Renormalization and homogenization of solutions of the inhomogeneous heat equation with a linear potential and of the related Burgers equation with random data. (English. Ukrainian original) Zbl 1004.60017

Leonenko, N. N.; Mel’nikova, O. O.

Theory Probab. Math. Stat. 62, 77-88 (2001); translation from Teor. Jmovirn. Mat. Stat. 62, 72-82 (2000).

One of the approaches to study the solutions of partial differential equations such as heat equation and connected with it (by Cole-Hopf changing) Burgers equation is their re-scaling (re-normalization) and homogenization. It allows to find the exact form for the spectrum of homogeneous Gaussian or non-Gaussian random fields which arise under respective limit. The limit random fields depend on the character of random processes which are the initial conditions of the respective Cauchy problem. An analogous approach is proposed in this paper to the nonhomogeneous heat equation and Burgers equation with random initial conditions. It is proved that the character of re-scaling is changed, but the limiting distributions are the same as for the respective homogeneous equations. New formulas which describe the exact solutions of inhomogeneous Burgers equation with linear or quadratic outer potential are found.

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

60F05	Central limit and other weak theorems
70L05	Random vibrations in mechanics of particles and systems
35Q53	KdV equations (Korteweg-de Vries equations)
60G15	Gaussian processes

Keywords:

heat equation; Burgers equation; linear potential; random initial data