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Stable patterns with jump discontinuity in systems with Turing instability and hysteresis
Rotationally symmetric solutions to the Cahn-Hilliard equation
1. | Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile |
2. | Departamento de Ingeniería Matemática and Centro, de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile |
This paper is devoted to construction of new solutions to the Cahn-Hilliard equation in $\mathbb R^d$. Staring from the Delaunay unduloid $D_τ$ with parameter $τ∈ (0,τ^*)$ we find for each sufficiently small $\varepsilon $ a solution $u$ of this equation which is periodic in the direction of the $x_d$ axis and rotationally symmetric with respect to rotations about this axis. The zero level set of $u$ approaches as $\varepsilon \to 0$ the surface $D_τ$. We use a refined version of the Lyapunov-Schmidt reduction method which simplifies very technical aspects of previous constructions for similar problems.
References:
[1] |
N. D. Alikakos, P. W. Bates and X. Chen,
Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rational Mech. Anal., 128 (1994), 165-205.
doi: 10.1007/BF00375025. |
[2] |
N. D. Alikakos, X. Chen and G. Fusco,
Motion of a droplet by surface tension along the boundary, Calc. Var. Partial Differential Equations, 11 (2000), 233-305.
doi: 10.1007/s005260000052. |
[3] |
N.D. Alikakos and G. Fusco,
The spectrum of the Cahn-Hilliard operator for generic interface in higher space dimensions, Indiana Univ. Math. J., 42 (1993), 637-674.
doi: 10.1512/iumj.1993.42.42028. |
[4] |
N.D. Alikakos and G. Fusco,
Slow dynamics for the Cahn-Hilliard equation in higher space dimensions: The motion of bubbles, Arch. Rational Mech. Anal., 141 (1998), 1-61.
doi: 10.1007/s002050050072. |
[5] |
N.D. Alikakos, G. Fusco and V. Stefanopoulos,
Critical spectrum and stability of interfaces for a class of reaction-diffusion equations, J. Differential Equations, 126 (1996), 106-167.
doi: 10.1006/jdeq.1996.0046. |
[6] |
P.W. Bates and G. Fusco,
Equilibria with many nuclei for the Cahn-Hilliard equation, J. Differential Equations, 160 (2000), 283-356.
doi: 10.1006/jdeq.1999.3660. |
[7] |
X. Chen,
Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces, Comm. Partial Differential Equations, 19 (1994), 1371-1395.
doi: 10.1080/03605309408821057. |
[8] |
X. Chen and M. Kowalczyk,
Existence of equilibria for the Cahn-Hilliard equation via local minimizers of the perimeter, Comm. Partial Differential Equations, 21 (1996), 1207-1233.
doi: 10.1080/03605309608821223. |
[9] |
S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, volume 251 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], SpringerVerlag, New York-Berlin, 1982. Google Scholar |
[10] |
H. Dang, P.C. Fife and L.A. Peletier,
Saddle solutions of the bistable diffusion equation, Z. Angew. Math. Phys., 43 (1992), 984-998.
doi: 10.1007/BF00916424. |
[11] |
M. del Pino, M. Kowalczyk, F. Pacard and J. Wei,
The Toda system and multiple-end solutions of autonomous planar elliptic problems, Adv. Math., 224 (2010), 1462-1516.
doi: 10.1016/j.aim.2010.01.003. |
[12] |
M. del Pino, M. Kowalczyk and J. Wei,
On De Giorgi's conjecture in dimension N ≥ 9, Ann. of Math. (2)(7), 174 (2011), 1485-1569.
doi: 10.4007/annals.2011.174.3.3. |
[13] |
M. del Pino, M. Kowalczyk and J. Wei,
Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature in ${{\mathbb{R}}^{3}}$, Journ. Diff. Geometry, 93 (2013), 67-131.
|
[14] |
M. del Pino, M. Kowalczyk and J. Wei,
Traveling waves with multiple and non-convex fronts for a bistable semilinear parabolic equation, Comm. Pure Appl. Math., 66 (2013), 481-547.
doi: 10.1002/cpa.21438. |
[15] |
M. del Pino, F. Pacard and M. Musso,
Solutions of the Allen-Cahn equation which are invariant under screw-motion, Manuscripta Math., 138 (2012), 273-286.
doi: 10.1007/s00229-011-0492-3. |
[16] |
C. Delaunay, Sur la surface de revolution dont la courbure moyenna est constante, J. Math. Pures Appl., 6 (1841), 309-320. Google Scholar |
[17] |
J. Eells,
The surfaces of Delaunay, Math. Intelligencer, 9 (1987), 53-57.
doi: 10.1007/BF03023575. |
[18] |
P. C. Fife, Models for phase separation and their mathematics, Electron. J. Differential Equations, 2000, pages No. 48, 26 pp. (electronic). Google Scholar |
[19] |
P.C. Fife,
Pattern formation in gradient systems, In Handbook of dynamical systems, NorthHolland, Amsterdam, 2 (2002), 677-722.
