A variational model for epitaxially-strained thin films on rigid substrates is derived both by {Gamma}-convergence from a transition-layer setting, and by relaxation from a sharp-interface description available in the literature for regular configurations. The model is characterized by a configurational energy that accounts for both the competing mechanisms responsible for the film shape. On the one hand, the lattice mismatch between the film and the substrate generate large stresses, and corrugations may be present because film atoms move to release the elastic energy. On the other hand, flatter profiles may be preferable to minimize the surface energy. Some first regularity results are presented for energetically-optimal film profiles.
{"title":"Derivation of a heteroepitaxial thin-film model","authors":"E. Davoli, Paolo Piovano","doi":"10.4171/ifb/435","DOIUrl":"https://doi.org/10.4171/ifb/435","url":null,"abstract":"A variational model for epitaxially-strained thin films on rigid substrates is derived both by {Gamma}-convergence from a transition-layer setting, and by relaxation from a sharp-interface description available in the literature for regular configurations. The model is characterized by a configurational energy that accounts for both the competing mechanisms responsible for the film shape. On the one hand, the lattice mismatch between the film and the substrate generate large stresses, and corrugations may be present because film atoms move to release the elastic energy. On the other hand, flatter profiles may be preferable to minimize the surface energy. Some first regularity results are presented for energetically-optimal film profiles.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79715956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
R. Cristoferi, I. Fonseca, Adrian Hagerty, Cristina Popovici
A homogenization problem arising in the gradient theory of ∞uid-∞uid phase transitions is addressed in the vector-valued setting by means of i-convergence.
在向量值条件下，利用i收敛方法解决了梯度理论中∞-∞流体相变的均匀化问题。
{"title":"A homogenization result in the gradient theory of phase transitions","authors":"R. Cristoferi, I. Fonseca, Adrian Hagerty, Cristina Popovici","doi":"10.4171/IFB/426","DOIUrl":"https://doi.org/10.4171/IFB/426","url":null,"abstract":"A homogenization problem arising in the gradient theory of ∞uid-∞uid phase transitions is addressed in the vector-valued setting by means of i-convergence.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74843487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by applications to image reconstruction, in this paper we analyse a emph{finite-difference discretisation} of the Ambrosio-Tortorelli functional. Denoted by $varepsilon$ the elliptic-approximation parameter and by $delta$ the discretisation step-size, we fully describe the relative impact of $varepsilon$ and $delta$ in terms of $Gamma$-limits for the corresponding discrete functionals, in the three possible scaling regimes. We show, in particular, that when $varepsilon$ and $delta$ are of the same order, the underlying lattice structure affects the $Gamma$-limit which turns out to be an anisotropic free-discontinuity functional.
{"title":"Quantitative analysis of finite-difference approximations of free-discontinuity problems","authors":"Annika Bach, Andrea Braides, C. Zeppieri","doi":"10.4171/ifb/443","DOIUrl":"https://doi.org/10.4171/ifb/443","url":null,"abstract":"Motivated by applications to image reconstruction, in this paper we analyse a emph{finite-difference discretisation} of the Ambrosio-Tortorelli functional. Denoted by $varepsilon$ the elliptic-approximation parameter and by $delta$ the discretisation step-size, we fully describe the relative impact of $varepsilon$ and $delta$ in terms of $Gamma$-limits for the corresponding discrete functionals, in the three possible scaling regimes. We show, in particular, that when $varepsilon$ and $delta$ are of the same order, the underlying lattice structure affects the $Gamma$-limit which turns out to be an anisotropic free-discontinuity functional.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72962470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A family of singular limits of reaction-diffusion systems of activator-inhibitor type in which stable stationary sharp-interface patterns may form is investigated. For concreteness, the analysis is performed for the FitzHugh-Nagumo model on a suitably rescaled bounded domain in R , with N ≥ 2. It is proved that when the system is sufficiently close to the limit the dynamics starting from the appropriate smooth initial data breaks down into five distinct stages on well-separated time scales, each of which can be approximated by a suitable reduced problem. The analysis allows to follow fully the progressive refinement of spatiotemporal patterns forming in the systems under consideration and provides a framework for understanding the pattern formation scenarios in a large class of physical, chemical, and biological systems modeled by the considered class of reaction-diffusion equations. ∗CMI Université d’Aix-Marseille, 39 rue Frédéric Joliot-Curie 13453 Marseille cedex 13, France †Laboratoire de Mathématiques, CNRS and University Paris-Sud Paris-Saclay, 91405 Orsay Cedex, France ‡Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA
