Scale renormalization and random solutions of the Burgers equation. (English) Zbl 0624.60071

Rosenblatt, M.

J. Appl. Probab. 24, 328-338 (1987).

This paper shows how various limit theorems could be useful to study the asymptotic behaviour (in space) of the solution of the Burgers’ equation \[ u_ t+uu_ x = \mu u_{xx},\quad \mu >0, \] with the initial data \(u_ 0(x)\) a continuous stationary stochastic process. It is well known that provided the growth condition \(\int^{x}_{0}u_ 0(\xi)d\xi =o(x^ 2)\) holds, for which the existence of the mean of \(u_ 0\) is enough, the above equation can be changed into the heat equation with a simple transformation. This allows to establish that u(x,t) is stationary in space for any \(t>0.\)
The main result of the paper is that if \(\sigma^ 2(R)=var(\int^{R}_{0}u_ 0(y)dy)\) diverges as \(R\to \infty\), and the renormalized process \[ \sigma (R)^{-1}\int^{Ry}_{0}\{u(t,x)-Eu_ 0(x)\}dx,\quad 0\leq y\leq 1, \] converges weakly as \(R\to \infty\) for \(t=0\), the same limit is obtained for any time \(t>0\). The nature of the limit depends on the correlation of \(u_ 0\). If this is strongly mixing the corresponding central limit theorem in I. Ibragimov, Teor. Veroyatn. Primen. 20, 134-141 (1975; Zbl 0335.60023), yields the Brownian motion. By using a noncentral limit theorem as in R. Dobrushin and P. Major, Z. Wahrscheinlichkeitstheor. Verw. Geb. 50, 27-52 (1979; Zbl 0397.60034), different limiting behaviours are exhibited by \(u_ 0(x)=h(G(x))\), where G is a Gaussian process with suitable long-range dependence.
Finally the extension of the main result to the case \(\mu\downarrow 0\) is discussed and a counterexample is given for which the growth condition does not hold.

Reviewer: M.Piccioni

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
35R60	PDEs with randomness, stochastic partial differential equations
35Q99	Partial differential equations of mathematical physics and other areas of application
60G10	Stationary stochastic processes
60F05	Central limit and other weak theorems

Keywords:

short-range dependence; asymptotic behaviour; Burgers’ equation; central limit theorem; long-range dependence

Citations:

Zbl 0335.60023; Zbl 0397.60034