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Explicit estimates on the torus for the sup-norm and the crest factor of solutions of the Modified Kuramoto-Sivashinky Equation in one and two space dimensions | | Michele V. Bartuccelli
; Jonathan H. Deane
; Guido Gentile
; | | Date: |
29 Sep 2016 | | Abstract: | We consider the Modified Kuramoto-Sivashinky Equation (MKSE) in one and two
space dimensions and we obtain explicit and accurate estimates of various
Sobolev norms of the solutions. In particular, by using the sharp constants
which appear in the functional interpolation inequalities used in the analysis
of partial differential equations, we evaluate explicitly the sup-norm of the
solutions of the MKSE. Furthermore we introduce and then compute the so-called
crest factor associated with the above solutions. The crest factor provides
information on the distortion of the solution away from its space average and
therefore, if it is large, gives evidence of strong turbulence. Here we find
that the time average of the crest factor scales like $lambda^{(2d-1)/8}$ for
$lambda$ large, where $lambda$ is the bifurcation parameter of the source
term and $d=1,2$ is the space dimension. This shows that strong turbulence
cannot be attained unless the bifurcation parameter is large enough. | | Source: | arXiv, 1609.9394 | | Services: | Forum | Review | PDF | Favorites | |
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