Global regularity for vortex patches. (English) Zbl 0771.76014

A vortex patch is a simply connected, open and bounded domain \(D\) of \(\mathbb{R}^ 2\) that moves with velocity \(v\) that is derived from the stream function \(\Psi(x)=(2\pi)^{-1}\omega_ 0\int_ D\log| x-y| dy\), where \(\omega_ 0\) is a constant. The paper seeks a scalar function \(\phi(x,t)\) governed by the equation \(\bigl({\partial\over \partial t}+v\cdot\text{grad}\bigr)\phi=0\), \(\phi(x,0)=\phi_ 0(x)\), where \(v\) is given the Biot-Savart formula \(2\pi\;v(x,t)=\omega_ 0\int_ D\nabla^ \perp_ x\log| x-y| dy\), with \(D=D(t)=\bigl\{x\in\mathbb{R}^ 2\mid\phi(x)>0\bigr\}\), and proves the main result that the boundary of a vortex patch remains smooth for all time if it is initially smooth.


76B47 Vortex flows for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
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