Existence of the free boundary in a diffusive ow in porous media - PowerPoint PPT Presentation

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 � � R 3 ; open bounded, � = @ � suf�ciently 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 ) � � Q s = f ( t; x ) 2 Q ; y ( t; x ) = y s g Q u = f ( t; x ) 2 Q ; y ( t; x ) < y s g 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 ) � � Q s = f ( t; x ) 2 Q ; y ( t; x ) = y s g Q u = f ( t; x ) 2 Q ; y ( t; x ) < y s g 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 in Q @t � � � ( t; x; y ) 3 f @� ( t; x; y ) on � ; � > 0 ( NE ) + �� ( t; x; y ) 3 0 @� in � : y (0 ; x ) = y 0 � : Q � ( �1 ; y s ] ! R ; � ( t; x; � ) 2 C 1 ( �1 ; y s ) ; r % y s � ( t; x; r ) = K � s ; a.e. ( t; x ) 2 Q: lim 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 in Q @t � � � ( t; x; y ) 3 f @� ( t; x; y ) on � ; � > 0 ; ( NE ) + �� ( t; x; y ) 3 0 @� in � : y (0 ; x ) = y 0 � : Q � ( �1 ; y s ] ! R ; � ( t; x; � ) 2 C 1 ( �1 ; y s ) ; r % y s � ( t; x; r ) = K � s ; a.e. ( t; x ) 2 Q: lim 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 ( R r � � ( t; x; s ) ds; r � y s ; j : Q � R ! ( �1 ; 1 ] ; j ( t; x; r ) = 0 otherwise + 1 ; ( t; x ) ! j ( t; x; r ) is measurable on Q for all r 2 ( �1 ; y s ] 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 j r j ; for any r � y s ; 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 ( !r � j ( t; x; r )) a.e. ( t; x ) 2 Q r 2 R 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: C 3 j ! j + C 0 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 ( !r � j ( t; x; r )) a.e. ( t; x ) 2 Q r 2 R 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: C 3 j ! j + C 0 3 � j � ( t; x; ! ) for any ! 2 R ; a.e. ( t; x ) 2 Q; C 3 > 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 in Q; @t � � � ( t; x; y ) 3 f @� ( t; x; y ) on � ; ( NE ) + �� ( t; x; y ) 3 0 @� in � : y (0 ; x ) = y 0 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 in Q @t � � � ( t; x; y ) 3 f @� ( t; x; y ) on � ( NE ) + �� ( t; x; y ) 3 0 @� in � : y (0 ; x ) = y 0 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 De�nition. Let f 2 L 1 ( Q ) ; 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 ( Q ) ; w 2 ( L 1 ( Q )) 0 ; w a 2 L 1 ( Q )) ( w = w a + w s ; w a ( t; x ) 2 @j ( t; x; y ) = � ( t; x; y ) a.e. on Q; w s 2 N D ( ' ) ( y ) Z Z Z yd ( � ) � dt dxdt + y 0 (0) + h w; A 0 ; 1 i ( L 1 ( Q )) 0 ;L 1 ( Q ) = f dxdt Q � Q for any 2 W 1 ; 1 ([0 ; T ]; L 1 (�)) \ L 1 (0 ; T ; D ( A 0 ; 1 )) ; ( T ) = 0 : 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 : 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 : L 1 ( Q ) � L 1 ( Q ) �Z � ( j ( t; x; y ( t; x )) + j � ( t; x; w ( t; x )) dxdt � y ( t; x ) w ( t; x )) dxdt Min ( y;w ) 2 U Q � � ( y; w ); y 2 L 1 ( Q ) ; y ( t; x ) � y s a.e., w 2 L 1 ( Q ) ; dy U = dt � � w = f; y (0) = y 0 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 : L 1 ( Q ) � ( L 1 ( Q )) 0 Z j ( t; x; y ( t; x )) dxdt + ' � ( t; x; w ) � w ( y ) J ( y; w ) = Q 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 ( L 1 ( Q )) 0 ; w = w a + w s L 1 ( Q ) 3 w a = " absolutely continuous component", w s = singular component Z ' : L 1 ( Q ) ! ( �1 ; 1 ] ; ' ( y ) = j ( t; x; y ( t; x )) dxdt proper, convex, lsc Q Z ' � : ( L 1 ( Q )) 0 ! ( �1 ; 1 ] ; ' � ( w ) = j � ( t; x; w a ( t; x )) dxdt + � � D ( ' ) ( w s ) ; Q � � D ( ' ) ( v ) = sup f v ( ); 2 D ( ' ) g = conjugate of the indicator function of D ( ' ) D ( ' ) = f y 2 L 1 ( Q ); ' ( y ) < + 1g = f y 2 L 1 ( Q ); y ( t; x ) � y s 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 approach 2 R Q j ( t; x; y ( t; x )) dxdt proper, convex, lsc ' : L 1 ( Q ) ! ( �1 ; 1 ] ' ( y ) = R ' � : ( L 1 ( Q )) 0 ! ( �1 ; 1 ] Q j � ( t; x; w a ( t; x )) dxdt + � � ' � ( w ) = D ( ' ) ( w s ) � � D ( ' ) ( v ) = sup f v ( ); 2 D ( ' ) g = conjugate of the indicator function of D ( ' ) D ( ' ) = f y 2 L 1 ( Q ); ' ( y ) < + 1g = f y 2 L 1 ( Q ); y ( t; x ) � y s a.e. g : 2 R.T. Rockafeller, Paci�c 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 approach 3 R Q j ( t; x; y ( t; x )) dxdt proper, convex, lsc ' : L 1 ( Q ) ! ( �1 ; 1 ] ' ( y ) = R ' � : ( L 1 ( Q )) 0 ! ( �1 ; 1 ] Q j � ( t; x; w a ( t; x )) dxdt + � � ' � ( w ) = D ( ' ) ( w s ) � � D ( ' ) ( v ) = sup f v ( ); 2 D ( ' ) g = conjugate of the indicator function of D ( ' ) D ( ' ) = f y 2 L 1 ( Q ); ' ( y ) < + 1g = f y 2 L 1 ( Q ); y ( t; x ) � y s a.e. g : 3 R.T. Rockafeller, Paci�c J. Math. J, 39, 2, 1971 Diffuse Interface Models - DIMO2013, September 10-13, 2013, Levico Terme, Italy