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Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media

Gabriela Marinoschi Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Bucharest

Existence of the free boundary in a diffusive ow in porous media 2

1 Problem presentation

Prove the existence of the solution to a two phase ow in a porous medium Compute the solution and the diffusive interface evolution G.M., J. Optimiz. Theory Appl., 2012

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 3

Problem presentation

R3; open bounded, = @ sufciently smooth

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 4

Problem presentation

Q = (0; T) ; T < 1; = (0; T) Qs = f(t; x) 2 Q; y(t; x) = ysg Qu = f(t; x) 2 Q; y(t; x) < ysg

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 5

Problem presentation

Q = (0; T) ; T < 1; = (0; T) Qs = f(t; x) 2 Q; y(t; x) = ysg Qu = f(t; x) 2 Q; y(t; x) < ysg

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 6

Problem presentation

@y @t (t; x; y) 3 f

in Q

@(t; x; y) @ + (t; x; y) 3 0

• n ; > 0

(NE)

y(0; x) = y0

in :

: Q (1; ys] ! R; (t; x; ) 2 C1(1; ys); lim

r%ys (t; x; r) = K s; a.e. (t; x) 2 Q:

Fast diffusion in porous media (G.M. Springer, 2006; A. Favini, G.M. Springer 2012)

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 7

Problem presentation

@y @t (t; x; y) 3 f

in Q

@(t; x; y) @ + (t; x; y) 3 0

• n ; > 0;

(NE)

y(0; x) = y0

in :

: Q (1; ys] ! R; (t; x; ) 2 C1(1; ys); lim

r%ys (t; x; r) = K s; a.e. (t; x) 2 Q:

Fast diffusion in porous media (G.M. Springer, 2006; A. Favini, G.M. Springer 2012)

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 8

Problem presentation

j : Q R ! (1; 1]; j(t; x; r) = ( R r

• (t; x; s)ds; r ys;

+1;

• therwise

(t; x) ! j(t; x; r) is measurable on Q for all r 2 (1; ys] j(t; x; ) proper convex l.s.c. a.e. (t; x) 2 Q @j(t; x; ) = (t; x; ) a.e. (t; x) 2 Q j(t; x; r) K

s jrj ; for any r ys; a.e. (t; x) 2 Q

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 9

Problem presentation Conjugate of j

j(t; x; !) = sup

r2R

(!r j(t; x; r)) a.e. (t; x) 2 Q

Legendre-Fenchel relations

j(t; x; r) + j(t; x; !) r! for all r 2 R; ! 2 R; a.e. (t; x) 2 Q; j(t; x; r) + j(t; x; !) = r! if and only if ! 2 @j(t; x; r), a.e. (t; x) 2 Q: C3 j!j + C0

3 j(t; x; !) for any ! 2 R; a.e. (t; x) 2 Q:

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 10

Problem presentation Conjugate of j

j(t; x; !) = sup

r2R

(!r j(t; x; r)) a.e. (t; x) 2 Q

Legendre-Fenchel relations

j(t; x; r) + j(t; x; !) r! for all r 2 R; ! 2 R; a.e. (t; x) 2 Q; j(t; x; r) + j(t; x; !) = r! if and only if ! 2 @j(t; x; r), a.e. (t; x) 2 Q: C3 j!j + C0

3 j(t; x; !) for any ! 2 R; a.e. (t; x) 2 Q; C3 > 0:

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 11

Problem presentation

@y @t (t; x; y) 3 f

in Q;

@(t; x; y) @ + (t; x; y) 3 0

• n ;

(NE)

y(0; x) = y0

in :

j singular potential, j minimal growth conditions, no time and space regularity m

Minimize J(y; w) for all (y; w) 2 U (P) connected with (NE) by j and j

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 12

Problem presentation

@y @t (t; x; y) 3 f

in Q

@(t; x; y) @ + (t; x; y) 3 0

• n

(NE)

y(0; x) = y0

in :

j singular potential, j minimal growth conditions, no time and space regularity m

Minimize J(y; w) for all (y; w) 2 U (P) connected with (NE) by j and j

1

1

• H. Brezis, I. Ekeland, C.R. Acad. Sci. Paris, 282, 971–974, 1197–1198, 1976

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 13

Problem presentation

I H. Brezis, I. Ekeland (C.R. Acad. Sci. Paris, 1976) I G. Auchmuty (NA, 1988) I N. Ghoussoub, L. Tzou ( Math. Ann. 2004) I N. Ghoussoub (Springer, 2009) I A. Visintin ( Adv. Math. Sci. Appl. 2008) I U. Stefanelli (SICON 2008, J. Convex Analysis 2009) I V. Barbu (JMAA 2011) I G. M. (JOTA 2012, 2013)

