The non-viscous Burgers equation associated with random positions in coordinate space: a threshold for blow up behaviour - Mathematics > Analysis of PDEsReport as inadecuate




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Abstract: It is well known that the solutions to the non-viscous Burgers equationdevelop a gradient catastrophe at a critical time provided the initial datahave a negative derivative in certain points. We consider this equationassuming that the particle paths in the medium are governed by a random processwith a variance which depends in a polynomial way on the velocity. Given aninitial distribution of the particles which is uniform in space and with theinitial velocity linearly depending on the position we show both analyticallyand numerically that there exists a threshold effect: if the power in the abovevariance is less than 1, then the noise does not influence the solutionbehavior, in the following sense: the mean of the velocity when we keep thevalue of position fixed goes to infinity outside the origin. If however thepower is larger or equal 1, then this mean decays to zero as the time tends toa critical value.



Author: Sergio Albeverio, Olga Rozanova

Source: https://arxiv.org/



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