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Some properties of Burgers turbulence with white or stable noise initial data. (English) Zbl 0984.60078
Barndorff-Nielsen, Ole E. (ed.) et al., Lévy processes. Theory and applications. Boston: Birkhäuser. 267-279 (2001).
The author reviews some qualitative and quantitative results on the weak solution to the inviscid Burgers equation \(\partial_t u+\partial(u^2/2)=0\) with random initial datum. (The solution is defined as the limit \(u_0=\lim_{\varepsilon\downarrow 0}u_\varepsilon\), where \(u_\varepsilon\) is a unique solution of the Burgers equation \(\partial_t u+\partial(u^2/2)=\varepsilon\partial^2_{xx}u\).) Specifically, he first considers the case where the initial datum is a white noise (the derivative in the Schwartz sense of a two-sided Brownian motion). Then he discusses some extensions to a stable noise (the derivative in the Schwartz sense of a stable Lévy process with index \(\alpha\in(1/2,2]\)).
For the entire collection see [Zbl 0961.00012].

60H30 Applications of stochastic analysis (to PDEs, etc.)
35Q53 KdV equations (Korteweg-de Vries equations)