Optimal System of Subalgebras for the Reduction of the Navier-Stokes Equations

The purpose of this paper is to find the admitted Lie group of the reduction of the Navier-Stokes equations               2 , , , , 2 , , 1 , , 2 0 yy t sy ss s U t s y U t s y y sU t s y y s U t s y sU       where s z y  using the basic Lie symmetry method. This equation is constructed from the Navier-Stokes equations rising to a partially invariant solutions of the Navier-Stokes equations. Two-dimensional optimal system is determined for symmetry algebras obtained through classification of their subalgebras. Some invariant solutions are also found.


Introduction
Mathematical modeling is a basis for analyzing physical phenomena.Almost all fundamental equations of mathematical physics are nonlinear, and in general, are very difficult to solve explicitly.Group analysis is a method for constructing exact solutions of differential equations.This method uses the symmetry properties for constructing exact solutions.There are two types of solutions, the class of invariant solutions and partially invariant solutions which can be obtained by group analysis.Constructing of invariant and partially invariant solutions consists of some steps: choosing a subgroup of the admitted group, finding a representation of solution, substituting the representation into the studied system of equations and the study of compatibility of the obtained (reduced) system of equations.
This paper is devoted to use the basic Lie symmetry method for finding the admitted Lie group of the reduction of the Navier-Stokes equations, where is a dependent variable and U , t s z y y  , are independent variables.This equation is constructed from the Navier-Stokes equations.Subgroups for studying are taken from the part of optimal system of subalgebras considered for the gas dynamics equations [1].One subgroup is not admitted the Navier-Stokes equations, par-tially invariant solutions can be found for the Navier-Stokes equations.These facts allow us to assume that one can construct partially invariant solution with respect to a Lie group, which is not necessary admitted.The proposed research will deal with two-dimensional optimal system of subalgebras for the reduction of the Navier-Stokes equations [1].It is determined for symmetry algebras obtained through classification of their subalgebras.Example of some invariant solutions are also found.They can return to new solutions of the Navier-Stokes equations.

Invariant and Partially Invariant Solutions
The notion of invariant solution was introduced by Sophus Lie [2].The notion of a partially invariant solution was introduced by Ovsiannikov [3].This notion of partially invariant solutions generalizes the notion of an invariant solution, and extends the scope of applications of group analysis for constructing exact solutions of partial differential equations.The algorithm of finding invariant and partially invariant solutions consists of the following steps.Let be a Lie algebra with the basis The defect is the number of the dependent functions which can not be found from the representation of partially invariant solution.The rank  and the defect  must satisfy the conditions where  is the maximum number of invariants which depends on the independent variables only.Note that for invariant solutions, 0 invariants and separate the universal invariant in two parts: , , ,
The number satisfies the inequality1 .The representation of the l l q m      , H   -solution is obtained by assuming that the first coordinates l J of the universal invariant are functions of the invariants J : Equation ( 2) form the invariant part of the representation of a solution.The next assumption about a partially invariant solution is that Equation (2) can be solved for the first dependent functions, for example, l   It is important to note that the functions are involved in the expressions for the functions .The functions are called superfluous.The rank and the defect of the   , the above algorithm is the algorithm for finding a representation of an invariant solution.If 0   , then Equation (3) do not define all dependent functions.Since a partially invariant solution satisfies the restrictions (2), this algorithm cuts out some particular solutions from the set of all solutions.After constructing the representation of an invariant or partially invariant solution (3), it has to be substituted into the original system of equations.The system of equations obtained for the functions and superfluous functions  is called the reduced system.This system is overdetermined and requires an analysis of compatibility.Compatibility analysis for invariant solutions is easier than for partially invariant solutions.Another case of partially invariant solutions which is easier than the general case occurs when J only depends on the independent variables , , .
In this case, a partially invariant solution is called regular, otherwise it is irregular.The number    is called the measure of irregularity.
The process of studying compatibility consists of reducing the overdetermined system of partial differential equations to an involutive system.During this process different subclasses of  , H    partially invariant solutions can be obtained.Some of these subclasses can be  

 
, H   -solu- tion is called reducible to an invariant solution.The problem of reduction to an invariant solution is important since invariant solutions are usually studied first.

