## Representation of a solution to the problem of evolution of a point vortex in an ideal fluid.(English. Russian original)Zbl 0859.76011

Sib. Math. J. 35, No. 2, 403-415 (1994); translation from Sib. Mat. Zh. 35, No. 2, 446-458 (1994).
The two-dimensional Euler equations for inviscid incompressible fluid are considered to describe the evolution of the vorticity $$\Omega(z,t)$$ and the velocity $$u(z,t)$$ with the initial data $$\Omega_0(z)=\omega_0(z) + \Gamma\delta(z-a_0)$$, $$z\in \mathbb{C}$$, concentrated at a point $$a_0\in \mathbb{C}$$, where $$\delta$$ is the Dirac function, $$\omega_0$$ is the regular part of vorticity, $$z=x+iy$$ is the complex variable, and $$\Gamma$$ is a real constant. An approximation $$\Omega_\varepsilon$$ is introduced with regular initial data $$\Omega_{0\varepsilon}(z)= \omega_0(z)+\varepsilon^{-2} \Gamma\chi_\varepsilon (z)$$, where $$\chi_\varepsilon$$ stands for the characteristic function of a ball with center $$a_0$$ and area $$\varepsilon^2$$. It is proved that $$\Omega_\varepsilon \to \omega(z,t)+\Gamma\delta(z-a(t)) \equiv\Omega$$ weakly, and $$u_\varepsilon\to v(z,t)+{\Gamma\over 2\pi i} {1\over z-a(t)}\equiv u$$ strongly as $$\varepsilon\to 0$$. The limit functions $$\Omega$$ and $$u$$ solve the Euler equations in a weak sense, and the function $$a(t)$$ solves the Cauchy problem $${d\over dt}a=v(a,t)$$, $$a(0)=a_0$$ in a weak sense, too.

### MSC:

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

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