# MATHEMATICAL ANALYSIS OF ONE-FLUID PROBLEMS

# MATHEMATICAL ANALYSIS OF ONE-FLUID PROBLEMS

This chapter focuses on the modelling of one-fluid magnetohydrodynamics problems. The crucial point under consideration is the coupling between hydrodynamics phenomena and electromagnetic phenomena. From a mathematical viewpoint, the coupling induces a nonlinearity, additional to the nonlinearities already present in the hydrodynamics. A series of difficult, thus interesting, problems follow. With a reasonable amount of theoretical efforts, these problems can be dealt with. For instance, it can be shown that a system coupling the time-dependent incompressible Navier-Stokes equations with a simplified form of the Maxwell equations (the so-called low-frequency approximation) is well-posed when the electromagnetic equation is taken time-dependent, in parabolic form. In contrast, the same model is likely to be ill-posed when the electromagnetic equation is taken time-independent, in elliptic form.

*Keywords:*
coupled problem, low-frequency approximation, energy inequalities, Navier-Stokes equations, Maxwell equations, a priori estimates

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