## 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.

### MSC:

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

### Keywords:

stream function; Biot-Savart formula
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### References:

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