Gradient flows of non convex functionals in Hilbert spaces and applications

Rossi, Riccarda; Savaré, Giusepp

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 3, pp. 564-614.

This paper addresses the Cauchy problem for the gradient flow equation in a Hilbert space $\mathscr{H}$

$$\left\{\begin{array}{cc}{u}^{\text{'}}\left(t\right)+{\partial }_{\ell }\phi \left(u\left(t\right)\right)\ni f\left(t\right)\hfill & \text{a.e.}\text{in}(0,T),\hfill \\ u\left(0\right)=u_0,\hfill \end{array}\right.$$

where $\phi :\mathscr{H}\to (-\infty ,+\infty ]$ is a proper, lower semicontinuous functional which is not supposed to be a (smooth perturbation of a) convex functional and ${\partial }_{\ell }\phi$ is (a suitable limiting version of) its subdifferential. We will present some new existence results for the solutions of the equation by exploiting a variational approximation technique, featuring some ideas from the theory of Minimizing Movements and of Young measures. Our analysis is also motivated by some models describing phase transitions phenomena, leading to systems of evolutionary PDEs which have a common underlying gradient flow structure: in particular, we will focus on quasistationary models, which exhibit highly non convex Lyapunov functionals.

Classification : 35A15, 35K50, 35K85, 58D25, 80A22
Mots clés : evolution problems, gradient flows, minimizing movements, Young measures, phase transitions, quasistationary models

[1] L. Ambrosio, Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995) 191-246. | Zbl

[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, Clarendon Press, Oxford (2000). | MR | Zbl

[3] L. Ambrosio, N Gigli and G. Savaré, Gradient flows.In metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag (2005). | MR | Zbl

[4] C. Baiocchi, Discretization of evolution variational inequalities, Partial differential equations and the calculus of variations, Vol. I, F. Colombini, A. Marino, L. Modica and S. Spagnolo, Eds., Birkhäuser Boston, Boston, MA (1989) 59-92. | Zbl

[5] E.J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory. SIAM J. Control Optim. 22 (1984) 570-598. | Zbl

[6] E.J. Balder, An extension of Prohorov's theorem for transition probabilities with applications to infinite-dimensional lower closure problems. Rend. Circ. Mat. Palermo 34 (1985) 427-447. | Zbl

[7] E.J. Balder, Lectures on Young measure theory and its applications in economics. Rend. Istit. Mat. Univ. Trieste 31 (2000) (Suppl. 1), 1-69, Workshop on Measure Theory and Real Analysis (Italian) (Grado, 1997). | Zbl

[8] J.M. Ball, A version of the fundamental theorem for Young measures, PDEs and continuum models of phase transitions (Nice 1988), Springer, Berlin. Lect. Notes Phys. 344 (1989) 207-215. | Zbl

[9] G. Bouchitté, Singular perturbations of variational problems arising from a two-phase transition model. Appl. Math. Optim. 21 (1990) 289-314. | Zbl

[10] A. Bressan, A. Cellina and G. Colombo, Upper semicontinuous differential inclusions without convexity. Proc. Amer. Math. Soc. 106 (1989) 771-775. | Zbl

[11] H. Brézis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, Contribution to Nonlinear Functional Analysis, in Proc. Sympos. Math. Res. Center, Univ. Wisconsin, Madison, 1971. Academic Press, New York (1971) 101-156. | Zbl

[12] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam (1973), North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). | MR | Zbl

[13] H. Brézis, Analyse fonctionnelle - Théorie et applications. Masson, Paris (1983). | MR | Zbl

[14] H. Brézis, On some degenerate nonlinear parabolic equations, Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 1, Chicago, Ill., 1968), Providence, R.I., Amer. Math. Soc. (1970) 28-38. | Zbl

[15] T. Cardinali, G. Colombo, F. Papalini and M. Tosques, On a class of evolution equations without convexity. Nonlinear Anal. 28 (1997) 217-234. | Zbl

[16] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions. Springer, Berlin-New York (1977). | MR | Zbl

[17] M.G. Crandall and T.M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces. Amer. J. Math. 93 (1971) 265-298. | Zbl

[18] M.G. Crandall and A. Pazy, Semi-groups of nonlinear contractions and dissipative sets. J. Functional Anal. 3 (1969) 376-418. | Zbl

[19] E. De Giorgi, New problems on minimizing movements, Boundary Value Problems for PDE and Applications, C. Baiocchi and J.L. Lions, Eds., Masson (1993) 81-98. | Zbl

