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Global small solutions for the Navier-Stokes-Maxwell system. (English) Zbl 1233.35157
Summary: We consider a full system of magnetohydrodynamic equations. The system does not enjoy any property of scaling invariance and, at least formally, has an energy estimate. Nevertheless, the existence of a global weak solution seems to remain an interesting open problem in both two and three space dimensions. In three dimensions, we show the existence of global small mild solutions. In two dimensions, we prove the same result in a space “close” to the energy space.

35Q30Stokes and Navier-Stokes equations
35Q61Maxwell equations
82C31Stochastic methods in time-dependent statistical mechanics
76D05Navier-Stokes equations (fluid dynamics)
76W05Magnetohydrodynamics and electrohydrodynamics
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