The Unsteady Navier-Stokes Equations
Unsteady motion of incompressible viscous fluid is governed by the Navier-Stokes equations , 0, where is the velocity field, p the fluid pressure, is  the gradient operator in the three-dimensional space , , , , x x x x y z   x and  is the Laplacian.A group classification of the Navier-Stokes equations in the three-dimensional case 1 was done in [5].The Lie group admitted by the Navier-Stokes equations is infinite.Its Lie algebra can be presented in the form of the direct sum , where the infinite-dimensional ideal is generated by the ope- with arbitrary functions . The subalgebra has the following basis: The Galilean algebra is contained in Several articles [7][8][9][10][11][12][13] are devoted to invariant solutions of the Navier-Stokes equations 3 .While partially invariant solutions of the Navier-Stokes equations have been less studied 4 , there has been substantial progress in studying such classes of solutions of inviscid gas dynamics equations [18][19][20][21][22][23][24][25].

The Reduction of the Navier-Stokes Equations
The reduction of the Navier-Stokes equations to partial differential equation in three independent variables is described.In this section analysis of compatibility of regular partially invariant solutions with defect 1 and rank 1 of the subalgebras   , , , is given.Note that the generator t x y z is not admitted by the Navier-Stokes equations.The groups are taken from the optimal system constructed for the gas dynamics equations [26].
The Navier-Stokes equations are used in the component form: , , The dependent variables and are functions of the space variables , , u v w , , p x y z and time .
t Invariants of the Lie group corresponding to subalgebra generated by   , , ,

, , v w p z y
The representation of the regular partially invariant solution is where s z y  .For the function

 
, , , u u t x y z  there is no restrictions.Substituting the representation of partially invariant solution (9) into the Navier-Stokes Equations ( 5)-( 8), we obtain Since and W only depend on V s , Equations ( 11) and ( 12) can be split with respect to : Solving Equation ( 15), we have Multiplying the first equation by s and combining it with the second equation of ( 14), we obtain . This means that and hence 5 .Substituting and W in Equation ( 13), we have Thus, there is a solution of the Navier-Stokes equations of the type where the function
In this case P  .Note that the Galilei transformation applied to and W , also change V s .Substituting V and W in Equation ( 13), we have or 0 Thus, there is a solution of the Navier-Stokes equations of the type 2 There is still no complete classification of the subalgebras of the Lie algebra 5 L L   .Classification of infinite-dimensional subalgebras of this algebra was studied in [6]. 3 Short reviews devoted to invariant solutions of the Navier-Stokes equations can be found in [14][15][16]. 4Firstly the approach of partially invariant solutions to the Navier-Stokes equations was applied in [17].
where the function

Admitted Group of Equation (16)
In this section, the Lie group admitted by Equation ( 16) is studied.It was obtained from the Navier-Stokes equations and gives rise to a partially invariant solutions of the Navier-Stokes equations The second prolongation of the operator X is The coefficients of the prolonged operator are defined by formulae Here we used the notations 1 The determining equations are All necessary calculations here were carried out on a computer using the symbolic manipulation program REDUCE.
The result of the calculations is the admitted Lie group with the basis of the generators: , , , , where   , , b t s y is an arbitrary solution of

Optimal System of Subalgebras
The problem is to construct subalgebras of the algebra , which can be a source of invariant solutions of Equation (1).The classification of subalgebras can be done relatively easy for small dimensions.The optimal system of subalgebras of the Lie algebra spanned by the generators Inner automorphisms [24] are constructed with the help of the table of commutators.
To construct inner automorphisms, one has to solve the Lie equations.For example, for the automorphism 1 A , one has the system of ordinary differential equations  : , , .
Also there is the involution

Decomposition of the Algebra L 9
Before constructing an optimal system, let us study the algebraic structure of the algebra .The algebra is decomposed as is a subalgebra.According to the algorithm for constructing an optimal system of the algebra , we use the two-step algorithm developed in [21].First, an optimal system of subalgebras of the algebra is obtained.The next step is to glue the subalgebras from the optimal system of subalgebras of the algebra and the ideal  , , Any subalgebra of a Lie algebra is completely defined by its basis generators.Any vector of the basis is a linear combination of the basis of generator of this Lie algebra.Hence, the subalgebra is completely defined by co-efficients of these linear combinations.For example, let be a -dimensional subalgebra of the algebra .Operators are Conditions for to be a subalgebra are For a classification of subalgebra, the coefficients ij C  have to be simplified by using the automorphism and subalgebra conditions.

Classification of the Algebra L 4
Let us classify the algebra .The table of commutators of the algebra is   Since the generator 6 X composes the center, the optimal system of subalgebras of can be easily constructed by classifying the subalgebra   , , L X X X  and gluing it with the center   6 X .The idea of construction is as follows.
Let a subalgebra of dimension be formed by the operators are arbitrary constants.
For the classification of we need to study two steps., it means that we will construct an optimal system of the subalgebra , , X 2) At least one of the coefficients of is not equal to zero.Let us study the first step, and construct an optimal system of the subalgebra .For convenience, we will denote the generators 3 L i X by i.

