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.
Convexity of Whitham's highest cusped wave
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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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Existence of analytic non-convex V-states
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.