Exact parabolic asymptotics for singular \(n\)-D Burgers' random fields: Gaussian approximation
An \(n\)-dimensional Burgers' equation with a random initial condition of a gradient type \[ \frac {\partial u}{\partial t} + (u,\nabla)u = \nu \Delta u, \quad u(0,\cdot)=\nabla \xi , \quad t\geq 0, \;x\in \mathbb R^{n}, \] is considered, where \(\nu >0\) and \(u=(u_{1},\ldots ,u_{n})\). The initial velocity potential \(\xi \) is assumed to be a Gaussian zero-mean, measurable, almost surely differentiable, homogeneous and isotropic random field on \(\mathbb R^{n}\) with \(E\xi ^{2}(x) =1\). Let its covariance function be of the form \(L(| x|)/| x| ^{\alpha}\), \(0<\alpha <n\), \(L\) being a locally bounded function slowly varying for large values of \(| x| \). The parabolic scaling limit for \(u\) is known, that is, it is known that the finite-dimensional distributions of the random field \(L^{-1/2}(\sqrt {t})t^{1/2+\alpha /4} u(t,a\sqrt {t})\), \(a\in \mathbb R^{n}\), converge weakly as \(t\to \infty \) to a certain homogeneous Gaussian random field \(X(a)\) whose covariance function may be described explicitly. In the paper under review, the rate of this convergence (in the uniform Kolmogorov distance) is studied.
scientific article; zbMATH DE number 1364491
scientific article