# American Institute of Mathematical Sciences

October  2018, 23(8): 3503-3534. doi: 10.3934/dcdsb.2018285

## Horizontal patterns from finite speed directional quenching

 School of Mathematics, University of Minnesota, 206 Church St. S.E., Minneapolis, MN, USA

Current address: Mathematics for Advanced Materials-OIL, AIST-Tohoku University, Sendai, Japan. Email: monteirodasilva-rafael@aist.go.jp

Received  August 2017 Revised  April 2018 Published  October 2018 Early access  August 2018

Fund Project: The author acknowledges partial support through NSF grants DMS-1612441 and DMS-1311740.

In this paper we study the process of phase separation from directional quenching, considered as an externally triggered variation in parameters that changes the system from monostable to bistable across an interface (quenching front); in our case the interface moves with speed $c$ in such a way that the bistable region grows. According to results from [9,10], several patterns exist when $c\underset{\tilde{\ }}{\mathop{>}}\,0$, and here we investigate their persistence for finite $c>0$. We find existence and nonexistence results of multidimensional horizontal stripped patterns, clarifying the selection mechanism relating their existence to the speed $c$ of the quenching front. We further illustrate our results by allying them to those of [9], hence obtaining the existence of a family of single interface patterns displaying different contact angles between their nodal lines and the quenching front; the existence of these patterns was known for small speeds $c> 0$ and here we show that they also exist in the range $0 < c < 2$.

Citation: Rafael Monteiro. Horizontal patterns from finite speed directional quenching. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3503-3534. doi: 10.3934/dcdsb.2018285
##### References:
 [1] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. [2] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. [3] P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, volume 28 of Lecture Notes in Biomathematics, Springer-Verlag, Berlin-New York, 1979. [4] E. M. Foard and A. J. Wagner, Survey of morphologies formed in the wake of an enslaved phase-separation front in two dimensions, Phys. Rev. E, 85 (2012), 011501. doi: 10.1103/PhysRevE.85.011501. [5] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977. [6] J. K. Hale, Ordinary Differential Equations, Robert E. Krieger Publishing Co., Inc., Huntington, N. Y., second edition, 1980. [7] L. Hörmander, The Analysis of Linear Partial Differential Operators. I, volume 256 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, second edition, 1990. Distribution theory and Fourier analysis. doi: 10.1007/978-3-642-61497-2. [8] M. Kolli and M. Schatzman, Approximation of a semilinear elliptic problem in an unbounded domain, ESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique, 37 (2003), 117-132.  doi: 10.1051/m2an:2003017. [9] R. Monteiro and A. Scheel, Contact angle selection for interfaces in growing domain, ZAMM, 98 (2018), 1096-1102, https://arXiv.org/abs/1705.00079. doi: 10.1002/zamm.201700119. [10] R. Monteiro and A. Scheel, Phase separation patterns from directional quenching, Journal of Nonlinear Science, 27 (2017), 1339-1378.  doi: 10.1007/s00332-017-9361-x. [11] Y. Nishiura, Far-from-equilibrium Dynamics, volume 209. American Mathematical Soc., 2002. [12] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV, Analysis of operators, Academic Press, New York-London, 1978. [13] H. L. Royden, Real Analysis, Macmillan Publishing Company, New York, third edition, 1988. [14] S. Thomas, I. Lagzi, F. Molnár and Z. Rácz, Helices in the wake of precipitation fronts, Phys. Rev. E, 88 (2013), 022141. doi: 10.1103/PhysRevE.88.022141. [15] J. M. Vega, Travelling wavefronts of reaction-diffusion equations in cylindrical domains, Comm. Partial Differential Equations, 18 (1993), 505-531.  doi: 10.1080/03605309308820939. [16] J.-L. Wang and H.-F. Li, Traveling wave front for the Fisher equation on an infinite band region, Appl. Math. Lett., 20 (2007), 296-300.  doi: 10.1016/j.aml.2006.04.011. [17] J. Zhu, M. Wilczek, M. Hirtz, J. Hao, W. Wang, H. Fuchs, S. V. Gurevich and L. Chi, Branch suppression and orientation control of Langmuir-Blodgett patterning on prestructured surfaces, Advanced Materials Interfaces, 3 (2016), 1600478. doi: 10.1002/admi.201600478.

