularity Of Solutions ◽

One Dimensional ◽

Stochastic Dynamical System ◽

Qualitative And Quantitative ◽

Stochastic Burgers Equation ◽

Quantitative Properties ◽

Turbulent Velocity Field

This work is devoted to investigating stochastic turbulence for the fluid flow in one-dimensional viscous Burgers equation perturbed by Lévy space-time white noise with the periodic boundary condition. We rigorously discuss the regularity of solutions and their statistical quantities in this stochastic dynamical system. The quantities include moment estimate, structure function and energy spectrum of the turbulent velocity field. Furthermore, we provide qualitative and quantitative properties of the stochastic Burgers equation when the kinematic viscosity [Formula: see text] tends towards zero. The inviscid limit describes the strong stochastic turbulence.

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Statistical analysis and simulation of random shocks in stochastic Burgers equation

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2014.0080 ◽

2014 ◽

Vol 470 (2171) ◽

pp. 20140080 ◽

Cited By ~ 7

Author(s):

Heyrim Cho ◽

Daniele Venturi ◽

George E Karniadakis

Keyword(s):

Shock Waves ◽

Burgers Equation ◽

Initial Conditions ◽

Statistical Properties ◽

Inviscid Limit ◽

High Dimensional ◽

Mathematical Framework ◽

Stochastic Burgers Equation ◽

Random Shock ◽

Random Initial Conditions

We study the statistical properties of random shock waves in stochastic Burgers equation subject to random space–time perturbations and random initial conditions. By using the response–excitation probability density function (PDF) method and the Mori–Zwanzig (MZ) formulation of irreversible statistical mechanics, we derive exact reduced-order equations for the one-point and two-point PDFs of the solution field. In particular, we compute the statistical properties of random shock waves in the inviscid limit by using an adaptive (shock-capturing) discontinuous Galerkin method in both physical and probability spaces. We consider stochastic flows generated by high-dimensional random initial conditions and random additive forcing terms, yielding multiple interacting shock waves collapsing into clusters and settling down to a similarity state. We also address the question of how random shock waves in space and time manifest themselves in probability space. The proposed new mathematical framework can be applied to different conservation laws, potentially leading to new insights into high-dimensional stochastic dynamical systems and more efficient computational algorithms.

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