%0 Journal Article
%T Semilinear geometric optics with boundary amplification
%+ Equations aux dérivées partielles
%+ Laboratoire d'Analyse, Topologie, Probabilités (LATP)
%+ Department of Mathematics [Chapel Hill]
%A Coulombel, Jean-François
%A Guès, Olivier
%A Williams, Mark
%< avec comité de lecture
%@ 2157-5045
%J Analysis & PDE
%I Mathematical Sciences Publishers
%V 7
%N 3
%P 551--625
%8 2014
%D 2014
%Z 1203.0479
%K Nash-Moser iteration
%K Hyperbolic systems
%K boundary conditions
%K weakly nonlinear geometric optics
%K Nash-Moser iteration.
%Z 35L50; 78A05
%Z Mathematics [math]/Analysis of PDEs [math.AP]Journal articles
%X We study weakly stable semilinear hyperbolic boundary value problems with highly oscillatory data. Here weak stability means that exponentially growing modes are absent, but the so-called uniform Lopatinskii condition fails at some boundary frequency $\beta$ in the hyperbolic region. As a consequence of this degeneracy there is an amplification phenomenon: outgoing waves of amplitude $O(\varepsilon^2)$ and wavelength $\varepsilon$ give rise to reflected waves of amplitude $O(\varepsilon)$, so the overall solution has amplitude $O(\varepsilon)$. Moreover, the reflecting waves emanate from a radiating wave that propagates in the boundary along a characteristic of the Lopatinskii determinant. An approximate solution that displays the qualitative behavior just described is constructed by solving suitable profile equations that exhibit a loss of derivatives, so we solve the profile equations by a Nash-Moser iteration. The exact solution is constructed by solving an associated singular problem involving singular derivatives of the form $\partial_{x'}+\beta\frac{\partial_{\theta_0}}{\varepsilon}$, $x'$ being the tangential variables with respect to the boundary. Tame estimates for the linearization of that problem are proved using a first-order calculus of singular pseudodifferential operators constructed in the companion article \cite{CGW2}. These estimates exhibit a loss of one singular derivative and force us to construct the exact solution by a separate Nash-Moser iteration. The same estimates are used in the error analysis, which shows that the exact and approximate solutions are close in $L^\infty$ on a fixed time interval independent of the (small) wavelength $\varepsilon$. The approach using singular systems allows us to avoid constructing high order expansions and making small divisor assumptions.
%G English
%2 https://hal.science/hal-00675979/document
%2 https://hal.science/hal-00675979/file/CGWII.pdf
%L hal-00675979
%U https://hal.science/hal-00675979
%~ UNIV-NANTES
%~ LMJL
%~ LATP
%~ CNRS
%~ UNIV-AMU
%~ FMPL
%~ INSMI
%~ I2M
%~ CHL
%~ TDS-MACS
%~ ANR
%~ NANTES-UNIVERSITE
%~ UNIV-NANTES-AV2022