[Estimations a priori pour le problème de l’évolution de la ligne de contact pour les équations de Saint-Venant $2D$ avec un obstacle partiellement immergé]
We consider the initial value problem for the $2D$ nonlinear shallow water model in the presence of a fixed partially immersed solid body. In this problem, we have a contact line where the solid body, the water, and the air meet, and whose projection on the horizontal plane moves freely even if the solid body is fixed. This wave-structure interaction problem reduces to an initial boundary value problem for the nonlinear shallow water equations in an exterior domain with a free boundary, for which we prove a priori energy estimates locally in time for solutions at the quasilinear regularity threshold under irrotationality and subcriticality assumptions. We use the weak dissipativity of the system, second order Alinhac good unknowns associated with a regularizing diffeomorphism, and a new type of hidden boundary regularity.
Nous étudions le problème de Cauchy pour les équations de Saint-Venant $2D$ en présence d’un objet fixe partiellement immergé. Nous sommes en présence d’une ligne de contact où le solide, le liquide et l’air se rencontrent et dont la projection sur le plan horizontal évolue librement même si le solide est immobile. Ce problème d’interaction vague-structure se réduit à un problème mixte pour les équations de Saint-Venant, dans un domaine extérieur avec frontière libre. Nous montrons des estimations d’énergie a priori, localement en temps, pour des solutions au seuil de régularité quasilinéaire, sous les hypothèses que l’écoulement est initialement irrotationnel et sous-critique. Nous utilisons la dissipativité faible du système, des inconnues d’Alinhac d’ordre $2$ associées à un difféomorphisme régularisant, ainsi qu’un nouveau type de régularité cachée au bord.
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Keywords: Wave-structure interactions, nonlinear hyperbolic initial boundary value problems, moving contact lines
Mots-clés : Interactions vague-structure, problème mixte hyperbolique non linéaire, dynamique de la ligne de contact
Tatsuo Iguchi 1 ; David Lannes 2
CC-BY 4.0
@article{JEP_2026__13__137_0,
author = {Tatsuo Iguchi and David Lannes},
title = {A priori estimates for the moving contact line problem for the {2\protect\emph{D}} nonlinear shallow~water equations with a~partially~immersed obstacle},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {137--202},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.325},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.325/}
}
TY - JOUR AU - Tatsuo Iguchi AU - David Lannes TI - A priori estimates for the moving contact line problem for the 2D nonlinear shallow water equations with a partially immersed obstacle JO - Journal de l’École polytechnique — Mathématiques PY - 2026 SP - 137 EP - 202 VL - 13 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.325/ DO - 10.5802/jep.325 LA - en ID - JEP_2026__13__137_0 ER -
%0 Journal Article %A Tatsuo Iguchi %A David Lannes %T A priori estimates for the moving contact line problem for the 2D nonlinear shallow water equations with a partially immersed obstacle %J Journal de l’École polytechnique — Mathématiques %D 2026 %P 137-202 %V 13 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.325/ %R 10.5802/jep.325 %G en %F JEP_2026__13__137_0
Tatsuo Iguchi; David Lannes. A priori estimates for the moving contact line problem for the 2D nonlinear shallow water equations with a partially immersed obstacle. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 137-202. doi: 10.5802/jep.325
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