Quantization of energy and weakly turbulent profiles of the solutions to some damped second order evolution equations
Résumé
We consider a second order equation with a linear ``elastic''
part and a nonlinear damping term depending on a power of the
norm of the velocity. We investigate the asymptotic behavior of
solutions, after rescaling them suitably in order to take into
account the decay rate and bound their energy away from zero.
We find a rather unexpected dichotomy phenomenon. Solutions with finitely many Fourier
components are asymptotic to solutions of the linearized equation
without damping, and exhibit some sort of equipartition of the
energy among the components. Solutions with infinitely many
Fourier components tend to zero weakly but not strongly. We show
also that the limit of the energy of solutions depends only on the
number of their Fourier components.
The proof of our results is inspired by the analysis of a
simplified model which we devise through an averaging procedure,
and whose solutions exhibit the same asymptotic properties as the
solutions to the original equation.
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