Optimal System of Subalgebras for the Reduction of the Navier-Stokes Equations ()
1. 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,
(1)
where 
is a dependent variable and 
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, partially invariant solutions can be found for the NavierStokes 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 NavierStokes 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 NavierStokes equations.
2. 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 universal invariant J consists of 
functionally independent invariants
where 
are the numbers of independent and dependent variables, respectively and 
is the total rank of the matrix composed by the coefficients of the generators
. If the rank of the Jacobi matrix
is equal to
, then one can choose the first 
invariants 
such that the rank of the Jacobi matrix 
is equal to
. A partially invariant solution is characterized by two integers: 
and
. These solutions are also called 
-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. 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, 
and
.
For constructing a representation of a 
solution one needs to choose 
invariants and separate the universal invariant in two parts:
The number 
satisfies the inequality
. The representation of the 
-solution is obtained by assuming that the first 
coordinates 
of the universal invariant are functions of the invariants
:
(2)
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,
(3)
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 
-solution are 
and
, respectively.
Note that if
, the above algorithm is the algorithm for finding a representation of an invariant solution. If
, 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 
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 
partially invariant solutions can be obtained. Some of these subclasses can be 
-solutions with subalgebra
. In this case
. The study of compatibility of partially invariant solutions with the same rank
, but with smaller defect 
is simpler than the study of compatibility for 
-solutions. In many applications, there is a reduction of a 
-solution to a 
solution. In this case the 
-solution is called reducible to an invariant solution. The problem of reduction to an invariant solution is important since invariant solutions are usually studied first.
3. The Unsteady Navier-Stokes Equations
Unsteady motion of incompressible viscous fluid is governed by the Navier-Stokes equations
(4)
where 
is the velocity field, 
is the fluid pressure, 
is the gradient operator in the three-dimensional space 
and 
is the Laplacian. A group classification of the Navier-Stokes equations in the three-dimensional case1 was done in [5]. The Lie group admitted by the NavierStokes 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 operators2
with arbitrary functions 
and
. The subalgebra 
has the following basis:
The Galilean algebra 
is contained in
. Several articles [7-13] are devoted to invariant solutions of the Navier-Stokes equations3. While partially invariant solutions of the Navier-Stokes equations have been less studied4, there has been substantial progress in studying such classes of solutions of inviscid gas dynamics equations [18-25].
4. 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 
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:
(5)
(6)
(7)
(8)
The dependent variables 
and 
are functions of the space variables 
and time 
Invariants of the Lie group corresponding to subalgebra generated by 
are
The representation of the regular partially invariant solution is
(9)
where
. For the function 
there is no restrictions. Substituting the representation of partially invariant solution (9) into the Navier-Stokes Equations (5)-(8), we obtain
(10)
(11)
(12)
(13)
Since 
and 
only depend on
, Equations (11) and (12) can be split with respect to
:
(14)
(15)
Solving Equation (15), we have
Multiplying the first equation by 
and combining it with the second equation of (14), we obtain
Let
, then
. This means that 
and hence
. Substituting 
and 
in Equation (13), we have
. It means that 
depend on 
or
. Equation (10) becomes
(16)
Thus, there is a solution of the Navier-Stokes equations of the type
where the function 
satisfies Equation (16).
If
, then 
In this case
. Note that the Galilei transformation applied to 
and
, also change
. Substituting 
and 
in Equation (13), we have 
or
. Equation (10) becomes
(17)
Thus, there is a solution of the Navier-Stokes equations of the type
where the function 
satisfies Equation (17).
These solutions are partially invariant solution with respect to the group which are not admitted Lie algebra
.
5. 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
where the function U depends on 
and
.
Assume that the generator has a representation of the form
The second prolongation of the operator 
is
The coefficients of the prolonged operator are defined by formulae
Here we used the notations 
and for the derivatives
The determining equations are
(18)
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:
(19)
where 
is an arbitrary solution of
6. 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 
are constructed here.
The table of commutators 
is
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
, one has the system of ordinary differential equations
and the initial values at 
Therefore, the automorphism 
only changes the coordinates 
and 
by the formulae
The remaining coordinates are unchanged.
In the same way, one obtains the automorphisms 
Also there is the involution
6.1. Decomposition of the Algebra 
Before constructing an optimal system, let us study the algebraic structure of the algebra
. The algebra 
is decomposed as
, where 
is an ideal and 
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 
together.
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 coefficients 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 
have to be simplified by using the automorphism and subalgebra conditions.
6.2. Classification of the Algebra 
Let us classify the algebra
. The table of commutators of the algebra 
is
Since the generator 
composes the center, the optimal system of subalgebras of 
can be easily constructed by classifying the subalgebra 
and gluing it with the center
. The idea of construction is as follows.
Let a subalgebra 
of dimension 
be formed by the operators
where 
are arbitrary constants.
For the classification of 
we need to study two steps.
1) All coefficients 
are zero, 
, it means that we will construct an optimal system of the subalgebra
.
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 
by i.
6.2.1. One-Dimensional Subalgebras of the Algebra 
Let 
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.
Case 1. Assume that
. Then one can divide 
by
. Hence, without loss of generality one can consider
By means of transformation
, it can transformed to an operator with
.
Case 1.1. Let
. By means of transformation
, one can transform it to
, where
.
Case 1.2. Let
, then the representative of the class is the operator
.
Case 2. Assume that
. Then one has
.
Case 2.1. Let
. Dividing the operator 
by
, one obtains
. By using the automorphism
, the operator 
is transformed to
.
Case 2.2. Let
, then
.