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 14

Problem presentation

• Denition. Let f 2 L1(Q);

y0 2 L1(); y0 ys a.e. on :

A weak solution to (NE) is a pair (y; w)

y 2 L1(Q); w 2 (L1(Q))0; (w = wa + ws; wa 2 L1(Q)) wa(t; x) 2 @j(t; x; y) = (t; x; y) a.e. on Q; ws 2 ND(')(y)

• Z

Q

yd dt dxdt + Z

• y0 (0) + hw; A0;1 i(L1(Q))0;L1(Q) =

Z

Q

f dxdt

() for any 2 W 1;1([0; T]; L1()) \ L1(0; T; D(A0;1)); (T) = 0:

A0;1 = ; A0;1 : D(A0;1) L1() ! L1(); D(A0;1) = f 2 W 2;1(); @ @ + = 0 on g:

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 15

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 16

2 A duality approach

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 17

A duality approach

J : L1(Q) L1(Q)

Min

(y;w)2U

Z

Q

(j(t; x; y(t; x)) + j(t; x; w(t; x))dxdt y(t; x)w(t; x)) dxdt

• U =
• (y; w); y 2 L1(Q); y(t; x) ys a.e., w 2 L1(Q); dy

dt w = f; y(0) = y0

• Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 18

A duality approach

J : L1(Q) (L1(Q))0 J(y; w) = Z

Q

j(t; x; y(t; x))dxdt + '(t; x; w) w(y)

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 19

A duality approach

w 2 (L1(Q))0; w = wa + ws L1(Q) 3 wa = "absolutely continuous component", ws = singular component ' : L1(Q) ! (1; 1]; '(y) = Z

Q

j(t; x; y(t; x))dxdt proper, convex, lsc ' : (L1(Q))0 ! (1; 1]; '(w) = Z

Q

j(t; x; wa(t; x))dxdt +

D(')(ws);

• D(')(v) = supfv( ); 2 D(')g = conjugate of the indicator function of D(')

D(') = fy 2 L1(Q); '(y) < +1g = fy 2 L1(Q); y(t; x) ys a.e. g:

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 20

A duality approach2

' : L1(Q) ! (1; 1] '(y) = R

Q j(t; x; y(t; x))dxdt proper, convex, lsc

' : (L1(Q))0 ! (1; 1] '(w) = R

Q j(t; x; wa(t; x))dxdt + D(')(ws)

• D(')(v) = supfv( ); 2 D(')g = conjugate of the indicator function of D(')

D(') = fy 2 L1(Q); '(y) < +1g = fy 2 L1(Q); y(t; x) ys a.e.g:

2 R.T. Rockafeller, Pacic J. Math. J, 39, 2, 1971 Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 21

A duality approach3

' : L1(Q) ! (1; 1] '(y) = R

Q j(t; x; y(t; x))dxdt proper, convex, lsc

' : (L1(Q))0 ! (1; 1] '(w) = R

Q j(t; x; wa(t; x))dxdt + D(')(ws)

• D(')(v) = supfv( ); 2 D(')g = conjugate of the indicator function of D(')

D(') = fy 2 L1(Q); '(y) < +1g = fy 2 L1(Q); y(t; x) ys a.e.g:

3 R.T. Rockafeller, Pacic J. Math. J, 39, 2, 1971 Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 22

A duality approach

J : L1(Q) (L1(Q))0 J(y; w) = Z

Q

j(t; x; y(t; x))dxdt + Z

Q

j(t; x; wa)dxdt +

D(')(ws) w(y)

In some good cases we have a Lemma of integration by parts:

w(y) = 1 2 ky(T)k2

(H1())0 1

2 ky0k2

(H1())0

Z

Q

fA1

0;1ydxdt

(**)

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 23

A duality approach

J : L1(Q) (L1(Q))0 J(y; w) = Z

Q

j(t; x; y(t; x))dxdt + Z

Q

j(t; x; wa)dxdt +

D(')(ws) w(y)

In some good cases we have a Lemma of integration by parts:

w(y) = 1 2 ky(T)k2

(H1())0 1

2 ky0k2

(H1())0

Z

Q

fA1

0;1ydxdt

(**)

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 24

A duality approach

J : L1(Q) (L1(Q))0 J(y; w) = Z

Q

j(t; x; y(t; x))dxdt + Z

Q

j(t; x; wa)dxdt +

D(')(ws) w(y)

In some good cases we have a Lemma of integration by parts:

dy dt w = f

• 1y

(**)

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 25

A duality approach Minimize (J) for all (y; w) 2 U (P)

J(y; w) = 8 > < > : R

Q j(t; x; y(t; x))dxdt +

R

Q j(t; x; wa(t; x))dxdt + D(')(ws)

+1

2 ky(T)k2 (H1())0 1 2 ky0k2 (H1())0

R

Q fA1 0;1ydxdt

if (y; w) 2 U;

