This (Diplom-) thesis deals with the particle trajectories of an incompressible and ideal fluid flow in 𝑛 ≥ 2 dimensions. It presents a complete and detailed proof of the surprising fact that the trajectories of a smooth solution of the incompressible Euler equations are locally analytic in time. In following the approach of P. Serfati, a complex ordinary differential equation (ODE) is investigated which can be seen as a complex extension of a partial differential equation, which is solved by the trajectories. The right hand side of this ODE is in fact given by a singular integral operator which coincides with the pressure gradient along the trajectories. Eventually, we may apply the Cauchy-Lipschitz existence theorem involving holomorphic maps between complex Banach spaces in order to get a unique solution for the above mentioned ODE. This solution is
real-analytic in time and coincides with the particle trajectories.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:17052 |
| Date | 22 January 2018 |
| Creators | Hertel, Tobias |
| Contributors | Universität Leipzig |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/acceptedVersion, doc-type:masterThesis, info:eu-repo/semantics/masterThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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