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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

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 $λ^{(2d-1)/8}$ for $λ$ large, where $λ$ 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.