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Volume 12, 2004

 

 

 

 

WAVE TYPE SOLUTIONS FOR THE SYSTEMS OF COMPRESSIBLE

ADIABATIC FLOW THROUGH POROUS MEDIA

 

Galina St. Panayotova

University of Burgas

Department of Mathematics

Burgas, Bulgaria

 

Abstract :

In this paper we study the problem for existence of wave type solution for the quasi-linear system of compressible adiabatic flow through porous media. They are often considered in a variety of physical problems, where one encounters source terms that are balanced by internal forces.

                It turns out that the problem for the existence of such solutions reduces to the problem of investigation of systems of Frobenius type, which is equivalent to a system of ordinary DEs and can be treated by standard methods.

 

            1991 AMS Subject Classification 35L40, 35L65

           

Key words and phrases: quasi-linear systems of PDEs, simple waves, Pfaff system

 

1.Introduction

 

In 1-dimensional porous media, the motion of compressible adiabatic flow can be modeled by the compressible Euler equations with frictional damping terms. For smooth solutions, we have the following balance laws:

 (1)

                                                           vt – ux = 0,

                                 ut + p(v,s)x = -au, a>0,

                                 st = 0. 

 

Here, v>0 is the specific volume, u is the velocity, s stands for entropy, p denotes the gas pressure with  pv(v,s)<0  for  v>0  and  a>0  is a constant. (See[4].)

            In this case we investigate the existence wave type solutions for the above model of hyperbolic system.

             

2.Preliminary notes and definitions

 

            Definition 1. (see e.g.[2]) We say that a solution (u(t,x),v(t,x),s(t,x)) of the system (1) is constructed by means of Riemann invariants if it is of the form

(2)

u = U(R(t,x)),   v = V(R(t,x)),   s = S(R(t,x)),

 

where U(R), V(R) and S(R) are functions of a single variable and  R(t,x) is a suitable function, called Riemann invariant .

We also say that solutions constructed by means of Riemann invariant are wave type solutions.

Note that the classical wave solutions are special case of the wave type  solutions and that the class of the wave type solutions is the minimal class of solutions, which contains the classical wave solutions and is invariant with respect to the changes of variables.

            Such solutions are called simple waves when the systems (1) is  homogeneous and simple states  when (1) is non-homogeneous.

            Lemma 1. [3]. The functions  u, v  and s  can be represented in the form (2) if and only if

utvx – uxvt = 0

stvx – sxvt = 0.

 

Note that if the condition of Lemma1 is satisfied and if  u  is nondegenerated, i.e.if   Ñu ¹ 0, then  u  is a Riemann invariant .

 

            3.The main result

 

Theorem 1.  (u,v,s) is a solution of the system (1), constructed by means of Riemann invariant, exactly when (u,v,s) is a solution of the system

(3)

  vt – ux = 0,

                                         ut + p(v,s)x = -au, a>0,

                                         st = 0, 

       utvx – uxvt = 0,

      stvx – sxvt = 0.

 

 

Proof. The proof of this theorem follows directly from Definition1 and Lemma1.

            Let  ux = y  and   K = K(u,v,s) = -au + p(v,s)x . Now (3) can be represented in the form

 

(4)

ux = y

                                                                       ut = K

                                                                       vx = y2/K

                                                                       vt  = y

sx = 0

st = 0.

 

            Therefore,  s = const. In the case of isentropic flow where s = const.,(1) takes the form

 

  vt – ux = 0,

                                 ut + p(v,s)x = -au, a>0.

                                  

 

For the investigation of (4) we will apply a method described in [6] and used in [3], [5] and [7].

            The associated with (4) Pfaff system in R5 has the form

(5)

w1(dz) = du – Kdt – ydx

    w2(dz) = dv – ydt – y2/Kdx

where dz = (dt,dx,dy,du,dv).

 

            Theorem 2.  1) The  system  (5)  has a solution if and only if  there exist vector fields h1, h2Î R5 such that  wi (hk) = 0  (I,k = 1,2)  and  [h1,h2]  belongs to the linear hull of  h1 and  h2  .

