We study the asymptotic behavior of the $N$-clock model, a nearest neighbors ferromagnetic spin model on the $d$-dimensional cubic $varepsilon$-lattice in which the spin field is constrained to take values in a discretization $mathcal{S}_N$ of the unit circle~$mathbb{S}^{1}$ consisting of $N$ equispaced points. Our $Gamma$-convergence analysis consists of two steps: we first fix $N$ and let the lattice spacing $varepsilon to 0$, obtaining an interface energy in the continuum defined on piecewise constant spin fields with values in $mathcal{S}_N$; at a second stage, we let $N to +infty$. The final result of this two-step limit process is an anisotropic total variation of $mathbb{S}^1$-valued vector fields of bounded variation.
我们研究了$N$ -时钟模型的渐近行为，这是一个在$d$维立方$varepsilon$ -晶格上的最近邻铁磁自旋模型，其中自旋场被约束为在由$N$等距点组成的单位圆$mathbb{S}^{1}$的离散化$mathcal{S}_N$中取值。我们的$Gamma$ -收敛分析包括两个步骤:首先，我们固定$N$并让晶格间距$varepsilon to 0$，得到在分段恒定自旋场上定义的连续统中的界面能量，其值为$mathcal{S}_N$;在第二阶段，我们让$N to +infty$。这两步极限过程的最终结果是有界变化的$mathbb{S}^1$值向量场的各向异性总变分。
{"title":"Coarse graining and large-$N$ behavior of the $d$-dimensional $N$-clock model","authors":"M. Cicalese, G. Orlando, M. Ruf","doi":"10.4171/ifb/456","DOIUrl":"https://doi.org/10.4171/ifb/456","url":null,"abstract":"We study the asymptotic behavior of the $N$-clock model, a nearest neighbors ferromagnetic spin model on the $d$-dimensional cubic $varepsilon$-lattice in which the spin field is constrained to take values in a discretization $mathcal{S}_N$ of the unit circle~$mathbb{S}^{1}$ consisting of $N$ equispaced points. Our $Gamma$-convergence analysis consists of two steps: we first fix $N$ and let the lattice spacing $varepsilon to 0$, obtaining an interface energy in the continuum defined on piecewise constant spin fields with values in $mathcal{S}_N$; at a second stage, we let $N to +infty$. The final result of this two-step limit process is an anisotropic total variation of $mathbb{S}^1$-valued vector fields of bounded variation.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"31 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84722701","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 prove existence and uniqueness of the motion by curvatureof networks in $mathbb{R}^n$ when the initial datum is of class $W^{2-frac{2}{p}}_p$, with triple junction where the unit tangent vectors to the concurring curves form angles of $120$ degrees. Moreover we investigated the regularization effect due to the parabolic nature of the system. An application of this wellposedness result is a new proof of Theorem 3.18 in "Motion by Curvature of Planar Networks" by Mantegazza-Novaga-Tortorelli where the possible behaviors of the solutions at the maximal time of existence are described. Our study is motivated by an open question proposed in "Evolution of Networks with Multiple Junctions " by Mantegazza-Novaga-Pluda-Schulze: does there exist a unique solution of the motion by curvature of networks with initial datum a regular network of class $C^2$? We give a positive answer.
{"title":"Existence and uniqueness of the motion by curvature of regular networks","authors":"Michael Gosswein, Julia Menzel, Alessandra Pluda","doi":"10.4171/ifb/477","DOIUrl":"https://doi.org/10.4171/ifb/477","url":null,"abstract":"We prove existence and uniqueness of the motion by curvatureof networks in $mathbb{R}^n$ when the initial datum is of class $W^{2-frac{2}{p}}_p$, with triple junction where the unit tangent vectors to the concurring curves form angles of $120$ degrees. Moreover we investigated the regularization effect due to the parabolic nature of the system. An application of this wellposedness result is a new proof of Theorem 3.18 in \"Motion by Curvature of Planar Networks\" by Mantegazza-Novaga-Tortorelli where the possible behaviors of the solutions at the maximal time of existence are described. Our study is motivated by an open question proposed in \"Evolution of Networks with Multiple Junctions \" by Mantegazza-Novaga-Pluda-Schulze: does there exist a unique solution of the motion by curvature of networks with initial datum a regular network of class $C^2$? We give a positive answer.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"50 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77703779","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 a coupled Stokes/Cahntextendash Hilliard system in a two dimensional, bounded and smooth domain, i.e., we consider the limiting behavior of solutions when a parameter $epsilon>0$ corresponding to the thickness of the diffuse interface tends to zero. We show that for sufficiently short times the solutions to the Stokes/Cahntextendash Hilliard system converge to solutions of a sharp interface model, where the evolution of the interface is governed by a Mullinstextendash Sekerka system with an additional convection term coupled to a twotextendash phase stationary Stokes system with the Young-Laplace law for the jump of an extra contribution to the stress tensor, representing capillary stresses. We prove the convergence result by estimating the difference between the exact and an approximate solutions. To this end we make use of modifications of spectral estimates shown by X. Chen for the linearized Cahn-Hilliard operator. The treatment of the coupling terms requires careful estimates, the use of the refinements of the latter spectral estimate and a suitable structure of the approximate solutions, which will be constructed in the second part of this contribution.
