The free boundary problem for Euler’s equations

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Johns Hopkins University
We consider the motion of a perfect uid body in vaccuum with no surface tension, in two settings. First, we study the motion of a compressible liquid occupying a bounded region of space which is subject to self-gravitational force. For this system, we construct a local-in-time solution in Sobolev spaces provided the Taylor sign condition holds initially and that the initial data satis es certain compatibility conditions. Second, we consider the motion of an incompressible uid subject to a uniform force of gravity which occupies an unbounded region (the water-waves system). We prove a long-time existence result for small, well-localized initial data with small initial vorticity which vanishes on the free boundary.
Fluid mechanics, free boundary problems, water waves