            2) If  FI (I=1,2,3)  denote  functionally  independent  solutions of  the system  hiF=0 (I=1,2) (whose existence follows from the condition 1) in view of Frobenius’Theorem ), then the system of equations  FI = CI  (I=1,2,3) , where CI are suitable constants determined by the initial  functions  yo = yo(t,x) , uo = uo(t,x) and  vo = vo(t,x), then namely the pair   (u,v)  is a solution of system (4).

           

            Proof. Theorem 2 is a particular case of the classical theory of Pfaff’s systems.

So our initial problem reduces to the problem of finding suitable vector fields  h1   and   h2   with properties, described in the conditions 1) of Theorem 2 . This is a particular case of the general Pfaff problem.

            Consider system (5) as an algebraic system of equations with respect to coordinates of the field  dz. Since its rank equals 2, the set of its solutions form a linear space ( depending on the point  z = (t,x,y,u,v) ), called distribution ; denote this distributions by  q(z).

            Definition 2 [6]. Every involutive subdistribution  qi(z)  of the distribution  q(z) is called resolving distribution for system (5).

            Comparing this definition with condition 1) of Theorem 2, we conclude that in order to reach our goals we must find a resolving distribution of dimension 2 for system (5).

            The following three linearly independent vector fields

(6)

            x1 = (0,0,1,0,0),           x2 = (0,1,0,y, y2/K),            x3 = (1,0,0,K,y)  

 

satisfy the Pfaff system (5), i.e. wi (xk) = 0  (i = 1,2; k = 1,2,3).

            The linear hull of xj (j = 1,2,3) determines a three-dimensional distribution q(z). If we choose a pair of linearly independent fields h1, h2Îq(z), then their linear hull determines a two-dimensional subdistribution  q1(z)Îq(z). Thus if  [h1, h2] Î q1(z), then  q1(z) is involutive and from the classical Frobenius’Theorem it follows that  q1(z) is completely integrable subdistribution of q(z). Therefore the system hiF =0 (i=1,2) possesses three functionally independent solutions  Fk (k=1,2,3) . So  our first task is to build a basis consisting of a pair of vector fields h1, h2,  generating an involutive two-dimensional subdistribution q1(z) of q(z).

            The vector fields x1 , x2 , x3  defined above form a basis of the distribution q(z). Then the following theorems give us a way to find a pair of suitable vector fields h1(z), h2(z).

            Theorem 3. There exists exactly one vector field  h1 , satisfying the system

     

(7)

                  wi (dz) = 0;                w1 (xj,dz) = 0              (I = 1,2; j = 1,2,3)   

 

and exactly one vector field  h2 , satisfying the system

(8)

                  wi (dz) = 0;                w2 (xj,dz) = 0              (I = 1,2; j = 1,2,3)

 

If the system (5) has 2-dimensional resolving distributions  q1(z) , then  h1Îq1(z)  and h2Îq1(z).  

Proof. For a proof of Theorem 3 see [5] and [6].

            Theorem 4. If the system (5) has a two-dimensional resolving distribution q1(z) , then the vector fields

 (9)

h1  = ( -y, K , 0,  0,  0 )

   h2  = ( 0, 1, ry/s, y, y2/K )

 

where    x1 ,  x2 ,  x3    are  defined  by  (6),  belong  to  q1(z)  and       s = K – yKy,    

  r = y(yKv + KKu)/K.

            Proof. We have  [x1 , x2 ]  =  yx2   =   (0, 0, 0, 1, 2y/K - y2Ky /K2)  ;  

[x1 , x3] = yx3 =  (0, 0, 0, Ky, 1 ) ;

[ x2 , x3] = x2x3  - x3 x2 = (0, 0, 0, r, yr /K).