{"title":"Sharp interface limit of a Stokes/Cahn–Hilliard system. Part I: Convergence result","authors":"H. Abels, A. Marquardt","doi":"10.4171/ifb/457","DOIUrl":"https://doi.org/10.4171/ifb/457","url":null,"abstract":"We consider the sharp interface limit of a coupled Stokes/Cahntextendash Hilliard system in a two dimensional, bounded and smooth domain, i.e., we consider the limiting behavior of solutions when a parameter $epsilon>0$ corresponding to the thickness of the diffuse interface tends to zero. We show that for sufficiently short times the solutions to the Stokes/Cahntextendash Hilliard system converge to solutions of a sharp interface model, where the evolution of the interface is governed by a Mullinstextendash Sekerka system with an additional convection term coupled to a twotextendash phase stationary Stokes system with the Young-Laplace law for the jump of an extra contribution to the stress tensor, representing capillary stresses. We prove the convergence result by estimating the difference between the exact and an approximate solutions. To this end we make use of modifications of spectral estimates shown by X. Chen for the linearized Cahn-Hilliard operator. The treatment of the coupling terms requires careful estimates, the use of the refinements of the latter spectral estimate and a suitable structure of the approximate solutions, which will be constructed in the second part of this contribution.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"251 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75839859","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 study a tumor growth model in two space dimensions, where proliferation of the tumor cells leads to expansion of the tumor domain and migration of surrounding normal tissues into the exterior vacuum. The model features two moving interfaces separating the tumor, the normal tissue, and the exterior vacuum. We prove local-in-time existence and uniqueness of strong solutions for their evolution starting from a nearly radial initial configuration. It is assumed that the tumor has lower mobility than the normal tissue, which is in line with the well-known Saffman-Taylor condition in viscous fingering.
{"title":"Interface dynamics in a two-phase tumor growth model","authors":"Inwon C. Kim, Jiajun Tong","doi":"10.4171/ifb/454","DOIUrl":"https://doi.org/10.4171/ifb/454","url":null,"abstract":"We study a tumor growth model in two space dimensions, where proliferation of the tumor cells leads to expansion of the tumor domain and migration of surrounding normal tissues into the exterior vacuum. The model features two moving interfaces separating the tumor, the normal tissue, and the exterior vacuum. We prove local-in-time existence and uniqueness of strong solutions for their evolution starting from a nearly radial initial configuration. It is assumed that the tumor has lower mobility than the normal tissue, which is in line with the well-known Saffman-Taylor condition in viscous fingering.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76820935","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 article, some curve shortening equation related to the evolution of grain boundaries is presented. The curve shortening equation with time-dependent mobility is derived from the grain boundary energy applying the maximum dissipation principle. Gradient estimates and large time asymptotic behavior of solutions are considered. A key ingredient is weighted monotonicity formula with time dependent mobility.
{"title":"A curve shortening equation with time-dependent mobility related to grain boundary motions","authors":"M. Mizuno, K. Takasao","doi":"10.4171/IFB/453","DOIUrl":"https://doi.org/10.4171/IFB/453","url":null,"abstract":"In this article, some curve shortening equation related to the evolution of grain boundaries is presented. The curve shortening equation with time-dependent mobility is derived from the grain boundary energy applying the maximum dissipation principle. Gradient estimates and large time asymptotic behavior of solutions are considered. A key ingredient is weighted monotonicity formula with time dependent mobility.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"85 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75004216","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 a discrete and a continuum model for the propagation of a curvature sensitive interface in a time independent random medium. In both cases we suppose that the medium contains obstacles that act on the propagation of the interface with an inhibitory or an acceleratory force. We show that the interface remains bounded for all times even when a small constant external driving force is applied. This phenomenon has already been known when only inhibitory obstacles are present. In this work we extend this result to the case of, for example, a random medium of random zero mean forcing.
