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Convexity of Whitham's highest cusped wave

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arxiv 1810.10935 v1 pith:RXJO5T2X submitted 2018-10-25 math.AP

Alberto Enciso , Javier Gómez-Serrano , Bruno Vergara This is my paper · ORCID
classification math.AP
keywords wavewhithamconvexitycuspedhighestbeenconjecturedconsecutive
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abstract

We prove the existence of a periodic traveling wave of extreme form of the Whitham equation that has a convex profile between consecutive stagnation points, at which it is known to feature a cusp of exactly $C^{1/2}$ regularity. The convexity of Whitham's highest cusped wave had been conjectured by Ehrnstr\"om and Wahl\'en.

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    A rigorous computer-assisted proof establishes the existence of non-convex analytic uniformly rotating vortex patches with 6-fold symmetry for the 2D Euler equation.

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