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  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 & 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.  Google Scholar

[2]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.  Google Scholar

[3]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, volume 28 of Lecture Notes in Biomathematics, Springer-Verlag, Berlin-New York, 1979.  Google Scholar

[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.  Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[6]

J. K. Hale, Ordinary Differential Equations, Robert E. Krieger Publishing Co., Inc., Huntington, N. Y., second edition, 1980.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Y. Nishiura, Far-from-equilibrium Dynamics, volume 209. American Mathematical Soc., 2002.  Google Scholar

[12]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV, Analysis of operators, Academic Press, New York-London, 1978.  Google Scholar

[13]

H. L. Royden, Real Analysis, Macmillan Publishing Company, New York, third edition, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.  Google Scholar

[2]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.  Google Scholar

[3]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, volume 28 of Lecture Notes in Biomathematics, Springer-Verlag, Berlin-New York, 1979.  Google Scholar

[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.  Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[6]

J. K. Hale, Ordinary Differential Equations, Robert E. Krieger Publishing Co., Inc., Huntington, N. Y., second edition, 1980.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Y. Nishiura, Far-from-equilibrium Dynamics, volume 209. American Mathematical Soc., 2002.  Google Scholar

[12]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV, Analysis of operators, Academic Press, New York-London, 1978.  Google Scholar

[13]

H. L. Royden, Real Analysis, Macmillan Publishing Company, New York, third edition, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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)
Figure 2.  Sketches of solutions for horizontal patterns; $\mathcal{H}_{\kappa}$ pattern (left) and $\mathcal{H}_{\infty}$ pattern (right)
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)
Figure 4.  Sketch of an unbalanced pattern with a contact angle; see Def. 1.3 or [9]
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 $\displaystyle{\lim_{x\to\infty}w(x) = 0}$
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)
$(1\leadsto 0)^{(c)}$ 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]

Jun Yang, Xiaolin Yang. Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains. Communications on Pure & Applied Analysis, 2013, 12 (1) : 303-340. doi: 10.3934/cpaa.2013.12.303

[2]

Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703

[3]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127

[4]

Christopher P. Grant. Grain sizes in the discrete Allen-Cahn and Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 127-146. doi: 10.3934/dcds.2001.7.127

[5]

Jie Shen, Xiaofeng Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1669-1691. doi: 10.3934/dcds.2010.28.1669

[6]

Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301

[7]

Alain Miranville, Wafa Saoud, Raafat Talhouk. On the Cahn-Hilliard/Allen-Cahn equations with singular potentials. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3633-3651. doi: 10.3934/dcdsb.2018308

[8]

Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015

[9]

Giorgio Fusco. On some elementary properties of vector minimizers of the Allen-Cahn energy. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1045-1060. doi: 10.3934/cpaa.2014.13.1045

[10]

Yan Hu. Layer solutions for an Allen-Cahn type system driven by the fractional Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 947-964. doi: 10.3934/cpaa.2016.15.947

[11]

Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679

[12]

Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure & Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577

[13]

Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319

[14]

Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009

[15]

Eleonora Cinti. Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 441-463. doi: 10.3934/dcdss.2018024

[16]

Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407

[17]

Hirokazu Ninomiya, Masaharu Taniguchi. Global stability of traveling curved fronts in the Allen-Cahn equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 819-832. doi: 10.3934/dcds.2006.15.819

[18]

Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4169-4190. doi: 10.3934/dcdsb.2019077

[19]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012

[20]

Irena Pawłow. Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1169-1191. doi: 10.3934/dcds.2006.15.1169

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (43)
  • HTML views (61)
  • Cited by (0)

Other articles
by authors

[Back to Top]