UNIFORM STABILITY AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF 2-DIMENSIONAL MAGNETOHYDRODYNAMICS EQUATIONS

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UNIFORM STABILITY AND ASYMPTOTIC BEHAVIOR OFSOLUTIONS OF 2-DIMENSIONAL MAGNETOHYDRODYNAMICS EQUATIONS

This paper is concerned with uniform stability and asymptoticbehavior for solutions of 2-dimensional Magnetohydrodynamicsequations. The author establishes the corresponding temporaldecay estimates when the initial data is in the following Sobolev spaces$H^2, L^1cap H^2$ with $int(u_0, A_0)dxnot= 0$, or $L^1cap H^2$with $int(u_0, A_0)dx=0$, respectively. Most of the decayrates in these estimates are optimal. Moreover, the author provesvarious uniform stability results, like $sup_{t>0}|(w, E, r)(t))|_Yle C|(w_0, E_0)|_X$, where $X$ and $Y$ are Sobolev spaces.It should be pointed out that the decay estimates of the solutions for the case$(u_0, A_0)in L^1cap H^2$ follow from the uniform stabilityestimates. The author utilizes the Fourier splitting methodinvented by Professor Schonbek and the new elaborate global energy estimates.