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Session S37 - New Developments in Mathematical Fluid Dynamics

Thursday, July 15, 12:00 ~ 12:25 UTC-3

Uniqueness and convexity of Whitham’s highest cusped wave II

Bruno Vergara

University of Barcelona, Spain   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloake727020ffe54ddffe2cfe401ef169abc').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addye727020ffe54ddffe2cfe401ef169abc = 'bv&#101;rg&#97;r&#97;' + '&#64;'; addye727020ffe54ddffe2cfe401ef169abc = addye727020ffe54ddffe2cfe401ef169abc + '&#117;b' + '&#46;' + '&#101;d&#117;'; var addy_texte727020ffe54ddffe2cfe401ef169abc = 'bv&#101;rg&#97;r&#97;' + '&#64;' + '&#117;b' + '&#46;' + '&#101;d&#117;';document.getElementById('cloake727020ffe54ddffe2cfe401ef169abc').innerHTML += '<a ' + path + '\'' + prefix + ':' + addye727020ffe54ddffe2cfe401ef169abc + '\'>'+addy_texte727020ffe54ddffe2cfe401ef169abc+'<\/a>';

Whitham’s equation is a nonlinear, nonlocal, very weakly dispersive shallow water wave model in one space dimension. In this talk we are concerned with non-smooth traveling wave solutions to this equation, which are often referred to as waves of extreme form. Their existence was conjectured by Whitham in 1967 and established by Ehrnström and Wahlén just a few years ago, who proved that there is a monotone traveling wave featuring a cusp of exactly $C^{1/2}$ Hölder regularity at the origin. Our objetive in this talk is to show that there is only one monotone traveling wave of extreme form and that, as widely believed in the community, its profile is in fact convex between crest and trough. This can be understood as the counterpart, in the case of the Whitham equation, of the landmark results on the uniqueness and convexity of traveling water waves of extreme form.

Joint work with Alberto Enciso (ICMAT, Spain) and Javier Gómez-Serrano (Brown University, USA - University of Barcelona, Spain).