6.2.2. Two-Dimensional Subalgebras of the Algebra 
Let a subalgebra be formed by the operators
where 
are arbitrary constants.
Note that the rank of the matrix 
is equal to two.
Case 1. Assume that
. We can divide 
by
. Hence, by subtracting the operator 
from
, one can assume 
and
.
Using the automorphisms
, the operator 
is transformed to
. The subalgebra condition gives
where 
and 
are arbitrary constants. Calculating the left hand side and comparing the coefficients on the left hand side with coefficients on the right hand side, one has
Therefore
Further consideration depends on values of the coefficients
. If
, then 
which is a contradiction to the condition
. Hence,
. One can assume that
. Therefore
, and
.
Case 1.1. If
, then using the automorphism
, the operators 
and 
are transformed to
,
.
Case 1.2. If
, then the operators 
and 
are
.
Case 2. Assume that
. If
, then by exchanging 
and
, this becomes the previous case. Hence, one can take
. Therefore, the operators are
. Because the rank of the matrix
is equal to 2, then by taking linear combinations of the operators 
and 
they can be transformed to 
and
.
6.2.3. Three-Dimensional Subalgebras of the Algebra 
Let a subalgebra be formed by these operators
where 
are arbitrary constants. Since the rank of the matrix
is equal to three, the basis if this subalgebra can be taken as
6.2.4. Optimal System of Subalgebras of the Algebra 
The result of classifying the algebra 
is the following:
where
.
6.3. Optimal System of Subalgebras of the Algebra 
Let us consider the second step where at least one of the coefficients 
is not equal to zero. Without loss of generality one can assume that
Using the conditions for 
to be a subalgebra, one obtains
Because 
is a subalgebra and the generator 6 forms the center, then
Comparing the coefficients, one obtains 
. 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.
Let us consider the subalgebra
. For constructing three-dimensional subalgebras of the algebra 
one considers
Since 
can be written as:
by forming a linear combination with 
and
, the operator 
can be taken in the form
. The subalgebra conditions gives
where 
and 
are arbitrary constants. Comparing the coefficients on the left side with the coefficients on the right side, one obtains
Thus, one obtains that
, 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
.
6.4. Optimal System of Subalgebras of the Algebra 
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 
consists of the following steps. In the first step, the vectors
are composed. 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
where the matrices A, B and C consist of the coefficients
In this step, the matrix A is arbitrary. The rank of the matrix
is equal to 
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.
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-dimensional 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
. The maximum possible dimension of a subalgebra of the algebra 
after gluing a subalgebra to the ideal 
is two. In this case, the matrix C is a 
matrix, the rank of which is equal to one:
By virtue of the automorphism
:
We can consider three cases:
1)
2)
3)
.
Case 1. By using the automorphism 
one can assume 
In this case, by means of linear combinations and by the automorphisms 
the table of coefficients is transformed to
The subalgebra conditions give
where the coefficients 
and 
are arbitrary constants. Comparing the coefficients, one obtains
Therefore, in this case the subalgebra is 
.
Case 2. Since
, or
. Because of
, by virtue of the automorphism 
one can take
. By means of linear combinations and by the automorphisms
, the coefficients are transformed to
The subalgebra condition gives
where the coefficients 
and 
are arbitrary constants. Comparing the coefficients, one obtains
This is a contradiction to
. Therefore, there exists no subalgebra in this case.
Case 3. Assume that 
and
, or
, 
, 
,
. Since
, without loss of generality one can choose
. By taking linear combinations and by virtue of the automorphism 
the table of coefficients can be transformed to
The subalgebra conditions give
which is satisfied with
Therefore, the subalgebra is
. Other elements of the optimal system of the algebra 
are constructed in the similar way.
The list of two-dimensional subalgebras of the optimal system of the algebra 
is presented in Table 1.
7. 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.
7.1. Subalgebra 7: 
The basis of this subalgebra is
Let a function
Table 1. Two-dimensional subalgebras of the optimal system of the algebra
.
be an invariant of the generator
. This means that
The general solution of this equation is
After substituting it into the equation 
, one obtains the equation
The characteristic system of the last equation is
Thus the universal invariant of this subalgebras consists of invariants
Hence, a representation of the invariant solution is
with arbitrary functions 
and
. After substituting this representation into Equation (1), one obtains the ordinary differential equation
The general solution of the last equation is
where 
are Whittaker functions and 
are arbitrary constants.
7.2. Subalgebra 16: 
The basis of this subalgebra consists of the generators
In order to find an invariant solution, one needs to find a universal invariant of this subalgebra. Let a function
be an invariant of the generator
. This means that
The characteristic system of the last equation is
The general solution of this equation is
After substituting it into the equation
one obtains the equation
The characteristic system of this equation is
Hence, the universal invariant of this subalgebras consists of invariants
A representation of the invariant solution of this subalgebra has the following form
with an arbitrary function
. After substituting the representation of the invariant solution into Equation (1), the functions 
has to satisfy the equation
The general solution of the last equation is
where 
is constant.
The two examples showed that there are solutions of the Navier-Stokes equations, which are partially invariant with respect to not admitted Lie algebra 
.
8. Conclusion
The algorithm of obtaining an optimal system of subalgebras was applied to the reduction of the NavierStokes 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.
9. Acknowledgements
This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
NOTES
2There is still no complete classification of the subalgebras of the Lie algebra
. Classification of infinite—dimensional subalgebras of this algebra was studied in [6].
3Short reviews devoted to invariant solutions of the Navier-Stokes equations can be found in [14-16].
4Firstly the approach of partially invariant solutions to the NavierStokes equations was applied in [17].