{"title":"Pinning of interfaces in a random medium with zero mean","authors":"P. Dondl, Martin Jesenko, M. Scheutzow","doi":"10.4171/ifb/455","DOIUrl":"https://doi.org/10.4171/ifb/455","url":null,"abstract":"We consider a discrete and a continuum model for the propagation of a curvature sensitive interface in a time independent random medium. In both cases we suppose that the medium contains obstacles that act on the propagation of the interface with an inhibitory or an acceleratory force. We show that the interface remains bounded for all times even when a small constant external driving force is applied. This phenomenon has already been known when only inhibitory obstacles are present. In this work we extend this result to the case of, for example, a random medium of random zero mean forcing.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"8 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74727780","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":"A nonlocal diffusion problem with a sharp free boundary","authors":"C. Cortázar, F. Quirós, N. Wolanski","doi":"10.4171/ifb/430","DOIUrl":"https://doi.org/10.4171/ifb/430","url":null,"abstract":"","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"13 1","pages":"441-462"},"PeriodicalIF":1.0,"publicationDate":"2019-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86027488","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}
This article is concerned with a porous medium equation whose pressure law is both nonlinear and nonlocal, namely $partial_t u = { nabla cdot} left(u nabla(-Delta)^{frac{alpha}{2}-1}u^{m-1} right)$ where $u:mathbb{R}_+times mathbb{R}^N to mathbb{R}_+$, for $0
本文研究压力律为非线性非局部的多孔介质方程，即$partial_t u = { nabla cdot} left(u nabla(-Delta)^{frac{alpha}{2}-1}u^{m-1} right)$，其中$u:mathbb{R}_+times mathbb{R}^N to mathbb{R}_+$为$0
{"title":"Regularity of solutions of a fractional porous medium equation","authors":"C. Imbert, R. Tarhini, Franccois Vigneron","doi":"10.4171/ifb/445","DOIUrl":"https://doi.org/10.4171/ifb/445","url":null,"abstract":"This article is concerned with a porous medium equation whose pressure law is both nonlinear and nonlocal, namely $partial_t u = { nabla cdot} left(u nabla(-Delta)^{frac{alpha}{2}-1}u^{m-1} right)$ where $u:mathbb{R}_+times mathbb{R}^N to mathbb{R}_+$, for $0<alpha<2$ and $mgeq2$. We prove that the $L^1cap L^infty$ weak solutions constructed by Biler, Imbert and Karch (2015) are locally Holder-continuous in time and space. In this article, the classical parabolic De Giorgi techniques for the regularity of PDEs are tailored to fit this particular variant of the PME equation. In the spirit of the work of Caffarelli, Chan and Vasseur (2011), the two main ingredients are the derivation of local energy estimates and a so-called \"intermediate value lemma\". For $alphaleq1$, we adapt the proof of Caffarelli, Soria and Vazquez (2013), who treated the case of a linear pressure law. We then use a non-linear drift to cancel out the singular terms that would otherwise appear in the energy estimates.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"27 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2019-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90362356","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 theoretical framework and numerical techniques to solve optimal control problems with a spatial trace term in the terminal cost and governed by regularized nonlinear hyperbolic conservation laws are provided. Depending on the spatial dimension, the set at which the optimum of the trace term is reached under the action of the control function can be a point, a curve or a hypersurface. The set is determined by geometric parameters. Theoretically the lack of a convenient functional framework in the context of optimal control for hyperbolic systems leads us to consider a parabolic regularization for the state equation, in order to derive optimality conditions. For deriving these conditions, we use a change of variables encoding the sensitivity with respect to the geometric parameters. As illustration, we consider the shallow-water equations with the objective of maximizing the height of the wave at the final time, a wave whose location and shape are optimized via the geometric parameters. Numerical results are obtained in 1D and 2D, using finite difference schemes, combined with an immersed boundary method for iterating the geometric parameters.
{"title":"Optimal control problem for viscous systems of conservation laws, with geometric parameter, and application to the Shallow-Water equations","authors":"Sébastien Court, K. Kunisch, Laurent Pfeiffer","doi":"10.4171/ifb/424","DOIUrl":"https://doi.org/10.4171/ifb/424","url":null,"abstract":"A theoretical framework and numerical techniques to solve optimal control problems with a spatial trace term in the terminal cost and governed by regularized nonlinear hyperbolic conservation laws are provided. Depending on the spatial dimension, the set at which the optimum of the trace term is reached under the action of the control function can be a point, a curve or a hypersurface. The set is determined by geometric parameters. Theoretically the lack of a convenient functional framework in the context of optimal control for hyperbolic systems leads us to consider a parabolic regularization for the state equation, in order to derive optimality conditions. For deriving these conditions, we use a change of variables encoding the sensitivity with respect to the geometric parameters. As illustration, we consider the shallow-water equations with the objective of maximizing the height of the wave at the final time, a wave whose location and shape are optimized via the geometric parameters. Numerical results are obtained in 1D and 2D, using finite difference schemes, combined with an immersed boundary method for iterating the geometric parameters.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"33 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2019-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74003843","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 answer a question left open in [Arch. Rat. Mech. Anal. 230 (1) (2018), 125-184] and [Arch. Rat. Mech. Anal. 230 (2) (2018), 783-784], by proving that the blow-up limit of minimizers $u$ of the lower dimensional obstacle problem is unique at generic point of the free-boundary. Moreover we show that at such points the only admissible frequencies are $2m-1+s$ and $2m$, $mgeq 1$.
{"title":"Almost everywhere uniqueness of blow-up limits for the lower dimensional obstacle problem","authors":"Maria Colombo, L. Spolaor, B. Velichkov","doi":"10.4171/ifb/452","DOIUrl":"https://doi.org/10.4171/ifb/452","url":null,"abstract":"We answer a question left open in [Arch. Rat. Mech. Anal. 230 (1) (2018), 125-184] and [Arch. Rat. Mech. Anal. 230 (2) (2018), 783-784], by proving that the blow-up limit of minimizers $u$ of the lower dimensional obstacle problem is unique at generic point of the free-boundary. Moreover we show that at such points the only admissible frequencies are $2m-1+s$ and $2m$, $mgeq 1$.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"26 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2019-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83105986","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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