%0 Journal Article %1 springerlink:10.1007/BF02104509 %A Avellaneda, Marco %A Weinan, E %C Berlin / Heidelberg %D 1995 %I Springer %J Communications in Mathematical Physics %K convex_minorant %P 13-38 %R 10.1007/BF02104509 %T Statistical properties of shocks in Burgers turbulence %U http://dx.doi.org/10.1007/BF02104509 %V 172 %X We consider the statistical properties of solutions of Burgers' equation in the limit of vanishing viscosity, , with Gaussian whitenoise initial data. This system was originally proposed by Burgers1 as a crude model of hydrodynamic turbulence, and more recently by Zel'dovichet al..12 to describe the evolution of gravitational matter at large spatio-temporal scales, with shocks playing the role of mass clusters. We present here a rigorous proof of the scaling relationP(s)∞s1/2,s≪1 whereP(s) is the cumulative probability distribution of shock strengths. We also show that the set of spatial locations of shocks is discrete, i.e. has no accumulation points; and establish an upper bound on the tails of the shock-strength distribution, namely 1−P(s)≤exp−Cs3 fors≫1. Our method draws on a remarkable connection existing between the structure of Burgers turbulence and classical probabilistic work on the convex envelope of Brownian motion and related diffusion processes.

@article{springerlink:10.1007/BF02104509, abstract = {We consider the statistical properties of solutions of Burgers' equation in the limit of vanishing viscosity, , with Gaussian whitenoise initial data. This system was originally proposed by Burgers[1] as a crude model of hydrodynamic turbulence, and more recently by Zel'dovichet al..[12] to describe the evolution of gravitational matter at large spatio-temporal scales, with shocks playing the role of mass clusters. We present here a rigorous proof of the scaling relationP(s)∞s1/2,s≪1 whereP(s) is the cumulative probability distribution of shock strengths. We also show that the set of spatial locations of shocks is discrete, i.e. has no accumulation points; and establish an upper bound on the tails of the shock-strength distribution, namely 1−P(s)≤exp{−Cs3} fors≫1. Our method draws on a remarkable connection existing between the structure of Burgers turbulence and classical probabilistic work on the convex envelope of Brownian motion and related diffusion processes.}, added-at = {2010-11-09T20:56:37.000+0100}, address = {Berlin / Heidelberg}, affiliation = {Courant Institute of Mathematical Sciences New York University 10012 New York N.Y. USA}, author = {Avellaneda, Marco and Weinan, E}, biburl = {https://www.bibsonomy.org/bibtex/27187ba525f3487f4ef1f64c9950d2a0e/pitman}, description = {SpringerLink - Communications in Mathematical Physics, Volume 172, Number 1}, doi = {10.1007/BF02104509}, interhash = {53b401ff6f40ea29c402546d072010bd}, intrahash = {7187ba525f3487f4ef1f64c9950d2a0e}, issn = {0010-3616}, issue = {1}, journal = {Communications in Mathematical Physics}, keyword = {Physics and Astronomy}, keywords = {convex_minorant}, pages = {13-38}, publisher = {Springer}, timestamp = {2010-11-09T20:56:37.000+0100}, title = {Statistical properties of shocks in Burgers turbulence}, url = {http://dx.doi.org/10.1007/BF02104509}, volume = 172, year = 1995 }