Delta-type solutions for a system of induction equations with discontinuous velocity field. (English) Zbl 1313.35335

The authors study the Cauchy problem for the equation \[ \frac{\partial B}{\partial t}+(V,\nabla )B-(B,\nabla )V=\varepsilon^2\mu \Delta B,\quad (\nabla,V)=(\nabla,B)=0,\quad t>0,x\in \mathbb R^n \] (\(n=2\, \text{or}\, 3\)). Here, \(B\) is a magnetic field, \(V\) is a given vector field (fluid velocity), and \(\varepsilon \to 0\). It is assumed that \(V\) changes rapidly in the vicinity of a surface. It is proved that the weak limit of the solution has a delta-type singularity on the surface. The Green matrix estimates and asymptotic series for solutions are obtained.


35Q60 PDEs in connection with optics and electromagnetic theory
35Q35 PDEs in connection with fluid mechanics