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Spectral Properties of Solutions of the Burgers Equation with Small Dissipation

Authors:	A È Biryuk

Institution:	(1) Moscow State University, Russia;(2) Heriot-Watt University, Edinburgh

Abstract:	We study the asymptotic behavior as $\delta \to 0$ of the Sobolev norm $\left\\| u \right\\|_m$ of the solution to the Cauchy problem for the one-dimensional quasilinear Burgers type equation $u_t + f\left( u \right)x = \delta u_{xx}$ (It is assumed that the problem is $C^\infty$ , the boundary conditions are periodic, and $f' \geqslant \sigma >0$ .) We show that the locally time-averaged Sobolev norms satisfy the estimate $c_m \delta ^{ - m + 1/2} < \left\langle {\left\\| u \right\\|_{_m }^2 } \right\rangle ^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} < c_m \delta ^{ - m + 1/2} \left( {m \geqslant 1} \right)$ . 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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