scholarly journals Burgers' turbulence problem with linear or quadratic external potential

2005 ◽  
Vol 42 (02) ◽  
pp. 550-565 ◽  
Author(s):  
O. E. Barndorff-Nielsen ◽  
N. N. Leonenko
Keyword(s):  
Burgers Equation ◽  
Initial Conditions ◽  
External Potential ◽  
Limit Laws ◽  
Random Initial Conditions ◽  
Burgers Turbulence

We consider solutions of Burgers' equation with linear or quadratic external potential and stationary random initial conditions of Ornstein-Uhlenbeck type. We study a class of limit laws that correspond to a scale renormalization of the solutions.

scholarly journals Burgers' turbulence problem with linear or quadratic external potential

2005 ◽  
Vol 42 (2) ◽  
pp. 550-565 ◽  
Author(s):  
O. E. Barndorff-Nielsen ◽  
N. N. Leonenko
Keyword(s):  
Burgers Equation ◽  
Initial Conditions ◽  
External Potential ◽  
Limit Laws ◽  
Random Initial Conditions ◽  
Burgers Turbulence

We consider solutions of Burgers' equation with linear or quadratic external potential and stationary random initial conditions of Ornstein-Uhlenbeck type. We study a class of limit laws that correspond to a scale renormalization of the solutions.

scholarly journals Random initial conditions for semi-linear PDEs

2019 ◽  
Vol 150 (3) ◽  
pp. 1533-1565
Author(s):  
Dirk Blömker ◽  
Giuseppe Cannizzaro ◽  
Marco Romito
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Burgers Equation ◽  
Initial Conditions ◽  
Stochastic Pdes ◽  
Well Posedness ◽  
Critical Spaces ◽  
Random Initial Conditions ◽  
Linear Pdes ◽  
Point Argument

AbstractWe analyse the effect of random initial conditions on the local well-posedness of semi-linear PDEs, to investigate to what extent recent ideas on singular stochastic PDEs can prove useful in this framework.In particular, in some cases, stochastic initial conditions extend the validity of the fixed-point argument to larger spaces than deterministic initial conditions would allow, but in general, it is never possible to go beyond the threshold that is predicted by critical scaling, as in our general class of equations we are not exploiting any special structure present in the equation.We also give a specific example where the level of regularity for the fixed-point argument reached by random initial conditions is not yet critical, but it is already sharp in the sense that we find infinitely many random initial conditions of slightly lower regularity, where there is no solution at all. Thus criticality cannot be reached even by random initial conditions.The existence and uniqueness in a critical space is always delicate, but we can consider the Burgers equation in logarithmically sub-critical spaces, where existence and uniqueness hold, and again random initial conditions allow to extend the validity to spaces of lower regularity which are still logarithmically sub-critical.

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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