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eling aspects and the analytical properties of the Allen–Cahn and other phase-field equations we refer the reader to the textbook [7] and the articles [1, 2, 4, 6, 10, 11]. The Allen–Cahn equation is the L2-gradient flow of the functional Iε(u) = 1 2 Ω |∇u|2 dx +ε−2 Ω F(u)dx.
- Sören Bartels
- 2015
The Allen–Cahn equation (after John W. Cahn and Sam Allen) is a reaction–diffusion equation of mathematical physics which describes the process of phase separation in multi-component alloy systems, including order-disorder transitions.
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- Existence and Regularity
- Orem 6.1
- Proof
- Arks 6.1
- Position 6.1
- Orem 6.2
- Stability Estimates
- Orem 6.3
- Ark 6.2
- Position 6.2
The existence of a unique solution ufollows, e.g., from a discretization in time and a subsequent passage to a limit.
(Existence) For every u_0\in L^2(\varOmega ) and T>0 there exists a weak solution u\in H^1({[0,T]};H^1(\varOmega )')\cap L^2({[0,T]};H^1(\varOmega )) that satisfies u(0)=u_0and for almost every t\in {[0,T]} and every v\in H^1(\varOmega ). If u_0\in H^1(\varOmega ), then we have u\in H^1({[0,T]};L^2(\varOmega )) \cap L^\infty ({[0,T]};H^1(\varOmega ...
The existence of a solution follows from an implicit discretization in time that leads to a sequence of well-posed minimization problems. Straightforward a-priori bounds, together with compact embeddings, then show the existence of a weak limit that solves the weak formulation. If u_0\in H^1(\varOmega ), then we may formally choose v=\partial _t ut...
(i) Stationary states for the Allen–Cahn equation are the constant functions u\equiv \pm 1 and u\equiv 0. The state u\equiv 0is unstable. (ii) For \varOmega =\mathbb {R}d a stationary solution is given by u(x) = \tanh (x\cdot a/ (\sqrt{2}\varepsilon )) for all x\in \mathbb {R}d and an arbitrary vector a\in \mathbb {R}d. This characterizes the profi...
(Maximum principle and uniqueness) If u is a weak solution of the Allen–Cahn equation and |u_0(x)|\le 1 for almost every x\in \varOmega , then |u(t,x)|\le 1 for almost every (t,x)\in {[0,T]}\times \varOmega . Solutions with this property are unique.
(Regularity) If the Laplace operator is H^2 regular in \varOmega and u_0\in H^1(\varOmega ), then u\in L^\infty ({[0,T]};H^2(\varOmega ))\cap H^2({[0,T]};H^1(\varOmega )')\cap H^1({[0,T]};H^2(\varOmega )) and there exists \sigma \ge 0such that If I_\varepsilon (u_0)\le c and \Vert \Delta u_0\Vert \le c \varepsilon ^{-2}, then we may choose \sigma =...
In the following stability result we assume that an approximate solution satisfies a maximum principle. This is satisfied for certain numerical approximations and the assumption can be weakened to a uniform L^\infty -bound. We recall that Gronwall’s lemma states that if a nonnegative function y\in C({[0,T]})satisfies for all T'\in {[0,T]} with a no...
(Stability) Let u\in H^1({[0,T]};H^1(\varOmega )')\cap L^\infty ({[0,T]};H^1(\varOmega )) be a weak solution of the Allen–Cahn equation with |u|\le 1 almost everywhere in {[0,T]}\times \varOmega . Let \widetilde{u}\in H^1({[0,T]};H^1(\varOmega )')\cap L^2({[0,T]};H^1(\varOmega )) satisfy |\widetilde{u}|\le 1 almost everywhere in {[0,T]}\times \varO...
The functional \widetilde{\fancyscript{R}} models the error introduced by a discretization of the equation so that we may assume that \Vert \widetilde{\fancyscript{R}}(t)\Vert _{H^1(\varOmega )'}^2 \le c \varepsilon ^{-\rho }(h^\alpha + \tau ^\beta ) for a mesh-size h>0 and a time-step size \tau >0, and parameters \alpha ,\beta ,\rho >0. If \Vert u...
(Generalized Gronwall lemma) Suppose that the nonnegative functions y_1\in C({[0,T]}), y_2,y_3 \in L1({[0,T]}), a\in L\infty ({[0,T]}), and the real number A\ge 0satisfy for all T'\in {[0,T]}. Assume that for B \ge 0, \beta >0, and every T'\in {[0,T]}, we have Set E =\exp \big (\int _0T a(t)\,{\mathrm d}t\big ) and assume that 8AE \le (8 B (1+T)E )...
- Sören Bartels
- bartels@mathematik.uni-freiburg.de
- 2015
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Dr. Kahn is a gastroenterologist specializing in caring for patients with esophageal disorders, with a particular expertise and clinical focus in Barrett's esophagus and esophageal cancer.
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Apr 20, 2022 · Allen Kahn passed on Aug. 30, 2021 in Omaha. Services were held Sept. 2, 2021 at Beth El Cemetery, and officiated by Rabbi Steven Abraham. He was preceded in death by his daughter, Linda Kahn. He was survived by his wife of 69 years, Esther (who passed on Dec. 18, 2021.)
