# Equation de Burgers g en eralis ée a force al éatoire et a viscosit é petite

Abstract : This Ph.D. thesis is concerned with studying solutions u of a generalised Burgers equation on the circle: u_t+f'(u)u_x=\nu u_{xx}+\eta,\ x \in S^1=\R/\Z. Here, $f$ is smooth, strongly convex, and satisfies some growth conditions. The constant 0<\nu << 1 corresponds to a viscosity coefficient. We will consider both the case \eta=0 and the case when \eta is a random force which is smooth in x and irregular ("kick" or white noise) in t. We obtain sharp bounds for Sobolev norms of u averaged in time and in ensemble of the type C \nu^{-\delta}, \delta >= 0, with the same value of \delta for upper and lower bounds. These results yield sharp bounds for small-scale quantities characterising turbulence, which confirm physical predictions. We are also concerned with the asymptotic behaviour of solutions: we prove hyperbolicity of minimizers for the action corresponding to the stochastic Hamilton-Jacobi equation, whose space derivative is the stochastic Burgers equation with \nu=0.
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Theses

Cited literature [57 references]

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Contributor : Alexandre Boritchev <>
Submitted on : Tuesday, October 9, 2012 - 5:02:39 AM
Last modification on : Wednesday, March 27, 2019 - 4:10:22 PM
Long-term archiving on: : Friday, December 16, 2016 - 9:45:52 PM

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• HAL Id : pastel-00739791, version 1

### Citation

Alexandre Boritchev. Equation de Burgers g en eralis ée a force al éatoire et a viscosit é petite. Equations aux dérivées partielles [math.AP]. Ecole Polytechnique X, 2012. Français. ⟨pastel-00739791⟩

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