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 
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/
@article{FAA_2001_35_1_a0,
     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},
     year = {2001},
     volume = {35},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2001_35_1_a0/}
}
TY  - JOUR
AU  - A. E. Biryuk
TI  - Spectral Properties of Solutions of the Burgers Equation with Small Dissipation
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2001
SP  - 1
EP  - 15
VL  - 35
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/FAA_2001_35_1_a0/
LA  - ru
ID  - FAA_2001_35_1_a0
ER  - 
%0 Journal Article
%A A. E. Biryuk
%T Spectral Properties of Solutions of the Burgers Equation with Small Dissipation
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2001
%P 1-15
%V 35
%N 1
%U http://geodesic.mathdoc.fr/item/FAA_2001_35_1_a0/
%G ru
%F FAA_2001_35_1_a0

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967 | MR

[2] Kruzhkov S. N., “Kvazilineinye uravneniya pervogo poryadka so mnogimi nezavisimymi peremennymi”, Matem. sb., 81:2 (1970), 228–255 | Zbl

[3] Kuznetsov N. N., “Tochnost nekotorykh priblizhennykh metodov rascheta slabykh reshenii kvazilineinogo uravneniya pervogo poryadka”, Zhurnal vych. matem. i matem. fiz., 16:6 (1976), 1489–1502 | MR | Zbl

[4] Kolmogorov A. N., “O neravenstvakh mezhdu verkhnimi granyami posledovatelnykh proizvodnykh proizvolnoi funktsii na beskonechnom intervale”, Izbrannye trudy. Matematika i mekhanika, Nauka, M., 1985, 252–263 | MR

[5] Hörmander L., Lectures on nonlinear hyperbolic differential equations, Springer-Verlag, Berlin, 1997 | MR | Zbl

[6] Lax P. D., “Hyperbolic systems of conservation laws, II”, Comm. Pure Appl. Math., 10 (1957), 537–566 | DOI | MR | Zbl

[7] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1989 | MR

[8] Frish U., Turbulentnost, Nasledie A. N. Kolmogorova, Biblioteka matematika, 4, Fazis, M., 1998

[9] Kuksin S., “Spectral Properties of Solutions for Nonlinear PDE's in the Turbulent Regime”, Geom. Funct. Anal., 9 (1999), 141–184 | DOI | MR | Zbl

[10] E. Weinan, Khanin K., Mazel A., Sinai Ya., “Invariant measures for Burgers equation with stochastic forcing”, Ann. of Math., 151:3 (2000), 877–960 | DOI | MR | Zbl