Lie Group Solutions of Magnetohydrodynamics Equations and Their Well-Posedness

and Applied Analysis 3 Applying prV to (3), we find φx1x1t 1 + φx2x2t 1 + φx2x2x2 1 ∂φ ∂x1 + φ x1x1x2 1 ∂φ ∂x1 − φx1x1x1 1 ∂φ ∂x2 − φ x1x2x2 1 ∂φ ∂x2 + η x1 1 ∂3φ ∂x3 2 − η x2 1 ∂3φ ∂x3 1 + ηx1 1 ∂ 3φ ∂x2 1∂x2 − η x2 2 ∂3φ ∂x2 2∂x1 = φ x2x2x2 2 ∂ψ ∂x1 + φx1x1x2 2 ∂ψ ∂x1 − φ x1x1x1 2 ∂ψ ∂x2 − φ x1x2x2 2 ∂ψ ∂x2 + ηx1 3 ∂ 3ψ ∂x3 2 − η x2 4 ∂3ψ ∂x3 1 + η x1 3 ∂3ψ ∂x2 1∂x2 − η x2 4 ∂3ψ ∂x2 2∂x1 , ∂ψ ∂x2 η x1 1 − ∂ψ ∂x1 η x2 2 − ∂φ ∂x2 η x1 3 + ∂φ ∂x1 η x4 2 + ηt 6 = 0. (13) Then merging similar terms coefficients, we have ηx1 1 = ηx2 2 = ηx1 3 = ηx2 4 = ηt 6 = 0. (14) According to the formula φJ α(x, u(n)) = DJ(φα − ∑pi=1 ξiuα i ) + ∑pi=1 ξiuα J,i, for example, we have ηt 6 = Dt (ξ5 − ξ1ψx1 − ξ2ψx2 − ξ3ψt) + ξ1ψx1t + ξ2ψx2t + ξ3ψtt = ξ5t − ξ1tψx1 − ξ1ψψx1ψt − ξ1φψx1ψt − ξ2tψx2 − ξ2ψψx2ψt − ξ2φψx2φt − ξ3tψt − ξ3φφtψt − ξ3ψψtψt + ξ5φφt + ξ5ψψt φx1x1x1 1 = D3 x1 (ξ4 − ξ1φx1 − ξ2φx2 − ξ3φt) + ξ1φx1x1x1x1 + ξ2φx1x1x1x2 + ξ3φx1x1x1t = D3 x1ξ4 − D3 x1ξ1φx1D x1ξ2φx2 − D3 x1ξ3φt − 3D2 x1ξ1φx1x1 − 3D2 x1ξ2φx1x2 − 3D2 x1ξ3φx1t − 3Dx1ξ1φx1x1x1 − 3Dx1ξ2φx1x1x2 − 3Dx1ξ3φx1x1t. (15)