U =

• (y; w); y 2 L1(Q); y(t; x) 2 [0; ys] a.e., y(T) 2 (H1())0; w 2 (L1(Q))0;

(y; w) veries ()g

• Z

Q

yd dt dxdt + Z

• y0 (0) + hw; A0;1 i(L1(Q))0;L1(Q) =

Z

Q

f dxdt

for any 2 W 1;1([0; T]; L1()) \ L1(0; T; D(A0;1)); (T) = 0:

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 26

A duality approach Minimize (J) for all (y; w) 2 U (P)

J(y; w) = 8 > < > : R

Q j(t; x; y(t; x))dxdt +

R

Q j(t; x; wa(t; x))dxdt + D(')(ws)

+1

2 ky(T)k2 (H1())0 1 2 ky0k2 (H1())0

R

Q fA1 0;1ydxdt

if (y; w) 2 U;

U =

• (y; w); y 2 L1(Q); y(t; x) 2 [0; ys] a.e., y(T) 2 (H1())0; w 2 (L1(Q))0;

(y; w) veries ()g

• Z

Q

yd dt dxdt + Z

• y0 (0) + hw; A0;1 i(L1(Q))0;L1(Q) =

Z

Q

f dxdt

() for any 2 W 1;1([0; T]; L1()) \ L1(0; T; D(A0;1)); (T) = 0:

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 27

A duality approach Theorem

J is proper, convex, l.s.c. on L1(Q) (L1(Q))0:

Problem (P) has at least a solution (y; w):

4

A solution to (P) is called a generalized solution to (NE)

4

Alaoglu theorem

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 28

A duality approach Theorem

J is proper, convex, l.s.c. on L1(Q) (L1(Q))0:

Problem (P) has at least a solution (y; w):

5

A solution to (P) is called a generalized solution to (NE)

5

Alaoglu theorem

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 29

A duality approach Theorem Let (y; w) be the null minimizer in (P); i:e:; J(y; w) = 0: Assume that

1 2 ky(T)k2

(H1())0 1

2 ky0k2

(H1())0

Z

Q

fA1

0;1ydxdt = w(y):

(**) Then, (y; w) is the weak solution to (NE); i.e.,

wa(t; x) 2 (t; x; y(t; x)) a.e. (t; x) 2 Q

and

ws 2 ND(')(y):

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 30

A duality approach Proof sketch

J(y; w) = Z

Q

j(t; x; y)dxdt + Z

Q

j(t; x; wa)dxdt +

D(')(ws)

+1 2 ky(T)k2

(H1())0 1

2 ky0k2

(H1())0

Z

Q

fA1

0;1ydxdt = 0

• D(')(ws) = supfws(z); z 2 D(')g ws(y)

Z

Q

j(t; x; y)dxdt + Z

Q

j(t; x; wa)dxdt Z

Q

waydxdt 0 ws(y) +

D(')(ws) = 0:

ws(y) ws(z) for any z 2 D(') = ) ws 2 ND(')(y):

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 31

A duality approach Proof sketch

J(y; w) = Z

Q

j(t; x; y)dxdt + Z

Q

j(t; x; wa)dxdt +

D(')(ws)

• Z

Q

waydxdt ws(y) = 0

• D(')(ws) = supfws(z); z 2 D(')g ws(y)

Z

Q

(j(t; x; y) + j(t; x; wa) way)dxdt = 0 = ) wa(t; x) 2 (t; x; y(t; x)) a.e. (t; x) 2 Q ws(y) +

D(')(ws) = 0:

ws(y) ws(z) for any z 2 D(') = ) ws 2 ND(')(y):

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 32

A duality approach Proof sketch

J(y; w) = Z

Q

j(t; x; y)dxdt + Z

Q

j(t; x; wa)dxdt +

D(')(ws)

• Z

Q

waydxdt ws(y) = 0

• D(')(ws) = supfws(z); z 2 D(')g ws(y)

Z

Q

(j(t; x; y) + j(t; x; wa) way)dxdt = 0 = ) wa(t; x) 2 (t; x; y(t; x)) a.e. (t; x) 2 Q ws(y) +

D(')(ws) = 0:

ws(y) ws(z) for any z 2 D(') = ) ws 2 ND(')(y):

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 33

A duality approach

dy dt( ) + wa(A0;1 ) + ws(A0;1 ) = f( );

for any 2 W 1;1([0; T]; L1()) \ L1(0; T; D(A0;1)); (T) = 0: If dy

dt 2 (L1(Q))0

dy dt

• a

( ) + dy dt

• s

( ) + wa(A0;1 ) + ws(A0;1 ) = f( ):

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 34

A duality approach If

y 2 intD(') = fy 2 L1(Q); ess sup y(t; x) < ysg = ) ND(')(y) = f0g = ) ws = 0 dy dt (t; x; y) = f; a.e. on Q

with initial condition and b.c.

