Horizontal patterns from finite speed directional quenching

Rafael Monteiro

doi:10.3934/dcdsb.2018285

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2018, Volume 23, Issue 8: 3503-3534. Doi: 10.3934/dcdsb.2018285

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Horizontal patterns from finite speed directional quenching

Rafael Monteiro

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

Early access: August 21, 2018

Published online: August 21, 2018

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

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Abstract

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 $0 < c < 2$.

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Mathematics Subject Classification: Primary: 35B36, 35B51, 35B50; Secondary: 35J47, 35B40.

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Figure 1. 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)

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Figure 2. Sketches of solutions for horizontal patterns; $\mathcal{H}_{\kappa}$ pattern (left) and $\mathcal{H}_{\infty}$ pattern (right)

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Figure 3. 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)

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Figure 4. Sketch of an unbalanced pattern with a contact angle; see Def. 1.3 or [9]

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Figure 5. 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 $\displaystyle{\lim_{x\to\infty}w(x) = 0}$

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Table 1. 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)

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