Nonlinear degenerate diffusion problems

DEGENERATE

DIFFUSION

PROBLEMS

Hongfei Zhang

TR diss

1647

NONLINEAR DEGENERATE DIFFUSION

PROBLEMS

PROEFSCHRIFT

ter verkrijging van de graad van

doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus,

Prof.dr. J.M. Dirken,

in het openbaar te verdedigen

ten overstaan van een commissie door

het college van Dekanen daartoe a a n g e w e z e n e ^

op dinsdag 21 juni 1988 ^ ^

< uA r

" °

C /

> ^

te 10.00 uur

door

HONGFEI ZHANG

geboren te Hunan, China,

Bachelor of Science in

Applied Mathematics.

TR diss

1647

Promotiecommissie:

Prof.dr. Ph.P.J.E. Clément Dr.ir. C.J. van Duijn Dr.ir. B.H. Gilding Prof.dr.ir. A . J . Hermans Prof.dr.ir. L.A. Peletier Prof. M. Primicerio Prof.dr.ir. J . W . Reyn

Dr.ir. C.J. van Duijn heeft als toegevoegd promotor in hoge mate bijgedragen aan het tot stand komen van het proefschrift. Het College van Dekanen heeft hem als zodanig aangewezen.

Nonlinear Degenerate Diffusion Problems

van

Hongfei Zhang

The first four theorems are related to the following initial value problem for u, A 6 IR

( fm _ 1f ) ' + - 2 - i j f ' = 0 on ( « , + ■ ) , (1)

f ( « ) = 0 , I(«,A)

fm-1f ' (W) = A ,

A function i{% u>, A) is said to be a solution of Problem I(w, A) if: i) f is nonnegative and absolutely continuous (a.c.) on (w,+oo); ii) f f' exists and is a.c. on (W,+<E); iii) equation (1) is satisfied a.e. on (u,+w) such that f(w) = 0 and fm~ f'(w) = A.

[1] Let m e (1, B>). Given any pair (u, A) e IR « IR+ U Of" * {0} problem l(u, A) has

a unique positive solution on (u,+a), which is bounded, strictly increasing and depends continuously on (w, A).

[2] Let m e (1, a>). Given 0 e IR there exists a function A = A(w, 0), defined on

[J*,+m) for some cJ* E IR-, such that for any u e [w*,+cr) the solution f(r/) of

problem I(w, A(w ; 0)) tends to 0 as T? -• w. Moreover, A(w*; 0) = 0, A(-; 0) is strictly increasing on [w*,+aj) and A ( - ; / 5 ) E C ' [w*,+a>).

Let m € (0,1). For a e R we define the set

Q(o) = {(w, A) € R~ . IR+: A1 - m ( - w )1 + m < a}

[3] Let m € (0, 1). Then there exists a a* e R , depending on m only, such that given any pair (w, A) 6 Q(a*) U DT"* R , problem I(w, A) has a unique positive solution on (w,+co), which is bounded strictly increasing and depends continuously on (w, A).

[4] Let m e (0,1). Given /? e R , there exists a positive function J = A"(w ; 0), defined on R, such that for any w e R, the solution {(TJ) of problem 1(a), J(u ; 0)) tends to 0 as 77 -• m. Moreover, A"(-; /?) is strictly increasing on IR,

Consider the following boundary value problem

(BVP) ( V ( l - V ) — — * ) ' - - ( Q V ) ' = 0 onR 1+V/Z

l V ( - , ) = 1 , V'(+.) = 0 ,

(2)

where Q(x) = Qg for x < 0, Q(x) = Qg + Qjx for 0 < x < a and Q(x) = Qg + Qp for

x > a, in which Q , Q, and a are positive constants.

A function V: R -> R is called a solution of (BVP) if i) V and V(l-V) ^ T J - are a.e. on R; ii) 0 < V < 1 on R and -1 < V' < 1 a.e. on R; iii) (2) is satisfied a.e. on R, V(-a>) = land V '(+■») = 0.

[5] If Q„ > Qf + 1/2, then problem (BVP) has a unique solution such that V(x) = 0 for x < a, V(x) is strictly decreasing on (0,a) and V(x) = QS/(QS+ aQ{) for x > a.

Consider the following free boundary problem

in DT= { ( x , t ) : 0 < x < s(t), 0 < t < T} r U. = U t x x (FP) u(x,0) = h ( x ) ux( 0 , t ) = f ( t u ( s ( t ) , t ) = 0 0 < x < s ( 0 ) = b € (0,1) 0 < t < T 0 < t < T - s ( t ) ux( s ( t ) , t ) = ( A - B | u ( x , t ) d x ) / ( l - s ( t ) ) , 0 < t < T .

where A and B are constants.

The model describes the saturated and unsaturated flow in porous medium when taking into account gravity.

[6] If h' and f are Holder continuous; h(b) = 0, h' (0) = f(0) and .b

h'(b) = (A-B j h(x))/(l-b), then there exists a unique triple {u, s, T} with

•'o

T > 0 which solves the problem in the sense that all the equations in (FP) are satisfied in classical sense. Moreover, ut is continuous up to x = s(t).

3^c m<u Aa/imdó

at

i

r

This thesis consists of four chapters, preceded by an introduction.

The first three chapters deal with a doubly nonlinear diffusion problem and the fourth chapter deals with the similarity solution of a degenerate diffusion problem.

In the introductory chapter an attempt is made to give a uniform description of the physical background from which the problems dealt with in the subsequent four chapters originate.

The four chapters were previously released as

C. J. van Duijn and Zhang Hongfei, Regularity properties of a doubly degenerate equation in hydrology, Comm. P.D.E., 13(1988), pp. 261-319. M. Bertsch, C. J. van Duijn, J. R. Esteban and Zhang Hongfei, Regularity of the free boundary in a doubly degenerate parabolic equation, Report of Dept. of Math, TU Delft.

M. Bertsch, J. R. Esteban and Zhang Hongfei, On the asymptotic behaviour of the solutions of a degenerate equation in hydrology, Report of Dept. of Math, TU Delft.

C. J. van Duijn, S. M. Gomes and Zhang Hongfei, On a class of similarity solutions of the e

Math, to appear.

Chapter 1 — Regularity properties of a doubly degenerate equation

in hydrology 38 Chapter 2 - Regularity of the free boundary in a doubly degenerate

parabolic equation 97 Chapter 3 - On the asymptotic behaviour of the solutions of a

degenerate equation in hydrology 123 Chapter 4 — On a class of similarity solutions of the equation

ut= : ( | u |m~1ux)xw i t h m > - l 137 Acknowledgement 161 Summary .' 162 English 162 Chinese ■ 163 Dutch ....: 164 Curriculum Vitae 165

Introduction

In this thesis we study the qualitative behaviour of some models which arise in the study of fresh and salt water flow in a porous medium. An important feature of these fluids is that the difference in viscosity is negligible compared with that in density, which is the major driving force for the flow.

1. Problems in Hydrology

Let us consider the flow of a fluid with constant viscosity and variable specific weight, through, a homogeneous and isotropic porous medium. In practice such flow situations occur, for instance, in coastal aquifers where fresh and salt groundwater meet.

The specific weight,, being the only nonconstant physical quantity, causes a movement of the fluid which involves a rotation. This follows immediately from the following momentum balance equation for porous media flow (Darcy Law), see e.g. Bear [11,12], Bear & Verruijt [13], Verruijt [44].

(1.1) ^ q + grad p + iez= 0,

where \i is the constant viscosity of the fluid and K is the permeability of the porous medium; q and p denote the specific discharge and the pressure of the flow respectively and j is the variable specific weight of the fluid. Finally, e denotes the unit vector in the positive z-direction.