doi: 10.1016/S1874-575X(02)80034-0. |
[20] |
Á. Hernández and M. Kowalczyk, Delaunay end solutions of the cahn-hilliard equation in, in ´ preparation. Google Scholar |
[21] |
W.-y. Hsiang and W.C. Yu,
A generalization of a theorem of Delaunay, J. Differential Geom., 16 (1981), 161-177.
|
[22] |
J.E. Hutchinson and Y. Tonegawa,
Convergence of phase interfaces in the van der {W}aals-Cahn-Hilliard theory, Calc. Var. Partial Differential Equations, 10 (2000), 49-84.
doi: 10.1007/PL00013453. |
[23] |
M. Jleli,
End-to-end gluing of constant mean curvature hypersurfaces, Ann. Fac. Sci. Toulouse Math. (6), 18 (2009), 717-737.
doi: 10.5802/afst.1222. |
[24] |
M. Jleli and F. Pacard,
An end-to-end construction for compact constant mean curvature surfaces, Pacific J. Math., 221 (2005), 81-108.
doi: 10.2140/pjm.2005.221.81. |
[25] |
R.V. Kohn and P. Sternberg,
Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 69-84.
doi: 10.1017/S0308210500025026. |
[26] |
R. Mazzeo and F. Pacard, Bifurcating nodoids, In Topology and geometry: Commemorating SISTAG, volume 314 of Contemp. Math. , pages 169-186. Amer. Math. Soc. , Providence, RI, 2002. Google Scholar |
[27] |
L. Modica,
The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142.
doi: 10.1007/BF00251230. |
[28] |
F. Pacard and M. Ritoré,
From constant mean curvature hypersurfaces to the gradient theory of phase transitions, J. Differential Geom., 64 (2003), 359-423.
|
[29] |
F. Pacard and J. Wei,
Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones, J. Funct. Anal., 264 (2013), 1131-1167.
doi: 10.1016/j.jfa.2012.03.010. |
[30] |
L.A. Peletier and J. Serrin,
Uniqueness of positive solutions of semilinear equations in $\textbf{R}^{n}$, Arch. Rational Mech. Anal., 81 (1983), 181-197.
doi: 10.1007/BF00250651. |
[31] |
P. Sternberg,
The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260.
doi: 10.1007/BF00253122. |
[32] |
J. Wei and M. Winter,
On the stationary Cahn-Hilliard equation: Bubble solutions, SIAM J. Math. Anal., 29 (1998), 1492-1518 (electronic).
doi: 10.1137/S0036141097320663. |
[33] |
J. Wei and M. Winter,
Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 459-492.
doi: 10.1016/S0294-1449(98)80031-0. |
show all references
References:
[1] |
N. D. Alikakos, P. W. Bates and X. Chen,
Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rational Mech. Anal., 128 (1994), 165-205.
doi: 10.1007/BF00375025. |
[2] |
N. D. Alikakos, X. Chen and G. Fusco,
Motion of a droplet by surface tension along the boundary, Calc. Var. Partial Differential Equations, 11 (2000), 233-305.
doi: 10.1007/s005260000052. |
[3] |
N.D. Alikakos and G. Fusco,
The spectrum of the Cahn-Hilliard operator for generic interface in higher space dimensions, Indiana Univ. Math. J., 42 (1993), 637-674.
doi: 10.1512/iumj.1993.42.42028. |
[4] |
N.D. Alikakos and G. Fusco,
Slow dynamics for the Cahn-Hilliard equation in higher space dimensions: The motion of bubbles, Arch. Rational Mech. Anal., 141 (1998), 1-61.
doi: 10.1007/s002050050072. |
[5] |
N.D. Alikakos, G. Fusco and V. Stefanopoulos,
Critical spectrum and stability of interfaces for a class of reaction-diffusion equations, J. Differential Equations, 126 (1996), 106-167.
doi: 10.1006/jdeq.1996.0046. |
[6] |
P.W. Bates and G. Fusco,
Equilibria with many nuclei for the Cahn-Hilliard equation, J. Differential Equations, 160 (2000), 283-356.
doi: 10.1006/jdeq.1999.3660. |
[7] |
X. Chen,
Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces, Comm. Partial Differential Equations, 19 (1994), 1371-1395.
doi: 10.1080/03605309408821057. |
[8] |
X. Chen and M. Kowalczyk,
Existence of equilibria for the Cahn-Hilliard equation via local minimizers of the perimeter, Comm. Partial Differential Equations, 21 (1996), 1207-1233.
