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
Existence of the free boundary in a diffusive ow in porous media 3 Problem presentation R 3 ; open bounded, = @ sufciently smooth
Existence of the free boundary in a diffusive ow in porous media 4 Problem presentation = (0; T ) ; T < 1; = (0; T ) s = f(t; x) 2 ; y(t; x) = y s g u = f(t; x) 2 ; y(t; x) < y s g
Existence of the free boundary in a diffusive ow in porous media 5 Problem presentation = (0; T ) ; T < 1; = (0; T ) s = f(t; x) 2 ; y(t; x) = y s g u = f(t; x) 2 ; y(t; x) < y s g
Existence of the free boundary in a diffusive ow in porous media 6 Problem presentation @y @t @(t; x; y) @ (t; x; y) 3 f in + (t; x; y) 3 0 on ; > 0 (NE) y(0; x) = y 0 in : : ( 1; y s ]! R; (t; x; ) 2 C 1 ( 1; y s ); lim r%ys (t; x; r) = K s; a.e. (t; x) 2 : Fast diffusion in porous media (G.M. Springer, 2006; A. Favini, G.M. Springer 2012)
Existence of the free boundary in a diffusive ow in porous media 7 Problem presentation @y @t @(t; x; y) @ (t; x; y) 3 f in + (t; x; y) 3 0 on ; > 0; (NE) y(0; x) = y 0 in : : ( 1; y s ]! R; (t; x; ) 2 C 1 ( 1; y s ); lim r%ys (t; x; r) = K s; a.e. (t; x) 2 : Fast diffusion in porous media (G.M. Springer, 2006; A. Favini, G.M. Springer 2012)
Existence of the free boundary in a diffusive ow in porous media 8 Problem presentation j : R! ( 1; 1]; j(t; x; r) = ( R r 0 (t; x; s)ds; r y s ; +1; otherwise (t; x)! j(t; x; r) is measurable on for all r 2 ( 1; y s ] j(t; x; ) proper convex l.s.c. a.e. (t; x) 2 @j(t; x; ) = (t; x; ) a.e. (t; x) 2 j(t; x; r) Ks jrj ; for any r y s ; a.e. (t; x) 2
Existence of the free boundary in a diffusive ow in porous media 9 Problem presentation Conjugate of j j (t; x;!) = sup(!r r2r j(t; x; r)) a.e. (t; x) 2 Legendre-Fenchel relations j(t; x; r) + j (t; x;!) r! for all r 2 R;! 2 R; a.e. (t; x) 2 ; j(t; x; r) + j (t; x;!) = r! if and only if! 2 @j(t; x; r), a.e. (t; x) 2 : C 3 j!j + C 0 3 j (t; x;!) for any! 2 R; a.e. (t; x) 2 :
Existence of the free boundary in a diffusive ow in porous media 10 Problem presentation Conjugate of j j (t; x;!) = sup(!r r2r j(t; x; r)) a.e. (t; x) 2 Legendre-Fenchel relations j(t; x; r) + j (t; x;!) r! for all r 2 R;! 2 R; a.e. (t; x) 2 ; j(t; x; r) + j (t; x;!) = r! if and only if! 2 @j(t; x; r), a.e. (t; x) 2 : C 3 j!j + C 0 3 j (t; x;!) for any! 2 R; a.e. (t; x) 2 ; C 3 > 0:
Existence of the free boundary in a diffusive ow in porous media 11 Problem presentation @y @t @(t; x; y) @ (t; x; y) 3 f in ; + (t; x; y) 3 0 on ; (NE) y(0; x) = y 0 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
Existence of the free boundary in a diffusive ow in porous media 12 Problem presentation @y @t @(t; x; y) @ (t; x; y) 3 f in + (t; x; y) 3 0 on (NE) y(0; x) = y 0 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
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)
