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Pré-Publication, Document De Travail Année : 2017

Quantization of energy and weakly turbulent profiles of the solutions to some damped second order evolution equations

Marina Ghisi
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  • PersonId : 986855
Massimo Gobbino
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  • PersonId : 986856
Alain Haraux
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  • PersonId : 836471

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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hal-01449019 , version 1 (29-01-2017)

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Marina Ghisi, Massimo Gobbino, Alain Haraux. Quantization of energy and weakly turbulent profiles of the solutions to some damped second order evolution equations. 2017. ⟨hal-01449019⟩
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