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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].

##### MSC:
 60H30 Applications of stochastic analysis (to PDEs, etc.) 35Q53 KdV equations (Korteweg-de Vries equations)
##### Keywords:
Burgers equation; white noise; stable process