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Smoothness of the trajectories of ideal fluid particles with Yudovich vorticities in a planar bounded domain. (English) Zbl 1248.76011

The author considers incompressible Euler equations in a bounded domain \(\Omega\subset\mathbb{R}^2\), which may be multiply connected. He studies the regularity of the flow map. Precisely speaking, he assumes that the initial value of the solution belongs to \(L^2 (\Omega )\), it has a bounded vorticity, and it is tangent to the boundary \(\partial\Omega\). He proves that if \(\partial\Omega\) is smooth, then the flow map is smooth in time. Furthermore, if \(\partial\Omega\) has Gevrey regularity of degree \(M\geq 1\), then the flow map has Gevrey regularity of degree \(M+2\). Similar results were proved also by J.Y. Chemin, P. Serfati, and P. Gamblin. The author also considers the case of initial value with Hölder regularity, in this case, a similar result is true, modifying the Gevrey degree slightly.
Some generalizations are also considered. First, the case is studied when the initial value has an unbounded vorticity. It is shown that in this case one can prove a simlar result, modifying the function space with respect to space variables. Secondly, the author studies the case when \(\partial\Omega\) has only \(C^2\) regularity. In this case, if the initial value has a constant vorticity outside of a compact subset of \(\Omega\), then the flow map has Gevrey regularity of a certain degree.

MSC:

76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76B47 Vortex flows for incompressible inviscid fluids
35Q31 Euler equations
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