Taking the curl of this equation results in

Here e and e are the unit vectors in the horizontal x and y directions, respectively. x y

This expression shows in fact that only the horizontal variation of the specific weight causes a rotation of the fluid.

To model the flow analytically, one of the methods is to find an equation for the pressure. Taking the divergence of equation (1.1) and using the incompressibility condition (1.3) gives

(1.4) -Ap = div(Tez) = g[.

Thus given a specific weight distribution on a flow domain and given appropriate boundary conditions, equation (1.4) may be used to obtain the corresponding pressure distribution. This pressure is substituted into Darcy's law to obtain the specific discharge field induced by the density variation and the boundary conditions.

When flow is two-dimensional, say, in the x-z plane, it is often convenient to use a stream function formulation. Because of (1.3) one introduces a function ip, called the stream function, which satisfies

(i.5) q = «in *=(-£££*).

Substituting this into expression (1.2) results , for two dimensions, in

(1-6) - ^ j j f e

A solution of this equation gives at once the corresponding specific discharge field, without the use of Darcy's law.

In the hydrology literature, equation (1.4) was first derived by Knudsen [36] and equation (1.6) by de Josselin de Jong [32].

Once the velocity field is known, one considers the transport of mass( i.e. the movement of the fluid ) through the porous medium. For this we use the mass—balance equation

(1.7) c | 2 + div(q7) = div (^grad7),

consisting of terms due to molecular diffusion and mechanical dispersion. If the scale of the problem is such that the influence of the hydraulic dispersion is small, one uses the equation

(1.8) £| 2 + d i v(q 7)=o .

Thus the underlying balance equations give a system of coupled partial differential equations. Generally speaking, such systems are difficult to solve. Only for a special choice of initial and boundary conditions explicit ( or almost explicit ) solutions are known , see Section 1.2 . In hydrology, simplified models were introduced under additional assumptions on the flow situation. These simplified models lead to differential equations which are not only simpler for practical purpose, but also of great interest from a mathematical point of view. Understanding these simplified situations is, without no doubt, useful for understanding more complicated situations.

1.1 -A sharp interface

Let a two—dimensional flow of fresh and salt water take place in a horizontally extended and vertically confined aquifer of constant thickness h, which we denote by fi = R x (0,h). We adopt a common assumption in hydrology that the scale of the problem is large with respect to the size of the transition zone between the fluids. Therefore we may assume that the two fluids do not mix and are separated by a sharp inclined interface with fresh water lying above and salt water below(see figure.1). Thus, there is an abrupt change in specific weight from fresh water with 7, to salt water with 7 ( 0 < y, < 7 < 00). Let for each time t > 0 the interface be parameterized as a function of the horizontal coordinate x : z = u(x,t) with x 6 R and t e R+.

Then the specific weight at each point is

(1.9) 7(x,z,t) = (7S - 7f) H(u(x,t) - z) + 7f for (x,z,t) e ïl * R+,

where H is the Heaviside function: H(s) = 1 if s > 0 and H(s) = 0 if s < 0. Inserting expression (1.9) into equation (1.6) gives

(1.10) -A^(x,z,t) = T |^H(u(x,t) - z) for (x,z,t) G fi * R+,

As a consequence the following jump conditions hold at. the interface( see also Chan-Hong et al. [20] ):

(1.11) [qj = [|f] = 0 at z = U(x,t),

(1.12) [ qs] = - [ ^ = r-sina, at z = u(x,t),

where n and s are the local orthogonal coordinates at the interface, s tangential and n normal, and where a is the angle of the tangent at the interface with the horizontal such that 2j£= tga.

Condition (1.12) implies that along the interface a shear flow exists of magnitude r • sin a, see also de Josselin de Jong [33].

In terms of the x—component we have

(1.13) [ql = r ft1/5* . atz = u(x,t). x l+(5u/ftc)2

z ( 0 , h )

Let Q denote the total discharge through the aquifer Q. Then Q is a given quantity depending on the time t only. We have

pu rh (1.14) Q = J qg x(x,z,t) dz + J qf x(x,z,t)dz,

where q and q, denote the restrictions of q to the salt water region and to the fresh water region, respectively.

To simplify the model, we use the Dupuit-approximation with respect to velocity field, which enables us to express the stream function ip in terms of u explicitly. In the Dupuit approximation, one assumes that the horizontal component of the specific discharge is constant over the height in each fluid and jumps across the interface (see figure 1.). One often uses this simplification in hydrology when the interface is flat enough. Numerical experiments [20] show that it is a reasonable

approximation for interfaces with an inclination angle of less than 45°.

Next we apply the approximation to (1.14) to obtain

(1.15) Q = u qs x(x,u,t) + ( h - u ) qf x(x,u,t).

We can now solve (1.13) and (1.15) for q and q, to obtain

(1.16) qs x( x , U , t ) = -r( \U) * / * + - * j - , s x n l+(<9u/dx)

and

Assuming that f = Ö on the lower boundary { z = 0 } we obtain

(1.18) #x,u,t) = - f V ( x , z , t ) d z = £ u ( h - u ) ft1/5* 9 - u

JO s x l+{da/dx.y

for (x,t) 6 R x K+.

Next we use equation (1.8) to determine the displacement of the interface. As in [20] this leads to

(1.19) e^(x,t) - ^{tf<x,u(x,t),t)} for (x,t) 6 R * OT. Finally we combine (1.18) and (1.19). Rescaling the variables according to

rt

x: = ( x - Q(s)ds/di)/h; t: = Tt/eh; u: = u/h,

J0

results in the equation

(

L20

> m = fe"(

1

-")

1

+

u

{*

/aK)

2>

for

<*•'>

e R x R+

1.2 Dispersion

In practical situations, due to mechanical dispersion and molecular diffusion,

the two fluids will mix, giving rise to a gradual transition zone and thus a gradual density distribution. The spreading process due to molecular diffusion and mechanical dispersion is described by equation (1.7).

Solving the coupled system consisting of equation (1.2), which describes the specific discharge rotation due to the gradual density distribution, and equation (1.7), which describes the dispersion caused by the specific discharge distribution, seems very complicated in the general case. In some circumstances, however, the analysis can be simplified. The example below is taken from de Josselin de Jong & van Duijn [34].

Consider the dispersion zone developing from an original flat, inclined interface, that extends in all directions to infinity. The boundary conditions of the infinite aquifer are assumed to be such that the flow remains constant in planes parallel to the original interface plane. Therefore, the plane flow situation occurs.

We choose coordinates £, q to be in the original interface plane with 77 horizontal and £ pointing upwards at an angle a with the horizontal and coordinate £ to be normal to the original interface plane and pointing upwards(see figure.2).

Fig.2

Then, the specific discharge component of velocity q=(q ,,q ,q^) satisfies (1-20) qv= qc= 0, q^= q^(C,t).

Initially, the interface is assumed to be sharp such that £ = 0 separates the region with fresh water above from the region with salt water below. Then the analysis of Section 1.1 shows that along the interface a shear flow exists, with its magnitude given by (1.12), i.e.

(1.21)

*f ^s r sina.

Superimposed on the shear flow is a constant specific discharge in ( direction of magnitude 0q, with 0 e R. Therefore, the initial flow conditions are

(1.22a) (1.22b)

7 = 7f, q^ = ( / ? + 5 ) q f o r C > 0 , t = 0,

7 = 7S, qf = ( 0 - j ) q for ( < 0, t = 0.

As time proceeds, the initial sharp transition from j ^ to 7g will spread by

mechanical dispersion and molecular diffusion so that the fresh and salt water will mix. Because of the plane flow character (1.20), we conclude that the specific weight will be a function of ( and t only, i.e.

and the specific discharge remain constant and equal to the initial value. Thus (1.24a) 7 =7 f, qe= ( / ? + s ) q f o r < ^ o o , t > 0 ,

and

(1.24b) 7= 7s, q^ = ( / * - £ ) q for ( - -*o, t > 0.

s

Because q and 7 satisfy (1.20) and (1.23) respectively, we obtain from equation (1.2)

(1.25) ^

=

_ f f g

8 i n o

.

