Burgers’ Topology on Random Point Measures

Abstract

The Burgers’ equation

$$\partial _tu+u\partial _xu=\mu\partial_x^2u,\;\;\;\;\;\;\;\;\;\;\;(1.1)$$

t < 0,x ∈ R, u = u(t,x),u(0,x) = u ₀(x), admits the well-known Hopf-Cole explicit solution

$$u(t,x)=\frac{\int _{-\infty }^\infty[(x-y)/t]\exp[(2\mu)^{-1}(\xi(y)-(x-y)^2/2t)]dy}{\int _{-\infty }^\infty \exp[(2\mu)^{-1}(\xi(y)-(x-y)^2/2t)]dy}\;\;\;\;\;\;\;\;\;\;(1.2)$$

where $\xi (x)=-\int_{x}^{-\infty } u_0(y)dy$ (see Hopf (1950)). It describes propagation of nonlinear hyperbolic waves, and has been considered as a model equation for hydrodynamic turbulence (see e.g. Chorin (1975)). Due to nonlinearity, the solution (1.2) enters several different stages, including that of shock waves’ formation, which are largely determined by the value of the Reynolds number R = σl/μ (see Gurbatov, Malakhov, Saichev (1991)). Here, μ < 0 is the viscosity parameter, while σ and l have the physical meaning of characteristic scale and amplitude of ξ(x), respectively.

Supported, in part, by grants from ONR and NSF.