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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.

MSC:

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