Since £ is the only independent variable, integration of (1.25) gives, together with (1.21) and (1.24),

(1.26) q^/q = [ 0 + ( Tf + 7 g- 2- l)l ( 2 (7 r 7f))],

a linear relationship between the specific discharge and the specific weight.

The mass transport equation (1.7) is written now as (see, for instance, [11], [12], [13] or [44]):

(1-27) ^ l ^ m o l ^ T ^ ^

where 3>mo\ is the molecular diffusion coefficient and a™ is the transverse dispersion

length in the £- direction. A special feature of the dispersion is that, not the specific discharge itself, but its absolute value is taken into account.

Using the linear relation (1.26) between 7 and q,, we finally arrive at an equation in terms of q , only

t dq^/dt = aT 3[(q.m + Iq^^/ÖCl/ÖC,

where m = e

^^/o^q-When molecular diffusion can be disregarded, we obtain, after a suitable rescaling, (1.28) dq(/dt = d[\q^\dqi/d(\/dC

2. Mathematical Background

Both (1.20) and (1.28) are parabolic equations of the form (2-1) ut= (D(u)^(ux))x,

where we use subscripts to denote partial differentiations. o

In the first example D(u)= u(l—u) and <f>(\i ) = u /(1+u ) and in the second example D(u) = | u | and (f>(u ) = u .

If the function D and <j> in equation (2.1) are such that both D and <f>' are strictly positive and uniformly bounded, then this equation is called uniformly parabolic, see Ladyzenskaja, Solonnikov and Ural'ceva [37]. If, on the other hand, D or <f>' vanishes for certain values of u and u , the equation (2.1) is called degenerate parabolic. Thus in the first example, equation (2.1) degenerates at points where u e {0,1} or u e {-1,+1} and in the second example it degenerates at points where u = 0.

Degenerate parabolic equations exhibit various properties which are not shared by uniformly parabolic equations. To make this explicit, consider for m > 1 and u > 0 the functions

D(u) = m um - 1 and <j>{ux) = u ,

which form the equation

(2-2) ut=- (u I\ x ' u > 0 .

For m > 1 this equation is known as the porous medium equation. It models the one-dimensional density distribution of a gas in a porous medium, see Muskat [38]. If m > 1, the equation degenerates at points where u = 0 and if m = 1, it reduces to the uniformly parabolic heat equation, see Cannon [19], Widder [45]. The difference in behaviour of solutions of equation (2.2) when m = 1 and m > 1 is clearly demonstrated by the point-source solution ïï: if equation (2.2) is considered in the set Q = IR x R+ and if at t = 0 one requires that

u(.,0) = U6 onR,

where 6 denotes the Dirac distribution at the origin and where M > 0 denotes the total mass, then for the solution ïï of this initial value problem one finds if

(2.3) u(X )t) = - ^ _Te x p { - |1- } ,

( 4 j r t )2 4 l and if

m > 1T

(2.4) ïï(x,t) = t"a{[A - B x V2^ }1/ ^ "1) ,

where [•] , = max{-,0}, a = /? = l/(m+l) and where B = (m-l)/2m(m+l) and

A = BM

2

(

m

-

1

)/(

m+1

)/4(/J(cosö)(

m+1

)/(

m

-

1

)dö)

2

.

The solution ïï(x,t) for m > 1 is called Barenblatt-Pattle solution [9], [10], [40]. Note

—a 3

that it is a self-similar solution of the form u(x,t)= t f(^) with T? = x/v.

While the solution for m = 1 is smooth and positive on the whole set Q, and satisfies equation (2.2) in a classical sense, the Barenblatt-Pattle solution is only smooth at points where it is positive: i.e. on the set

(2.5) P(ïï) = { ( x , t ) 6 Q | ï ï ( x , t ) > 0 } . If y. [0,oo) -»R is given by

(2.6)

7

(t) = AM/B*,

then from (2.4) it follows that

(2.7) P(ïï) = {(x,t) € Qjt > 0, |x| < 7(t)}, i.e. the smooth curves x = ±7(t) form the boundary of the support of ïï.

Across the boundary of P(ïï), the derivative of ü with respect to the x-coordinate does not exist. In fact one verifies directly that the following interface equations hold: (2-8) Hm - ^ ( ï ï ™ -1) (x,t) = ±7(t).

(x,t) 6 P(ïï) x

x -• ± 7 ( t )

One refers to (2.7) and (2.8) as the finite speed of propagation property. Similar results for the general initial value problem of equation (2.2) (with m > 1) were proven during the last two decades.

In order to incorporate possible singularities at points where the equation degenerates one introduced the concept of generalized solution. This was first done by Oleinik, Kalashnikov and Czhou [39}. To be definite consider the initial value problem

(P)

ut= ( u I\ x f o r ( x , t ) 6 Q = 0 U R+

. u( • , t) = UQ( •) for x e R , where un is a given nonnegative function.

A continuous nonnegative bounded function u = u(x,t): R x R -» (0,00) is said to be a generalized solution of (2.2) if (u ) exists and is bounded in the sense of distributions, and if for every T > 0,

| u t ft- *x( um)x + jtf(x,0)u0 = 0,

R*(0,T)

for all smooth test functions tf such that ^ = 0 for t = T and for |x| large.

Then it is proven in [39] that given a nonnegative bounded continuous function uQ

such that (um) is bounded , there exists a unique generalized solution of (2.2).

Moreover, u 6 C°°(P(u)), where

P(u) = {u(x,t) 6 Q: u(x,t) > 0}.

Here, for convenience, we take uQ to be such that (um) is bounded. In fact, using

nonlinear semigroup theory, Benilan, Brezis and Crandall [14] have developed the existence and uniqueness theory for (2.2) with the initial function UQ 6 L (R). More general existence and uniqueness results with the initial data taken on in the sense of

consider the initial value problem (P) with u« > 0 satisfying (2.9) UQ > 0 iff x 6.(a,, a2) for some a, < a, e R.

Then the set P(u) can be characterized as

P(u) = {(x,t) 6 Q: d ( t ) < x < C2(t)},

where (•: R -> R, i=l,2, are two Lipschitz continuous functions of t such that (. is nonincreasing and £2 is nondecreasing. The functions C(t) are usually referred to as

the free boundaries induced by the degeneracy.

(2.10)

Let u be the solution of Problem (P) and let

This expression is usually denoted as the pressure. If one writes equation (2.2) in the form of the mass-balance equation

ut + (uq)x = 0,

then q = —v is the local velocity of gas. One expects that the free boundary will move with the value of this velocity near u = 0. Thus formally one expects the following interface equation to hold

(2.11) C f ( 0 = Mm ' -vx(x,t) i=l,2.

x - C j ( t ) (x,t)6P(u)

An important role in studying the solution of the porous medium equation and its free boundary is played by the following Aronson—Benilan estimate[5], which states the semiconvexity of the pressure

(2-12) ^ [ m T T J t ( °r vx x ^ - ( m + T ) t ) i n *<«)•

In [35] Knerr has shown that equation (2.11) holds in the sense that (2.13) D+C;(t)= l i m -^(nm-\(^),

x - ( j ( t )

( x , t ) e P ( u )

for all t 6 (0,T), where D denotes the right hand derivative.

He also showed that for i=l,2 there exist numbers t | 6 [0,T] such that C;(t) = aj for t 6 [0, t | ] ,

and

(—lj'CjOO is strictly increasing for t £ [t|, T].

