Geodesic
Spectral Properties of Solutions of the Burgers Equation with Small Dissipation
Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 1, pp. 1-15

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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 $c_m\delta^{-m+1/2}\langle\|u\|_m^2\rangle^{1/2}$ ($m\ge1$). The estimates obtained as a consequence for the Fourier coefficients justify Kolmogorov's spectral theory of turbulence for the case of the Burgers equation. 
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     author = {A. E. Biryuk},
     title = {Spectral {Properties} of {Solutions} of the {Burgers} {Equation} with {Small} {Dissipation}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {1--15},
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     volume = {35},
     number = {1},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2001_35_1_a0/}
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A. E. Biryuk. Spectral Properties of Solutions of the Burgers Equation with Small Dissipation. Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 1, pp. 1-15. http://geodesic.mathdoc.fr/item/FAA_2001_35_1_a0/