# Stability Theorems

## Abstract

In this chapter we face the problem of determining conditions under which the minimizing-movement scheme commutes with *Γ*-convergence. Let \(F_{\varepsilon }\) *Γ*-converge to *F* with initial data \(x_{\varepsilon }\) converging to *x* _{0}. In Sect. 8.2 it is proved that by suitably choosing \(\varepsilon =\varepsilon (\tau )\) the minimizing movement along the sequence \(F_{\varepsilon }\) from \(x_{\varepsilon }\) converges to a minimizing movement for the limit *F* from *x* _{0}. A further issue is whether, by assuming some further properties on \(F_{\varepsilon }\) we may deduce that the same thing happens for any choice of \(\varepsilon\). In order to give an answer we will use results from the theory of gradient flows recently elaborated by Ambrosio, Gigli and Savaré, and by Sandier and Serfaty.

## References

- 1.Ambrosio, L., Gigli, N.: A user’s guide to optimal transport. In: Piccoli, B., Rascle, M. (eds.) Modelling and Optimisation of Flows on Networks. Lecture Notes in Mathematics, pp. 1–155. Springer, Berlin (2013)CrossRefGoogle Scholar
- 2.Braides, A., Chiadò Piat, V., Piatnitski, A.: Discrete double porosity models. Preprint Scuola Normale Superiore, Pisa (2013). http://cvgmt.sns.it/paper/2237/
- 3.Jerrard, R.L., Sternberg, P.: Critical points via
*Γ*-convergence: general theory and applications. J. Europ. Math. Soc.**11**, 705–753 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Sandier, E., Serfaty, S.: Gamma-convergence of gradient flows and application to Ginzburg–Landau. Comm. Pure Appl. Math.
**57**, 1627–1672 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Serfaty, S.: Stability in 2D Ginzburg–Landau passes to the limit. Indiana Univ. Math. J.
**54**, 199–222 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Serfaty, S.: Gamma-convergence of gradient flows and applications to Ginzburg–Landau vortex dynamics. In: Braides, A., Chiadò Piat, V. (eds.) Topics on concentration phenomena and problems with multiple scales. Lect. Notes Unione Mat. Ital. vol. 2, pp. 267–292. Springer, Berlin (2006)CrossRefGoogle Scholar