The numbers t* are called the waiting times. It is possible to, have t? > 0 if the initial value is sufficiently flat near a- [35]. Caffarelli and Friedman [18] proved that Cj 6 C ^ t ï , T) and ( - l )1^ ' > 0 on (tï, T). Their results imply that (2.11) actually

holds for t e(t?,T), i=l,2. A similar result was proven by Aronson, Caffarelli and Kamin in [7]. There they also considered estimates on the waiting times t ï . Aronson and Vazquez [8] showed that (• e C°°(t|,oo). Using nonlinear semigroup theory, Angenent [1] proved that £., i=l,2, are actually analytic functions on (max(tï,t;p, oo).

With respect to the large time behaviour of the solution Vazquez [43] proved that the solution converges to the Barenblatt-Pattle solution with the same mass and the same center of mass. In addition the corresponding free boundaries converge to the free boundaries of the limiting profile. His result is the following.

Let uQ satisfy (2.9) and let

fa2 1 fa2

M = u0dx and xQ = w x u^dx.

. J SL. J a..

Then, as t -> oo

and

Cj(t)

- x

0

+

HHt).

Here v(x,t) is the pressure defined by (2.10) and v(x,t) is the corresponding pressure of the Barenblatt-Pattle solution (2.4) with mass M. The function 7(t) is given by (2.6).

For survey papers on this subject, see Peletier [42] and Aronson [5]. Another example of a degenerate diffusion equation is

(2.14) ut= =(u rlux lP~l ux ) x u- ° o n R x l R +

where p e (0,oo) and p + r e (0,oo).

Equation (2.14) occurs in the study of water infiltration through porous medium. When p = 1 , it reduces to the porous medium equation; when p -M, the equation may degenerate or .become singular at points where u = 0 or u = 0. Again this equation does not necessarily have classical solutions and assumes the finite speed of propagation property. See Esteban & Vazquez ([25], [26]) and their references.

2.1 O n ut= ( D ( u ) 0 ( ux) )x

The first three chapters of this thesis are devoted to the study of the qualitative behaviour of solutions of a class of degenerate equations which include the model equation (1.20). In particular, we will study the initial value problem

(2.15) ut.=(D(u)0(ux))x onQ=R*R+,

. u( - . * ) = u_an (_0^ -) o n R '

where D: [0,1] -♦ R is such that D(s) > 0 for s 6 (0,1) and D(0) = D(l) = 0;

</>: [—1,1] -> 1R satisfies 0'(s) > 0 for s e (-1,1) and 0 ' ( - l ) = 4>'{+l) = 0 and uQ is

chosen to be such that UQ - H e L (R) with H being the Heaviside function. The precise assumptions on the smoothness of D, </> and u„ are given in Chapter 1.

When D(s)= s(l-s), equation (2.15) admits a class of almost explicit self-similar solutions. Let u : Q-» [0,1] be defined by

s 1 (2.16) us(x,t) = 1 X > + 27[tJ' ~T+ f ( t )'x -27[tJ < x < +2 7 [ t ? 0 x < where f(t) is the solution of the initial value problem

■2T[tJ' ' f' =-2-4>(f)r

f(o) - f0e(o,i].

for t > 0,

Then u (x,t) satisfies equation (2.15) almost everywhere. The solution has the form of a line, rotating around the point x = a, u = 1/2, with initial slope f^.

Motivated by this self—similar solution, the notion of generalized solution for the initial value problem of (2.15) was introduced by van Duijn and Hilhorst in [22]. There they also proved the existence of a unique generalized solution which is continuous in Q. The precise statements are given in Chapter 1, where we also list the precise assumptions on D, <p and u~.

As indicated by the solution (2.16) and also by (1.13), one may expect that the interface will rotate in time and become flatter. Indeed, in Chapter 1 we prove a bound on the spatial derivative which decreases in time. To be precise, we show that for each t > 0

(2.17)

where f(t) is the solution of

| ux( - , t ) | < f ( t ) a.e. on R, f' = - C - 0 ( f ) r on (0,a f(0) f ₀ ess sup | un| R u with C = sup{ - D " l > 0. (0,1)

This estimate is sharp when D(s)= s(l-s). Since f(t) is strictly decreasing and f(t) < 1 when t > 0, we conclude from (2.17) that the degeneracy in the spatial derivative vanishes instantaneously and consequently u satisfies equation (2.15) in the classical sense at points where u e (0,1).

We are interested in the behaviour of the free boundaries induced by the degeneracy at u = 0 and at u = 1. For this purpose, we consider the solution u with the initial value u„ satisfying

UQ(X) = 0 for x < a,, and 0 < UQ(X) < 1 for a, < x < a„,

UQ(X) = 1 for x > a„ for some a, < a„ E R.

Then, it is proven in Chapter 1 that the set

M(u)={(x,t) 6 Q: 0 < u(x,t) < 1} can be characterized by

M(u)={(x,t) E Q: Cj(t) < x < C2(t)}

where £•: [0,oo) -» IR (i=l,2) are two Lipschitz continuous functions satisfying for all t > 0 u((\(t),t) = 0 and u((2(t)>t) = 1- Moreover, (-l)'(.(t) is nondecreasing on (0,oo)

and there exist numbers T? > 0 called waiting times, such that the free boundaries C stay stationary in [0,T|] and are strictly monotone on (T|, oo). We show that if UQ(X) is sufficient flat near x = a-, then the waiting time Tt is positive.

From a physical point of view we expect that (, moves at the speed of the salt water near that point and that (« moves at the speed of the fresh water near that point. Looking back to the derivation of equation (1.20) this means that the following interface equation should hold

(2.18) Cf(t)= Km V.(x,t) (i=l,2), X - C j ( t )

(x,t)eM(u) where V. are the local velocities defined by

■ ■ Vj = - 2 M 0(ux), V2 = 5 M 0(ux) When (x,t) e M(u)

(2.19) and

To verify (2.18), we first derive an estimate which assures the existence of the limit appearing on the right hand side of (2.18). In fact we prove an estimate which is similar to the Aronson—Benilan estimate for the porous medium equation.

Let

Nj = {(x,t) € Q | - » < x < Cx(t) + sQ/4, t > 0}

and

N2 = {(x,t) e Q | C2(t) - (l-s0)/4 < x < oo, t > 0}

where sQ satisfies D'(s0) = 0.

Then, under some additional assumptions on u„, we have (2.20) and

(0(u

x

))

x

> —f" i n - Z ^ )

(<P(\))X<-+-T- i n ^ ( N2)

for some positive constant K > 0.

One of the consequences of (2.20) is that the following limits exist lim # u ) (i=l,2).

x - C j ( t ) X

( x , t ) e M ( u )

Therefore, the right hand side of (2.18) is well defined. In Chapter 2 we prove that

(2.21) C - e C ^ T j » for i=l,2, and that equation (2.18) holds except possibly at t=T?.

The proof of (2.21) is based on a differential inequality for £.: (2.22) ■ (_i)i{C..+ 2g . C(}<0 in ^(0,oo),

(2.23) u(x,t) = ur(x-Cj(t0), t-tQ)+ o(|x-C}(t0)| + | t - *0| )

where 7. = D+C;(tn) and u is the local linear front 1 'iv u' 7.

u7(x,t) = <?j[x-7 it]+

with 0. and 7. being related by 7.= - D ' ( i - l ) <j>{0X i=l,2.

For the proof of inequality (2.22) we introduce a new coordinate which transforms the free boundaries C constant in time. For this purpose we write (when dealing with the left free boundary) the differential equation (2.15) as

(2.24) ut+ ( u Vx)x = 0 in M(u),

where V, is defined in (2.19). Then V, can be considered as the local velocity in horizontal direction of the salt-water ( see also (1.16)). It generates the Lagrangian coordinate (y,t) through

(2.25)

x = x ( t ; y ) , xt=Vx(x,t).