Hence using the identity w(X,Y) = Xw(Y) – Yw(X) – w([X,Y]), we obtain:

w1(x1 , x2) = -w1([x1 , x2]) = -1;                

w1(x1 , x3) = -w1([x1 , x3]) = -Ky ;

w1(x2 , x3) = -w1([x2 , x3]) = -r ; 

w2(x1 , x2) = -w2([x1 , x2]) = -2y/K + y2Ky /K2;  

w2(x1 , x3) = -w2([x1 , x3]) = -1 ;   

w2(x2 , x3) = -w2([x2 , x3]) = -yr/K. 

Therefore, solving (7) and (8), we get:  

h10 = rx1 - Kyx2+ x3 ;   h20 =(yr/K)x1 +  x2 + (-2y/K + y2Ky /K2)x3 .

According to Theorem 3, h10 and h20  belong to  q1(z).

We define  h1 = yh10 -h2   ; h2 = h10 + (K/y)h20   , which can be written in coordinate basis as (9).

 

            If [h1, h2] Îq1(z), then q1(z) is involutive  and from Frobenius’ theorem it follows that q1(z) is a completely integrable subdistribution  of  q(z).

            The proof of Theorem 4 is complete.

            Theorem 5. The subdistribution  q1(z)Îq(z)   defined as a linear hull of the vectorial fields hi(x) (i =1,2)  is involutive and therefore the system (1) has simple wave solutions.

            Proof. The subdistribution q1(x) spanned by the pair of vectorial fields  h1  and h2  is invilutive if and only if there exists a linear dependence between h1, h2 and [h1, h2] i.e. if the rank of the matrix M º (h1, h2 , [h1, h2] ) equals 2. Indeed  [h1, h2] =  (1, -Ky, -s/y, 0, 0 ) . Therefore the rank of the  M  is 2.

The proof of Theorem 5 is complete.

            Having in mind the classical Frobenius Theorem we conclude that wave type solutions of the problem (1) do always exist. The Frobenius system is:

         

(10)

                   h1F = Ft - ayFy - auFu + yFv = 0

h2F = -yFt - auFx = 0

 

            Further we continue with the special case when the system (1) is of the form

(11)

  vt – ux = 0

ut = -au,

            The fields h1 and  h2 in this case are

 

h1  = ( 1, 0, yKu, K, y )

 h2  = ( -y, K , 0,  0,  0 )

 

            They lead to the Frobenius system

(12)

                   h1F = Ft - ayFy - auFu + yFv = 0

h2F = -yFt - auFx = 0

 

which has the following three functionally independent solutions Fi = Ci  (i=1,2,3), where CI  are suitable constants determined by the initial values.

            The system (12) has two particular integrals

 

y + av = C1        and              y = C2u

 

Therefore  we have       C2u + av = C1 .   The second integral gives us a connection between y and u, i.e. u = eC2X+C(t)  , where  C(t) is an arbitrary differentiable function of t. For  v  we obtain  of (11)    v = (-1/a) eC2X. The solutions (u,v) is a general wave type solutions for the system (11).

 

                References

 

[1]. Grundland A. “Riemann invariants. In: Wave phenomena: Modern theory and applications” , North-Holand Math. Studies 97, Elsevier, New York, 1984.

            [2]. Jeffrey A. “Equations of evolution and waves” In: :Wave phenomena modern theory and applications” ,North-Holand Math. Studies 97, Elsevier, New York, 1984.

[3]. Kolev D. “Simple states of non-linear electric field model  in R2. Mathematika Balkanika, 12 (1998), 315-320.

[4]. Pan R. ”Boundary effects and large time behavior for the system of compressible adiabatic flow through porous media” Michigan Math.J.49, 2001, 519-540. 

[5]. Panayotova G. ”Simple waves for certain hyperbolic systems of PDEs” “International Journal  of  Differential Equations and Applications” Volume 6 No.3  2002.

 [6]. Tabov J. “Simple waves and simple states in R2” Jornal of Math. Analysis and Applications”, 214(1997), 613-632.

[7]. Tabov J. and  Kolev D. “Solvability of quasi-linear systems in R2.with time-depending coefficients”. Compt. rend.  de  l’Acad. bulg. des sci.,49(1996), 9-10, 35-38.

 

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