Scalar conservation laws with white noise initial data

• 成果类型: Article
• 署名作者: Ouaki, Mehdi
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-021-01083-z
• 发表日期: 2022
• 页码: 955-998
• 关键词: burgers turbulence; brownian-motion; shocks

摘要

The statistical description of the scalar conservation law of the form rho(t) = H(rho)(x) with H : R -> R a smooth convex function has been an object of interest when the initial profile rho(., 0) is random. The special case when H(rho) = rho(2)/2 (Burgers equation) has in particular received extensive interest in the past and is now understood for various random initial conditions. We prove in this paper a conjecture on the profile of the solution at any time t > 0 for a general class of Hamiltonians H and show that it is a stationary piecewise-smooth Feller process. Along the way, we study the excursion process of the two-sided linear Brownian motionW belowany strictly convex function phi with superlinear growth and derive a generalized Chernoff distribution of the random variable argmax(z is an element of R)(W(z)- phi(z)). Finally, when rho(., 0) is a white noise derived from an abrupt Levy process, we show that the structure of shocks of the solution is a.s discrete at any fixed time t > 0 under some mild assumptions on H.