Indeed, if we define for y > 0, the function x( • ;y): [0,oo) -+ R by rx(t;y)

y = u(s,t)ds (mass of salt water to the left of x),

J—00

then differentiation with respect to t, for fixed y, and the substitution of equation (2.24) gives equation (2.25.). Clearly for each t > 0,

C1(t) = limx(t;y).

1

yio

Thus if we consider now the solution u as a function of y and t, then the free boundary corresponds to y = 0: for all t > 0, u(0,t) = 0 and u(y,t) > 0 for y > 0. The idea of introducing a Lagrange coordinate has been used by many authors to study nonlinear problems, e.g. Gurtin, MacCamy and Socolovsky [29]. In case of the porous medium equation (2.2) the transformation has also been used to track the free boundary numerically, see Bertsch and dal Passo [16].

A direct consequence of (2.22) is that for each t > 0, D —(. (t) exist and

that

(2.26) (-l/D-^t) > (-ïyD+CjW 0 = 1,2).

Consequently the C -regularity of (. in (T|,oo) follows if one can prove that (2.27) D~Cj(t) = D+<j(t) for i = 1,2 and for t > Tj.

This equality is a consequence of the linearizing property (2.23), which in turn follows from a blow—up procedure introduced by Aronson, Friedman and Kamin [7] in the study of the free boundary of the porous medium equation .

We also demonstrate in Chapter 2 that u and u, have limits as (x,t) -»((;(tn),tQ) with (x,t) e M(u) and tQ > T Ï . Moreover these limits satisfy, on

the set {x=(j(t), t > T | } , the equations (1=1,2)

(2.28) C{(t)=-D'(i-l)0(ux),

and

(2.29) ut= D ' ( i - l ) ux0 ( ux) . 1=1,2

Chapter 3 is devoted to the study of the large time behaviour of the initial value problem of (2.15) when D(s)=s(l-s). In [22], it was proven that if the initial function u„ can be "sandwiched" by two self-similar solutions, that is, if there exists

c, < c2 6 R such that

u (x-c, ,0) < u0(x) < u (x-c2,0) for x e R,

then, for any c e IR and for any initial slope L 6 (0,1]

| u(x,t)-uc(x-c,t) | = 0(f(t)) as t -»oo.

By specifying a particular c e K, we improve the rate of convergence to be o(f(t)). Our result is the following:

Let XQ 6 R be the unique point for which

(2.30) r°(u0(x)-us(x-x0,0)) = 0.

I u(x,t)-ug(x-x0,t) I = o(f(t) ) as t -. oo

for any f„ € (0,1]. In addition, if fQ > ess sup |uQ' |, then and IK

C

x

(t)

-

^ ( t )

T o-

C

2

(t)

-

?

2

(

l

)

i ° ■

M t

-

«"•

where £.(t) are the free boundaries of the self-similar solution u (x-x0,t), i.e.

*1 W = x0 _ 27[tJ a n d ^ W = x0+

2T[tJ-Our proof is elementary and is geometrically clear. We use the fact that the estimate (2.17) for the spatial derivative is sharp in this case.

2.2 On u ^ d u l1 " - ^

In Chapter 4 we study an initial value problem involving a differential equation that generalizes equation (1.28). We consider

(2.31)

(P)

f ut=(]u j m 1ux)x inQ = RxR+,

. u(-,0) = u0(-) onR,

where uQ(x) = a for x > 0 and uQ(x) = 0 for x < 0, see also (1.24).

By a rescaling, we may assume that a - fi = 1 and that a 6 [0,1].

First we observe that if u(x,t) satisfies (2.31), so does u(kx,k2t) for any k > 0.

Hence by uniqueness

u(kx,k2t) = u(x,t) on Q,

since this equality is achieved initially. This means that u is a self-similar solution of the form u(x,t)=f(j/) with r/ = x/t'.

Due to this similarity, Problem P reduces to a boundary value problem for an ordinary differential equation:

(2.32)

(P*)

( | f | m - lr y +_^_rjr = 0 o n ( R j

f(-00) = o - l , f(+oo) = a.

By a solution of Problem P*, we mean a function f: R -» R such that i) f is absolutely continuous on R, ii) |f|m— f' exists and is absolutely continuous on R and. iii)

|f|m~ f' and f' satisfy (2.32) almost everywhere on R such that f(+oo) = a,

f (-to) = a - 1 .

It is possible to use a matching argument (see, e.g. van Duijn and Peletier [23]) to study Problem P*. However here we propose to study the boundary value problem

(2.33) (P**) y . y » = - 2 | f y ( a - l ) = y ( a ) = 0 , y > o m - l _{on (a-1,a),} on (a-1,a),. which yields a solution f of Problem P* via the transformation

y'(f(7,))=-T7.

A solution of Problem P** is understood in the sense of Problem P*. Our method applies to the following two cases:

i) a 6 [0,1] when m e (0>), ii) a = 1 when m 6 (-1,0].

In each of the two cases, we prove that there exists a unique solution y e C (a-1,a) D C[a~l,a] of Problem P**. This implies the existence of a solution f of Problem P*.

case i) (2.34) l i m - , f <1-, n) /2( , ) = ( 2 . } ± g ) * T)-> — oo and (2.35) t(rj) - 1 = 0(erfc(-r)/2)) T) -» oo and for case ii) (2.36) f ( j?) - a = 0(er/c(-»?/2am~1)) 7/-00 and (2.37) f ( r / ) - ( o ^ l ) = 0 ( e r / c ( + 7 / / 2 | a ^ l |m - 1) ) jj-.-».

The result (2.34) is obtained in Chapter 4. The other results, (2.35)-(2.37), follow from an argument due to Peletier [41], see also Gilding [28].

The uniqueness of solutions for Problem P* follows from Brezis and Crandall [17] in case i) and from Rodriguez, Esteban and Vazquez [24] in case ii)uniqueness result from [24] concerns maximal solutions satisfying for m 6 (—1,0)

(2.38) um(x,t) = o ( | x | ) a s | x | - o o ,

and for m = 0

(2.39) -logu(x,t) < o(IxI) as|x|->oo uniformly for t 6 (r,T) for all r > 0.

Expression (2.34) in terms of the original variables x, t and u yields lim ( - x ) u (1-m) /2( x , t ) = ( 2 t . ^ ±m- ) ^

x -• - 0 0

for each t > 0, which, a fortiori, satisfies (2.38) or (2.39). Thus the self-similar solution represents a unique maximal solution of Problem P.

Finally, special attention in Chapter 4 is given to the sign change solution of Problem P and to the occurrence of the free boundary and the behaviour of the solution near the free boundary.

3. Numerical results

To compute a numerical approximation of a solution u of the equation (3-1) ut= (D(uMux))x >

we adapt an algorithm introduced by Hoff [30] for computing the solution and the free boundaries of the equation

(3.2) ut= (u(l-u) ux)x.

The main feature of the scheme is that both the solution and the free boundaries are computed simultaneously. The idea of discretizing both the equation and the free boundary equation was first introduced by Hüber [31] in approximating the solution of a one—dimensional two—phase Stefan type problem, see also Fasano and Primicerio [27]. In our case the free boundary condition is given by (2.28), i.e.

