Representation of a solution to the problem of evolution of a point vortex in an ideal fluid
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.
scientific article; zbMATH DE number 916371
scientific article