{"id":273,"date":"2019-02-13T12:04:45","date_gmt":"2019-02-13T11:04:45","guid":{"rendered":"https:\/\/davidlannes.wordpress.com\/?p=273"},"modified":"2021-12-01T21:42:04","modified_gmt":"2021-12-01T20:42:04","slug":"ss","status":"publish","type":"post","link":"https:\/\/david-lannes.perso.math.cnrs.fr\/?p=273","title":{"rendered":"Waves interacting with a partially immersed obstacle in the Boussinesq regime"},"content":{"rendered":"\n<p class=\"has-medium-font-size wp-block-paragraph\">D. Bresch, D. Lannes, G. M\u00e9tivier, to appear in Analysis &amp; PDE, Analysis &amp; PDE, <strong>14<\/strong> (2021), 1085-1124    <a href=\"https:\/\/david-lannes.perso.math.cnrs.fr\/wp-content\/uploads\/2019\/02\/transmission-final.pdf\">Download<\/a>                                       <\/p>\n\n\n\n<p class=\"has-small-font-size wp-block-paragraph\">This paper is devoted to the derivation and mathematical analysis of a wave-structure interaction problem which can be reduced to a transmission problem for a Boussinesq system. Initial boundary value problems and transmission problems in dimension d= 1 for 2 x 2 hyperbolic systems are well understood. However, for many applications, and especially for the description of surface water waves, dispersive perturbations of hyperbolic systems must be considered. We consider here a conguration where the motion of the waves is governed by a Boussinesq system (a dispersive perturbation of the hyperbolic nonlinear shallow water equations), and in the presence of a fixed partially immersed obstacle. We shall insist on the differences and similarities with respect to the standard hyperbolic case, and focus our attention on a new phenomenon, namely, the apparition of a dispersive boundary layer. In order to obtain existence and uniform bounds on the solutions over the relevant time scale, a control of this dispersive boundary layer and of the oscillations in time it generates is necessary. This analysis leads to a new notion of compatibility condition that is shown to coincide with the standard hyperbolic compatibility conditions when the dispersive parameter is set to zero. To the authors&rsquo; knowledge, this is the rst time that these phenomena (likely to play a central role in the analysis of initial boundary value problems for dispersive perturbations of hyperbolic systems) are exhibited.<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" width=\"1526\" height=\"854\" src=\"https:\/\/david-lannes.perso.math.cnrs.fr\/wp-content\/uploads\/2019\/02\/fig2.png?w=1024\" alt=\"\" class=\"wp-image-278\" srcset=\"https:\/\/david-lannes.perso.math.cnrs.fr\/wp-content\/uploads\/2019\/02\/fig2.png 1526w, https:\/\/david-lannes.perso.math.cnrs.fr\/wp-content\/uploads\/2019\/02\/fig2-300x168.png 300w, https:\/\/david-lannes.perso.math.cnrs.fr\/wp-content\/uploads\/2019\/02\/fig2-768x430.png 768w, https:\/\/david-lannes.perso.math.cnrs.fr\/wp-content\/uploads\/2019\/02\/fig2-1024x573.png 1024w\" sizes=\"auto, (max-width: 1526px) 100vw, 1526px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><br><\/p>\n","protected":false},"excerpt":{"rendered":"<p>D. Bresch, D. Lannes, G. M\u00e9tivier, to appear in Analysis &amp; PDE, Analysis &amp; PDE, 14 (2021), 1085-1124 Download This paper is devoted to the derivation and mathematical analysis of a wave-structure interaction problem which can be reduced to a transmission problem for a Boussinesq system. Initial boundary value problems and transmission problems in dimension [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-273","post","type-post","status-publish","format-standard","hentry","category-recent"],"_links":{"self":[{"href":"https:\/\/david-lannes.perso.math.cnrs.fr\/index.php?rest_route=\/wp\/v2\/posts\/273","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/david-lannes.perso.math.cnrs.fr\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/david-lannes.perso.math.cnrs.fr\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/david-lannes.perso.math.cnrs.fr\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/david-lannes.perso.math.cnrs.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=273"}],"version-history":[{"count":2,"href":"https:\/\/david-lannes.perso.math.cnrs.fr\/index.php?rest_route=\/wp\/v2\/posts\/273\/revisions"}],"predecessor-version":[{"id":435,"href":"https:\/\/david-lannes.perso.math.cnrs.fr\/index.php?rest_route=\/wp\/v2\/posts\/273\/revisions\/435"}],"wp:attachment":[{"href":"https:\/\/david-lannes.perso.math.cnrs.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=273"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/david-lannes.perso.math.cnrs.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=273"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/david-lannes.perso.math.cnrs.fr\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=273"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}