Localized asymptotic solutions of the magneto dynamo equation in abc fields
One of the problems that arise in the linear theory of a magneto dynamo is to determine whether it is possible for the solution to increase without bound as \(t\to\infty\) for large Reynolds numbers. The mathematical statement of the problem involves investigating the Cauchy problem \[ {\partial H\over\partial t}+(v\cdot\nabla)H=(H\cdot\nabla)v+\nu\Delta H,\;x\in\mathbb{R}^3_x,\;H\bigl|_{t=0}=H_0(x),\text{ div }H_0=0.\tag{1} \] Here, \(v(x,t)\) is a specified vector function (flow rate of liquid), which is determined from the Euler equations and, consequently, \(\text{div }v=0\). The condition \(\text{div }H_0=0\) is, as may be easily verified, preserved by virtue of (1), and the field \(H(x,t)\) is also divergence free. The parameter \(\nu\) is equal to \(1/\text{Re}\), where Re is the Reynolds number, which is usually assumed to be small. We say that there exists a rapid dynamo in \(v\) if there exists a solution of (1) when \(\nu\ll 1\) that is at least exponentially increasing as \(t\to\infty\).
scientific article; zbMATH DE number 687520
scientific article