ON THE DIFFERENTIAL SYSTEM GOVERNING FLOWS IN MAGNETIC FIELD WITH DATA

In this paper we study the system governing flows in the magnetic field within the earth. The system is similar to the magnetohydrodynamic (MHD) equations. For initial data in space Lp, we obtained the local in time existence and uniqueness ofweak solutions ofthe system subject to appropriate initial and boundary conditions.


INTRODUCTION
We consider in this work the following differential system arising from geophysics (cf.Hide [1]), which governs the flow of an electrically-conducting fluid in the presence of a magnetic field, when referred to a frame which rotates with angular velocity ft relative to an inertial frame: cob AAb + V x (v x b) 1 Vq + g(x), (1.2) div v 0; div b 0, (1.3)   where v is the Eulerian flow velocity, p is the density, b is the magnetic field, p is the pressure, v, # are respectively constants of kinematical viscosity, magnetic permeability, A with electrical resistivity % and f(z), g(z) are volume forces.Let K be an open bounded subset of R" with boundary F. The initial and boundary conditions are as follows respectively: v(z, O) vo; b(x, O) bo for xeK (1.5)   where n is the outward unit normal on F. The existence of solutions of systems (1.1)-(1.5) in L 2 has been proved in Qu et al. [2].Some regularity properties and large time behaviors of the solutions for a similar system, the MHD equations, are obtained in Sermange [3] and Temam [4].In this work we consider the initial value problem for the above system in infinite cylinder ST (0, T) x R with initial data v0, b0 E L. Following Fabes et al. [5], we consider the solution of (1.1)-(1.5) in weak form.And we prove the local existence and uniqueness of weak solution of the system in Lp space.This article is arranged in the following order.In Section 2, we first introduce some notations and definitions and formulate the setting of our problem.Then we prove that solving the system (1.1)-(1.5) is equivalent to solving a nonlinear integral equation.Applying a well-known imbedding theorem, we prove in Section 3 the uniqueness of the solution for all T and the existence ofthe solution for small value of T.

NOTATIONS AND AN INTEGRAL EQUATION
In this section, we introduce some notations and define the weak solution of differential system (1.1)-(1.5).Then reduce the system to an integral equation.We prove later in this section that the differential system (1.1)-(1.5)and the integral equation are equivalent.
PROOF.We prove the theorem for the case of f t7 0 (the proof for the case when .f : 0, g 0 is similar).
Let us first assume that u (v,b) is a solution of the integral equation (2.10).Set v B(v,v),v B(b,b),b B2(v,b),b2 B2(b,v).Following the argument in the proof of E(A,x,t) 9=-I(a(E,.:)),where a(x) a(Ax)," is the Fourier transform taken in x variables.For b,(t) p() and fixed x, t, we use b,x (Z/, 8) p(8 + 2)b,(t )g,CA, x /,t 8) as test functions in Definition 2.1 and following Fabes et al. [5] we can showthat u is a solution of(2.10). [-l

ON THE INTEGRAL EQUATION
In this section we study the integral equation (2.0).We will show that it has a unique solution in "q (ST) with initial data in P,q (ST).In the following, it is assumed that F 0 (the case F -0 can be treated similarly).
for (3.2), we have In the following, we use the above lemma to prove the existence of solutions of the integral equations (2.10).TIOREM 3.2 (i).Suppose -/ 1 wM < p < oo.Then tere exists constant 6'0 depending on ,, suc out when there exists T > 0 suc tt the integral equation: z + A(,) + W() o (3.11)   has a solution m ', (ST).
(ii).Suppose + < I with n < p < oo.Then there exists T > 0 such that the integral equation (3.11) has a solution in P'q (ST).
PROOF.Let T(u, v) A(u, v) + D(u) + u.By Theorem 3.1, we have Let c CT1/2(--), c2 C'T and IIFII IIll<s).Then if we chose Ilullz,.,(s) and T small enough in case (i) and T small enough in case (ii), conditions in Lemma 3.1 are satisfied.Brown's fixed point theorem implies the existence of solution of integral equation (3.11).I-I The following theorem assures the uniqueness ofthe solution of (3.11) THEOREM 3.3.Suppose p, q satisfy the conditions of Theorem 3.1.Then the solution of (3.1 I) is unique in the space ffP'q (ST) for any T > O.
PROOF.If ut, ,22 P'q(T) are tWO solutions of the integral equation with the same initial data uo (vo, bo), then 2 [A(, t 2) + A( , 2)1 D( ).By Theorem 3.1, for -< T, Choose " so small that So we see that u u2 in S,.Repeating the same procedure, we can cdnclude that u u2 in $2,.Cominuing repeating the procedure, we can prove finally that ttt= u2 in ST. 71 Under the conditions of Theorem 3.1, we can easily translate the existence and uniqueness results for the integral equation (2.10) with F 0 to respectively existence and uniqueness results for the system 1.1)-( 1.5) because ofthe equivalence result of Theorem 2.1.