Spectral Properties of Solutions of the Burgers Equation with Small Dissipation

We study the asymptotic behavior as $\delta\to0$ of the Sobolev norm $\|u\|_m$ of the solution to the Cauchy problem for the one-dimensional quasilinear Burgers type equation $u_t+f(u)_x=\delta u_{xx}$ (It is assumed that the problem is $C^{\infty}$, the boundary conditions are periodic, and $f''\ge\sigma>0$.) We show that the locally time-averaged Sobolev norms satisfy the estimate $c_m\delta^{-m+1/2}<\langle\|u\|_m^2\rangle^{1/2}<C_m\delta^{-m+1/2}$ ($m\ge1$). The estimates obtained as a consequence for the Fourier coefficients justify Kolmogorov's spectral theory of turbulence for the case of the Burgers equation.