Some observations of bispectral behavior of large ensembles of exact solutions to the Burgers equation for random initial conditions

1992 ◽  
Vol 4 (4) ◽  
pp. 845-848
Author(s):  
Weiguo Zheng ◽  
X. B. Reed
Keyword(s):  
Exact Solutions ◽  
Burgers Equation ◽  
Initial Conditions ◽  
Random Initial Conditions ◽  
Large Ensembles

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 Exact solutions for the formation of stagnant caps of insoluble surfactant on a planar free surface

2021 ◽  
Vol 131 (1) ◽  
Author(s):  
Darren G. Crowdy
Keyword(s):  
Exact Solutions ◽  
Peclet Number ◽  
Burgers Equation ◽  
Initial Conditions ◽  
Implicit Representation ◽  
Péclet Number ◽  
New Exact Solutions ◽  
Kinetics Modelling ◽  
Complex Burgers Equation ◽  
Insoluble Surfactant

AbstractA class of exact solutions is presented describing the time evolution of insoluble surfactant to a stagnant cap equilibrium on the surface of deep water in the Stokes flow regime at zero capillary number and infinite surface Péclet number. This is done by demonstrating, in a two-dimensional model setting, the relevance of the forced complex Burgers equation to this problem when a linear equation of state relates the surface tension to the surfactant concentration. A complex-variable version of the method of characteristics can then be deployed to find an implicit representation of the general solution. A special class of initial conditions is considered for which the associated solutions can be given explicitly. The new exact solutions, which include both spreading and compactifying scenarios, provide analytical insight into the unsteady formation of stagnant caps of insoluble surfactant. It is also shown that first-order reaction kinetics modelling sublimation or evaporation of the insoluble surfactant to the upper gas phase can be incorporated into the framework; this leads to a forced complex Burgers equation with linear damping. Generalized exact solutions to the latter equation at infinite surface Péclet number are also found and used to study how reaction effects destroy the surfactant cap equilibrium.

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.

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.

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