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Local well-posedness of the EPDiff equation: a survey. (English) Zbl 1362.58003

Summary: This article is a survey on the local well-posedness problem for the general EPDiff equation. The main contribution concerns recent results on local existence of the geodesics on \(\mathrm{Diff}^{\infty}(\mathbb{T}^d)\) and \(\mathrm{Diff}_{H^{\infty}}(\mathbb{R}^d)\) when the inertia operator is a non-local Fourier multiplier.

MSC:

58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
35Q35 PDEs in connection with fluid mechanics
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