研究了一类激活剂-抑制剂型反应扩散体系的奇异极限，在这些体系中可以形成稳定的、固定的锐界面图案。具体而言，在R中N≥2的适当重新标度的有界域上对FitzHugh-Nagumo模型进行分析。证明了当系统足够接近极限时，从适当的光滑初始数据开始的动力学在分离良好的时间尺度上分解为五个不同的阶段，每个阶段都可以用一个适当的约简问题来近似。该分析允许完全遵循所考虑的系统中形成的时空模式的逐步细化，并为理解由所考虑的一类反应扩散方程建模的大类物理，化学和生物系统中的模式形成场景提供了框架。* CMI法国艾克斯-马赛大学，39 rue frsamdsamric jolio - curie 13453 Marseille cedex 13;法国国家科学研究中心和巴黎- sud Paris-Saclay大学，91405 Orsay cedex, France;美国新泽西理工学院数学科学系，Newark, NJ 07102
{"title":"A multiple scale pattern formation cascade in reaction-diffusion systems of activator-inhibitor type","authors":"M. Henry, D. Hilhorst, C. Muratov","doi":"10.4171/IFB/403","DOIUrl":"https://doi.org/10.4171/IFB/403","url":null,"abstract":"A family of singular limits of reaction-diffusion systems of activator-inhibitor type in which stable stationary sharp-interface patterns may form is investigated. For concreteness, the analysis is performed for the FitzHugh-Nagumo model on a suitably rescaled bounded domain in R , with N ≥ 2. It is proved that when the system is sufficiently close to the limit the dynamics starting from the appropriate smooth initial data breaks down into five distinct stages on well-separated time scales, each of which can be approximated by a suitable reduced problem. The analysis allows to follow fully the progressive refinement of spatiotemporal patterns forming in the systems under consideration and provides a framework for understanding the pattern formation scenarios in a large class of physical, chemical, and biological systems modeled by the considered class of reaction-diffusion equations. ∗CMI Université d’Aix-Marseille, 39 rue Frédéric Joliot-Curie 13453 Marseille cedex 13, France †Laboratoire de Mathématiques, CNRS and University Paris-Sud Paris-Saclay, 91405 Orsay Cedex, France ‡Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74132834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Bubbles and droplets in a singular limit of the FitzHugh–Nagumo system","authors":"Chao-Nien Chen, Yung-Sze Choi, Xiaofeng Ren","doi":"10.4171/IFB/400","DOIUrl":"https://doi.org/10.4171/IFB/400","url":null,"abstract":"","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81700897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the local well-posedness of the one-dimensional nonisentropic Euler equations with moving physical vacuum boundary condition. The physical vacuum singularity requires the sound speed to be scaled as the square root of the distance to the vacuum boundary. The main difficulty lies in the fact that the system of hyperbolic conservation laws becomes characteristic and degenerate at the vacuum boundary. Our proof is based on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term. Then we construct the solutions to this degenerate parabolic problem and establish the estimates that are uniform with respect to the artificial viscosity parameter. Solutions to the compressible Euler equations are obtained as the limit of the vanishing artificial viscosity. Different from the isentropic case cite{Coutand4, Lei}, our momentum equation of conservation laws has an extra term $p_{S}S_x$ that leads to some extra terms in the energy function and causes more difficulties even for the case of $gamma=2$. Moreover, we deal with this free boundary problem starting from the general cases of $2leqgamma<3$ and $1
{"title":"Well-posedness of non-isentropic Euler equations with physical vacuum","authors":"Yong-cai Geng, Yachun Li, Dehua Wang, Runzhang Xu","doi":"10.4171/IFB/422","DOIUrl":"https://doi.org/10.4171/IFB/422","url":null,"abstract":"We consider the local well-posedness of the one-dimensional nonisentropic Euler equations with moving physical vacuum boundary condition. The physical vacuum singularity requires the sound speed to be scaled as the square root of the distance to the vacuum boundary. The main difficulty lies in the fact that the system of hyperbolic conservation laws becomes characteristic and degenerate at the vacuum boundary. Our proof is based on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term. Then we construct the solutions to this degenerate parabolic problem and establish the estimates that are uniform with respect to the artificial viscosity parameter. Solutions to the compressible Euler equations are obtained as the limit of the vanishing artificial viscosity. \u0000Different from the isentropic case cite{Coutand4, Lei}, our momentum equation of conservation laws has an extra term $p_{S}S_x$ that leads to some extra terms in the energy function and causes more difficulties even for the case of $gamma=2$. Moreover, we deal with this free boundary problem starting from the general cases of $2leqgamma<3$ and $1<gamma<2 $ instead of only emphasizing the isentropic case of $gamma=2$ in cite{Coutand4, jang1, Lei}.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84659522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the sharp interface limit of the Allen-Cahn equation with homogeneous Neumann boundary condition in a two-dimensional domain $Omega$, in the situation where an interface has developed and intersects $partialOmega$. Here a parameter $varepsilon>0$ in the equation, which is related to the thickness of the diffuse interface, is sent to zero. The limit problem is given by mean curvature flow with a $90$textdegree-contact angle condition and convergence using strong norms is shown for small times. Here we assume that a smooth solution to this limit problem exists on $[0,T]$ for some $T>0$ and that it can be parametrized suitably. With the aid of asymptotic expansions we construct an approximate solution for the Allen-Cahn equation and estimate the difference of the exact and approximate solution with the aid of a spectral estimate for the linearized Allen-Cahn operator.
