Delta-type solutions for a system of induction equations with discontinuous velocity field (Q2877331)

The authors study the Cauchy problem for the equation NEWLINE\[NEWLINE \frac{\partial B}{\partial t}+(V,\nabla )B-(B,\nabla )V=\varepsilon^2\mu \Delta B,\quad (\nabla,V)=(\nabla,B)=0,\quad t>0,x\in \mathbb R^n NEWLINE\]NEWLINE (\(n=2\, \text{or}\, 3\)). 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.