One-Dimensional Subalgebras of the Algebra
8 which forms a one-dimensional subalgebra of the algebra .The process of simplification of the coefficients of the operator is separated into the following cases.
Let us consider the second step where at least one of the coefficients 3 i is not equal to zero.Without loss of generality one can assume that Because is a subalgebra and the generator 6 forms the center, then Comparing the coefficients, one obtains 0;, . Because of these results and since the algebra has already been classified, therefore this allows simplifying the process of constructing the optimal system of the algebra .This process construct by using the result of the optimal system of algebra : we have to classify each optimal system of subalgebras of together with the generator . Here we give one example of this process.Other elements of the optimal system of the algebra are constructed in the similar way.


, ,8 Let us consider the subalgebra   ,   1 8 1 4 .For constructing three-dimensional subalgebras of the algebra one considers Since can be written as: 1 1 4 by forming a linear combination with and , the operator 1 can be taken in the form . The subalgebra conditions gives Thus, one obtains that 1 Y  6 , and the subalgebra is .

 
The result of calculation is an optimal system of subalgebras of the algebra which is where  is an arbitrary real parameter and 1    .

Optimal System of Subalgebras of the Algebra L 9
After constructing an optimal system of subalgebras of the algebra , the next step is the construction of an optimal system of subalgebras of the algebra , by gluing subalgebras from the optimal system of subalgebras of the algebra and the ideal together.As it was seen for the algebra , the process of constructing an optimal system of subalgebras of the algebra by gluing the algebra and the ideal Here the vectors are basis elements from one of the k-dimensional subalgebras of the optimal system of the algebra .In matrix form, this step can be explained by the construction of the matrix , .
In this step, the matrix A is arbitrary.The rank of the matrix is equal to k s  and this is the dimension of the subalgebra of the algebra .The matrix C is chosen to be the simplest by taking linear combinations of it columns and has to take all possible values of the given rank s.Note also that the matrix A can be simplified with the help of the matrix C. 9 L The next step is the process of checking the subalgebra conditions and checking linear dependence of commutators on the basis generators of the subalgebra.
In this manuscript, we study only two-dimensional subalgebras of the algebra , because the two-dimen-sional subalgebras allow obtaining invariant solutions which reduce the initial system of partial differential equations to a system of ordinary differential equations.Let us give an example for constructing two-dimensional subalgebras, using the subalgebra    1 8 .The maximum possible dimension of a subalgebra of the algebra after gluing a subalgebra to the ideal 4 L I is two.In this case, the matrix C is a matrix, the rank of which is equal to one: ,    The subalgebra conditions give


which is satisfied with 0, 0.

   
Therefore, the subalgebra is   ,   1 8 9 .Other elements of the optimal system of the algebra are constructed in the similar way.9

L
The list of two-dimensional subalgebras of the optimal system of the algebra is presented in Table 1. 9 L

Invariant Solutions of Equation (1)
Invariant solutions of Equation ( 1) are presented in this section.Analysis of invariant solutions is presented in details for two examples.

 
The basis of this subalgebra is Hence, a representation of the invariant solution is The general solution of the last equation is

Conclusion
The algorithm of obtaining an optimal system of subalgebras was applied to the reduction of the Navier-Stokes equations.Some exact invariant solutions corresponding to the optimal system are presented.Examples given in the manuscript showed that this algorithm can be applied to groups, which are not admitted.These possibilities extend an area of using group analysis for constructing exact solutions.

H
J consists of s m n r     functionally independent invariants   -solutions.The number  is called the rank of a partially invariant solution.This number gives the number of the independent variables in the representation of the partially invariant solution.The number  is called the defect of a partially invariant solution.
-solutions.In many ap- plications, there is a reduction of a

4 L 1 ) All coefficients 3 i
are zero,

.
and  are arbitrary constants.Comparing the coefficients on the left side with the coefficients on the right side, one obtains

4 L 4 L 9 L
I consists of the following steps.In the first step, the vectors the matrices A, B and C consist of the coefficients   , , , . These solutions are partially invariant solution with

Table 1 . Two-dimensional subalgebras of the optimal sys- tem of the algebra . L 9
In order to find an invariant solution, one needs to find a universal invariant of this subalgebra.Let a function The two examples showed that there are solutions of the Navier-Stokes equations, which are partially invariant with respect to not admitted Lie algebra .
t  .After substituting the C y y 