(3.3) C|(t)=-D'(i-lMux)(Ci(t)+,t) (i=l,2)

We will discretize both equation (3.1) and (3.3). Description of algorithm:

Let At and Ax be increments in t and x, and let t = nAt for n e 1 U {0}, x, = kAx for k e 1. We denote the approximations to u(x, ,t ), (i(t ) and Co(tn) by

.n An

uj\ CV and Co respectively. The starting values are set to be

Define and and set K? = min { k: xk_1 > <J } K3 - max { k: xk + 1 < ^ }, s11 - x - (n s2 ~ 4 X

K§-Then £? and dj a r e computed from the following discretized version of

equation (3.3)

(3.4) C j+ 1 = C? - A t D ' ( 0 ) ^ ( ^ | — ) ,

(3.5) C5+1 = <2 - A t D ' ( l ) ^ ( ^ | 2 - ) . S2

For KÏ < k < K§ we compute uP from the finite difference equation

un + 1- u11 Aun + 1 Au11

(3.6)-!5 ! l _ = D(u?)0'(Vu?) 1—+ e 2 -+ D> k ) Wu! > k '

At K K A x^ A x^ k K k

where A is the difference operator .n :

Auk: = uk - l -2 uk + uk+l'

and where VuJ1 is the central difference approximation for the derivative u at point

x = xk

n n Vu": = " k + l ^ - l , .

K 2Ax

Further e > 0 is an artificial viscosity parameter introduced in order to overcome the difficulty caused by degeneracy at u = 0 and u = 1.

For convenience we set

0 = At/Ax2,

and write (3.6) in the form

(3.7) u £+ 1 = u£ + 0 D(u°)*'(Vu£) A u £+ 1 + 0 Au£ + At D'(uJ) #VuJ)VuJ,

f o r K Ï < k < K 5 a n d n e Z+u { 0 } .

For x, n < x, < («> uv 's computed from the linear interpolation

Ji+1

n + l _ , n + l ^2 xk , xk ~ xK g

Cn+1 - x

Kg s2 XK§

<3-8) uk ~uK g - J Ï + 1 " T^nTT Y

and for £n < x, < xK ? , u? is computed from

x - Cn+1

ft 9Ï un + 1 - u n + 1 k ^

(J-9) uk - uK ? j .

XK? M

This scheme, consisting of (3.4), (3.5) and (3.7) - (3.9), leads to a linear tridiagonal system of equations. It was implemented on the IBM 3083—JX1 computer at Delft University of Technology and the NAG library was used to solve the linear tridiagonal system. The purpose of the computation is to determine quantitatively the behaviour of the solution of the degenerate diffusion equation. In addition they show the accuracy of the scheme involved in dealing with degenerate diffusion problems. The following phenomena which have been proven theoretically are observed from the result of computation.

The initial function u«(x) is chosen to be

u

n

M

0 0 . 1 ( x + 0 . 7 5 )2 7(x+0. 5)/8+0.05 1-0. l ( x - 0 . 7 5 )2 1 f x < - 0 . 7 5 f -0.75 < x < -0.50 f -0.50 < x < 0.50 f 0.50 < x < 0.75 f x > 0 . 7 5

where we take a, = -0.75 and a« = 0.75 to be the starting point of the free boundaries. The initial value u„(x) behaves as a parabola near the points a* and a„. The result of the computation is shown in figure 3, below. It clearly demonstrates the waiting time phenomena.

I N T E R F A C E C U R V E

- Ü . I 5 - 0 . 5 - 0 . £ S O . Z S Q . S Ü . / 5

Large time behaviour

We take the initial function to be such that

u

0

M

=

0 x + 1 .00 1 - 0.7x 0.7x + 0.3 1 f x < - 1 . 0 0 f -1.00 < x < 0 . 0 0 f 0 . 0 0 < x < 0 . 5 0 f 0.50 < x < 1.00 f x > 1 . 0 0

Here we allow the initial function to have inner degeneracy point. The numerical result is shown in figure 4. Observe that the inner point of degeneracy vanishes instantaneously and that the numerical approximation also converges toward the rotating self—similar solution.

Fig.4: convergence towards the self-similar solution.

At several times the computed solution is compared with the self-similar solution u s given in section 2.1. The values of the computed free boundary are given below

C(t) £(t)(computed) 1.1226 2.2547 3.1037 3.7642 1.0104 1.4094 1.6595 1.8342 1.0119 1.4095 1.6621 1.8369

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Chapter 1

REGULARITY PROPERTIES OF A DOUBLY DECENERATE

EQUATION IN HYDROLOCY

C.J. van Duijn Zhang Hongfei*

Department of Mathematics & Informatics Delft University of Technology

Delft, The Netherlands

I. INTRODUCTION

In this paper we study the regularity of solutions of the ini­

tial value problem

(C)

fU(. = (D(u)(|>(ux))x i n Q = K » K+, (1.1)

u(-,0) = uQ(-) o n » , (1.2)

*) This author is on leave from the Department of Applied Mathe­ matics, Tsinghua University, Beijing, People's Republic of China. His research was sponsored by funds of the Dutch Minis­ try of Education and Sciences.

261

262 VAN DUIJN AND HONCFEI

where t and x denote, respectively, a time and a space coordinate

and where the subscripts t and x denote partial differentiation

with respect to these variables. Further

D: [0,1] -♦ [0,°o) satisfies

D(s) > 0 for s € (0,1) and D(0) = D(l) = 0 (1.3)

and

<(>: [-1,13 ->1R satisfies

4>'(s) > 0 for s £ (-1,1) and (j>'(-l) = <(i'(+l) = 0. (1.4)

Observe that equation (1.1) is only of parabolic type at points

where the solution u £ (0,1) and its derivative u € (-1,1). Since

D(s) vanishes at s = 0,1 and (|>'(s) at s = ± 1 , the equation degenerates

at points where u = 0,1 or u = ±1.

An example of a differential equation which has the form of

(1.1) with D and <|) satisfying (1.3) and (1.4), respectively, is

found in a model from hydrology. In this model one studies the

evolution in time of the interface between fresh and salt

ground-water under certain simplifying assumptions on the flow field.

This model was first derived by de Josselin de Jong [15]. In Sec­

tion 2 we give a brief description of its background and of its

D(s) = s(l - s) and <f>(s) = s/(l + s2) . (1.5)

Problem C was first studied by Van Duijn & Hilhorst [9]. They

introduced a class of solutions and they showed that if D,<f> and

un satisfy the hypotheses:

HD1: (i) D 6 c'CO.Ü fl C2(0,1);

(ii) D > 0, D" < 0 on (0,1) and D(0) = D(l) = 0;

H<)>l: (i) $ € C C-1,1] 0 C2(-l,l);

(ii) Q' > 0 on (-1,1) with <J>'(±1) = 0 and $(0) = 0;

HuQ: (i) uQ € W1 , O°0R);

(ii) 0 C u < 1 and -1 < u' < 1 a.e. onlR;

(iii) u - H € L(1R), where H denotes the Heaviside func­

tion: H(s) = 1 when s > 0 and H(s) = 0 when s < 0;

then Problem C has a unique solution u on Q.

In Section 3 we recall some of their results for later use. In

Section 4 we introduce subsolution and supersolution for Problem C

and we prove a comparison principle.

To prove regularity properties for solutions of Problem C, we

have to impose additional assumptions on the functions D and <J>.

264 VAN DUIJN AND HONCFEI

HD2: (i) There exist c,d £ 1K+ such that -d ^ D" < -c = supU"

(0,1) on (0,1);

(ii) D E C +Ml(0,l) for some a £ (0,1) and D"'- D' > 0 on

(0,1).

and

Hd>2: (i) s<|)"(s) ^ 0 and <Ms) = -<|>(-s) for s £ (-1,1);

(ii) ^ E C (-1,1) for some a £ (0,1).

Observe that the functions D and <|>, given by (1.5), satisfy all

the imposed conditions.

In Section 5 we use HD2 and Hi|)2 to obtain a pointwise bound on

the derivative of u with respect to x, which is sharp in the case

where D(s) = s(l - s ) . It follows from this bound that for all

t > 0

u (• ,t) £ (-1,1) a.e. o n K

which means that the degeneracy in the derivative vanishes instan­

taneously for t > 0. This in turn implies that the solutions are

classical solutions in the region where u 6 (0,1).

