On the well-posedness of the incompressible Euler equation

Inci, Hasan. On the well-posedness of the incompressible Euler equation. 2013, University of Zurich, Faculty of Science.

Abstract

In this thesis we prove that the homogeneous incompressible Euler equation of hydrodynamics on the Sobolev spaces H s (Rn ), n ≥ 2 and s > n/2 + 1, can be ex- pressed as a geodesic equation on an inﬁnite dimensional manifold. As an application of this geometric formulation we prove that the solution map of the incompressible Euler equation, associating intial data in H s (Rn ) to the corresponding solution at time t > 0, is nowhere locally uniformly continuous and nowhere diﬀerentiable.

Abstract

In this thesis we prove that the homogeneous incompressible Euler equation of hydrodynamics on the Sobolev spaces H s (Rn ), n ≥ 2 and s > n/2 + 1, can be ex- pressed as a geodesic equation on an inﬁnite dimensional manifold. As an application of this geometric formulation we prove that the solution map of the incompressible Euler equation, associating intial data in H s (Rn ) to the corresponding solution at time t > 0, is nowhere locally uniformly continuous and nowhere diﬀerentiable.

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