Abstract
We study the effect of directional quenching on patterns formed in simple bistable systems such as the Allen–Cahn and the Cahn–Hilliard equation on the plane. We model directional quenching as an externally triggered change in system parameters, changing the system from monostable to bistable across a trigger line. We are then interested in patterns forming in the bistable region, in particular as the trigger progresses with small speed and increases this bistable region. We find existence and nonexistence results of single interfaces and striped patterns. For zero speed, we find stripes parallel or perpendicular to the trigger line and exclude stripes with an oblique orientation. Single interfaces are always perpendicular to the trigger line. For small positive speed, striped patterns can align perpendicularly. Other orientations are excluded in Allen–Cahn for all nonnegative speeds. Single interfaces for positive trigger speeds are excluded for Cahn–Hilliard and align perpendicularly in Allen–Cahn.
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Notes
Notice that this is not direct, since this function does not solve the PDE in the classical sense and distributions are applied to the space of smooth compactly supported functions. However, we know that distributions with finite order (say, order k) can be extended to the space of \({\mathscr {C}}_0^k\) functions (cf. Hörmander 1990, §2).
Monotonicity in y is restricted to \(y\in [0,\kappa /2]\).
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Acknowledgements
R.M and A.S. are grateful to the University of Münster, Germany, where part of this work is carried out. R.M. also would like to thank Itsván Lagzi and Zoltán Rácz from Ëotvos University, Hungary, for stimulating discussions. R.M. acknowledges financial support through a DAAD Research Grant. A.S. acknowledges partial support through NSF Grants DMS-1612441 and DMS-1311740.
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Communicated by Mary Silber.
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Monteiro, R., Scheel, A. Phase Separation Patterns from Directional Quenching. J Nonlinear Sci 27, 1339–1378 (2017). https://doi.org/10.1007/s00332-017-9361-x
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DOI: https://doi.org/10.1007/s00332-017-9361-x