A duality variational approach to time-dependent nonlinear diffusion equations

Transcription

1 A duality variational approach to time-dependent nonlinear diffusion equations Gabriela Marinoschi Institute of Mathematical Statistics and Applied Mathematics, Bucharest, Romania

2 A duality variational approach to time-dependent nonlinear diffusion equations 2 Presentation outline 1. Overview Problem presentation Problem history Connection with physical processes 2. Main results Functional framework Continuous case with polynomial growth General continuous case Discontinuous case 3. Comments and applications

3 A duality variational approach to time-dependent nonlinear diffusion equations 3 1 Overview Formulation of nonlinear time-dependent diffusion problems as convex minimization problems

4 A duality variational approach to time-dependent nonlinear diffusion equations 4 Problem presentation - (t; x; y) 3 f in Q = (0; T ) (t; x; y) = (t; x; y) on = (0; T ) ; > 0; (ODP y(0; x) = y 0 in

5 A duality variational approach to time-dependent nonlinear diffusion equations 5 Problem presentation - (t; x; y) 3 f in Q = (0; T ) (t; x; y) = (t; x; y) on = (0; T ) ; > 0; (ODP y(0; x) = y 0 in (t; x; ) is a maximal monotone graph on R, a.e. (t; x) 2 x; ) = (t; x; ) a.e. (t; x) 2 Q:

6 A duality variational approach to time-dependent nonlinear diffusion equations 6 Problem presentation j(t; x; ) : R! ( 1; 1] proper convex l.s.c. a.e. (t; x) 2 Q; (t; x)! j(t; x; r) is measurable on Q for any r 2 R:

7 A duality variational approach to time-dependent nonlinear diffusion equations 7 Problem presentation Conjugate of j j(t; x; ) : R! ( 1; 1] proper convex l.s.c. a.e. (t; x) 2 Q; (t; x)! j(t; x; r) is measurable on Q for any r 2 R: j (t; x;!) = sup(!r r2r j(t; x; r)) a.e. on Q

8 A duality variational approach to time-dependent nonlinear diffusion equations 8 Problem presentation j(t; x; r) + j (t; x;!) r! for any r;! 2 R; a.e. (t; x) 2 Q; j(t; x; r) + j (t; x;!) = r! if and only if! x; r), a.e. (t; x) 2 Q: W. Fenchel, 1953

9 A duality variational approach to time-dependent nonlinear diffusion equations 9 Problem history Brezis-Ekeland principle (C.R. Acad. Sci. Paris, 1976) du dt 3 f a.e. t; u(0) = u 0 m Minimize J(v) for all v 2 K J(v) = Z T 0 '(v) + ' f dv dt (f; v) dt kv(t )k ;

10 A duality variational approach to time-dependent nonlinear diffusion equations 10 Problem history Brezis-Ekeland principle (C.R. Acad. Sci. Paris, 1976) I G. Auchmuty (Nonlinear Anal. 1988) I N. Ghoussoub, L. Tzou ( Math. Ann. 2004) I A. Visintin ( Adv. Math. Sci. Appl. 2008)

11 A duality variational approach to time-dependent nonlinear diffusion equations 11 Problem presentation (ODP ) can be reduced to a minimization problem + * Minimize Z Q (j(t; x; y(t; x)) + j (t; x; w(t; x)) w(t; x)y(t; x)) dxdt, for (y; w) 2 U; (P ) y and w connected by a certain equation!

12 A duality variational approach to time-dependent nonlinear diffusion equations 12 Problem presentation This work: (P ) optimal control problem in terms of j and j 1. Case 1. j(t; x; ) continuous on R; a.e. on Q; and has a polynomial growth 2. Case 2. j(t; x; ) continuous on R; a.e. on Q; j(t; x; r) jrj! 1; j (t; x;!) jrj!1 j!j! 1; j!j!1 3. Case 3. D(j) = ( 1; y s ] and j is not continuous at y s :

13 A duality variational approach to time-dependent nonlinear diffusion equations 13 2 Main results

14 A duality variational approach to time-dependent nonlinear diffusion equations Continuous case with polynomial growth Hypotheses j : R! ( 1; 1] continuous C 1 jrj m + C 0 1 j(t; x; r) C 2 jrj m + C 0 2; a.e. (t; x) 2 Q; m 2

15 A duality variational approach to time-dependent nonlinear diffusion equations 15 Connection with physical processes Diffusion with time-dependent diffusion coefcients (t; x; r) in heterogeneous media (t; x; r) = Z r 0 (t; x; s)ds; j(t; x; r) = Z r 0 (t; x; s)ds; r 2 R continuous diffusion coefcient (t; x; )