Q = Qu = f(t; x); y(t; x) < ysg

unsaturated ow

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 35

A duality approach If

y 2 @D(') = fy 2 L1(Q); ess sup y(t; x) = ysg = ) Qs = f(t; x); y(t; x) = ysg Qu = f(t; x); y(t; x) < ysg

• dy

dt

• s ws 3 0; in D0(
• Qs)
• dy

dt

• a (t; x; y) = f; a.e. on Qu

saturated unsaturated

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 36

A duality approach If

y 2 @D(') = fy 2 L1(Q); ess sup y(t; x) = ysg = ) Qs = f(t; x); y(t; x) = ysg Qu = f(t; x); y(t; x) < ysg

• dy

dt

• s ws 3 0; in D0(
• Qs)
• dy

dt

• a (t; x; y) = f; a.e. on Qu

saturated unsaturated

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 37

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 38

3 A direct approach

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 39

A direct approach: regularity for (t; x; r)

@y @t (t; x; y) + r K0(t; x; y) 3 f

in Q

@(t; x; y) @ + (t; x; y) 3 0

• n

(NE)

y(0; x) = y0

in :

(; ; r) 2 W 2;1(Q); for any r 2 (1; ys]; ((t; x; r) (t; x; r))(r r) (r r)2; for any r; r 2 (1; ys]; > 0 K0(t; x; y) = a(t; x)K(y); aj 2 W 1;1(Q); K Lipschitz and bounded.

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 40

A direct approach: regularity for (t; x; r)

• Denition. Let f 2 L2(0; T; V 0); y0 2 L1(); y0 ys a.e.

We call a solution to (NE) a pair (y; w) such that

y 2 C([0; T]; L2()) \ L2(0; T; V ) \ W 1;2([0; T]; V 0) w 2 L2(0; T; V ); w(t; x) 2 (t; x; y(t; x)) a.e. on Q; y ys; a.e. (t; x) 2 Q;

which satises the equation

Z T dy dt(t); (t)

• V 0;V

dt + Z

Q

(rw K0(t; x; y)) rdxdt = Z T hf(t); (t)iV 0;V dt

for any 2 L2(0; T; V ); and the condition y(0) = y0:

V = H1(); V 0 = (H1())0

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 41

A direct approach: regularity for (t; x; r) For each t

A(t) : D(A(t)) (H1())0 ! (H1())0; D(A(t)) =

• z 2 L2(); 9w 2 V; w(x) 2 (t; x; z)) a.e. x 2
• ;

by the relation

hA(t)z; iV 0;V := Z

• (rw K0(t; x; y)) r dx; for any 2 V:

dy dt(t) + A(t)y(t) 3 f(t); a.e. t 2 (0; T); y(0) = y0:

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 42

A direct approach: regularity for (t; x; r)

• Step. 1 Prove the existence to the regularized Cauchy problem6

dy dt(t) + A"(t)y(t) = f(t); a.e. t 2 (0; T); y(0) = y0:

(a) The domain of A"(t) is independent of t and D(A"(t)) = D(A"(0)) = V: (b) For each " > 0 and t 2 [0; T] xed, the operator A"(t) is quasi m-accretive on V 0: (c) For 2 V and 0 s; t T we have

kA"(t) A"(s)kV 0 jt sj g (kkV 0) (kA(t)kV 0 + 1);

where g : [0; 1) ! [0; 1) is a nondecreasing function. Step 2. passing to the limit.

6 M.G. Crandall, A. Pazy, Proceedings of The American Mathematical Society, 1979

• C. Dafermos, M. Slemrod, J. Functional Anal.,1973

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 43

4 Numerical results

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 44

Numerical results

dy dt(t) + Ay(t) = f(t); a.e. t 2 (0; T); y(0) = y0: yh

i+1 yh i

h + Ayh

i+1 = f h i+1; i = 1; :::; n 1

yh

0 = y0

Stability and convergence Algorithm based on the proof of the m-accretivity of Ah = A + 1

hI

1 hI + Ah

• = g

for any g 2 V 0; has a solution 2 D(A):

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 45

Numerical results

= f(x; y); x 2 (0; 5); y 2 (0; 5)g; T = 20 (r) = (

(c1)r (cr) ; r 2 [0; 1);

[1; 1) ; r = 1: ; K(r) = (c 1)r2 c r ; r 2 [0; 1] 0(x; y) = 0:01; f = 0; = 108; rain = exp(30 x2)= exp(30); c = 1:02 " " " " rain

Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy

Existence of the free boundary in a diffusive ow in porous media 46

Numerical results

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Existence of the free boundary in a diffusive ow in porous media 47

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Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy