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Asymptotic L-p-stability for strong solutions of magneto-hydrodynamics equations in the R-3 space
Abstract
Asymptotic L-p-stability of global strong solutions of Magneto-Hydrodynamics (MHD) equations in the whole R-3 space will be studied in this short note. It will be shown that the global strong solutions belonging to L-infinity(0,infinity; L-2(R-3) boolean AND Lp+2(R-3)) (p > 3) are asymptotically stable. At the same time the algebraic L-p-decay rate of difference of two global strong solutions belonging to the above class will also be presented if the difference of initial values is small.
We investigate the stability of two-dimensional periodic solutions to magnetohydrodynamics equations in the class of three-dimensional periodic solutions. We show the existence of global, strong three-dimensional solutions to magnetohydrodynamics equations, which are close to two-dimensional solutions. The advantage of our approach is that neither these solutions nor the external forces have to vanish at infinity.
The initial boundary value problem for the evolution system describing geophysical flow in three-dimensional domains was considered. The existence and uniqueness of global strong solution to the evolution system were proved under assumption on smallness of data. Moreover, solvable compatibility conditions of initial data and boundary values which guarantee the existence and uniqueness of global strong solution were discussed. Copyright © 2003 John Wiley & Sons, Ltd.
Some questions relating to the large time behavior of the solutions of MHD equations for a viscous incompressible resistive fluid are investigated. The physical system is briefly described and the functional setting of the equations, a flow in a bounded domain or in whole space with a space periodicity property in all directions. The main existence and uniqueness results for weak and strong solutions of the MHD equations are recalled. Regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular. The squeezing property of the trajectories are stated and it is shown that any functional invariant set for the MHD equations, and in particular any attractor, has a finite Haussdorf dimension. The flow is found to be totally determined for large dimensions by a finite number of parameters.
