Mathematics – Analysis of PDEs
Scientific paper
2007-08-17
Mathematical Models and Methods in Applied Sciences Vol. 19, No. 5 (2009) 1-19
Mathematics
Analysis of PDEs
12 pages, submitted Minor corrections to the text and algebra, added references
Scientific paper
It is well known that the solutions to the non-viscous Burgers equation develop a gradient catastrophe at a critical time provided the initial data have a negative derivative in certain points. We consider this equation assuming that the particle paths in the medium are governed by a random process with a variance which depends in a polynomial way on the velocity. Given an initial distribution of the particles which is uniform in space and with the initial velocity linearly depending on the position we show both analytically and numerically that there exists a threshold effect: if the power in the above variance is less than 1, then the noise does not influence the solution behavior, in the following sense: the mean of the velocity when we keep the value of position fixed goes to infinity outside the origin. If however the power is larger or equal 1, then this mean decays to zero as the time tends to a critical value.
Albeverio Sergio
Rozanova Olga
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