{"id":451,"date":"2022-04-12T10:00:19","date_gmt":"2022-04-12T08:00:19","guid":{"rendered":"https:\/\/david-lannes.perso.math.cnrs.fr\/?p=451"},"modified":"2022-04-12T10:03:40","modified_gmt":"2022-04-12T08:03:40","slug":"initial-boundary-value-problems-for-hyperbolic-systems-and-dispersive-perturbations","status":"publish","type":"post","link":"https:\/\/david-lannes.perso.math.cnrs.fr\/?p=451","title":{"rendered":"Initial boundary value problems for hyperbolic systems, and dispersive perturbations"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">D. Lannes, <em>Lecture notes volume of the Winter School on Fluid Dynamics, Dispersive Equations and Quantum Fluids<\/em> (Bressanone, december 2018), to appear.  <a href=\"https:\/\/david-lannes.perso.math.cnrs.fr\/wp-content\/uploads\/2022\/04\/Bressanone_lannes.pdf\">Download<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The goal of these notes is to point out similarities and differences between\ntwo kinds of initial boundary value problems in dimension one. The first one concerns\nhyperbolic systems (such as the nonlinear shallow water equations) while the second\none concerns dispersive perturbations of such systems (such as Boussinesq systems).\nIn the absence of a boundary, that is, for the initial value problem, the link between\nboth classes is quite obvious but in the presence of a boundary, the situation is\nmore complex and dispersive boundary layers must be understood if one wants to\nunderstand the links between both classes of problems. After reviewing several types\nof initial boundary value problems (some of which being free boundary problems)\narising in the study of waves in shallow water, we sketch the general theory for\nhyperbolic initial boundary value problems developed in [18] and that encompasses\nall of the above examples that involve hyperbolic systems. Such a general theory\ndoes not exist for dispersive perturbations of hyperbolic systems, but we treat two\nimportant examples involving Boussinesq systems. In the first one, we show that\nthe nature of the initial boundary value problem shares little in common with the\nhyperbolic configuration. For instance, the problem has the structure of an ODE and\nno higher order compatibility conditions on the data are required to have solutions\nof high regularity. These differences naturally raise the questions of the control of\nthe time of existence and of the dispersionless limit; they are addressed in a second\nexample motivated by a wave-structure interaction problem. We explain the approach\ndeveloped in [13] to treat this problem, pointing out the role played by dispersive\nboundary layers.\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>D. Lannes, Lecture notes volume of the Winter School on Fluid Dynamics, Dispersive Equations and Quantum Fluids (Bressanone, december 2018), to appear. Download The goal of these notes is to point out similarities and differences between two kinds of initial boundary value problems in dimension one. The first one concerns hyperbolic systems (such as the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-451","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\/451","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=451"}],"version-history":[{"count":3,"href":"https:\/\/david-lannes.perso.math.cnrs.fr\/index.php?rest_route=\/wp\/v2\/posts\/451\/revisions"}],"predecessor-version":[{"id":454,"href":"https:\/\/david-lannes.perso.math.cnrs.fr\/index.php?rest_route=\/wp\/v2\/posts\/451\/revisions\/454"}],"wp:attachment":[{"href":"https:\/\/david-lannes.perso.math.cnrs.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=451"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/david-lannes.perso.math.cnrs.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=451"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/david-lannes.perso.math.cnrs.fr\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=451"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}