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On pressureless gases driven by a strong inhomogeneous magnetic field. (English) Zbl 1145.35095
The authors study the asymptotic behavior of a fluid submitted to a strong inhomogeneous magnetic field. The simplified magnetohydrodynamics model is used: Euler system for pressureless gas dynamics. The flow is driven by the external magnetic field \(b(\vec x)\varepsilon^{-1}\), but no internal force (i.e. pressure) is present. The external magnetic field has constant direction. Thus the problem is considered in the plane orthogonal to the direcion of the field. The magnetic field intensity \(b(\vec x)\) is separated from zero and obeys certain regularity conditions in \(\mathbb R^2\). Under these assumptions the local well-posedness is proved by means of the fixed-point argument. The authors study the lifespan of the solution as \(\varepsilon\rightarrow 0\). They prove the existence and uniqueness of the solution on the uniform time interval.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
76U05 General theory of rotating fluids
76W05 Magnetohydrodynamics and electrohydrodynamics
35B20 Perturbations in context of PDEs
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