On a free boundary problem in a nonlinear diffusive–convective system

Physics Letters A 310 (2003) 25–29 www.elsevier.com/locate/pla

On a free boundary problem in a nonlinear diffusive–convective system S. De Lillo a,b,∗ , M.C. Salvatori a , G. Sanchini a,b a Dipartimento di Matematica e Informatica, Università di Perugia, Via Vanvitelli, 1, Perugia, Italy b Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, Perugia, Italy

Received 2 October 2002; received in revised form 6 January 2003; accepted 8 January 2003 Communicated by A.R. Bishop

Abstract A free boundary problem for a nonlinear diffusion–convection equation is considered. The problem is reduced to a nonlinear integral equation in time which is shown to admit a unique solution for small time. An explicit traveling wave solution is also obtained, which travels with the same velocity as that of the free boundary.  2003 Elsevier Science B.V. All rights reserved. PACS: 02.60.LJ Keywords: Burgers; Nonlinear; Diffusion–convection; Free-boundary

Free Boundary Problems (FBP) are relevant both from the mathematical and the physical [1–4] point of wiew. Technically speaking, they are boundary value problems defined over a domain with a moving boundary [5]. The motion of such a boundary is unknown and has to be determined as a part of the solution. In this sense the moving boundary is called a free boundary. On the other hand, FBP arise in several physical applications, e.g., the so-called Stefan problems [6] associated with the time evolution of the free boundary during a change of state, the internal motion of the free boundary between two immiscible fluids, the motion of a membrane constrained to remain above a solid * Corresponding author.

E-mail address: [email protected] (S. De Lillo).

obstacle (obstacle problem) [7] while being forced downward by pressure, etc. More recently, some FBP for nonlinear evolution equations were considered. In [8] a class of Stefan problems in nonlinear conduction was analyzed and the exact solution was constructed in parametric form. In [9] a 1-phase Stefan problem for the Burgers equation was studied and shown to admit a unique solution for small time; it was also proven [10] that the system admits a shock solution which travels with the same velocity as that of the free boundary and is stable. In [11] a Stefan problem in nonlinear heat conduction was formulated and the existence of a weak solution was proven. In this Letter we consider a 1-phase free boundary problem for the Rosen–Fokas–Yorstos equation: ϑt = ϑ 2 (ϑxx − ϑx ),

0375-9601/03/$ – see front matter  2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00068-9

ϑ = ϑ(x, t)

(1)

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S. De Lillo et al. / Physics Letters A 310 (2003) 25–29

over the domain x ∈ (−∞, s(t)) with initial datum θ (x, 0) = θ0 (x) > 0, θ0 (b) = β2 > 0,

where, in virtue of (4b) and (4c), z0 is given by x

x ∈ (−∞, b), s(0) = b > 0,

(2)

and boundary conditions

z = z(x, t)

1 ∂z = , ∂x ϑ(x, t) ∂z = ϑ(x, t) − ϑx (x, t), ∂t

(3b)

with

(3c)

(4a)

(4b) (4c) 2

∂ z z whose compatibility, ∂x∂t = ∂t∂ ∂x is guaranteed by (1). The free boundary problem for Eq. (1), specified by conditions (2) and (3), takes the form

−∞ < z < z¯ (t)

(5)

which is the Burgers equation for the dependent variable ψ(z, t) with initial datum ψ(z, 0) ≡ ψ(z0 ) = θ0 (x),

0



 z¯ (t) ≡ z s(t), t = s(t) − b + f s(t) ,

with

ψt = ψzz − 2ψz ψ,

(6b)

(3a)

Eq. (1) is a well-known nonlinear diffusion– convection equation modeling the flow of two immiscible fluids through a porous medium [12,13]. The function s(t) describes the motion of the free boundary between the two fluids; it is unknown and has to be determined together with the solution ϑ(x, t). (3c) is a condition on the flux at the free boundary, steaming from energy balance considerations. In the following we outline a method for solving the free boundary problem given by (1) with (2) and (3). We reduce the problem to a nonlinear integral equation in one variable (time) which admits a unique solution for small time. Moreover, we show that the system admits an exact travelling wave solution which propagates with the same velocity s˙ (t) of the free boundary. We start our analysis by introducing the change of the independent variable

2

1 dx , ϑ0 (x )

and

ϑ(−∞, t) = β1 > 0 (β1 > β2 ),
 ϑ s(t), t = β2 ,

 ϑx s(t), t − ϑ s(t), t = −˙s (t).