16 A duality variational approach to time-dependent nonlinear diffusion equations 16 Connection with physical processes Diffusion with time-dependent diffusion coefcients (t; x; r) in heterogeneous media (t; x; r) = Z r 0 (t; x; s)ds; j(t; x; r) = Z r 0 (t; x; s)ds; r 2 R discontinuous diffusion coefcient (t; x; )

17 A duality variational approach to time-dependent nonlinear diffusion equations 17 Functional framework Let m > 1 1 m m = 1 X m := n 2 W + = 0 on o A 0;m : D(A 0;m ) = X m L m ()! L m (); A 0;m = ;

18 A duality variational approach to time-dependent nonlinear diffusion equations 18 Functional framework Let m > 1 1 m m = 1 X m := n 2 W + = 0 on o X 0 m is the dual of X m A 0;m : D(A 0;m ) = X m L m ()! L m () A m : D(A m ) = L m () X 0 m! X 0 m A 0;m = ha m y; i X 0 m ;X m 0 = (y; A 0;m 0 ) Lm (Q);L m0 (Q) V = H 1 (); V 0 = (H 1 ()) 0

19 A duality variational approach to time-dependent nonlinear diffusion equations 19 Denition. Let f 2 L m0 (0; T ; Xm 0 0); y 0 2 V 0 : A weak solution to (ODP ) (y; ); (t; x) x; y(t; x)) a.e. on Q; which satises Z T dy (t); (t) dt 0 y 2 L m (Q) \ W 1;m0 ([0; T ]; X 0 m 0); 2 Lm0 (Q); X 0 m 0 ;X m dt Z Q dxdt = Z T for any 2 L m (0; T ; X m ); and the initial condition y(0; x) = y 0 : 0 hf(t); (t)i X 0m 0 ;X m dt

20 A duality variational approach to time-dependent nonlinear diffusion equations 20 Variational formulation J 0 : L m (Q) L m0 (Q)! ( 1; 1] J 0 (y; w) = R Q (j(t; x; y) + j (t; x; w) wy) dxdt, if (y; w) 2 U 0 ; +1; otherwise U 0 = R (y; w); y 2 L m (Q); w 2 L m0 (Q); Q j(t; x; y)dxdt < 1; R Q j(t; x; y)dxdt < 1; (y; w) satises ( ) dy dt (t) + A m 0 w(t) = f(t) a.e. t 2 (0; T ); y(0) = y 0 : ()

21 A duality variational approach to time-dependent nonlinear diffusion equations 21 Variational formulation Minimization problem Minimize J 0 (y; w) for all (y; w) 2 U 0 : (P 0 ) m (ODP )

22 A duality variational approach to time-dependent nonlinear diffusion equations 22 Some results J 0 is proper, convex, l.s.c. on L m (Q) L m0 (Q) J 0 (y; w) = Z Q (j(t; x; y) + j (t; x; w))dxdt Z wydxdt Q

23 A duality variational approach to time-dependent nonlinear diffusion equations 23 Some results J 0 is proper, convex, l.s.c. on L m (Q) L m0 (Q) J 0 (y; w) = Z Q (j(t; x; y) + j (t; x; w))dxdt nky(t )k 2V 0 ky 0 k 2V 0 o Z t 0 f(); A 1 0;m y() X 0 m 0 ;X m d:

24 A duality variational approach to time-dependent nonlinear diffusion equations 24 Theorem Problem (P 0 ) has at least a solution (y ; w ).

25 A duality variational approach to time-dependent nonlinear diffusion equations 25 Proof sketch: (y n ; w n ) a minimizing sequence d J 0 (y n ; w n ) d + 1 n ; y n * y w n * w =) (y n ) n and (w n ) n are bounded in L m (Q) and L m0 (Q) - in L m (Q) in L m0 (Q) dy n dt * dy dt in L m0 (0; T ; X 0 m 0 ) =) (y ; w ) satises () J 0 (y ; w ) = d + y n (T ) * y (T ) in V 0 Ascoli-Arzelá

26 A duality variational approach to time-dependent nonlinear diffusion equations 26 Theorem Let (y ; w ) be a solution to (P 0 ): Then, inf (P 0 ) = J 0 (y ; w ) = 0 (y;w)2u 0 and (y ; w ) is the unique weak solution to (ODP ):

