Gradient flows of non convex functionals in Hilbert spaces and applications
ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 3, p. 564-614

This paper addresses the Cauchy problem for the gradient flow equation in a Hilbert space $ℋ$ $\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}\left(0,T\right),\hfill \\ u\left(0\right)={u}_{0},\hfill \end{array}\right\$ where $\phi :ℋ\to \left(-\infty ,+\infty \right]$ 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.

DOI : https://doi.org/10.1051/cocv:2006013
Classification:  35A15,  35K50,  35K85,  58D25,  80A22
Keywords: evolution problems, gradient flows, minimizing movements, Young measures, phase transitions, quasistationary models
@article{COCV_2006__12_3_564_0,
author = {Rossi, Riccarda and Savar\'e, Giusepp},
title = {Gradient flows of non convex functionals in Hilbert spaces and applications},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {12},
number = {3},
year = {2006},
pages = {564-614},
doi = {10.1051/cocv:2006013},
zbl = {1116.34048},
mrnumber = {2224826},
language = {en},
url = {http://www.numdam.org/item/COCV_2006__12_3_564_0}
}

Rossi, Riccarda; Savaré, Giusepp. Gradient flows of non convex functionals in Hilbert spaces and applications. ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 3, pp. 564-614. doi : 10.1051/cocv:2006013. http://www.numdam.org/item/COCV_2006__12_3_564_0/

[1] L. Ambrosio, Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995) 191-246. | Zbl 0957.49029

[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, Clarendon Press, Oxford (2000). | MR 1857292 | Zbl 0957.49001

[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 2129498 | Zbl 1090.35002

[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 0677.65068

[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 0549.49005

[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 0606.60006

[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 1032.91007

[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 0991.49500

[9] G. Bouchitté, Singular perturbations of variational problems arising from a two-phase transition model. Appl. Math. Optim. 21 (1990) 289-314. | Zbl 0695.49003

[10] A. Bressan, A. Cellina and G. Colombo, Upper semicontinuous differential inclusions without convexity. Proc. Amer. Math. Soc. 106 (1989) 771-775. | Zbl 0698.34014

[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 0278.47033

[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 348562 | Zbl 0252.47055

[13] H. Brézis, Analyse fonctionnelle - Théorie et applications. Masson, Paris (1983). | MR 697382 | Zbl 0511.46001

[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 0231.47034

[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 0880.47045

[16] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions. Springer, Berlin-New York (1977). | MR 467310 | Zbl 0346.46038

[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 0226.47038

[18] M.G. Crandall and A. Pazy, Semi-groups of nonlinear contractions and dissipative sets. J. Functional Anal. 3 (1969) 376-418. | Zbl 0182.18903

[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 0851.35052

[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 0465.47041

[21] C. Dellacherie and P.A. Meyer, Probabilities and potential. North-Holland Publishing Co., Amsterdam (1978). | MR 521810 | Zbl 0494.60001

[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 0915.35120

[23] T. Kato, Perturbation theory for linear operators. Springer, Berlin (1976). | MR 407617 | Zbl 0342.47009

[24] Y. Kōmura, Nonlinear semi-groups in Hilbert space. J. Math. Soc. Japan 19 (1967) 493-507. | Zbl 0163.38302

[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 0449.49015

[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 0734.35159

[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 0681.49012

[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 0699.49015

[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 1012.74054

[31] L. Modica, Gradient theory of phase transitions and minimal interface criterion. Arch. Rational Mech. Anal. 98 (1986) 123-142. | Zbl 0616.76004

[32] L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza. Boll. Un. Mat. Ital. B 14 (1977) 285-299. | Zbl 0356.49008

[33] B. Sh. Mordukhovich, Nonsmooth analysis with nonconvex generalized differentials and conjugate mappings. Dokl. Akad. Nauk BSSR 28 (1984) 976-979. | Zbl 0557.49007

[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 1021.65047

[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 0802.35165

[36] R.T. Rockafellar, Convex analysis. Princeton University Press, Princeton (1970). | MR 274683 | Zbl 0193.18401

[37] R.T. Rockafellar and R.J.B. Wets, Variational analysis. Springer-Verlag, Berlin (1998). | MR 1491362 | Zbl 0888.49001

[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. | Zbl pre05019614

[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. | Zbl pre05058751

[40] J. Rulla, Error analysis for implicit approximations to solutions to Cauchy problems. SIAM J. Numer. Anal. 33 (1996) 68-87. | Zbl 0855.65102

[41] G. Savaré, Weak solutions and maximal regularity for abstract evolution inequalities. Adv. Math. Sci. Appl. 6 (1996) 377-418. | Zbl 0858.35073

[42] G. Savaré, Compactness properties for families of quasistationary solutions of some evolution equations. Trans. Amer. Math. Soc. 354 (2002) 3703-3722. | Zbl 1008.47065

[43] R. Schätzle, The quasistationary phase field equations with Neumann boundary conditions. J. Differential Equations 162 (2000) 473-503. | Zbl 0963.35188

[44] L. Simon, Lectures on geometric measure theory, in Proc. Centre for Math. Anal., Australian Nat. Univ. 3 (1983). | MR 756417 | Zbl 0546.49019

[45] M. Valadier, Young measures, Methods of nonconvex analysis (Varenna, 1989). Springer, Berlin (1990) 152-188. | Zbl 0738.28004

[46] A. Visintin, Differential models of hysteresis. Appl. Math. Sci. 111, Springer-Verlag, Berlin (1994). | MR 1329094 | Zbl 0820.35004

[47] A. Visintin, Models of phase transitions. Progress in Nonlinear Differential Equations and Their Applications 28, Birkhäuser, Boston (1996). | MR 1423808 | Zbl 0882.35004

[48] A. Visintin, Forward-backward parabolic equations and hysteresis. Calc. Var. Partial Differential Equations 15 (2002) 115-132. | Zbl 1010.35056