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##### References:
 [1] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. [2] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. [3] P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, volume 28 of Lecture Notes in Biomathematics, Springer-Verlag, Berlin-New York, 1979. [4] E. M. Foard and A. J. Wagner, Survey of morphologies formed in the wake of an enslaved phase-separation front in two dimensions, Phys. Rev. E, 85 (2012), 011501. doi: 10.1103/PhysRevE.85.011501. [5] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977. [6] J. K. Hale, Ordinary Differential Equations, Robert E. Krieger Publishing Co., Inc., Huntington, N. Y., second edition, 1980. [7] L. Hörmander, The Analysis of Linear Partial Differential Operators. I, volume 256 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, second edition, 1990. Distribution theory and Fourier analysis. doi: 10.1007/978-3-642-61497-2. [8] M. Kolli and M. Schatzman, Approximation of a semilinear elliptic problem in an unbounded domain, ESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique, 37 (2003), 117-132.  doi: 10.1051/m2an:2003017. [9] R. Monteiro and A. Scheel, Contact angle selection for interfaces in growing domain, ZAMM, 98 (2018), 1096-1102, https://arXiv.org/abs/1705.00079. doi: 10.1002/zamm.201700119. [10] R. Monteiro and A. Scheel, Phase separation patterns from directional quenching, Journal of Nonlinear Science, 27 (2017), 1339-1378.  doi: 10.1007/s00332-017-9361-x. [11] Y. Nishiura, Far-from-equilibrium Dynamics, volume 209. American Mathematical Soc., 2002. [12] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV, Analysis of operators, Academic Press, New York-London, 1978. [13] H. L. Royden, Real Analysis, Macmillan Publishing Company, New York, third edition, 1988. [14] S. Thomas, I. Lagzi, F. Molnár and Z. Rácz, Helices in the wake of precipitation fronts, Phys. Rev. E, 88 (2013), 022141. doi: 10.1103/PhysRevE.88.022141. [15] J. M. Vega, Travelling wavefronts of reaction-diffusion equations in cylindrical domains, Comm. Partial Differential Equations, 18 (1993), 505-531.  doi: 10.1080/03605309308820939. [16] J.-L. Wang and H.-F. Li, Traveling wave front for the Fisher equation on an infinite band region, Appl. Math. Lett., 20 (2007), 296-300.  doi: 10.1016/j.aml.2006.04.011. [17] J. Zhu, M. Wilczek, M. Hirtz, J. Hao, W. Wang, H. Fuchs, S. V. Gurevich and L. Chi, Branch suppression and orientation control of Langmuir-Blodgett patterning on prestructured surfaces, Advanced Materials Interfaces, 3 (2016), 1600478. doi: 10.1002/admi.201600478.
Sketches of solutions for pure phase selection $1 \leadsto 0$; solution $\theta^{(c)}(x)$ (left) and contour plot for $(x, y)\in\mathbb{R}^2$ (right)
Sketches of solutions for horizontal patterns; $\mathcal{H}_{\kappa}$ pattern (left) and $\mathcal{H}_{\infty}$ pattern (right)
Existence diagram for parameters $c\geq0$ (speed of the front) and $\kappa > \pi$ ($y$-periodicity of the patterns); the dashed curve represents the critical case $\mathcal{P}(c, \kappa) = 1$ (see Thm. 1.2)
Sketch of an unbalanced pattern with a contact angle; see Def. 1.3 or [9]
Sketch of solutions to $\partial_x^2w(x) + c\partial_xw(x) + w(x) - w^3(x) = 0$ for $0 < c < 2$ (left) and $c\geq 2$(right) satisfying $\displaystyle{\lim_{x\to-\infty}w(x) = 1}$ and $\displaystyle{\lim_{x\to\infty}w(x) = 0}$
A grasshoppers guide to the existence and nonexistence of patterns $(1\leadsto 0)^{(c)}$ and $\mathcal{H}_{\kappa}$ ($\pi < \kappa \leq \infty$). Critical quantity $\mathcal{P}(c;\kappa)$ defined in (8), where $c$ denotes the speed of the quenching front and $2\kappa$ denotes the $y$-period of the pattern (see Fig. 2)
 $(1\leadsto 0)^{(c)}$ and $\mathcal{H}_{\infty}$ problems $\mathcal{H}_{\kappa}$ problem ($\pi < \kappa < \infty$) $0 \leq c< 2$ $c \geq2$ $\mathcal{P}(c;\kappa)< 1$ $\mathcal{P}(c;\kappa) = 1$ $\mathcal{P}(c;\kappa) >1$ Yes Thms. 1.1 & 1.2 No Thm. 1.1 & Obs. 4.1 Yes Thm. 1.2(ⅰ) Not known No Thm. 1.2(ⅱ)
 $(1\leadsto 0)^{(c)}$ and $\mathcal{H}_{\infty}$ problems $\mathcal{H}_{\kappa}$ problem ($\pi < \kappa < \infty$) $0 \leq c< 2$ $c \geq2$ $\mathcal{P}(c;\kappa)< 1$ $\mathcal{P}(c;\kappa) = 1$ $\mathcal{P}(c;\kappa) >1$ Yes Thms. 1.1 & 1.2 No Thm. 1.1 & Obs. 4.1 Yes Thm. 1.2(ⅰ) Not known No Thm. 1.2(ⅱ)
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