27 A duality variational approach to time-dependent nonlinear diffusion equations 27 Theorem Let (y ; w ) be a solution to (P 0 ): Then, w (t; x) 2 (t; x; y ) a:e: (t; x) 2 Q and (y ; w ) is the unique weak solution to (ODP ): dy dt (t) + A m 0 w (t) = f(t) a.e. t 2 (0; T ); y(0) = y 0 : ()

28 A duality variational approach to time-dependent nonlinear diffusion equations 28 Proof sketch Let (y ; w ) be a solution to (P 0 ): J 0 (y ; w ) J 0 (y; w) for any (y; w) 2 U 0 :

29 A duality variational approach to time-dependent nonlinear diffusion equations 29 Proof sketch Let (y ; w ) be a solution to (P 0 ): J 0 (y ; w ) J 0 (y; w) for any (y; w) 2 U 0 : Let > 0 y = y + Y; w = w + W dy dt (t) + A 0;mW (t) = 0 a.e. t 2 (0; T ); Y (0) = 0:

30 A duality variational approach to time-dependent nonlinear diffusion equations 30 Proof sketch A m 0 Z T t ( w )(s)ds + ( ) 1 (t; x; w ) y = 0:

31 A duality variational approach to time-dependent nonlinear diffusion equations 31 Proof sketch A m 0 Z T t ( w )(s)ds + ( ) 1 (t; x; w ) y = 0: We denote p(t; x) = ( ) 1 (t; x; w ) y p t + A m 0(w ) = 0; p(t ) = 0: =) p(t) = 0 for any t 2 [0; T ]

32 A duality variational approach to time-dependent nonlinear diffusion equations 32 Uniqueness is proved using (ODP ):

33 A duality variational approach to time-dependent nonlinear diffusion equations General continuous case j(t; x; r) jrj! 1; as jrj! 1; j (t; x;!) j!j! 1; as j!j! 1 uniformly w.r. (t; x) 2 Q

34 A duality variational approach to time-dependent nonlinear diffusion equations 34 Let f 2 L 1 (Q); y 0 2 V 0 D eay; E ea : L 1 () X 0! X 0 ; X 0 = (D(A 0;1 )) 0 X 0 ;X = hy; A 0;1i L1 ();L 1 () ; for any 2 X = D(A 0;1):

35 A duality variational approach to time-dependent nonlinear diffusion equations 35 J 1 (y; w) = 8 >< >: R Q (j(t; x; y) + j (t; x; w)) dxdt ky(t )k2 V ky 0k 2 V 0 R Q ya 1 0;m 0 fdxdt if (y; w) 2 U 1 ; +1; otherwise, U 1 = (y; w); y 2 L 1 (Q); w 2 L 1 (Q) R Q j(t; x; y)dxdt < 1; R Q j (t; x; w)dxdt < 1; y(t ) 2 V 0 ; (y; w) veries ( ) dy dt (t) + Aw(t) e = f(t) a.e. t 2 (0; T ); y(0) = y 0 : ()

36 A duality variational approach to time-dependent nonlinear diffusion equations 36 J 1 (y; w) = 8 >< >: R Q (j(t; x; y) + j (t; x; w)) dxdt ky(t )k2 V ky 0k 2 V 0 R Q ya 1 0;m 0 fdxdt if (y; w) 2 U 1 ; +1; otherwise, U 1 = (y; w); y 2 L 1 (Q); w 2 L 1 (Q) R Q j(t; x; y)dxdt < 1; R Q j (t; x; w)dxdt < 1; y(t ) 2 V 0 ; (y; w) veries ( ) dy dt (t) + Aw(t) e = f(t) a.e. t 2 (0; T ); y(0) = y 0 : () The function J 1 is the closure of the function J 0 in U 0 :

37 A duality variational approach to time-dependent nonlinear diffusion equations 37 Variational formulation Minimize J 1 (y; w) for all (y; w) 2 U 1 : (P 1 ) + A solution to (P 1 ) is called a generalized solution to (ODP )

38 A duality variational approach to time-dependent nonlinear diffusion equations 38 Theorem Problem (P 1 ) has at least a solution (y ; w ). If j is strictly convex the solution is unique.