{"title":"Convergence of the Allen–Cahn equation to the mean curvature flow with 90o-contact angle in 2D","authors":"H. Abels, M. Moser","doi":"10.4171/ifb/425","DOIUrl":"https://doi.org/10.4171/ifb/425","url":null,"abstract":"We consider the sharp interface limit of the Allen-Cahn equation with homogeneous Neumann boundary condition in a two-dimensional domain $Omega$, in the situation where an interface has developed and intersects $partialOmega$. Here a parameter $varepsilon>0$ in the equation, which is related to the thickness of the diffuse interface, is sent to zero. The limit problem is given by mean curvature flow with a $90$textdegree-contact angle condition and convergence using strong norms is shown for small times. Here we assume that a smooth solution to this limit problem exists on $[0,T]$ for some $T>0$ and that it can be parametrized suitably. With the aid of asymptotic expansions we construct an approximate solution for the Allen-Cahn equation and estimate the difference of the exact and approximate solution with the aid of a spectral estimate for the linearized Allen-Cahn operator.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74481405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we prove the global existence, uniqueness, optimal large time decay rates, and uniform gain of analyticity for the exponential PDE $h_t=Delta e^{-Delta h}$ in the whole space $mathbb{R}^d_x$. We assume the initial data is of medium size in the critical Wiener algebra $Delta h in A(mathbb{R}^d)$. This exponential PDE was derived in (Krug, Dobbs, and Majaniemi in 1995) and more recently in (Marzuola and Weare 2013).
本文证明了指数函数PDE $h_t=Delta e^{-Delta h}$在整个空间$mathbb{R}^d_x$上的全局存在性、唯一性、最优大时间衰减率和均匀可解析性增益。我们假设初始数据在临界维纳代数$Delta h in A(mathbb{R}^d)$中具有中等大小。指数偏微分方程是由(Krug, Dobbs, and Majaniemi, 1995)和(Marzuola and Weare, 2013)导出的。
{"title":"Global stability for solutions to the exponential PDE describing epitaxial growth","authors":"Jian‐Guo Liu, Robert M. Strain","doi":"10.4171/IFB/417","DOIUrl":"https://doi.org/10.4171/IFB/417","url":null,"abstract":"In this paper we prove the global existence, uniqueness, optimal large time decay rates, and uniform gain of analyticity for the exponential PDE $h_t=Delta e^{-Delta h}$ in the whole space $mathbb{R}^d_x$. We assume the initial data is of medium size in the critical Wiener algebra $Delta h in A(mathbb{R}^d)$. This exponential PDE was derived in (Krug, Dobbs, and Majaniemi in 1995) and more recently in (Marzuola and Weare 2013).","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74166881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract. We consider the quasi-static evolution of a prescribed cohesive interface: dissipative under loading and elastic under unloading. We provide existence in terms of parametrized BV evolutions, employing a discrete scheme based on local minimization, with respect to the Hnorm, of a regularized energy. Technically, the evolution is fully characterized by: equilibrium, energy balance and Karush-Kuhn-Tucker conditions for the internal variable. Catastrophic regimes (discontinuities in time) are described by gradient flows of visco-elastic type.
{"title":"Approximation and characterization of quasi-static $H^1$-evolutions for a cohesive interface with different loading-unloading regimes","authors":"M. Negri, E. Vitali","doi":"10.4171/IFB/396","DOIUrl":"https://doi.org/10.4171/IFB/396","url":null,"abstract":"Abstract. We consider the quasi-static evolution of a prescribed cohesive interface: dissipative under loading and elastic under unloading. We provide existence in terms of parametrized BV evolutions, employing a discrete scheme based on local minimization, with respect to the Hnorm, of a regularized energy. Technically, the evolution is fully characterized by: equilibrium, energy balance and Karush-Kuhn-Tucker conditions for the internal variable. Catastrophic regimes (discontinuities in time) are described by gradient flows of visco-elastic type.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79556762","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Oulhaj, C. Cancès, C. Chainais-Hillairet, Philippe Laurencçot
We study the large time behavior of the solutions to a two phase extension of the porous medium equation, which models the so-called seawater intrusion problem. The goal is to identify the self-similar solutions that correspond to steady states of a rescaled version of the problem. We fully characterize the unique steady states that are identified as minimizers of a convex energy and shown to be radially symmetric. Moreover, we prove the convergence of the solution to the time-dependent model towards the unique stationary state as time goes to infinity. We finally provide numerical illustrations of the stationary states and we exhibit numerical convergence rates.
{"title":"Large time behavior of a two phase extension of the porous medium equation","authors":"A. Oulhaj, C. Cancès, C. Chainais-Hillairet, Philippe Laurencçot","doi":"10.4171/IFB/421","DOIUrl":"https://doi.org/10.4171/IFB/421","url":null,"abstract":"We study the large time behavior of the solutions to a two phase extension of the porous medium equation, which models the so-called seawater intrusion problem. The goal is to identify the self-similar solutions that correspond to steady states of a rescaled version of the problem. We fully characterize the unique steady states that are identified as minimizers of a convex energy and shown to be radially symmetric. Moreover, we prove the convergence of the solution to the time-dependent model towards the unique stationary state as time goes to infinity. We finally provide numerical illustrations of the stationary states and we exhibit numerical convergence rates.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80562108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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