[20] E. De Giorgi, A. Marino and M. Tosques, Problems of evolution in metric spaces and maximal decreasing curve. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 68 (1980) 180-187. | Zbl

[21] C. Dellacherie and P.A. Meyer, Probabilities and potential. North-Holland Publishing Co., Amsterdam (1978). | MR | Zbl

[22] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998) 1-17 (electronic). | Zbl

[23] T. Kato, Perturbation theory for linear operators. Springer, Berlin (1976). | MR | Zbl

[24] Y. Kōmura, Nonlinear semi-groups in Hilbert space. J. Math. Soc. Japan 19 (1967) 493-507. | Zbl

[25] A.Ja. Kruger and B. Sh. Mordukhovich, Extremal points and the Euler equation in nonsmooth optimization problems. Dokl. Akad. Nauk BSSR 24 (1980) 684-687, 763. | Zbl

[26] S. Luckhaus, Solutions for the two-phase Stefan problem with the Gibbs-Thomson Law for the melting temperature. Euro. J. Appl. Math. 1 (1990) 101-111. | Zbl

[27] S. Luckhaus, The Stefan Problem with the Gibbs-Thomson law. Preprint No. 591 Università di Pisa (1991) 1-21.

[28] S. Luckhaus, The Gibbs-Thompson relation within the gradient theory of phase transitions. Arch. Rational Mech. Anal. 107 (1989) 71-83. | Zbl

[29] A. Marino, C. Saccon and M. Tosques, Curves of maximal slope and parabolic variational inequalities on nonconvex constraints. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 (1989) 281-330. | Numdam | Zbl

[30] A. Mielke, F. Theil and V.I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Ration. Mech. Anal. 162 (2002) 137-177. | Zbl

[31] L. Modica, Gradient theory of phase transitions and minimal interface criterion. Arch. Rational Mech. Anal. 98 (1986) 123-142. | Zbl

[32] L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza. Boll. Un. Mat. Ital. B 14 (1977) 285-299. | Zbl

[33] B. Sh. Mordukhovich, Nonsmooth analysis with nonconvex generalized differentials and conjugate mappings. Dokl. Akad. Nauk BSSR 28 (1984) 976-979. | Zbl

[34] R.H. Nochetto, G. Savaré and C. Verdi, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations. Comm. Pure Appl. Math. 53 (2000) 525-589. | Zbl

[35] P.I. Plotnikov and V.N. Starovoitov, The Stefan problem with surface tension as the limit of a phase field model. Differential Equations 29 (1993) 395-404. | Zbl

[36] R.T. Rockafellar, Convex analysis. Princeton University Press, Princeton (1970). | MR | Zbl

[37] R.T. Rockafellar and R.J.B. Wets, Variational analysis. Springer-Verlag, Berlin (1998). | MR | Zbl

[38] R. Rossi and G. Savaré, Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces. Ann. Sc. Norm. Sup., Pisa 2 (2003) 395-431.

[39] R. Rossi and G. Savaré, Existence and approximation results for gradient flows. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (9) Mat. Appl. 15 (2004) 183-196.

[40] J. Rulla, Error analysis for implicit approximations to solutions to Cauchy problems. SIAM J. Numer. Anal. 33 (1996) 68-87. | Zbl

[41] G. Savaré, Weak solutions and maximal regularity for abstract evolution inequalities. Adv. Math. Sci. Appl. 6 (1996) 377-418. | Zbl

[42] G. Savaré, Compactness properties for families of quasistationary solutions of some evolution equations. Trans. Amer. Math. Soc. 354 (2002) 3703-3722. | Zbl

[43] R. Schätzle, The quasistationary phase field equations with Neumann boundary conditions. J. Differential Equations 162 (2000) 473-503. | Zbl

[44] L. Simon, Lectures on geometric measure theory, in Proc. Centre for Math. Anal., Australian Nat. Univ. 3 (1983). | MR | Zbl

[45] M. Valadier, Young measures, Methods of nonconvex analysis (Varenna, 1989). Springer, Berlin (1990) 152-188. | Zbl

[46] A. Visintin, Differential models of hysteresis. Appl. Math. Sci. 111, Springer-Verlag, Berlin (1994). | MR | Zbl

[47] A. Visintin, Models of phase transitions. Progress in Nonlinear Differential Equations and Their Applications 28, Birkhäuser, Boston (1996). | MR | Zbl

[48] A. Visintin, Forward-backward parabolic equations and hysteresis. Calc. Var. Partial Differential Equations 15 (2002) 115-132. | Zbl