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English

Sharp Estimates for Turbulence in White-Forced Generalised Burgers Equation

Published inGeometric and Functional Analysis, vol. 23, no. 6, p. 1730-1771
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  • Open Access - Licence nationale Springer 
Publication date2013
Abstract

Here f is strongly convex and satisfies a growth condition, ν is small and positive, while η is a random forcing term, smooth in space and white in time. For any solution u of this equation we consider the quasi-stationary regime, corresponding to t ≥ T1, where T1 depends only on f and on the distribution of η. We obtain sharp upper and lower bounds for Sobolev norms of u averaged in time and in ensemble. These results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence. All our bounds do not depend on the initial condition or on t for t ≥ T1, and hold uniformly in ν. Estimates similar to some of our results have been obtained by Aurell, Frisch, Lutsko and Vergassola on a physical level of rigour; we use an argument from their article.

Citation (ISO format)
BORITCHEV, Alexandre. Sharp Estimates for Turbulence in White-Forced Generalised Burgers Equation. In: Geometric and Functional Analysis, 2013, vol. 23, n° 6, p. 1730–1771. doi: 10.1007/s00039-013-0245-4
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ISSN of the journal1016-443X
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