The inviscid Burgers equation with Brownian initial velocity. (English) Zbl 0917.60063

Bertoin, Jean

Commun. Math. Phys. 193, No. 2, 397-406 (1998).

Since the early 90’s several authors have been studied the inviscid Burgers equation with random initial velocity, among them Ya. G. Sinaj [ibid. 148, No. 3, 601-621 (1992; Zbl 0755.60105)], Z.-S. She, E. Aurell and U. Frisch [ibid. 148, No. 3, 623-641 (1992; Zbl 0755.60104)] and M. Avellaneda and W. E [ibid. 172, No. 1, 13-18 (1995; Zbl 0844.35144)]. The author of the present paper studies the case of a Brownian initial velocity for which he characterizes completely the distribution of the inverse Lagrangian function \(a(x, t)\), and thus also that of the Hopf-Cole solution \(u(x,t)= t^{-1}(x-a(x,t))\) of the inviscid Burgers equation. In particular, he obtains that the inverse Lagrangian function and the Hopf-Cole solution have independent and homogeneous increments. As examples of applications, the author studies the smoothness of the solution, the statistical distribution of the shocks, he determines the exact Hausdorff function of the Lagrangian regular points and investigates the existence of the Lagrangian regular points in a fixed Borel set.

Reviewer: R.Buckdahn (Brest)

MSC:

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

Keywords:

inviscid Burgers equation; Hopf-Cole solution; inverse Lagrangian function; first passage process

Citations:

Zbl 0755.60105; Zbl 0755.60104; Zbl 0844.35144