Bertoin, Jean Structure of shocks in Burgers turbulence with stable noise initial data. (English) Zbl 0943.60055 Commun. Math. Phys. 203, No. 3, 729-741 (1999). The author considers the Hopf-Cole solution \(u=u_0\) of the one-dimensional inviscid equation with the initial velocity given by a stable noise (the derivative in the Schwartz sense of a stable Lévy process) with index \(\alpha\in(1/2,2]\). (The solution is defined as the limit \(u_0=\lim_{\varepsilon\downarrow 0}u_\varepsilon u\), where \(u_\varepsilon\) is a unique solution of the Burgers equation \(\partial_t u+\partial(u^2/2)=\varepsilon\partial^2_{xx}u\).) It is proved that Lagrangian regular points exist (i.e., there are fluid particles that have not participated in shocks in the time interval \([0,t)\)) if and only if the noise is not completely asymmetric; otherwise, the shock structure is discrete. Moreover, in the Cauchy case \(\alpha=1\), it is shown that there are no rarefaction intervals, i.e., at any time \(t\), the locations of the fluid particles form an everywhere dense set a.s. Reviewer: V.Mackevičius (Vilnius) Cited in 5 Documents MSC: 60H30 Applications of stochastic analysis (to PDEs, etc.) 35Q53 KdV equations (Korteweg-de Vries equations) Keywords:Burgers equation; stable process; Hopf-Cole solution PDFBibTeX XMLCite \textit{J. Bertoin}, Commun. Math. Phys. 203, No. 3, 729--741 (1999; Zbl 0943.60055) Full Text: DOI