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We consider the Modified Kuramoto-Sivashinky Equation (MKSE) in one and twospace dimensions and we obtain explicit and accurate estimates of variousSobolev norms of the solutions. In particular, by using the sharp constantswhich appear in the functional interpolation inequalities used in the analysisof partial differential equations, we evaluate explicitly the sup-norm of thesolutions of the MKSE. Furthermore we introduce and then compute the so-calledcrest factor associated with the above solutions. The crest factor providesinformation on the distortion of the solution away from its space average andtherefore, if it is large, gives evidence of strong turbulence. Here we findthat the time average of the crest factor scales like $\lambda^{(2d-1)/8}$ for$\lambda$ large, where $\lambda$ is the bifurcation parameter of the sourceterm and $d=1,2$ is the space dimension. This shows that strong turbulencecannot be attained unless the bifurcation parameter is large enough.

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