scholarly journals Stochastic turbulence for Burgers equation driven by cylindrical Lévy process

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.

scholarly journals On a stochastic Burgers equation with Dirichlet boundary conditions

2003 ◽  
Vol 2003 (43) ◽  
pp. 2735-2746 ◽  
Author(s):  
Ekaterina T. Kolkovska
Keyword(s):  
Weak Solution ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Burgers Equation ◽  
Martingale Problem ◽  
Weak Limit ◽  
Dirichlet Boundary Conditions ◽  
One Dimensional ◽  
Stochastic Burgers Equation ◽  
The One

We consider the one-dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.

scholarly journals Exact statistical properties of the Burgers equation

2000 ◽  
Vol 417 ◽  
pp. 323-349 ◽  
Author(s):  
L. FRACHEBOURG ◽  
Ph. A. MARTIN
Keyword(s):  
White Noise ◽  
Large Distance ◽  
Burgers Equation ◽  
Higher Order ◽  
Statistical Properties ◽  
Inviscid Limit ◽  
Initial Condition ◽  
Spatial Correlations ◽  
One Dimensional ◽  
The One

The one-dimensional Burgers equation in the inviscid limit with white noise initial condition is revisited. The one- and two-point distributions of the Burgers field as well as the related distributions of shocks are obtained in closed analytical forms. In particular, the large distance behaviour of spatial correlations of the field is determined. Since higher-order distributions factorize in terms of the one- and two- point functions, our analysis provides an explicit and complete statistical description of this problem.

scholarly journals Averaging principle for one dimensional stochastic Burgers equation

2018 ◽  
Vol 265 (10) ◽  
pp. 4749-4797 ◽  
Author(s):  
Zhao Dong ◽  
Xiaobin Sun ◽  
Hui Xiao ◽  
Jianliang Zhai
Keyword(s):  
Burgers Equation ◽  
Averaging Principle ◽  
One Dimensional ◽  
Stochastic Burgers Equation

scholarly journals Gated Coupling of Dopamine and Neuropeptide Signaling Underlies Perceptual Processing of Appetitive Odors in Drosophila

2017 ◽  
Author(s):  
Yuhan Pu ◽  
Melissa Megan Masserant Palombo ◽  
Yiwen Zhang ◽  
Ping Shen
Keyword(s):  
Cognitive Processing ◽  
Neural Circuit ◽  
Perceptual Processing ◽  
Olfactory Neurons ◽  
Excitatory Response ◽  
One Dimensional ◽  
Qualitative And Quantitative ◽  
Gating Mechanism ◽  
Olfactory Stimuli ◽  
Quantitative Properties

AbstractHow olfactory stimuli are perceived as meaningful cues for specific appetitive drives remains poorly understood. Here we show that despite their enormous diversity, Drosophila larvae can discriminatively respond to food-related odor stimuli based on both qualitative and quantitative properties. Perceptual processing of food scents takes place in a neural circuit comprising four dopamine (DA) neurons and a neuropeptide F (NPF) neuron per brain hemisphere. Furthermore, these DA neurons integrate and compress inputs from second-order olfactory neurons into one-dimensional DA signals, while the downstream NPF neuron assigns appetitive significance to limited DA outputs via a D1-type DA receptor (DoplR1)-mediated gating mechanism. Finally, Dop1R, along with a Gβ13F/Irk2-mediated inhibitory and a Gαs-mediated excitatory pathway, underlie a binary precision tuning apparatus that restricts the excitatory response of NPF neurons to DA inputs that fall within an optimum range. Our findings provide fresh molecular and cellular insights into cognitive processing of olfactory cues.

scholarly journals A Meshless Method for Burgers’ Equation Using Multiquadric Radial Basis Functions With a Lie-Group Integrator

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 113 ◽  
Author(s):  
Muaz Seydaoğlu
Keyword(s):  
Lie Group ◽  
Kinematic Viscosity ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Time Integration ◽  
Group Scheme ◽  
Basis Functions ◽  
One Dimensional ◽  
Radial Basis ◽  
The One

An efficient technique is proposed to solve the one-dimensional Burgers’ equation based on multiquadric radial basis function (MQ-RBF) for space approximation and a Lie-Group scheme for time integration. The comparisons of the numerical results obtained for different values of kinematic viscosity are made with the exact solutions and the reported results to demonstrate the efficiency and accuracy of the algorithm. It is shown that the numerical solutions concur with existing results and the proposed algorithm is efficient and can be easily implemented.

scholarly journals Statistical analysis and simulation of random shocks in stochastic Burgers equation

2014 ◽  
Vol 470 (2171) ◽  
pp. 20140080 ◽  
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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