Spectral Compactness and Critical Lorentz Defects for Maxwell--Ohm Evolution with Prescribed Velocity

Abstract

We establish a spectral weak-to-strong stability criterion and a sharp Sobolev dichotomy for Maxwell--Ohm evolution with prescribed velocity on $\T^d$, $d=2,3$. Uniform spatial spectral tightness, together with weak convergence of each fixed spatial Fourier mode and strong convergence of the electromagnetic initial states, yields strong convergence of the electromagnetic fields, weak convergence of the currents, and distributional convergence of the Lorentz forces. In particular, weak convergence of the velocity coefficients in $L^2_tH^s_x$ suffices when $s>d/2$. At the critical index $s=d/2$, this conclusion fails sharply. We construct smooth divergence-free velocities converging strongly to zero in the critical Sobolev space, together with electromagnetic initial states converging strongly to zero, while the corresponding terminal fields remain of order one and the currents stay bounded. In two dimensions the construction is based on a moving logarithmic core, whereas in three dimensions it uses a Fourier-capacity core built from Leray-projected lattice modes and a polarized six-component Maxwell packet with exact magnetic divergence constraint. In both dimensions, the Lorentz forces converge to an explicit nonzero measure supported on a single ray with fixed unit direction.