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完整後設資料紀錄
DC 欄位 | 值 | 語言 |
---|---|---|
dc.contributor.advisor | 陳俊全 | |
dc.contributor.author | Hung-Yu Chien | en |
dc.contributor.author | 簡鴻宇 | zh_TW |
dc.date.accessioned | 2021-06-17T07:20:47Z | - |
dc.date.available | 2019-07-24 | |
dc.date.copyright | 2019-07-24 | |
dc.date.issued | 2019 | |
dc.date.submitted | 2019-07-05 | |
dc.identifier.citation | [1] S. Alama, L. Bronsard, and C. Gui. Stationary layered solutions in R2 for an Allen-Cahn system with multiple well potential. Calculus of Variations and Partial Differential Equations, 5(4):359–390, 1997.
[2] F. Alessio, A. Calamai, and P. Montecchiari. Saddle-type solutions for a class of semilinear elliptic equations. Adv. Differential Equations, 12(4):361– 380, 2007. [3] N. D. Alikakos. On the structure of phase transition maps for three or more coexisting phases. In Geometric Partial Differential Equations proceedings, pages 1–31, Pisa, 2013. Scuola Normale Superiore. [4] N. D. Alikakos and G. Fusco. Entire Solutions to Equivariant Elliptic Sys- tems with Variational Structure. Archive for Rational Mechanics and Analysis, 202(15):567–597, 2011. [5] N. D. Alikakos and N. I. Katzourakis. Heteroclinic travelling waves of gra- dient diffusion systems. Transactions of the American Mathematical Society, 363(03):1365–1365, 2011. [6] S. M. Allen and J. W. Cahn. A Microscopic Theory for Antiphase Boundary Motion and Its Application to Antiphase Domain Coarsening. Acta Metallur- gica, 27(6):1085–1095, 1975. [7] S. Baldo. Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids. Annales de l’I.H.P. Analyse non linéaire, 7(2):67–90, 1990. [8] F. Béthuel, H. Brezis, and F. Hélein. Ginzburg-Landau Vortices. Birkhäuser Boston, 1994. [9] F. Béthuel and G. Orlandi. Ginzburg-Landau functionals, phase transitions and vorticity. In Noncompact problems at the intersection of geometry, analysis, and topology, pages 35–47. American Mathematical Society, Providence, RI, 2004. [10] A. Braides. Gamma Convergence for Beginners. Oxford University Press, 2002. [11] L. Bronsard, C. Gui, and M. Schatzman. A three-layered minimizer in R2 for a variational problem with a symmetric three-well potential. Communications on Pure and Applied Mathematics, 49(7):677–715, 1996. [12] L. Bronsard and F. Reitich. On three-phase boundary motion and the singu- lar limit of a vector-valued Ginzburg-Landau equation. Archive for Rational Mechanics and Analysis, 124(4):355–379, 1993. [13] J. W. Cahn and J. E. Hilliard. Free Energy of a Nonuniform System. I. Inter- facial Free Energy. The Journal of Chemical Physics, 28, 1958. [14] C.-N. Chen, C.-C. Chen, and C.-C. Huang. Traveling waves for the FitzHugh–Nagumo system on an infinite channel. Journal of Differential Equa- tions, 261(6):3010–3041, 2016. [15] C.-N. Chen and Y. S. Choi. Traveling pulse solutions to FitzHugh–Nagumo equations. Calculus of Variations and Partial Differential Equations, 54(1):1–45, 2015. [16] X. Chen. Generation and propagation of interfaces for reaction-diffusion equations. Journal of Differential Equations, 96(1):116–141, 1992. [17] X. Chen, J. S. Guo, F. Hamel, H. Ninomiya, and J. M. Roquejoffre. Trav- eling waves with paraboloid like interfaces for balanced bistable dynamics. Annales de l’Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis, 24(3):369–393, 2007. [18] H. Dang, P. C. Fife, and L. A. Peletier. Saddle solutions of the bistable dif- fusion equation. Zeitschrift für angewandte Mathematik und Physik ZAMP, 43(6):984–998, 1992. [19] E. De Giorgi. Converence problems for functional and operators. In Ennio De Giorgi Selected Papers, pages 487–516. Springer, 2006. [20] E. De Giorgi and T. Franzoni. On a type of variational convergence. In Ennio De Giorgi Selected Papers, pages 517–526. Springer, 2006. [21] P. de Mottoni and M. Schatzman. Geometrical Evolution of Developed In- terfaces. Transactions of the