Stability of invariant manifolds in one and two dimensions

Reference:

Bellettini, G., De Masi, A., Dirr, N. and Presutti, E., 2007. Stability of invariant manifolds in one and two dimensions. Nonlinearity, 20 (3), pp. 537-582.

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Abstract

We consider the gradient flow associated with a nonlocal 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 an L-2 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.

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