Structure of shocks in Burgers turbulence with stable noise initial data
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.
scientific article; zbMATH DE number 1379945