Remarks on the Burgers Equation

Abstract

Periodic and aperiodic solutions of the Burgers equation u_t + uu_x = μu_xx, μ > 0, are studied in this paper. A harmonic analysis of the solutions is carried out and the form of the spectrum is estimated for large time. Corresponding estimates of energy decay are also made. In Burgers' work on this equation, the case in which μ ↓ 0 with t fixed, and one then lets t → ∞, is studied. In our investigation, a fixed value of μ > 0 is taken and then one lets t → ∞. A similar analysis is also carried out for an irrotational solution of a similar 3‐dimensional system of equations. For large time and moderate wavenumbers there is, to the first order, a drift of spectral mass from low wavenumbers to higher wavenumbers. Comments are also made on the asymptotic distribution of a class of random solutions.