American Mathematical Society, 347(5):1533–1589, 1995. [22] P. C. Fife and J. B. McLeod. The approach of solutions of nonlinear diffu- sion equations to travelling front solutions. Archive for Rational Mechanics and Analysis, 65(4):335–361, 1975. [23] D. Gilbarg and N. S. Trudinger. Ellitic Partial Differential Equations of Second Order. Springer, 1998 edition, 2001. [24] C. Gui and M. Schatzman. Symmetric quadruple phase transitions. Indiana University Mathematics Journal, 57(2):781–836, 2008. [25] M. E. Gurtin. On phase transitions with bulk, interfacial, and boundary En- ergy. Archive for Rational Mechanics and Analysis, 96(3):243–264, 1986. [26] M. E. Gurtin and H. Matano. On the structure of equilibrium phase transi- tions within the gradient theory of fluids. Quarterly of Applied Mathematics, 46:301 – 317, 1988. [27] F. Hamel. Bistable transition fronts in RN. Advances in Mathematics, 289:279– 344, 2016. [28] F. Hamel, R. Monneau, and J.-M. Roquejoffre. Existence and qualitative properties of multidimensional conical bistable fronts. Discrete and Continu- ous Dynamical Systems, 13(4):1069–1096, 2005. [29] S. Heinze. A Variational Approach to Traveling Waves. Technical Report 85, Max Planck Institute for Mathematical Sciences, 2001. [30] R. Kohn and P. Sternberg. Local minimisers and singular perturbations. Pro- ceedings of the Royal Society of Edinburgh, 111A:69–84, 1989. [31] C.D.LevermoreandJ.X.Xin.MultidimensionalStabilityofTravelingWaves in a Bistable Reaction-Diffusion Equation, II. Communications in Partial Dif- ferential Equations, 17(11-12):1901–1924, 1992. [32] M. Lucia, C. B. Muratov, and M. Novaga. Existence of Traveling Waves of Invasion for Ginzburg-Landau-type Problems in Infinite Cylinders. Archive for Rational Mechanics and Analysis, 188(3):475–508, 2008. [33] F. Maggi. Sets of Finite Perimeter and Geometric Variational Problems:An Intro- duction to Geometric Measure Theory. Cambridge University Press, 2012. [34] H. Matano, M. Nara, and M. Taniguchi. Stability of Planar Waves in the Allen-Cahn Equation. Communications in Partial Differential Equations, 34(9):976–1002, 2009. [35] L.Modica.Thegradienttheoryofphasetransitionsandtheminimalinterface criterion. Arch. Ration. Mech. Anal., 98:123–142, 1987. [36] Y. Morita and H. Ninomiya. Monostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space. Bulletin of the Institute of Mathematics Academia Sinica (New Series), 3(4):567–584, 2008. [37] Y. Morita and H. Ninomiya. Traveling wave solutions and entire solutions of reaction-diffusion equations. Sugaku Expositions, 23:213–233, 2010. [38] C. B. Muratov. A global variational structure and propagation of distur- bances in reaction-diffusion systems of gradient type. Discrete and Continuous Dynamical Systems - Series B, 4:867 – 892, 2004. [39] H. Ninomiya and M. Taniguchi. Existence and global stability of traveling curved fronts in the Allen-Cahn equations. Journal of Differential Equations, 213(1):204–233, 2005. [40] R. S. Palais. The principle of symmetric criticality. Communications in Mathe- matical Physics, 69(1):19–30, 1979. [41] P. Sternberg. Vector-Valued Local Minimizers of Nonconvex Variational Problems. Rocky Mountain Journal of Mathematics, 21(2):799–807, 1991. [42] P. Sternberg and W. P. Zeimer. Local minimisers of a three-phase partition problem with triple junctions. Proc. Roy. Soc. Edinburgh Sect. A, 124(6):1059– 1073, 1994. [43] M. Taniguchi. Traveling Fronts of Pyramidal Shapes in the Allen–Cahn Equations. SIAM Journal on Mathematical Analysis, 39(1):319–344, 2007. [44] M. Taniguchi. The uniqueness and asymptotic stability of pyramidal trav- eling fronts in the Allen–Cahn equations. Journal of Differential Equations, 246(5):2103–2130, 2009. [45] M. Taniguchi. An (N − 1)-Dimensional Convex Compact Set Gives an N- Dimensional Traveling Front in the Allen-Cahn Equation. SIAM Journal on Mathematical Analysis, 47(1):455–476, 2015. [46] M. Taniguchi. Convex compact sets in RN−1 give traveling fronts of cooperation–diffusion systems in RN. Journal of Differential Equations, 260(5):4301–4338, 2016. [47] M. S. Trumper. Existence of a solution to a vector-valued ginzburg-landau equation with a three well potential. Indiana University Mathematics Journal, 58(1):213–267, 2009. | |
dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/73172 | - |
dc.description.abstract | 針對 Allen-Cahn 類型的反應擴散系統,在高維度空間中我們好奇其解的非平 面結構以及多相性。我們目標是在無界區域如全平面或者⻑條狀定義域上找到此 類系統的駐波 (standing wave) 以及行波 (traveling wave) 解。
我們有兩種嘗試,第一種是利用 Γ-收斂理論,將方程式的解寫成一系列變分 問題的解,並討論其收斂性。我們假設了系統的勢 (potential) 滿足一種簡單的對 稱不變性 — 這個條件比過去部分文獻上假設的在特定對稱群下的不變性還要弱, 以及考慮勢能的低點同時也是系統的常數穩定態 (constant equilibrium),其中相 對稱的兩點之間有唯一的相變穩定解 (stationary phase transition, 連接此兩個相 的一維方程式穩定解)。最後可以得到一個全平面上的穩定解,而前述的唯一條件 保證了最後得到的解不是常數的退化解。 另外考慮同樣的假設,並多假設了此方程之勢能中的單一相點附近是嚴格上 凹,則可以證明存在一「三相行波解」,連結此「單一相點」與前述「兩點間相變 函數的近似解」。最後根據這個解,我們考慮將⻑條狀定義域拉寬成全平面,相對 應到的解也會存在子序列收斂至一全平面解,由對行波速度的估計可以得知速度 會隨著寬度遞減至零,可知最後收斂得到的全平面解亦是一穩定解。 | zh_TW |
dc.description.abstract | In this paper we aim to find standing and traveling wave solutions, i.e. w(z, y) = u(x, y, t) with z = x − ct, to the reaction-diffusion gradient system with a triple- well potential ∂tu = ∆u − ∇W(u) on an entire domain R2 or a cylindrical domain R×(−l,l).
Firstly by the theory of Γ-convergence, standing wave solutions (i.e. stationary solutions) are obtained under a condition that the potential W is invariant under a simple reflection. This symmetry assumption is weaker than the invariance under a general symmetric group, which is assumed by some literatures. And also, under the same condition on symmetry, via a variational method, we can show the existence of a traveling wave solution that connects the three constant equilibria in an approximate sense on a cylindrical domain. We propose a convexity condition on one of the equilibria of W to ensure the asymptotical convergence to this equilibrium of the traveling wave solutions at z = −∞. | en |
dc.description.provenance | Made available in DSpace on 2021-06-17T07:20:47Z (GMT). No. of bitstreams: 1 ntu-108-D01221005-1.pdf: 2565593 bytes, checksum: 4e1894af694287635e4900003646dfd3 (MD5) Previous issue date: 2019 | en |
dc.description.tableofcontents | 口試委員審定書 i
誌謝 ii 中文摘要 iii 英文摘要 iv 1 Introduction 1 2 Construction of a Non-Planar Entire Stationary Solution by the Theory of Γ-Convergence 10 2.1 The Concepts of Γ-Convergence and a Kind of Local Minimizers to 3-PhasePartitionProblem 11 2.2 Construction of an Isolated Local Minimizer for Our Need 15 2.3 Existence of a Non-Trivial Stationary Solution on R2 Obtained by Γ-Convergence 22 3 The Existence of 1-D Stationary Solutions that Approximate the b1-b2 Phase Transition 25 4 The Existence of a Traveling Wave Solution on the Infinite Strip Dl 37 4.1 The Modified Formulation of the Variational Structure 38 4.2 Determine the Traveling Wave Speed 44 4.3 Minimizer for the Constrained Problem 46 4.4 Proof of Theorem 1.3 51 4.5 Proof of Corollary1.5 56 5 An Example of a Potential Satisfying (H1) to (H4) 57 Reference 60 | |
dc.language.iso | en | |
dc.title | 具三井位能之反應擴散梯度系統的三相行波解及穩定解之變分研究 | zh_TW |
dc.title | Variational Approaches to Three-Phase Standing and Traveling Waves of Reaction-Diffusion-Gradient Systems with a Triple-Well Potential | en |
dc.type | Thesis | |
dc.date.schoolyear | 107-2 | |
dc.description.degree | 博士 | |
dc.contributor.oralexamcommittee | 崔茂培,郭忠勝,陳兆年,陳建隆,王振男 | |
dc.subject.keyword | 反應擴散方程,行波解,三相行波解,伽馬收斂, | zh_TW |
dc.subject.keyword | reaction diffusion system,3-phase traveling wave solution,Gamma convergence, | en |
dc.relation.page | 65 | |
dc.identifier.doi | 10.6342/NTU201901241 | |
dc.rights.note | 有償授權 | |
dc.date.accepted | 2019-07-05 | |
dc.contributor.author-college | 理學院 | zh_TW |
dc.contributor.author-dept | 數學研究所 | zh_TW |
顯示於系所單位: | 數學系 |
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