39 A duality variational approach to time-dependent nonlinear diffusion equations 39 Theorem Let (y; w) be the minimizer such that J 1 (y; w) = 0: If y 2 [y m ; y M ] then and (y; w) is the weak solution to (ODP ): w(t; x) 2 (t; x; y(t; x)) a.e. on Q

40 A duality variational approach to time-dependent nonlinear diffusion equations Discontinuous case j not continuous : Q ( 1; y s )! (0; +1)

41 A duality variational approach to time-dependent nonlinear diffusion equations 41 Connection with physical processes j not continuous=) saturated-unsaturated ow (inltration in porous media) (t; x; r) = R r 0 (t; x; s)ds; r < y s; a.e. on Q; 2 [K s(t; x); +1); r = y s ; a.e. on Q;

42 A duality variational approach to time-dependent nonlinear diffusion equations 42 Connection with physical processes j(t; x; r) = ( R r 0 (t; x; s)ds; r < y s ; a.e. on Q; +1; r = y s ; a.e. on Q

43 A duality variational approach to time-dependent nonlinear diffusion equations 43 Connection with physical processes (a) fast diffusion K s s diffusion coefcient s

44 A duality variational approach to time-dependent nonlinear diffusion equations 44 Connection with physical processes (b) slow diffusion (t; x; y s ) = s (t; x) a.e. (t; x) 2 Q diffusion coefcient

45 A duality variational approach to time-dependent nonlinear diffusion equations 45 Preliminaries w 2 (L 1 (Q)) 0 ; w = w a + w s L 1 (Q) 3 w a = continuous component, w s = singular component

46 A duality variational approach to time-dependent nonlinear diffusion equations 46 Preliminaries w 2 (L 1 (Q)) 0 ; w = w a + w s L 1 (Q) 3 w a = continuous component, w s = singular component Z ' : L 1 (Q)! ( 1; 1]; '(y) = j(t; x; y(t; x))dxdt: Q

47 A duality variational approach to time-dependent nonlinear diffusion equations 47 Preliminaries w 2 (L 1 (Q)) 0 ; w = w a + w s L 1 (Q) 3 w a = continuous component, w s = singular component Z ' : L 1 (Q)! ( 1; 1]; '(y) = j(t; x; y(t; x))dxdt: ' : (L 1 (Q)) 0! ( 1; 1] Z ' (w) = j (t; x; w a (t; x))dxdt + C(w s ); C(w) = supfw( ); 2 Cg: C = fy 2 L 1 (Q); '(y) < +1g Rockafeller, Pacic J. Math. J., 1971 Q Q

48 A duality variational approach to time-dependent nonlinear diffusion equations 48 J 1 (y; w) = 8 >< R J 1 : L m (Q) (L 1 (Q)) 0! ( 1; 1]; Q j(t; x; y)dxdt ky(t )k2 V ky 0k 2 V 0 R T 0 hf(t); A 0;my(t)i X 0 m 0 ;X m dt + R Q j (t; x; w a )dxdt + C(w s ) if (y; w) 2 U 1 ; U 1 = >: +1; otherwise, (y; w); y 2 L m (Q); y(t ) 2 V 0 ; w 2 (L 1 (Q)) 0 ; R Q j(t; x; y)dxdt < 1; ' (w) = R Q j (t; x; w a )dxdt + C(w s ) < 1; (y; w) veries () Z Q y d dt dxdt + hy 0; (0)i V 0 ;V + hw; A 0;1 i (L1 (Q)) 0 ;L 1 (Q) = for any 2 W 1;2 ([0; T ]; L m0 ()) \ L 1 (0; T ; X); (T ) = 0: Z T 0 y(0) = y 0 ; hf(t); (t)i X 0 m ;X m dt()

49 A duality variational approach to time-dependent nonlinear diffusion equations 49 Variational formulation Minimize (J 1 ) for all (y; w) 2 U 1 (P 1 ) + A solution to (P 1 ) is called a generalized solution to (ODP )

50 A duality variational approach to time-dependent nonlinear diffusion equations 50 Theorem Problem (P 1 ) has a unique solution (y ; w ).

51 A duality variational approach to time-dependent nonlinear diffusion equations 51 Theorem Let (y; w) be the minimizer such that J 1 (y; w) = 0: If y 2 L 1 (Q) then (y; w) is the weak solution to (ODP ), w a (t; x) 2 (t; x; y(t; x)) a.e. (t; x) 2 Q w s 2 N C (y):

52 A duality variational approach to time-dependent nonlinear diffusion equations 52 Interpretation The solution (y; w) satises the equations If y 2 int C= fy 2 L 1 (Q); ess sup y(t; x) < y s a.e. on Qg =) N C (y) = f0g =) w s = 0 dy (t; x; y) 3 f; a.e. on Q dt with initial condition and b.c.

