Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle
نویسندگان
چکیده
In this paper, we study the geodesic flow of a right-invariant metric induced by a general Fourier multiplier on the diffeomorphism group of the circle and on some of its homogeneous spaces. This study covers in particular right-invariant metrics induced by Sobolev norms of fractional order. We show that, under a certain condition on the symbol of the inertia operator (which is satisfied for the fractional Sobolev norm H for s ≥ 1/2), the corresponding initial value problem is well-posed in the smooth category and that the Riemannian exponential map is a smooth local diffeomorphism. Paradigmatic examples of our general setting cover, besides all traditional Euler equations induced by a local inertia operator, the Constantin-Lax-Majda equation, and the EulerWeil-Petersson equation.
منابع مشابه
Right-invariant Sobolev Metrics of Fractional Order on the Diffeomorphisms Group of the Circle
In this paper we study the geodesic flow of a right-invariant metric induced by a general Fourier multiplier on the diffeomorphisms group of the circle and on some of its homogeneous spaces. This study covers in particular right-invariant metrics induced by Sobolev norms of fractional order. We show that, under a certain condition on the symbol of the inertia operator (which is satisfied for th...
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