{"id":261,"date":"2022-08-10T17:41:11","date_gmt":"2022-08-10T21:41:11","guid":{"rendered":"https:\/\/dev.eventos.cmm.uchile.cl\/pdeseminar\/?p=261"},"modified":"2022-08-17T16:49:40","modified_gmt":"2022-08-17T20:49:40","slug":"on-the-fractional-zakharov-kuznetsov-equation","status":"publish","type":"post","link":"https:\/\/dev.eventos.cmm.uchile.cl\/pdeseminar\/2022\/08\/on-the-fractional-zakharov-kuznetsov-equation\/","title":{"rendered":"On the fractional Zakharov-Kuznetsov equation"},"content":{"rendered":"

Speaker: Argenis Mendez<\/b>\u00a0<\/a><\/h3>\n

Pontificia Universidad Cat\u00f3lica de Valpara\u00edso<\/h3>\n

Date: August 16, 2022 at 12 Santiago time<\/h3>\n

Abstract:<\/b> In this talk, we will present some new results related to the regularity properties of the initial value problem (IVP) for the equation
\n\\begin{equation}\\label{eq1}
\n\\left\\{
\n\\begin{array}{ll}
\n\\partial_{t}u-\\partial_{x_{1}}(-\\Delta)^{\\alpha\/2} u+u\\partial_{x_{1}}u=0, \\quad 0< \\alpha< 2, & \\\\
\nu(x,0)=u_{0}(x),x=(x_{1},x_{2},\\dots,x_{n})\\in \\mathbb{R}^{n},n\\geq 2,& t\\in\\mathbb{R}, \\\\
\n\\end{array}
\n\\right.
\n\\end{equation}
\nwhere $(-\\Delta)^{\\alpha\/2}$ denotes the $n-$dimensional fractional Laplacian<\/em>.<\/p>\n

In the case that \\(\\alpha=2,\\) the equation is known as the Zakharov-Kuznetsov<\/em>-(ZK) equation, Zakharov and Kuznetsov proposed it as a model to describe the propagation of ion-sound waves in magnetic fields in dimension n=3.<\/p>\n

A property that enjoys the solutions of the ZK equation is Kato’s smoothing effect<\/em>. Roughly speaking, the solution to the initial value problem is, locally, one derivative smoother (in all directions) in comparison to the initial data.<\/p>\n

The goal of this talk is to show that despite the non-local character of the operator \\((-\\Delta)^{\\frac{\\alpha}{2}}\\), the solution of the equation (IVP) is locally smoother. It becomes \\(\\frac{\\alpha}{2}-\\) smoother in all directions.<\/p>\n

As a byproduct, we show the applicability of this result in establishing the propagation of localized regularity of the solutions of (IVP) in a suitable Sobolev space.<\/p>\n

Venue:<\/strong> Online via Zoom \/ Sala de seminarios John Von Neumann, Beauchef 851,\u00a0 piso 7
\nChair:<\/strong> Claudio Mu\u00f1oz<\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

Argenis Mendez (Puc-Valpara\u00edso, Chile)<\/p>\n","protected":false},"author":85,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[5],"tags":[],"class_list":["post-261","post","type-post","status-publish","format-standard","hentry","category-past-seminar"],"_links":{"self":[{"href":"https:\/\/dev.eventos.cmm.uchile.cl\/pdeseminar\/wp-json\/wp\/v2\/posts\/261","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/dev.eventos.cmm.uchile.cl\/pdeseminar\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/dev.eventos.cmm.uchile.cl\/pdeseminar\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/dev.eventos.cmm.uchile.cl\/pdeseminar\/wp-json\/wp\/v2\/users\/85"}],"replies":[{"embeddable":true,"href":"https:\/\/dev.eventos.cmm.uchile.cl\/pdeseminar\/wp-json\/wp\/v2\/comments?post=261"}],"version-history":[{"count":4,"href":"https:\/\/dev.eventos.cmm.uchile.cl\/pdeseminar\/wp-json\/wp\/v2\/posts\/261\/revisions"}],"predecessor-version":[{"id":301,"href":"https:\/\/dev.eventos.cmm.uchile.cl\/pdeseminar\/wp-json\/wp\/v2\/posts\/261\/revisions\/301"}],"wp:attachment":[{"href":"https:\/\/dev.eventos.cmm.uchile.cl\/pdeseminar\/wp-json\/wp\/v2\/media?parent=261"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/dev.eventos.cmm.uchile.cl\/pdeseminar\/wp-json\/wp\/v2\/categories?post=261"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/dev.eventos.cmm.uchile.cl\/pdeseminar\/wp-json\/wp\/v2\/tags?post=261"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}