Abstract. In this note we provide new non-uniqueness examples for the continuity equation by constructing infinitely many weak solutions with prescribed energy.

Magnetometry and visualization of very small magnetic fields are vital for a large variety of the areas ranging from magnetocardiography and encephalography to nondistractive defectoscopy and ultra-low-frequency communications. It is very advantageous to measure magnetic fields using exchange-coupled spins in magnetically ordered media (flux-gate magnetometry).Here we introduce and demonstrate a novel concept of a roomtemperature magnetoplasmonic magnetic field sensor with high sensitivity and spatial resolution. It is based on the advanced fluxgate technique in which magnetization of the fully saturated magnetic film is rotated in the film plane and the monitored magnetic field is measured by detecting variation of transmittance through the sensing element: a magnetoplasmonic crystal. The experimental study revealed that such an approach allows one to reach the nT sensitivity level, which was limited by the noise of the laser. Moreover, we propose an approach to improve the sensitivity up to fT/Hz 1/2 and reach micrometer spatial resolution. Therefore, the demonstrated magnetoplasmonic magnetometry method is promising for mapping and visualization of ultrasmall magnetic fields.

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