{"657159":{"#nid":"657159","#data":{"type":"event","title":"PhD Defense by Xiao Liu","body":[{"value":"\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003ETitle: Capillary Gravity Water Wave Linearized at Monotone Shear Flows: Eigenvalues and Inviscid Damping\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EXiao Liu\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EMathematics Ph.D. Candidate\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EGeorgia Institute of Technology\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EEmail: \u003Ca href=\u0022mailto:xliu458@gatech.edu\u0022\u003Exliu458@gatech.edu\u003C\/a\u003E\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EDate: Friday, April 8, 2022\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003ETime: 09:30 AM to 11:30 AM (EST)\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EMeeting Link:\u0026nbsp;\u003Ca href=\u0022https:\/\/bluejeans.com\/421317143\/2787\u0022\u003Ehttps:\/\/bluejeans.com\/421317143\/2787\u003C\/a\u003E\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003ECommittee:\u003C\/p\u003E\r\n\r\n\u003Cp\u003EDr. Chongchun Zeng(Advisor)\u0026mdash;School of Mathematics, Georgia Tech\u003C\/p\u003E\r\n\r\n\u003Cp\u003EDr. Ronghua Pan\u0026mdash;School of Mathematics, Georgia Tech\u003C\/p\u003E\r\n\r\n\u003Cp\u003EDr. Zhiwu Lin\u0026mdash;School of Mathematics, Georgia Tech\u003C\/p\u003E\r\n\r\n\u003Cp\u003EDr. Rafael De La Llave\u0026mdash;School of Mathematics, Georgia Tech\u003C\/p\u003E\r\n\r\n\u003Cp\u003EDr. Yao Yao\u0026mdash;Department of Mathematics, National University of Singapore\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EAbstract:\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EThis work is concerned with the two dimensional capillary gravity water waves of finite depth $x_2 \\in (-h, 0)$ linearized at a uniformly monotonic shear flow $U(x_2)$. We focus on the eigenvalue distribution and linear inviscid damping. Unlike the linearized Euler equation in a fixed channel at a shear flow where eigenvalues exist only in low wave numbers $k$ of the horizontal variable $x_1$, we first prove that the linearized capillary gravity wave has two branches of eigenvalues $-ik c^\\pm (k)$, where the wave speeds $c^\\pm (k) = O(\\sqrt{|k|})$ for $|k|\\gg1$ have the same asymptotics as the those of the linear irrotational capillary gravity waves. Under the additional assumption of $U\u0026#39;\u0026#39;\\ne 0$, we obtain the complete continuation of these two branches, which are all the eigenvalues of the linearized capillary gravity waves in this (and some other) case(s). In particular, $-ik c^-(k)$ could bifurcate into unstable eigenvalues at $c^-(k)=U(-h)$. In general the bifurcation of unstable eigenvalues from inflection values of $U$ is also obtained. Assuming there are no singular modes, i.e. no embedded eigenvalues for any horizontal wave number $k$, linear solutions $(v(t, x), \\eta(t, x_1))$ are considered in both periodic-in-$x_1$ and $x_1\\in\\R$ cases, where $v$ is the velocity and $\\eta$ the surface profile. Each solution can be split into $(v^p, \\eta^p)$ and $(v^c, \\eta^c)$ whose $k$-th Fourier modes in $x_1$ correspond to the eigenvalues \u0026nbsp;and the continuous spectra of the wave number $k$, respectively. The component $(v^p, \\eta^p)$ is governed by a (possibly unstable) dispersion relation given by the eigenvalues, which is simply $k \\to k c^\\pm (k)$ in the case of $x_1 \\in \\R$ and is conjugate to the linear irrotational capillary gravity waves under certain conditions. The other component $(v^c, \\eta^c)$ satisfies the linear inviscid damping as fast as $|v_1^c|_{L_x^2}, |\\eta^c|_{L_2^x} = O(\\frac 1{|t|})$ and $|v_2^c|_{L_x^2}=O(\\frac 1{t^2})$ as $|t| \\to \\infty$. Furthermore, additional decay of $tv_1^c, t^2 v_2^c$ in $L_x^2 L_t^q$, $q\\in (2, \\infty]$, is obtained after leading asymptotic terms are singled out, which are in the forms of $t$-dependent translations in $x_1$ of certain functions of $x$. \u0026nbsp;The proof is based on detailed analysis of the Rayleigh equation.\u0026nbsp;\u003C\/p\u003E\r\n","summary":null,"format":"limited_html"}],"field_subtitle":"","field_summary":"","field_summary_sentence":[{"value":"Capillary Gravity Water Wave Linearized at Monotone Shear Flows: Eigenvalues and Inviscid Damping"}],"uid":"27707","created_gmt":"2022-04-11 15:25:56","changed_gmt":"2022-04-11 15:25:56","author":"Tatianna Richardson","boilerplate_text":"","field_publication":"","field_article_url":"","field_event_time":{"event_time_start":"2022-04-08T10:30:00-04:00","event_time_end":"2022-04-08T12:30:00-04:00","event_time_end_last":"2022-04-08T12:30:00-04:00","gmt_time_start":"2022-04-08 14:30:00","gmt_time_end":"2022-04-08 16:30:00","gmt_time_end_last":"2022-04-08 16:30:00","rrule":null,"timezone":"America\/New_York"},"extras":[],"groups":[{"id":"221981","name":"Graduate Studies"}],"categories":[],"keywords":[{"id":"100811","name":"Phd Defense"}],"core_research_areas":[],"news_room_topics":[],"event_categories":[{"id":"1788","name":"Other\/Miscellaneous"}],"invited_audience":[{"id":"78761","name":"Faculty\/Staff"},{"id":"78771","name":"Public"},{"id":"78751","name":"Undergraduate students"}],"affiliations":[],"classification":[],"areas_of_expertise":[],"news_and_recent_appearances":[],"phone":[],"contact":[],"email":[],"slides":[],"orientation":[],"userdata":""}}}