# ASYMPTOTICAL METHODS FOR HELMHOLTZ OR NAVIER-STOKES TYPE EQUATIONS

Abstract : In this thesis, we study two diﬀerential problems which depend on a small parameter ε. We study the asymptotic of the solutions when ε tends to 0. The ﬁrst problem deals with the high-frequency Helmholtz equation. We construct a non-trapping potential which does not satisfy the refocusing condition introduced by F. Castella. We prove that the Hamiltonian tra jectories (associated with this potential) issued from 0 which go back to the origin form a submanifold of dimension d − 1, where d denotes the space dimension. We show that the solution converges to a perturbation of the out-going solution with coeﬃcients frozen at 0. Then we study a Navier-Stokes type equation forced by a polarised and oscillating source. We exhibit a family of exact solutions to the problem. We study the stability of this solution when we perturb it at the initial time. We construct an approximated solution of this problem thanks to a boundary layer in time in t = 0. In particular, it shows that interactions of oscillating waves, which propagate at diﬀerent scales, can be modelised at macroscopic scales by some creation of dissipation. Finally, we justify the convergence of the approximated solution towards the exact solution by performing some energy methods.
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https://tel.archives-ouvertes.fr/tel-00606023
Contributor : Aurélien Klak <>
Submitted on : Tuesday, July 5, 2011 - 11:26:51 AM
Last modification on : Thursday, November 15, 2018 - 11:56:11 AM
Long-term archiving on : Monday, November 12, 2012 - 10:10:10 AM

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• HAL Id : tel-00606023, version 1

### Citation

Aurélien Klak. ASYMPTOTICAL METHODS FOR HELMHOLTZ OR NAVIER-STOKES TYPE EQUATIONS. Mathematics [math]. Université Rennes 1, 2011. English. ⟨tel-00606023⟩

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