Abstract
In this paper we consider a nonlocal evolution equation in one dimension, which describes the dynamics of a ferromagnetic system in the mean field approximation. In the presence of a small magnetic field, it admits two stationary and homogeneous solutions, representing the stable and metastable phases of the physical system. We prove the existence of an invariant, one dimensional manifold connecting the stable and metastable phases. This is the unstable manifold of a distinguished, spatially nonhomogeneous, stationary solution, called the critical droplet.(4, 10) We show that the points on the manifold are droplets longer or shorter than the critical one, and that their motion is very slow in agreement with the theory of metastable patterns. We also obtain a new proof of the existence of the critical droplet, which is supplied with a local uniqueness result.
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REFERENCES
P. W. Bates, P. C. Fife, X. Ren, and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal. 138:105-136 (1997).
P. Buttà, On the validity of an Einstein relation in models of interface dynamics, J. Stat. Phys. 72:1401-1406 (1993).
X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations 2:125-160 (1997).
A. Chmaj and X. Ren, Homoclinic solutions of an integral equation: existence and stability, J. Differential Equation 155:17-43 (1999).
J. Carr and R. Pego, Metastable patterns in solutions of u t=ɛ2 u xx−f(u), Proc. Roy. Soc. Edinburgh Sect. A 116:133-160 (1990).
J. Carr and R. Pego, Invariant manifolds for metastable patterns in u t=ɛ2 u xx−f(u), Comm. Pure Appl. Math. 42:523-576 (1989).
M. Cassandro, A. Galves, E. Olivieri, and E. Vares, Metastable behavior of stochastic dynamics: A pathwise approach, J. Stat. Phys. 35:603-634 (1984).
A. De Masi, T. Gobron, and E. Presutti, Traveling fronts in non-local evolution equations, Arch. Rational Mech. Anal. 132:143-205 (1995).
A. De Masi, E. Olivieri, and E. Presutti, Spectral properties of integral operators in problems of interface dynamics and metastability, Markov Process. Related Fields 4: 27-112 (1998).
A. De Masi, E. Olivieri, and E. Presutti, Critical droplet for a nonlocal mean field equation, Markov Process. Related Fields 6:439-471 (2000).
A. De Masi, E. Orlandi, E. Presutti, and L. Triolo, Glauber evolution with Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamics, Nonlinearity 7:1-67 (1994).
A. De Masi, E. Orlandi, E. Presutti, and L. Triolo, Stability of the interface in a model of phase separation, Proc. Roy. Soc. Edinburgh Sect. A 124:1013-1022 (1994).
A. De Masi, E. Orlandi, E. Presutti, and L. Triolo, Uniqueness and global stability of the instanton in nonlocal evolution equations, Rend. Mat. Appl. (7) 14:693-723 (1994).
J.-P. Eckmann and J. Rougemont, Coarsening by Ginzburg-Landau dynamics, Comm. Math. Phys. 199:441-470 (1998).
G. Fusco and J. K. Hale, Slow motion manifolds, dormant instability and singular perturbations, J. Dynam. Differential Equations 1:75-94 (1989).
A. Galves, E. Olivieri, and E. Vares, Metastability for a class of dynamical systems subject to small random perturbations, Ann. Probab. 15:1288-1305 (1987).
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer Lectures Notes in Mathematics, Vol. 840 (Springer-Verlag, Berlin/New York, 1981).
M. Kac, G. Uhlenbeck, and P. C. Hemmer, On the van der Waals theory of vapor-liquid equilibrium. I. Discussion of a one dimensional model, J. Math. Phys. 4:216-228 (1963); II. Discussion of the distribution functions, J. Math. Phys. 4:229-247 (1963); III. Discussion of the critical region, J. Math. Phys. 5:60-74 (1964).
J. L. Lebowitz and O. Penrose, Rigorous treatment of the van der Waals Maxwell theory of the liquid vapour transition, J. Math. Phys. 7:98-113 (1966).
J. L. Lebowitz and O. Penrose, Rigorous treatment of metastable states in the van der Waals Maxwell theory, J. Stat. Phys. 3:211-236 (1971).
E. Olivieri and E. Scoppola, Markov chains with exponentially small transition probabilities: first exit problem from a general domain. I. The reversible case, J. Stat. Phys. 79:613-647 (1995).
E. Orlandi and L. Triolo, Travelling fronts in nonlocal models for phase separation in an external field, Proc. Roy. Soc. Edinburgh Sect. A 127:823-835 (1997).
H. Spohn, Interface motion in models with stochastic dynamics, J. Stat. Phys. 71:1081-1132 (1993).
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Buttà, P., de Masi, A. & Rosatelli, E. Slow Motion and Metastability for a Nonlocal Evolution Equation. Journal of Statistical Physics 112, 709–764 (2003). https://doi.org/10.1023/A:1023832210342
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DOI: https://doi.org/10.1023/A:1023832210342