Stability of invariant manifolds in one and two dimensions

Giovanni Bellettini, Anna De Masi, Nicolas Dirr, and Errico Presutti

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Submission date: 17. Mar. 2006
Pages: 56
published in: Nonlinearity, 20 (2007) 3, p. 537-582 
DOI number (of the published article): 10.1088/0951-7715/20/3/002
Bibtex
MSC-Numbers: 82C05
Keywords and phrases: nonlocal evolution equation, invariant manifold, Ising model with Kac-potential
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Abstract:
We consider the gradient flow associated with a non local free energy functional and extend to such a case results obtained for the Allen-Cahn equation on ``slow motions on invariant manifolds''. The manifolds in question are time-invariant one-dimensional curves in a space which connect the two ground states (interpreted as the pure phases of the system) to the first excited state (interpreted as a diffuse interface). Local and structural stability of the manifolds are proved and applications to the characterization of optimal tunnelling are discussed.