In a Tokamak the plasma is confined inside a toroidal vacuum vessel by the magnetic field. In the present Tokamak modeling the plasma boundary is a flux line, which is a free boundary and in contact with a given limiter. We gire a model for the plasma equilibrium in a Tokamak with a limiter anda nonlinear plasma current. And we prove, under suitable conditions on the nonlinearity of the plasma current, the existence of such an equilibrium for given external current. w IntroductionWe shall considera problem which has its origin in present plasma physics. It stems from the model describing the equilibrium of plasma in the Tokamak, where the plasma is confined in the vacuum vessel by the magnetic field due to the external currents. And the hotter the plasma is, the better. But ir super hot plasma touched the wall of the vessel, it would melt down, creating panic. Therefore the volume and the shape of the plasma must be controlled, anda limiter is provided in the vacuum vessel in the Tokamak to prevent the plasma from touching the vacuum vessel. But the volume of plasma should be as large as possible, therefore the desired plasma boundary must touch the limiter and satisfy the contact condition.The mathematical modeling of the plasma problem is formulated asa kind of nonlinear eigenvalue problem: the plasma region is the positive region of the solution of the equations and the plasma boundary is a free boundary.'And in the case with the limiter, the positive region must be controlled to satisfy the contact condition with the limiter. This gives a new feature to the problems. Many papers have been devoted to the equilibrium problem without a limiter. In particular R. Temam [7], H. Berestycki-H. Brezis [1,2] and J. P. Puel [6] have established the existence results. And in the case with a limiter J. Blum, T. Gallouet and J. Simon [3] recently formulated the problem and proved the existence of such an equilibrium for a model case with the simple plasma current j = Mvu +. The various types of plasma current can be obtained by magnetic induction from an external electric circuit made of poloidal field coils, so in experiments and numerical simulations of the Tokamak plasma, various nonlinear forms of the current are considered.
Abstract:The influence of the magnetic field on the heat convection is an important issue in various industrial productions. The mathematical formulation of such viscous magneto hydrodynamic fluid is given by the magnetic Bénard system. We here show the existence of attractors, which is finite dimensional and characterize the long time dynamics of the system. Furthermore, we estimate their dimensions in terms of the magnetic components as controlling parameters.
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