TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 305, Number 1, January 1988 LONGTIME DYNAMICS OF A CONDUCTIVE FLUID IN THE PRESENCE OF A STRONG MAGNETIC FIELD C. BARDOS, C. SULEM AND P. L. SULEM ABSTRACT. We prove existence in the large of localized solutions to the MHD equations for an ideal conducting fluid subject to a strong magnetic field. We show that, for large time, the dynamics may reduce to linear Alfven waves. 1. Introduction. In contrast with a uniform velocity field, a uniform magnetic field has a significant dynamical effect on a conducting flow. This is easily seen by writing the (ideal) MHD equations: &v „ „ , „, — + v ■ V«j = -Vp + b ■ V6, at at V -v = V-6 = 0 in terms of the Elsässer variables (1.2) Z+=v + b, Z~=v-b which satisfy dZ+ dt (1.3) dZ~ + Z- - VZ+ = -Vp, + Z+ ■ VZ~ = -Vp, dt V • Z+ = V •z- = 0. When equations (1.3) are linearized around the static solution with a constant magnetic field Bo, one obtains that the fluctuations z± = Z± =p Bo propagate along the B0 magnetic field in opposite directions. This suggests that in the orig- inal nonlinear problem, a strong enough magnetic field will reduce the nonlinear interactions [1] and inhibit formation of strong gradients. This effect was observed in direct numerical simulations of equations (1.3) with periodic boundary conditions [2]. These calculations showed that in the presence of a strong enough magnetic field, solutions remain analytic in a strip whose width is bounded from below. In this paper, we consider the problem in the entire 722 or 723; we prove that for large B0 and small enough localized initial fluctuations z±, the solution of the MHD equations (1.3) remain smooth for all time and that the nonlinear interactions become asymptotically negligible when t —> +oo. Received by the editors March 11, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 35Q20; Secondary 76E25. ©1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page 175 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use