Unit:
Κατεύθυνση Εφαρμοσμένα ΜαθηματικάLibrary of the School of Science
Supervisors info:
Αλικάκος Νικόλαος Καθηγητής Ε.Κ.Π.Α.
Original Title:
Μελέτη Δυναμικών Συστημάτων-Η κινητική συμπεριφορά μιας φυσαλίδας(bubble) προς το σύνορο για την Cahn-Hilliard εξίσωση.
Summary:
In this thesis is the study of solutions of the Cahn-Hilliard equation
exhibiting slow motion and are called bubble solutions.
Specifically, we consider the dynamics of these solutions are spherical
surfaces, that are moving too slowly toward the border without changing their
shape. Turns on the Cahn-Hilliard equation, that the bubble entrainment to the
closest point on the boundary, with the condition that is very small. Similar
behaviour is observed in non-local Allen-Cahn equation when we maintain mass.
The company seems to be drifting exponentially slowly to nearest point on the
boundary of the field without changing shape.
This is why the overall evolution takes place so that the free energy can be
monotonous to t, and specifically applies.
In the case of non-local Allen-Cahn with conservation of mass, the company sees
only the nearest point on the boundary and directed toward this following
section the minimum distance. In the case of the Cahn-Hilliard equation,
solution-company interacts with the whole border and moves towards it by
following a path that depends on the entire boundary and drastically changes
the size of the bubble.
Only bubbles very small size under Cahn-Hilliard forcefully moved to the
section with the shortest distance as in the case of non-local Allen-Cahn.
Minimise energy consumption in a given time, the company will have to cross the
border at right angles and should be asymptotically near the arc of the circle
which encloses the required mass. Then the company will move in quick time in
the direction where the magnitude of the curvature increases at most. This
process reduces the surface energy until you reach a local minimum. a minimum
likely to reach near the boundary where the curvature is a local maximum. If
the border contains some sections where the curvature is constant, the bubble
may collapse along the boundary and need further asymptotic analysis
(asymptotic metastability analysis).
Keywords:
Bubble, Slow motion, Boundary, Doubble well potential, Binary alloy
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