Introduction
Two-dimensional ideal incompressible magnetohydrodynamics (MHD) equation can be described by a set of two scalar equations for the vorticity  and the magnetic stream function ; namely [1], Due to the divergence freedom of the magnetic field , it is possible to define a magnetic stream function  via  = ∇ ⊥ .
In the incompressible case, ∇ ⋅  = 0, the velocity stream function  and velocity  are connected in the same way, and  = ∇ ⊥ .Vorticity and current density are defined as the Laplacian of the stream functions,  = Δ and  = Δ.
The magnetic stream function is convected with the flow field [2][3][4]; that means the ideal MHD equations do not allow for magnetic reconnection in contrast to the dissipative version of the above equations [5].Note that in contrast to the two-dimensional incompressible Euler equations case, there is not a production term on the right-hand side of (1).The equations show a tendency to develop fine structures, namely, current sheets.Analytically the problem about the regularity of solutions is still an open problem [6,7].
It is interesting to seek the solutions of MHD in mathematics and physics for a long time [8].In order to construct solutions of MHD, many effective methods have been put forward, such as the inverse scattering method, Backlund transformation, Hirota method, and homogeneous balance method [3].In the branches of mathematics and physics, Lie Group theory [9][10][11] was often used extensively.Ever since the 1970s Bluman and Col proposed similarity theory for differential equations, the Lie Group theory has been developed indifferential equations.The main idea of Lie Group method is to use the prolongation formulae, providing an effective computational procedure for finding the most general symmetry group of almost any system of partial differential equations of interest.To the best of our knowledge, related classical Lie Group method has not been preformed to the MHD equation.
Many mathematicians are devoted to studying the MHD equations.For example, Duvant and Lions proved the existence and uniqueness of the global strong solutions of twodimensional MHD equations with initial-boundary value problem.They also proved existence and uniqueness of locally strong solutions and the existence of the global weak solutions of three-dimensional MHD equations [6,12].As we all know, the studying of uniqueness and stability of MHD equations is based on some certain conditions or some assumptions.But, in this paper, we study a special class of According to the method of determining the infinitesimal generator of nonlinear partial differential equation, we take the infinitesimal generator of equation as follows: It is a vector field defined on an open subset  ⊂  × ; the th prolongation of V is the vector field defined on the corresponding jet space  () ⊂  ×  () , the second summation being overall multi-indices  = ( 1 ,  2 ,  3 , . . .,   ), with 1 ≤   ≤ , 1 ≤  ≤ .The coefficient functions    of pr () V are given by the following formula: where    =   /  and   , =    /  (in this paper  1 =  1 ,  2 =  2 , and  3 = ).
The coefficient functions    of pr (3) V are given by the following formula: where    =   /  and   , =    /  ( 1 =  1 ,  2 =  2 ,  3 = ): Applying pr (3) V to (3), we find Then merging similar terms coefficients, we have According to the formula , , for example, we have So we have Similarly we can find the determining equations for the symmetry group of the equations to be the following: As usual, subscripts indicate derivatives.The solution of the determining equations is elementary.According to ( 16) and (17) we have Finally, solving the above differential equations, we conclude that the most general infinitesimal symmetry of (3) has coefficient functions of the form where , , , , , and  are arbitrary constants and () is an arbitrary function of  only.Thus the Lie algebra of infinitesimal symmetries of the MHD equations is spanned by the six vector fields: And the infinite-dimensional subalgebra where () is an arbitrary function of  only.So we have The entries give the transformed point exp (  )( 1 ,  2 , , , ) = (x 1 , x2 , t, φ, ψ): If  = ( 1 ,  2 , ) and  = ( 1 ,  2 , ) are known solutions of (3), then using the above groups   ,  = (1, 2, . . ., 6), the corresponding new solutions   and   can be obtained, respectively, as follows: where  is a real number and () is an arbitrary function of  only.

Theorem 2. One assumes that the solutions have forms of 𝜑 = 𝜑(𝑎
One has the following: is arbitrary function.

The Uniqueness of Wave Solutions
In this section we give some nonzero solutions by considering the wave solutions to the two-dimensional MHD equations [13][14][15].Firstly we could give the initial-boundary value of three conditions in Theorem 2. ( Theorem 3. Assume that Ω ⊂  2 is a bounded domain.One can make the solutions  1 ,  1 like the condition in Theorem 2 and satisfying Then the following initial-boundary value problem: has a unique smooth solution , . Proof.To prove the uniqueness we consider two smooth solution pairs, say  1 ,  1 and , .Let their difference be  =  −  1 ,  =  −  1 .Then subtracting the equations from each other in (26), we have Multiplying the first and second equations by , , respectively, integrating over Ω, we obtain It is easy to see that Similarly we have Notice that Multiplying equation by Δ, integrating over Ω, we obtain It is easy to see that Therefore there exists a unique solution in the sense of  2 ((( 2 (Ω)) 2 ; 0, )), ∀ > 0.

The Lyapunov Stability of Steady State Solution
In this section we discuss the stability of the steady state solutions, in  2 (Ω) for problem (41).
Definition 4. A steady state solution  is said to be stable if and only if  in any one of the neighborhood , there is a neighborhood  of , making any solutions (, ⋅) with the initial condition (0, ⋅) ∈  satisfy (, ⋅) ∈  (∀ ≥ 0).

Conclusions
In this paper, we studied the symmetry groups by using the classical Lie Group method to structural equation solutions.First, we perform Lie symmetry analysis for the MHD equation and get its infinitesimal generator.Then, we obtain many solutions by it.It can be seen by the results of this paper that the Lie Group method is an effective method for studying nonlinear partial differential equations.