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Structured phase transitions on a finite interval. (English) Zbl 0564.76075
Consider a mass M of a compressible van der Waals fluid, i.e. a fluid whose free energy per unit volume E depends on density u and its gradient u’ as \(E=W(u)+\epsilon^ 2u'{}^ 2\) and W’(u) has the well-known undulating dependence on u. Consider the fluid contained in a, say, cylindrical vessel of length 2L. Consider density distributions u(x) depending only on the axial variable x and respecting the constraint of conservation of mass. Then among these distributions there are only two (specular) minimizers of the total free energy and they are monotonic (single-interface solutions), provided that the average density M/2L is contained within the Maxwell interval and \(\epsilon\) is sufficiently small.
The proof of this theorem of uniqueness (modulo reversals) is difficult and long (it extends well over 20 pages); it is achieved by transforming the associated Euler problem (with constraint) into a set of two integral equations, the study of which is delicate because of singularities present.
The result is an important contribution to the study of phase transitions within a finite domain.
Reviewer: G.Capriz

MSC:
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
49J27 Existence theories for problems in abstract spaces
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