Existence of the free boundary in a diffusive ow in porous media 14 Problem presentation Denition. Let f 2 L 1 (); y 0 2 L 1 (); y 0 y s a.e. on : A weak solution to (NE) is a pair (y; w) y 2 L 1 (); w 2 (L 1 ()) 0 ; (w = w a + w s ; w a 2 L 1 ()) w a (t; x) 2 @j(t; x; y) = (t; x; y) a.e. on ; w s 2 N D(') (y) y d dt dxdt + y 0 (0) + hw; A 0;1 i (L1 ()) 0 ;L 1 () = for any 2 W 1;1 ([0; T ]; L 1 ()) \ L 1 (0; T ; D(A 0;1 )); (T ) = 0: f dxdt () A 0;1 = ; A 0;1 : D(A 0;1 ) L 1 ()! L 1 (); D(A 0;1 ) = f 2 W 2;1 (); @ @ + = 0 on g:
Existence of the free boundary in a diffusive ow in porous media 15
Existence of the free boundary in a diffusive ow in porous media 16 2 A duality approach
Existence of the free boundary in a diffusive ow in porous media 17 A duality approach U = Min (y;w)2u J : L 1 () L 1 () (j(t; x; y(t; x)) + j (t; x; w(t; x))dxdt (y; w); y 2 L 1 (); y(t; x) y s a.e., w 2 L 1 (); dy dt y(t; x)w(t; x)) dxdt w = f; y(0) = y 0
Existence of the free boundary in a diffusive ow in porous media 18 A duality approach J(y; w) = J : L 1 () (L 1 ()) 0 j(t; x; y(t; x))dxdt + ' (t; x; w) w(y)
Existence of the free boundary in a diffusive ow in porous media 19 A duality approach w 2 (L 1 ()) 0 ; w = w a + w s L 1 () 3 w a = "absolutely continuous component", w s = singular component ' : L 1 ()! ( 1; 1]; '(y) = ' : (L 1 ()) 0! ( 1; 1]; ' (w) = j(t; x; y(t; x))dxdt proper, convex, lsc j (t; x; w a (t; x))dxdt + D(') (w s); D(') (v) = supfv( ); 2 D(')g = conjugate of the indicator function of D(') D(') = fy 2 L 1 (); '(y) < +1g = fy 2 L 1 (); y(t; x) y s a.e. g:
Existence of the free boundary in a diffusive ow in porous media 20 A duality approach 2 ' : L 1 ()! ( 1; 1] '(y) = R j(t; x; y(t; x))dxdt proper, convex, lsc ' : (L 1 ()) 0! ( 1; 1] ' (w) = R j (t; x; w a (t; x))dxdt + D(') (w s) D(') (v) = supfv( ); 2 D(')g = conjugate of the indicator function of D(') D(') = fy 2 L 1 (); '(y) < +1g = fy 2 L 1 (); y(t; x) y s a.e.g: 2 R.T. Rockafeller, Pacic J. Math. J, 39, 2, 1971
Existence of the free boundary in a diffusive ow in porous media 21 A duality approach 3 ' : L 1 ()! ( 1; 1] '(y) = R j(t; x; y(t; x))dxdt proper, convex, lsc ' : (L 1 ()) 0! ( 1; 1] ' (w) = R j (t; x; w a (t; x))dxdt + D(') (w s) D(') (v) = supfv( ); 2 D(')g = conjugate of the indicator function of D(') D(') = fy 2 L 1 (); '(y) < +1g = fy 2 L 1 (); y(t; x) y s a.e.g: 3 R.T. Rockafeller, Pacic J. Math. J, 39, 2, 1971
Existence of the free boundary in a diffusive ow in porous media 22 A duality approach J(y; w) = J : L 1 () (L 1 ()) 0 j(t; x; y(t; x))dxdt + j (t; x; w a )dxdt + D(') (w s) w(y) In some good cases we have a Lemma of integration by parts: w(y) = 1 2 ky(t 1 )k2 (H 1 ()) 0 2 ky 0k 2 (H 1 ()) fa 1 0 0;1ydxdt (**)
Existence of the free boundary in a diffusive ow in porous media 23 A duality approach J(y; w) = J : L 1 () (L 1 ()) 0 j(t; x; y(t; x))dxdt + j (t; x; w a )dxdt + D(') (w s) w(y) In some good cases we have a Lemma of integration by parts: w(y) = 1 2 ky(t 1 )k2 (H 1 ()) 0 2 ky 0k 2 (H 1 ()) fa 1 0 0;1ydxdt (**)
Existence of the free boundary in a diffusive ow in porous media 24 A duality approach J(y; w) = J : L 1 () (L 1 ()) 0 j(t; x; y(t; x))dxdt + j (t; x; w a )dxdt + D(') (w s) w(y) In some good cases we have a Lemma of integration by parts: dy w = f dt 1 y (**)