doi: 10.1080/03605309608821223. |
[9] |
S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, volume 251 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], SpringerVerlag, New York-Berlin, 1982. Google Scholar |
[10] |
H. Dang, P.C. Fife and L.A. Peletier,
Saddle solutions of the bistable diffusion equation, Z. Angew. Math. Phys., 43 (1992), 984-998.
doi: 10.1007/BF00916424. |
[11] |
M. del Pino, M. Kowalczyk, F. Pacard and J. Wei,
The Toda system and multiple-end solutions of autonomous planar elliptic problems, Adv. Math., 224 (2010), 1462-1516.
doi: 10.1016/j.aim.2010.01.003. |
[12] |
M. del Pino, M. Kowalczyk and J. Wei,
On De Giorgi's conjecture in dimension N ≥ 9, Ann. of Math. (2)(7), 174 (2011), 1485-1569.
doi: 10.4007/annals.2011.174.3.3. |
[13] |
M. del Pino, M. Kowalczyk and J. Wei,
Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature in ${{\mathbb{R}}^{3}}$, Journ. Diff. Geometry, 93 (2013), 67-131.
|
[14] |
M. del Pino, M. Kowalczyk and J. Wei,
Traveling waves with multiple and non-convex fronts for a bistable semilinear parabolic equation, Comm. Pure Appl. Math., 66 (2013), 481-547.
doi: 10.1002/cpa.21438. |
[15] |
M. del Pino, F. Pacard and M. Musso,
Solutions of the Allen-Cahn equation which are invariant under screw-motion, Manuscripta Math., 138 (2012), 273-286.
doi: 10.1007/s00229-011-0492-3. |
[16] |
C. Delaunay, Sur la surface de revolution dont la courbure moyenna est constante, J. Math. Pures Appl., 6 (1841), 309-320. Google Scholar |
[17] |
J. Eells,
The surfaces of Delaunay, Math. Intelligencer, 9 (1987), 53-57.
doi: 10.1007/BF03023575. |
[18] |
P. C. Fife, Models for phase separation and their mathematics, Electron. J. Differential Equations, 2000, pages No. 48, 26 pp. (electronic). Google Scholar |
[19] |
P.C. Fife,
Pattern formation in gradient systems, In Handbook of dynamical systems, NorthHolland, Amsterdam, 2 (2002), 677-722.
doi: 10.1016/S1874-575X(02)80034-0. |
[20] |
Á. Hernández and M. Kowalczyk, Delaunay end solutions of the cahn-hilliard equation in, in ´ preparation. Google Scholar |
[21] |
W.-y. Hsiang and W.C. Yu,
A generalization of a theorem of Delaunay, J. Differential Geom., 16 (1981), 161-177.
|
[22] |
J.E. Hutchinson and Y. Tonegawa,
Convergence of phase interfaces in the van der {W}aals-Cahn-Hilliard theory, Calc. Var. Partial Differential Equations, 10 (2000), 49-84.
doi: 10.1007/PL00013453. |
[23] |
M. Jleli,
End-to-end gluing of constant mean curvature hypersurfaces, Ann. Fac. Sci. Toulouse Math. (6), 18 (2009), 717-737.
doi: 10.5802/afst.1222. |
[24] |
M. Jleli and F. Pacard,
An end-to-end construction for compact constant mean curvature surfaces, Pacific J. Math., 221 (2005), 81-108.
doi: 10.2140/pjm.2005.221.81. |
[25] |
R.V. Kohn and P. Sternberg,
Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 69-84.
doi: 10.1017/S0308210500025026. |
[26] |
R. Mazzeo and F. Pacard, Bifurcating nodoids, In Topology and geometry: Commemorating SISTAG, volume 314 of Contemp. Math. , pages 169-186. Amer. Math. Soc. , Providence, RI, 2002. Google Scholar |
[27] |
L. Modica,
The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142.
doi: 10.1007/BF00251230. |
[28] |
F. Pacard and M. Ritoré,
From constant mean curvature hypersurfaces to the gradient theory of phase transitions, J. Differential Geom., 64 (2003), 359-423.
|
[29] |
F. Pacard and J. Wei,
Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones, J. Funct. Anal., 264 (2013), 1131-1167.
doi: 10.1016/j.jfa.2012.03.010. |
[30] |
L.A. Peletier and J. Serrin,
Uniqueness of positive solutions of semilinear equations in $\textbf{R}^{n}$, Arch. Rational Mech. Anal., 81 (1983), 181-197.
doi: 10.1007/BF00250651. |
[31] |
P. Sternberg,
The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260.
doi: 10.1007/BF00253122. |
[32] |
J. Wei and M. Winter,
On the stationary Cahn-Hilliard equation: Bubble solutions, SIAM J. Math. Anal., 29 (1998), 1492-1518 (electronic).
doi: 10.1137/S0036141097320663. |
[33] |
J. Wei and M. Winter,
Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 459-492.
doi: 10.1016/S0294-1449(98)80031-0. |
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