THE REGULARIZATION OF SOLUTION FOR THE COUPLED NAVIER-STOKES AND MAXWELL EQUATIONS

. The purpose of this paper is to build the existence of time-spatial global regular solution to the coupled Navier-Stokes and Maxwell equations.


1.
Introduction. It is customary to describe the dynamics of homogeneous, incompressible, conducting fluids under the influence of body forces and applied currents by the system see, e.g., [4,9]. Here v denotes the velocity field, p the pressure, f the known body force, j the known applied current, and E and H the electric and magnetic fields, respectively. µ, σ and ν are the magnetic permeability, the electric conductivity and the kinematic viscosity. The density of the fluid is assumed to equal to one. In this paper, we consider the standard case where µ, σ and ν are all positive constants. The displacement current is proportional to E t and is assumed to be negligible. According to [5] generalized solutions of the system with three different initialboundary value problems are exist. For the unique solvability of those problems, some results that analogous to the results of A.Kiselev and O.Ladyzhenskaya [3] for the three-dimensional Navier-Stokes system were derived. Furthermore, in [2], it was shown that the problem is globally uniquely generalized solution to the coupled modified Navier-Stokes and Maxwell equations in three dimensional case of initialboundary value problems.
In order to investigate the regularity of solutions of the coupled modified Navier-Stokes and Maxwell equations, we concern the coupled Navier-Stokes and Maxwell equations with nonlinear dispersive long waves as the same as considering the mathematical models for the unidirectional propagation of long waves in nonlinear and dispersive systems u t + u x + uu x + u xxx = 0.
The term u xxx represents dispersive and dissipative effects when they are considered. The term −u xxt we chosen instead of u xxx not only obviates the difficulties in question but also makes the model more suitably posed for long waves systems; see, e.g., [1,7]. Therefore in this paper we consider the following equation: As discussing the long wave propagation equation by the following equation we first consider the coupled Navier-Stokes and Maxwell equations with nonlinear dispersive long waves Next we consider the regularization of solution of the coupled Navier-Stokes and Maxwell equations over bounded spatial-time domain. Finally we show the existence of regular solution over global spatial-time space. The regularization of the initial-boundary value problem in a bounded, threedimensional domain with fixed perfectly conducting boundaries is discussed for this system with the smooth boundary condition by energy method and inductive method. In the rest of this section, we establish some notation that will be used throughout the paper and then provide a description of the problems we consider.
1.1. Notation. In this section, we introduce the notation that will be used throughout the paper. Vector-valued function will be denoted in bold-face, i, e., u = (u 1 , u 2 , ..., u l ) ∈ R l and furthermore, u · v = l k=1 u k v k , we also denote by u · v = u k v k unless explicitly noted. Points in Euclidean space R l are denoted by x = (x 1 , ..., x l ) and k times partial derivatives are denoted by Let Ω ⊂ R l denotes an open domain with boundary ∂Ω. For x ∈ ∂Ω, n = n(x) denotes an outward unit normal of ∂Ω and τ = τ (x) denotes unit tangential vectors of ∂Ω.
The Lebesgue spaces are denoted by L p (Ω) and have norms The inner product in L 2 (Ω) is denoted by (φ, ϕ) = Ω φϕdx. Sobolev spaces are denoted by W k,p (Ω) and W k,p 0 (Ω) is defined to be the closure of W k,p (Ω) with zero boundary.
The set of all infinitely differentiable functions with compact support with respect to Ω is denoted by C ∞ c (Ω). We then introduce the set and the subspace of L 2 (Ω) where div v=0 is understood in the sense of distributions, i.e., Then, K 0 (Ω) is defined to be the closure of K ∞ (Ω) in the norms of L 2 (Ω) and we also define K k,p (Ω) = W k,p (Ω) ∩ K(Ω) and K k,p 0 (Ω), the closure of K ∞ (Ω) in the norms of W k,p (Ω) which are given by where i 1 , i 2 , ..., i are non-negative integers and |i| = j=1 i j .
Finally, C, C 1 , C 2 will denote positive constants whose value changes with context and some other notation in this paper will explain appropriately.   Theorem 1.1. Suppose that Ω is a bounded domain in R 3 with smooth boundary and let Q T = Ω × (0, T ) with some fixed times T < ∞. v 0 ∈ W k+1,2 (Ω), H 0 ∈ W k,2 (Ω), f, curlj ∈ L 2 (0, T ; W k−1,2 (Ω)), divj = 0, (k = 0, 1, 2, ...), and suppose that v x k and H x k equal to 0 on ∂Ω in the trace sence. Then the system has generalized solution v, H and the generalized solution has the properties for every k.

2.
Estimates. In this section, we divide into three parts: three order for the estimates at most, some additional estimates and higher order estimates under the appropriate conditions.
we will use the integral identities for any η, ζ ∈ K 0 (Ω). For the identity (2.1) with η = v and (2.2) with ζ = H, we get the energy relation 1 2 In order to obtain the identity (2.3), we have used the identity (1.2), (1.4) and the boundary conditions. Furthmore, we have used the relation that holds for arbitrary elements u, v, w of W 1,2 (Ω) satisfying the conditions The result of intergrating (2.4) over t and using the Gronwall's inequality, we can yield the relation Here, Φ is a continuous function.
For the estimation of A 1 , we apply the Hölder inequality, the multiplicative inequalities, the imbedding inequalities and the Young's inequalities. In more detail, where ,Ω + v x 2,Ω . It is easy to obtain the estimations of A 2 and A 3 analogous to A 1 . In more detail, ,Ω ) for a continuous function Φ 1 that depends on the arguments of Φ from (2.5).
2.2. Some additional estimates. In (2.1), let η = v t (this is allowable since div v t = 0 and v t | S T = 0), see,e.g., [5] and we have the relation From the result of (2.11) and integrating over t in (2.12), we can obtain the inequality ∆v t 2,Q T ≤ D 1 with D 1 under our control. It is also easy to get estimates for H t , ∇p analogous to estimates in [2].

2.3.
Higher order estimates. In this section, in order to get higher order estimates, we have to get the three order relation and corresponding estimates first. Let η = ∆ 2 v in (2.1) and ζ = ∆ 2 H in (2.2), then we can obtain the relation ,Ω in (2.13), we apply the Hölder inequalities, the multiplicative inequalities, the imbedding inequalities and the Young's inequalities. In more detail, (2.14) In (2.13), we can obtain the estimations of ∇(H · ∇H)  (2.17) Using the inequalities (2.14)-(2.17) with sufficiently small ε, ε 1 , ε 2 and ε 3 , we can conclude from (2.13) that 1 2 So when η = (−∆) k+1 v in (2.1) and ζ = (−∆) k+1 H in (2.2), we can obtain the relation To majorize the terms in the right-hand in the side of (2.17), we just calculate four terms in B. In more detail, We can obtain the estimates of B 1 , B 2 , B 3 , B 4 from the the previous arguments and hypothesis. In more detail, From the previous hypothesis and arguments, we know that if j + 2 or i + 1 ≤ k + 1, then v x j+2 4 2,Ω or v x i+1 4 2,Ω is uniform bound for arbitrary t ∈ [0, T ] and if j + 2 or i + 1 ≤ k, then H x j+2 4 2,Ω or H x i+1 4 2,Ω is uniform bound for any t ∈ [0, T ]. From (2.19) and the estimates of B, as well as sufficiently small ε, ε 1 , ε 2 and ε 3 , we gain the relation 1 2