ϑ(x, t) = ψ(z, t),

z0 ≡ z0 (x) =

(6a)


 f s(t) =

s(t )

(6c)

1 dx . ϑ0 (x )

(6d)

0

Under the transformation (4a)–(4c) the boundary conditions (3a)–(3c) take the form ψ(−∞, t) = β1 ,
 ψ z¯ (t), t = β2 ,
 ψz z¯ (t), t = −β2 s˙ (t) + β22 .

(7a) (7b) (7c)

We have then obtained a 1-phase free boundary problem for the Burgers equation (5), characterized by the initial condition (6a) and by the set of boundary data (7a)–(7c). It is now expedient to use the Galilean transformation  z → z − 2β2 t, (8) ψ → ψ − β2 which leaves Eq. (5) invariant while changing (7a)– (7c) into ψ(−∞, t) = β1 − β2 > 0,
 ψ F (t), t = 0,
 ψz F (t), t = −β2 s˙ (t) + β22 ,

(9a) (9b) (9c)

where F (t) = z¯ (t) − 2β2 t.

(9d)

Next we introduce the generalized Hopf–Cole transformation [14] ψ(z, t) = −

v(z, t) z , C(t) + F (t ) v(z , t) dz  z

v(z, t) = C(t)ψ(z, t) exp − F (t )

(10a) 

ψ(z , t) dz ,

(10b)

S. De Lillo et al. / Physics Letters A 310 (2003) 25–29

with the initial condition C(0) = 1.

(10c)

Under this transformation Eq. (5) is mapped into the linear heat equation vt = vzz ,

(11)

with the compatibility condition
 ˙ = −vz F (t), t . C(t)

(13c)

27

and integrate Green’s identity

 ∂K ∂v ∂ ∂ −v (Kv) K = ∂ξ ∂ξ ∂ξ ∂τ

(16)

over the domain −∞ < ξ < F (τ ) and 0 < ε < τ < t − ε and let ε → 0. Using v (F (τ ), τ ) = 0 and K(z − ξ, 0) = δ(z − ξ ), we get the solution of the Stefan problem (11), (12a), (12b), (13a), (13b) as F (0)

K(z − ξ, t)v0 (ξ ) dξ

v(z, t) = −∞

t

Moreover, from (6a) and (9b), (9c), we obtain the following set of initial and boundary data for Eq. (11) 

F

v0 (z) ≡ v(z, 0) = ψ0 (z) exp −



ψ(z ) dz , (12a)

where, due to (6c), (6d) F (0) = f (b) =



ϑ(x )

−1

dx

(12b)

0

and

 v F (t), t = C(t)ψ F (t), t = 0,

 vz F (t), t = C(t) −β2 s˙ (t) + β22 .



 K z − F (τ ), t − τ vz F (τ ), τ dτ.

0



F (0)

b

+

(17) Eq. (17) implies that v(z, t) is known once vz (F (t), t) is known. We then take the z-derivative of both sides in (17) and evaluate it as z F (t). By putting vz (F (t), t) = G(t) and using [15]. ∂ Lim z F (t ) ∂z

t


 K z − F (τ ), t − τ G(τ ) dτ

0

(13a) (13b)

The original FBP for the nonlinear diffusion– convection equation (1) is reduced to a 1-phase Stefan problem for the linear heat equation (11) with initial datum (12a), (12b) and boundary conditions at the free boundary given by (13a), (13b). This problem is now solved by observing that (13b) and (13c) imply  
C(t) = exp β2 s(t) − b − β2 t , (14a) which is readily inverted as   t
 1 s(t) = b + β2 t + ln 1 − vz F (t ), t dt . β2 0

(14b) We now consider the fundamental kernel of the heat equation

2 1 1 z K(z, t) = √ √ exp − (15) 4t 2 π t

1 = G(t) + 2

t


 Kz F (t) − F (τ ), t − τ G(τ ) dτ,

0

we get F (0)


 K F (t) − ξ, t v0 (ξ ) dξ

G(t) = 2 −∞

t +2


 Kz F (t) − F (τ ), t − τ G(τ ) dτ,

0

(18a)

with

  t
 1 F (t) = ln 1 − G(t ) dt − β2 t + f s(t) , β2 0

(18b) where (14b), (9d), (6c) have been used and f (s(t)) is given by (6d). Thus the solution of the Stefan problem for the linear heat equation (11) has been reduced to the solution of the nonlinear integral equations (18a) and (18b), for only one independent variable t.