Existence of the free boundary in a diffusive ow in porous media 25 A duality approach J(y; w) = 8 >< >: R Minimize (J) for all (y; w) 2 U (P ) j(t; x; y(t; x))dxdt + R j (t; x; w a (t; x))dxdt + D(') (w s) + 1 2 ky(t )k2 1 (H 1 ()) 0 2 ky 0k 2 R (H 1 ()) 0 fa 0;1 1 ydxdt if (y; w) 2 U; U = (y; w); y 2 L 1 (); y(t; x) 2 [0; y s ] a.e., y(t ) 2 (H 1 ()) 0 ; w 2 (L 1 ()) 0 ; (y; w) veries ()g y d dt dxdt + y 0 (0) + hw; A 0;1 i (L1 ()) 0 ;L 1 () = for any 2 W 1;1 ([0; T ]; L 1 ()) \ L 1 (0; T ; D(A 0;1 )); (T ) = 0: f dxdt
Existence of the free boundary in a diffusive ow in porous media 26 A duality approach J(y; w) = 8 >< >: R Minimize (J) for all (y; w) 2 U (P ) j(t; x; y(t; x))dxdt + R j (t; x; w a (t; x))dxdt + D(') (w s) + 1 2 ky(t )k2 1 (H 1 ()) 0 2 ky 0k 2 R (H 1 ()) 0 fa 0;1 1 ydxdt if (y; w) 2 U; U = (y; w); y 2 L 1 (); y(t; x) 2 [0; y s ] a.e., y(t ) 2 (H 1 ()) 0 ; w 2 (L 1 ()) 0 ; (y; w) veries ()g y d dt dxdt + y 0 (0) + hw; A 0;1 i (L1 ()) 0 ;L 1 () = for any 2 W 1;1 ([0; T ]; L 1 ()) \ L 1 (0; T ; D(A 0;1 )); (T ) = 0: f dxdt ()
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 L 1 () (L 1 ()) 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
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 L 1 () (L 1 ()) 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
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 1 )k2 (H 1 ()) 0 2 ky 0k 2 (H 1 ()) fa 1 0 0;1ydxdt = w(y): (**) Then, (y; w) is the weak solution to (NE); i.e., and w a (t; x) 2 (t; x; y(t; x)) a.e. (t; x) 2 w s 2 N D(') (y):
Existence of the free boundary in a diffusive ow in porous media 30 A duality approach Proof sketch J(y; w) = j(t; x; y)dxdt + + 1 2 ky(t )k2 (H 1 ()) 0 1 2 ky 0k 2 (H 1 ()) 0 j (t; x; w a )dxdt + D(') (w s) fa 1 0;1 ydxdt = 0 D(') (w s) = supfw s (z); z 2 D(')g w s (y) j(t; x; y)dxdt + j(t; x; w a )dxdt w a ydxdt 0 w s (y) + D(') (w s) = 0: w s (y) w s (z) for any z 2 D(') =) w s 2 N D(') (y):
Existence of the free boundary in a diffusive ow in porous media 31 A duality approach Proof sketch J(y; w) = j(t; x; y)dxdt + j (t; x; w a )dxdt + D(') (w s) w a ydxdt w s (y) = 0 D(') (w s) = supfw s (z); z 2 D(')g w s (y) (j(t; x; y) + j (t; x; w a ) w a y)dxdt = 0 =) w a (t; x) 2 (t; x; y(t; x)) a.e. (t; x) 2 w s (y) + D(') (w s) = 0: w s (y) w s (z) for any z 2 D(') =) w s 2 N D(') (y):
Existence of the free boundary in a diffusive ow in porous media 32 A duality approach Proof sketch J(y; w) = j(t; x; y)dxdt + j (t; x; w a )dxdt + D(') (w s) w a ydxdt w s (y) = 0 D(') (w s) = supfw s (z); z 2 D(')g w s (y) (j(t; x; y) + j (t; x; w a ) w a y)dxdt = 0 =) w a (t; x) 2 (t; x; y(t; x)) a.e. (t; x) 2 w s (y) + D(') (w s) = 0: w s (y) w s (z) for any z 2 D(') =) w s 2 N D(') (y):
Existence of the free boundary in a diffusive ow in porous media 33 A duality approach dy dt ( ) + w a(a 0;1 ) + w s (A 0;1 ) = f( ); for any 2 W 1;1 ([0; T ]; L 1 ()) \ L 1 (0; T ; D(A 0;1 )); (T ) = 0: If dy dt 2 (L1 ()) 0 dy ( ) + dt a dy ( ) + w a (A 0;1 ) + w s (A 0;1 ) = f( ): dt s
Existence of the free boundary in a diffusive ow in porous media 34 A duality approach If y 2 intd(') = fy 2 L 1 (); ess sup y(t; x) < y s g =) N D(') (y) = f0g =) w s = 0 dy (t; x; y) = f; a.e. on dt with initial condition and b.c. = u = f(t; x); y(t; x) < y s g unsaturated ow