53 A duality variational approach to time-dependent nonlinear diffusion equations 53 Interpretation The solution (y; w) satises the equations If y 2 int C= fy 2 L 1 (Q); ess sup y(t; x) < y s a.e. on Qg =) N C (y) = f0g =) w s = 0 dy (t; x; y) 3 f; a.e. on Q dt with initial condition and b.c. If y fy 2 L 1 (Q); ess sup y(t; x) = y s a.e.g then there exist Q s = f(t; x); y(t; x) = y s g and Q u = f(t; x); y(t; x) < y s g dy (t; x; y) 3 f; a.e. on Q u dt a dy dt s w s 3 0; a.e. on Q s :

54 A duality variational approach to time-dependent nonlinear diffusion equations 54 3 Comments and applications

55 A duality variational approach to time-dependent nonlinear diffusion equations 55 Advantages prove of the well-posedness of (ODP ) when other methods fail

56 A duality variational approach to time-dependent nonlinear diffusion equations 56 Advantages prove of the well-posedness of (ODP ) when other methods fail dy (t) + A(t)y(t) 3 f(t) a.e. t 2 (0; T ); dt y(0) = y 0 :

57 A duality variational approach to time-dependent nonlinear diffusion equations 57 Advantages prove of the well-posedness of (ODP ) when other methods fail dy (t) + A(t)y(t) 3 f(t) a.e. t 2 (0; T ); dt y(0) = y 0 : Theory of evolution equations with m-accretive operators Lions theorem M.G. Crandall, A. Pazy, Proc. AMS, 1979 (f = 0)

58 A duality variational approach to time-dependent nonlinear diffusion equations 58 Advantages prove of the well-posedness of (ODP ) when other methods fail solving a minimization convex problem involving a linear state equation numerical facilities.

59 A duality variational approach to time-dependent nonlinear diffusion equations 59 e () 3 f in Q Pollutant propagation in porous saturated media u is the absobtion-desorbtion coefcient, is the pollutant concentration

60 A duality variational approach to time-dependent nonlinear diffusion equations 60 e () 3 f in Q Pollutant propagation in porous saturated media u is the absobtion-desorbtion coefcient, is the pollutant concentration Inltration processes in porous media (water inltration in soils) u is the medium porosity, is the water saturation in pores

61 A duality variational approach to time-dependent nonlinear diffusion equations 61 e () 3 f in Q Pollutant propagation in porous saturated media u is the absobtion-desorbtion coefcient, is the pollutant concentration Inltration processes in porous media (water inltration in soils) u is the medium porosity, is the water saturation in pores Previous results (for u very regular) u constant G.M u(t) G.M u(x) degenerate case A. Favini, G.M. 2008, 2009

62 A duality variational approach to time-dependent nonlinear diffusion equations 62 e () 3 f in Q + y(t; x) = u(t; x)(t; x); u (t; x; y) 3 f in Q:

63 A duality variational approach to time-dependent nonlinear diffusion equations 63 Future interests Case with transport Control problems Numerical methods

64 A duality variational approach to time-dependent nonlinear diffusion equations 64 [1] G. Auchmuty, Variational principles for operator equations and initial value problems, Nonlinear Anal. 12 (1988) [2] V. Barbu, Nonlinear Differential Equations of Monotone Type in Banach Spaces, Springer, New York, [3] H. Brezis, I. Ekeland, Un principe variationnel associé à certaines equations paraboliques. Le cas independant du temps, C.R. Acad. Sci. Paris Sér. A 282 (1976) [4] H. Brezis, I. Ekeland, Un principe variationnel associé à certaines equations paraboliques. Le cas dependant du temps, C.R. Acad. Sci. Paris Sér. A 282 (1976) [5] W. Fenchel, Convex Cones, Sets and Functions., Princeton Univ., [6] N. Ghoussoub, L. Tzou, A variational principle for gradient ows, Math. Ann. 330 (2004) [7] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, [8] G. Marinoschi, Functional Approach to Nonlinear Models of Water Flow in Soils, Mathematical Modelling: Theory and Applications, volume 21, Springer, Dordrecht, [9] Gabriela Marinoschi, Well-posedness for a non-autonomous model of fast diffusion, Applied Analysis and Differential Equations (Editors: O. Carja, I. Vrabie), , World Sci. Publ., Hackensack, NJ, [10] R.T. Rockafeller, Pacic J. Math. J, 39, 2, [11] J.J. Telega, Extremum principles for nonpotential and initial-value problems, Arch. Mech. 54 (2002) [12] A. Visintin, Extension of the Brezis-Ekeland-Nayroles principle to monotone operators, Adv. Math. Sci. Appl. 18 (2008),

65 A duality variational approach to time-dependent nonlinear diffusion equations 65 Thank you for your attention