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S. De Lillo et al. / Physics Letters A 310 (2003) 25–29

The above reduction is similar to the one obtained in [15] for the classical Stefan problem. Nevertheless, there is a significant difference between the two, namely Eq. (18b) implies that in the present case the relation between the motion of the boundary F (t) and the function G(t) in nonlinear. In [15] the corresponding relation is instead linear. The difference is of course due to the fact that the starting point of our analysis is a free boundary problem for a nonlinear equation. The existence and the uniqueness of G(t) for small times can now be established following the lines of [9] and we will report on it elsewhere. Here we wish to point out that once G(t) is shown to exist and to be unique, from (17), (10a) and (4a) there follows that v(z, t) and ψ(z, t) exist and are unique together with the solution ϑ(x, t) of the free boundary problem for nonlinear diffusion–convection equation (1) characterized by the set of initial and boundary data (2), (3). We now turn our attention to a particular solution ϑ(x, t) of our problem. Namely, we show that there exists a solution ϑ(x, t) corresponding to a shock solution of the FBP for the Burgers equation (5) specified by the boundary conditions (7a)–(7c). The shock travels with a velocity which is proportional to the velocity s˙ (t) of the free boundary. We consider the shock solution of Eq. (5) compatible with (7a) ψ(z, t) = u2 +

β 1 − u2 , 1 + exp(β1 − u2 )(z − V t − z0 )

On the other hand, when (6c) and (6d) are used, one obtains 

1 + β2 . z¯˙ = s˙(t) 1 + (21) β2 The above relations imply that the shock solution (19a) is moving with a velocity V = z¯˙(t), proportional to the velocity of the free boundary of the nonlinear diffusion–convection problem. Moreover, equations (20a), (20b) and (21), together with (19b), fix the value of the constant u2 and the velocity V u2 =

β2 (β22 + β1 ) , β1 + β1 β2 − β2

(22a)

V=

β12 + β2 β12 + β23 > 0. β1 + β1 β2 − β2

(22b)

Finally, the solution of the one-phase free boundary problem for the nonlinear diffusion–convection equation (1) is given in parametric form by

−1 ∂z , ϑ(x, t) = (23a) ∂x where, in virtue of (4b), z solves z x=

ψ(z , t) dz ,

with ψ(z, t) given by (19a) together with (22a), (22b). (19a)

with V = β 1 + u2 ,

β 1 > u2

(19b)

where u2 is a constant to be determined. We use this solution on the interval −∞ < z < z¯ (t) and require ψ(¯z(t), t) = β2 together with ψ(z, t) = β2 , z > z¯ (t). It can be seen that this solution satisfies the boundary conditions (7b) and (7c) provided z¯ (t) − V t = z0 + γ

(20a)

(note that z¯ (0) = f (b) = z0 + γ ) and s˙ (t) =

(β2 − u2 )(β1 − β2 ) + β2 . β2

(23b)

0

(20b)

Fig. 1. The shock solution ψ(z, t).

S. De Lillo et al. / Physics Letters A 310 (2003) 25–29

29

obtained via the inversion of the relation (23b) by using the function ψ(z, t) of Fig. 1.

References

Fig. 2. Graphical representation of the function z(x, t).

The functions ψ(z, t) and z(x, t) corresponding to the above mentioned solution are shown in Figs. 1 and 2, respectively. We notice that ψ(z, t) exhibits the characteristic profile of the travelling wave solution of the Burgers equation, obtained in this case for values of the parameters u2 , V compatible with the conditions (22a) and (22b). On the other hand the profile of the function z(x, t) shown in Fig. 2 is

[1] A. Friedman, Variational Principles and Free-Boundary Problems, Wiley, New York, 1982. [2] D. Kinderleher, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. [3] C.M. Elliot, J.R. Ockendon, Weak and Variational Methods for Moving Boundary Problems, in: Research Notes in Mathematics, Vol. 59, Pitman, New York, 1982. [4] J. Crank, Free and Moving Boundary Problems, Oxford Science Publications, 1984. [5] A.S. Fokas, B. Pelloni, Phys. Rev. Lett. 84 (2000) 4785. [6] L.I. Rubinstein, The Stefan Problem, in: Am. Math. Soc. Transl., Vol. 27, 1971. [7] A. Friedman, Notices Am. Math. Soc. 47 (2000) 854. [8] C. Rogers, J. Nonlinear Mech. 21 (1986) 249. [9] M.J. Ablowitz, S. De Lillo, Nonlinearity 13 (2000) 471. [10] M.J. Ablowitz, S. De Lillo, Phys. Lett. A 271 (2000) 273. [11] S. De Lillo, M.C. Salvatori, J. Nonlinear Math. Phys. 9 (2002) 446. [12] G. Rosen, Phys. Rev. Lett. 49 (1982) 1844. [13] A.S. Fokas, Y.C. Yorstos, SIAM J. Appl. Math. 42 (1982) 318. [14] F. Calogero, S. De Lillo, Nonlinearity 2 (1989) 37. [15] A. Friedman, J. Math. Mech. 8 (1959) 499.