Existence of the free boundary in a diffusive ow in porous media 35 A duality approach If y 2 @D(') = fy 2 L 1 (); ess sup y(t; x) = y s g =) s = f(t; x); y(t; x) = y s g dy dt s w s 3 0; in D 0 ( s ) saturated u = f(t; x); y(t; x) < y s g dy dt a (t; x; y) = f; a.e. on u unsaturated
Existence of the free boundary in a diffusive ow in porous media 36 A duality approach If y 2 @D(') = fy 2 L 1 (); ess sup y(t; x) = y s g =) s = f(t; x); y(t; x) = y s g dy dt s w s 3 0; in D 0 ( s ) saturated u = f(t; x); y(t; x) < y s g dy dt a (t; x; y) = f; a.e. on u unsaturated
Existence of the free boundary in a diffusive ow in porous media 37
Existence of the free boundary in a diffusive ow in porous media 38 3 A direct approach
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 K 0 (t; x; y) 3 f @(t; x; y) @ in + (t; x; y) 3 0 on (NE) y(0; x) = y 0 in : (; ; r) 2 W 2;1 (); for any r 2 ( 1; y s ]; ((t; x; r) (t; x; r))(r r) (r r) 2 ; for any r; r 2 ( 1; y s ]; > 0 K 0 (t; x; y) = a(t; x)k(y); a j 2 W 1;1 (); K Lipschitz and bounded.
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 L 2 (0; T ; V 0 ); y 0 2 L 1 (); y 0 y s a.e. We call a solution to (NE) a pair (y; w) such that y 2 C([0; T ]; L 2 ()) \ L 2 (0; T ; V ) \ W 1;2 ([0; T ]; V 0 ) w 2 L 2 (0; T ; V ); w(t; x) 2 (t; x; y(t; x)) a.e. on ; y y s ; a.e. (t; x) 2 ; which satises the equation T dy (t); (t) dt + dt 0 V 0 ;V (rw K 0 (t; x; y)) rdxdt = for any 2 L 2 (0; T ; V ); and the condition y(0) = y 0 : T 0 hf(t); (t)i V 0 ;V dt V = H 1 (); V 0 = (H 1 ()) 0
Existence of the free boundary in a diffusive ow in porous media 41 A direct approach: regularity for (t; x; r) For each t by the relation A(t) : D(A(t)) (H 1 ()) 0! (H 1 ()) 0 ; D(A(t)) = z 2 L 2 (); 9w 2 V; w(x) 2 (t; x; z)) a.e. x 2 ; ha(t)z; i V 0 ;V := (rw K 0 (t; x; y)) r dx; for any 2 V: dy (t) + A(t)y(t) 3 f(t); a.e. t 2 (0; T ); dt y(0) = y 0 :
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 problem 6 dy dt (t) + A "(t)y(t) = f(t); a.e. t 2 (0; T ); y(0) = y 0 : (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)k V 0 jt sj g (kk V 0) (ka(t)k V 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
Existence of the free boundary in a diffusive ow in porous media 43 4 Numerical results
Existence of the free boundary in a diffusive ow in porous media 44 Numerical results dy (t) + Ay(t) = f(t); a.e. t 2 (0; T ); dt y(0) = y 0 : y h i+1 h y h i + Ay h i+1 = f h i+1; i = 1; :::; n 1 y h 0 = y 0 Stability and convergence Algorithm based on the proof of the m-accretivity of A h = A + 1 h I 1 h I + Ah = g for any g 2 V 0 ; has a solution 2 D(A):
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) = ( (c 1)r (c r) ; r 2 [0; 1); [1; 1) ; r = 1: ; K(r) = (c 1)r2 ; r 2 [0; 1] c r 0 (x; y) = 0:01; f = 0; = 10 8 ; rain = exp(30 x 2 )= exp(30); c = 1:02 " " " " rain
Existence of the free boundary in a diffusive ow in porous media Numerical results 46
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