The rest of the paper is devoted to the study of the free

boundaries induced by the degeneracy of the equation at u = 0 and

u = I. To avoid unnecessary complications we assume that there

exist real numbers -eo < af < a < «> such that

UQ = 0 on (-°°,a ]

"0 £ (0,0 on (a,,a ) (1.6)

L e t

M(u) = { ( x , t ) £ Q I u ( x , t ) e ( 0 . 1 ) } .

In Section 6 we follow Knerr [16] to show that there exist

functions £,.,£„: C0,°°) -*TR such that the set M(u) can be written

M(u) = {(x,t) € Q k , ( t ) < x < C2( t ) , t > 0 } . (1.7) F u r t h e r , we show t h a t e a c h f u n c t i o n s a t i s f i e s : T h e r e e x i s t s t * € CO.oo) s u c h t h a t C - ( t ) = a . f o r t € C O . t f ] , ( - 1 ) C i s n o n d e c r e a s i n g on C0,<») w i t h l i m ( - 1 ) t . . ( t ) = +°° and C. € C0 , 1I I O , c o ) . I n w h a t f o l l o w s we r e f e r t o t, and r a s t h e i n t e r f a c e s of P r o b l e m C. They s e p a r a t e t h e r e g i o n s w h e r e u = 0 , u 6 ( 0 , 1 ) and u = 1. N e x t , l e t v , : = - ^ - <(>(u ) and v , : = r ^ - è ( u ) on H ( u ) , ( 1 . 8 ) 1 u x 2 l - u T x

In terms of the hydrological model, v represents the x component of

the salt water velocity and v the x component of the fresh water

velocity, see (2.16) and (2.17) after suitable rescaling. Based on

inter-266 VAN DUIJN AND HONGFEI

faces is equal to the velocity of the fluids in the corresponding

points, see also (2.18) and (2.19). Thus at the interfaces X,. one

expects for t > 0 the differential equations

C!(t) = lira v.(x.t) i = 1,2. (1.9)

1 x-d(t) L

(x,t)£M(u)

In order to establish these equations, we first derive an

Aronson-Benilan C23 type estimate for (<j>(u )) , near the inter­

faces. As in the case of the porous medium equation and other re­

lated degenerate equations, this appears to be the crucial esti­

mate for the regularity theory, e.g. see Vazquez C20],or Esteban

& Vazquez ClOÜ.

For solutions of Problem C we prove in Section 7 first an

interior estimate on M(u). Let (x tn) £ M(.u) such that

u(xQ,t0) € (5,1-6) for some 6 6 (0,1/2) and let |u^| < 1 - e a.e.

2

o n K for some e > 0 and u„ 6 H, (a,,a„). Then there exists a con-0 ■ loc 1 2

stant K > 0, depending on <5 and £, such that

i(*("x))x(x0,t0)| < K t -, / 2. (1..0)

After that we prove one-sided bounds near the interfaces. For this

we need

and

and the sets

N, = |(x,t) £ Q ^o < x < C,(t) + s0/4, t > OJ (1.12)

N2 = |(x,t) € Q c2(t) - (1 - sQ)/4 < x < o», t > o | (1.13)

where s_ is the zero of D'. The result is the following. Let

|u'| < v < o a.e. onlR and let u_ € H. (a , a _ ) . Then there exist

constants K. > 0, i = 1,2, such that

l

(<t>(ux))x > - K ^ t ' + t"'/2) in P'CN,) (1.14)

and

(<t>(ux))x < +K2(t"' + t"l / 2) in P ' ( N2) . (1.15)

In establishing the interior estimate we use an energy type

inequality, also given by Hoff Cl4], combined with a Bernstein esti­

mate. The inequalities (1.14) and (1.15) follow from a maximum

principle argument.

2 We observe here that for the special case <\>(s) = s/(l + s ) , the

constant O from (1.11) is given by

O = 0.505

From estimates (1.14) and (1.15) it follows that for every t > 0

is nonincreasing when x + t.(t) and

>(ux)(x,t) - K2 (t ' + t 1 / 2) x

is nonincreasing when x + C-(t). Since both expressions are bound­ ed, this means that lira <))(u ) (x,t) exists and consequently,

x->rai(t) x

(x,t)€M(u)

! im u (x.t) = u (C.(t),t) exists.

(x,t)€M(u)

In Section 8, we then show that the interfaces are right

dif-ferentiable for each t > 0 and that

D ^ ( t ) = -D'(i - l)<Kux(?.(t),t)), (1.16)

for i = 1,2 and for all t > 0. We also show that (-1) C.(t) is

strictly increasing for t > t*.

The differentiability of the interfaces is much more involved, see

e.g. Caffarelli & Friedman C63 and Aronson , Caffarelli & Kamin C3].

We leave it for a future paper.

2. THE MODEL

Consider the two-dimensional flow of a fluid of variable den­

sity and constant viscosity through a homogeneous porous medium.

vertical and pointing upwards. If the flow is incompressible, then

the underlying equations are, see Bear [53:

Momentum balance (Darcy law)

7- q + grad p + ye = 0, (2.1)

and

div q = 0. (2.2)

Here q is the specific discharge, p the pressure, y the specific

weight and |i the dynamic viscosity of the fluid. Further, tc denotes

the intrinsic permeability of the porous medium and e is the unit

vector in positive z-direction. Introducing the stream function 4>:

q = curl i(/

V dz'ZxJ'

and taking the curl of (2.1), one finds the equation

-**-£&•

(2

-

3)

see also Chan-Hong et al. [73, where these manipulations are made

precise.

In our model we consider the specific case where the flow takes

place in a coastal aquifer which is vertically confined and hori­

270 VAN DUIJN AND HONGFEI

Fig. 1. Example of a fresh/salt distribution in a hori­ zontal aquifer.

water, separated by an abrupt transition - an interface. Our ob­ ject is to derive an equation for t h i s fresh-salt interface, which has the form of equation (1.1). We follow the derivation given in [7] and the original work of de Josselin de Jong [ 1 5 ] .

Suppose that for each time t the interface can be parametrized by the horizontal space coordinate x, see Fig. 1, and that i t s height with respect to the z = 0 plane is given by £ = C ( x , t ) . In terms of £, the specific weight becomes

Y(x,z,t) = (Y " YF)H(f;(x,t) " z) + ye ( 2 . 4 ) .

s i I

where Y . and Yr ar e constants ( Y > Yf) denoting the specific

weight of the salt and the fresh water and where H is the Heaviside

I n s e r t i n g ( 2 . 4 ) i n t o ( 2 . 3 ) r e s u l t s i n t h e f o l l o w i n g jump c o n d i t i o n s

a t t h e i n t e r f a c e ( s e e C 7 ] ) :

I,„l - [ » ] - 0. <2.5>

-m

C O = - I T £ I - r - s i n a , r - * ■ ( Ys" Yf) . ( 2 . 6 )

Here s and n are local orthogonal coordinates at the interface, s

tangential and n normal, and a denotes the angle of the tangent at

the interface with the horizontal.

From (2.5) and (2.6) we find for the jump in the x-component of the

specific discharge at the interface

Tqx] = qf (x.C.t) " qs (x,£,t) = T t a"a 2 ■ (2.7)

x 'x 1 + tan a

If t h e a q u i f e r has a c o n s t a n t t h i c k n e s s R and i f i t i s bounded

below and above by an impervious boundary, then the t o t a l fresh

and s a l t water d i s c h a r g e through the a q u i f e r in the p o s i t i v e

x-d i r e c t i o n i s given by rR Qf ( x , t ) x JC ( x , t ) x and qf ( x , z , t ) d z , (2.8)

s

_x ( x

'

t }

- [

J0 C ( x , t ) Q ( x , t ) = q ( x , z , t ) d z . ( 2 . 9 ) S" i n Sx

272 VAN DUIJN AND HONGFEI

Also the following fresh and salt water continuity equations hold:

x

c 34, <

2

-

1 0 ) Dx 3t' and 3 Qs * = -e K (2.11) 3x 3t'

when E denotes the porosity of the porous material.

Consequently, the total discharge

Q = Qc + Q (2.12)

f s

X X

is' constant in space and is considered here as a given quantity.

Next we make an assumption with respect to the discharge field,

which in hydrology is called the Dupuit-approximation, see also

Bear [5]. We assume that the horizontal component of the specific

discharge is constant over the height in each fluid such that

q£ ( x . z . t ) » qf ( x . C . t ) , C < z < B ( 2 . 1 3 )

X X

and

qs ( x , z , t ) « qg ( x , F , , t ) , 0 < z < C ( 2 . 1 4 )

X X

interfaces, see C73, where numerical computations were carried out.

Substituting these simplifications into (2.8) and (2.9), gives for

(2.12)

Q = qf (x,C,t)(B - 5) + q £. (2.15)

Next the unknowns q. and q are solved from (2.7) and (2.15). x x This gives x 1 + tan a and % =Ï Ï " Ï Ï ( B" « t a"a 2 • (2-17) x 1 + tan a

These equations give the discharges in both fluids in terms of the

interface. From (2.16) we find for the velocity of the fresh water

particles in the top of the interface:

■ qf

_ x Q r tana . .

v

f-~r

=

IB

+

? " r -

( 2

-

l 8 )

1 + tan a

Similarly, for the salt water particles in the toe we find

v = — * - -S- - I t a n a ( 2,9)

vs e E B e , 2 - u.i»;

1 + tan a

274 VAN DUIJN AND HONGFEI

(2.10) and (2.16) to obtain

e g - f ( ( B - 6 ) q + r ( B - o e 8 g / 3 x 2| -Dt 3 x (■ l + o 5 / 3 x rJ

Finally we apply the rescaling

x := (x - i f Q(s)ds\ /B; t := Tt/e; u := £/B,

which results in the equation

8" 3 I „ ^ 3u/8x \

■jr— = -5— < U ( 1 " U ) rjt.

8 t 3 x I 1 + (3u/3x)2j

3. PREVIOUS RESULTS

In this section we recall some results from [9] with respect

to Problem C.

Definition 3.1. A function u: Q -»1R is called a solution of Problem

C, if it satisfies

(i) u € ^ ( ( O . o o ) ^1' ^ ) ) , ut £ L2((-R,R)x(0,T)) for all

R,T > 0;

(ii) u 6 [0,1], u € C-1,1] a.e. in Q;

(iii) u(-,0) = u0( O ;

Theorem 3.2. Suppose HDl, H<(> 1 and Hu. are satisfied. Then Problem

C has a unique solution with the following properties:

(i) u(t) - H € L ' ( R ) for all t > 0; (ii) u € C ( Q ) ;

(iii) Let u. and u. be two solutions of Problem C with initial

functions u~. and u_9, respectively. Then

lU j(t) - "2( t ) «L,( m ) < H u0 | - u0 2lL l ( ] R ) for all t > 0;

(iv) Let u and u, be two solutions of Problem C with initial

functions u. < u . Then u (t) < u (t) for all t > 0.

The existence part of this theorem is proved by parabolic

regularization: given a un which satisfies H un, one constructs a

sequence {u } _ c C (M) such that for sufficiently large n £ IN

u. € (-, 1 - - ) , u' € (-1 + - , 1 - - ) onlR, On n n ' On n' n

u' (x) = 0 for |x| > n,

and u -► u„ uniformly on compact subsets of 1R.

Then for n € IN (sufficiently large) and for any T > 0 consider the

problem

(

V

ut = (D(u)<j.(ux))x in Q^ := (-n,n)x (0,T],

u (-n,t) = u (n,t) = 0 for 0 < t S T ,

276 VAN DUIJN AND HONGFEI

Problem C has a unique classical solution u £ C (Qm) , for each n n ^T

a € (0,1), and u £ {-, 1 - -) and u £ (-1 + -, 1 - - ) . A solution

n n n nx n n

u of Problem C is now obtained as the limit of a subsequence of

{u ) . The convergence is uniform on compact subsets of Q. The

uniqueness part follows from the contraction property (iii), which

in turn follows from the accretivity in L (TR) of the operator

Au := -(D(u)<(>(u')) ' when defined on a suitable domain.

For the special case when D(s) = s(l - s) a self-similar solu­

tion of Problem C can be constructed. It has the form

u (x,t) = s(n) and r\ = xf(t), (3.1)

where the function s: TR -» [ 0 , l 3 i s given by

s(n)

0 . -=o < n < - i

i i < n < co

and where the function f: C0,e») -»R is determined by the problem

(S)

f'(t) = -24>(f(t))f2(t) t > 0,

f(0) = fQ.

A solution f of Problem S, with f„ £ [-1,13, induces a solution u

of Problem C which has the form of a rotating line: this line

rotates clockwise when f_ £ (0,13 and counter clockwise when

It was shown that a solution of Problem C, with uQ satisfying (1.6),

converges to a self-similar solution with fn > 0. Also an estimate

for the rate of convergence was given.

4. SUB- AND SUPERSOLUTIONS

Definition 4.1. A function u is called a sub-(super)solution of

Problem C on a domain S =ffi»(0,T) if there exists a T £ (0,°»]

such that

(i) u £ LO°((0,T); W1' " ( « ) ) ;

(ii) u £ CO,13, u £ C-1,13 a.e. in S^;

(iii) u , (D(u)<t>(ux))x £ L2((-R,R)*(0,T')) for all R > 0 and

for all finite 0 < T' < T; •

(ivj D(u(-,t))£ L'(1R) for all t € (0,T);

(v) ut - (D(u)<Kux))x < (>) 0 a.e. in S^ and u(-,0) < (»)

uQ(') o n » .

Remark 4.2. In C93 it was proven that if u is any solution of

Problem C with D and ij> bounded functions and u. - H £ L (K) , then

u(t) - H E L'(1R) for all t > 0. This implies D(u(',t)) £ L'(1R) for

all t > 0. Therefore under these conditions, a solution u is both

a subsolution and a supersolution of Problem C on S = Q.

Theorem 4.3. (Comparison principle). Let D € CC0,|3 satisfy (1.3)

and let 4> £ CC-l,l3 be strictly increasing. Further let IJ and u

be respectively a subsolution and a supersolution of Problem C on

278 VAN DUIJN AND HONCFEI

Proof. From (v) of Definition 4.1 we have

(u - u) < (D(u)<Ku )) - (D(u)<J)(u )) a.e. in S .

t XX XX L

M u l t i p l y i n g both s i d e s by H(u - u), where H i s the Heaviside func­

t i o n , and. i n t e g r a t i n g with r e s p e c t t o x over (-R.R) for every

R > 0 gives

fR - - f+R

-(u - u) H-(u - u) ^ {D-(u)(Mu. ) " D-(u)<Mu )) H(ii - u)

J_R c j _ R x x x

for a . e . t € ( 0 , T ) .

Arguing exactly as in C 9 ] , we obtain from this inequality

fR

(u - u) H(u - u) < {|D(u)<(>(u ) | + |D(u)(()(u )|} (R,t)

j _D t x x

-R

+ {|D(u)((>(ux)| + |D(G)((.(ux)|} (-R,t) (4.1)

for a.e. t £ (0,T).

Since (u - ü) € Wl,1((0,T'); L (-R.R)) for any finite T' *ï T we

have

H(u - u) £ (u - u) =_{— at — at —} 4ï <u - ' " ) * °n (°.T)

with (s) = max (s,0), see Crandall 4 Pierre f.8, Lemma a.l].