Burgers Turbulence and Random Lagrangian Systems

Iturriaga, Renato; Khanin, Konstantin

doi:10.1007/s00220-002-0748-6

Cited by 71 publications

(119 citation statements)

References 30 publications

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“…It is of particular interest to analyze the effects of non-compactness of the domain and the intermediate time asymptotics in higher dimensions. The notion of main shock is then replaced by that of topological shock [9,10], which are no more isolated points, but spatially extended objects. The natural problem in this setting is to study geometrical and statistical properties of the T -global shocks and to find the asymptotic behavior of their age distribution.…”

Section: Discussionmentioning

confidence: 99%

“…In the case of smooth spatially periodic forcing potential F (x, t), it was shown in the one-dimensional case [8] and for higher dimensions [9] that a statistical steady state is reached at large times by the solution to the Burgers equation. Taking the initial time at −∞, the velocity field is periodic in space and uniquely determined by the realization of the forcing.…”

Section: Periodic Forcingmentioning

confidence: 99%

“…The statement about uniqueness of the global minimizer holds under some natural conditions on the forcing potential (see, e.g., Refs. [8,9]). Shocks are generically born at some finite time and then grow and merge.…”

Section: Periodic Forcingmentioning

confidence: 99%

“…(1.1) in the limit of vanishing * Electronic address: bec@obs-nice.fr † Electronic address: kk262@newton.cam.ac.uk viscosity (ν → 0) displays after a finite time dissipative singularities, namely shocks, corresponding to discontinuities in the velocity field. In the presence of large-scale forcing, it was recently stressed for the one-dimensional case [7,8] and also for higher dimensions [9,10], that the global topological structure of such singularities is strongly related to the boundary conditions associated to the equation. More precisely, when, for instance, space periodicity is assumed, a generic topological shock structure can be outlined.…”

Section: Introductionmentioning

confidence: 99%

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Bec

Khanin

2003

Journal of Statistical Physics

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The inviscid Burgers equation with random and spatially smooth forcing is considered in the limit when the size of the system tends to infinity. For the one-dimensional problem, it is shown both theoretically and numerically that many of the features of the space-periodic case carry over to infinite domains as intermediate time asymptotics. In particular, for large time T we introduce the concept of T -global shocks replacing the notion of main shock which was considered earlier in the periodic case (1997, E et al., Phys. Rev. Lett. 78, 1904. In the case of spatially extended systems these objects are no anymore global. They can be defined only for a given time scale and their spatial density behaves as ρ(T ) ∼ T −2/3 for large T . The probability density function p(A) of the age A of shocks behaves asymptotically as A −5/3 . We also suggest a simple statistical model for the dynamics and interaction of shocks and discuss an analogy with the problem of distribution of instability islands for a simple first-order stochastic differential equation.

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Section: Discussionmentioning

confidence: 99%

Section: Periodic Forcingmentioning

confidence: 99%

Section: Periodic Forcingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bec

Khanin

2003

Journal of Statistical Physics

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“…(1.4) would be the well-known twodimensional Burgers equations [17], which is transformed by the Cole-Hopf transformation and is an integrable generalization of the well-known Burgers equation [4]. Some researches have been done in physical study and the mathematical analysis of the two-dimensional Burgers equations, such as the stationary solutions [13], the exact solutions [1], the numerical solutions [2,20], and so on. Meanwhile, if there is no coupling terms with v in Eq.…”

Section: )mentioning

confidence: 99%

Global existence and attractors for the two-dimensional Burgers-Ginzburg- Landau equations

Guo¹,

Fang²

2017

J. Nonlinear Sci. Appl.

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This paper investigates the periodic initial value problem for the two-dimensional Burgers-Ginzburg-Landau (2D Burgers-GL) equations, which can be derived from the so-called modulated modulation equations (MME) that govern the dynamics of the modulated amplitudes of some periodic critical modes. The well-posedness of the solutions and the global attractors for the 2D Burgers-GL equations are obtained via delicate a priori estimates, the Galerkin method, and operator semigroup method.

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Stationary measures for a randomly forced burgers equation

Suidan

2005

Comm. Pure Appl. Math.

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We study a randomly forced Burgers equation and its corresponding HamiltonJacobi equation on the line. The forcing is of the form of a randomly modulated and shifted potential. We prove the existence of invariant probability measures and provide examples for which these measures are not unique. These measures do exhibit certain ergodic behavior. The methods we use are closely connected to the Lax-Oleȋnik variational principle.

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Burgers Turbulence and Random Lagrangian Systems

Cited by 71 publications

References 30 publications

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Global existence and attractors for the two-dimensional Burgers-Ginzburg- Landau equations

Stationary measures for a randomly forced burgers equation

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