In this paper, we study the asymptotic behaviour of the solutions to a degenerate reaction–diffusion system. This system admits a continuum of discontinuous stationary solutions due to the effect of a hysteresis process, but only one discontinuous stationary solution is compatible with a principle of preservation of locally invariant regions. Using a macroscopic mass effect which guarantees that fast particles help slow particles to displace, we establish a novel result of convergence of a non trivial set of trajectories towards a discontinuous pattern.
{"title":"How hysteresis produces discontinuous patterns in degenerate reaction–diffusion systems","authors":"Guillaume Cantin","doi":"10.3233/asy-221818","DOIUrl":"https://doi.org/10.3233/asy-221818","url":null,"abstract":"In this paper, we study the asymptotic behaviour of the solutions to a degenerate reaction–diffusion system. This system admits a continuum of discontinuous stationary solutions due to the effect of a hysteresis process, but only one discontinuous stationary solution is compatible with a principle of preservation of locally invariant regions. Using a macroscopic mass effect which guarantees that fast particles help slow particles to displace, we establish a novel result of convergence of a non trivial set of trajectories towards a discontinuous pattern.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42232938","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 study the least energy sign-changing solutions to the following nonlinear Kirchhoff equation − ( a + b ∫ V | ∇ u | 2 d μ ) Δ u + c ( x ) u = f ( u ) on a locally finite graph G = ( V , E ), where a, b are positive constants. We use the constrained variational method to prove the existence of a least energy sign-changing solution u b of the above equation if c ( x ) and f satisfy certain assumptions, and to show the energy of u b is strictly larger than twice that of the least energy solutions. Moreover, if we regard b as a parameter, as b → 0 + , the solution u b converges to a least energy sign-changing solution of a local equation − a Δ u + c ( x ) u = f ( u ).
{"title":"Existence and convergence of the least energy sign-changing solutions for nonlinear Kirchhoff equations on locally finite graphs","authors":"Guofu Pan, Chao Ji","doi":"10.3233/asy-221819","DOIUrl":"https://doi.org/10.3233/asy-221819","url":null,"abstract":"In this paper, we study the least energy sign-changing solutions to the following nonlinear Kirchhoff equation − ( a + b ∫ V | ∇ u | 2 d μ ) Δ u + c ( x ) u = f ( u ) on a locally finite graph G = ( V , E ), where a, b are positive constants. We use the constrained variational method to prove the existence of a least energy sign-changing solution u b of the above equation if c ( x ) and f satisfy certain assumptions, and to show the energy of u b is strictly larger than twice that of the least energy solutions. Moreover, if we regard b as a parameter, as b → 0 + , the solution u b converges to a least energy sign-changing solution of a local equation − a Δ u + c ( x ) u = f ( u ).","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47289591","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 both stationary and time-dependent solutions of the 3-D Navier–Stokes equations (NSE) on a multi-connected bounded domain Ω ⊂ R 3 with inhomogeneous boundary values on ∂ Ω = Γ; here Γ is a union of disjoint surfaces Γ 0 , Γ 1 , … , Γ l . Our starting point is Leray’s classic problem, which is to find a weak solution u ∈ H 1 ( Ω ) of the stationary problem assuming that on the boundary u = β ∈ H 1 / 2 ( Γ ). The general flux condition ∑ j = 0 l ∫ Γ j β · n d S = 0 must be satisfied due to compatibility considerations. Early results on this problem including the initial results in (J. Math. Pures Appl. 12 (1933) 1–82) assumed the more restrictive flux condition ∫ Γ j β · n d S = 0 for each j = 1 , … , l. More recent results, of which those in (An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. II 1994 Springer–Verlag) and (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) are particularly representative, assume only the general flux condition in exchange for size restrictions on the data. In this paper we also assume only the general flux condition throughout, and for virtually the same size restrictions on the data as in (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) we obtain the existence of a weak solution that matches that found in (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) when the assumptions imposed here and those assumed in (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) are both met; additionally we demonstrate that this solution is unique. For slightly stronger size restrictions we obtain the existence and uniqueness of solutions of both Leray’s problem and global mild solutions of the corresponding time-dependent problem, while showing that both the stationary and time-dependent solutions we construct are a bit stronger than weak solutions. The settings in which we establish our results allow us to culminate our discussion by showing that our time-dependent solutions converge to each other exponentially in time, so that in particular our stationary solutions are asymptotically stable. We also discuss additional features which allow for data of increased size on certain domains, including those which are thin in a generalized sense.
我们考虑了三维Navier–Stokes方程（NSE）在多连通有界域Ω⊂R3上的定常解和含时解，该域在ΓΩ=Γ上具有非齐次边值；这里Γ是不相交曲面Γ0，Γ1，…，Γl的并集。我们的出发点是Leray的经典问题，即假定在边界u=β∈H1/2（Γ）上，找到平稳问题的弱解u∈H1（Ω）。由于兼容性的考虑，必须满足一般通量条件∑j=0 lΓjβ·n d S=0。关于这个问题的早期结果，包括（J.Math.Pures Appl.12（1933）1–82）中的初始结果，假设每个J=1，…，l都有更严格的通量条件ΓΓJβ·n d S=0。II 1994 Springer–Verlag）和（在非线性偏微分方程分析讲座2013 237–290 Int.Press）特别具有代表性，仅假设一般通量条件来交换数据的大小限制。在本文中，我们还假设整个过程中只有一般的通量条件，并且对于与（在《非线性偏微分方程分析讲座》2013 237–290 Int.Press中）中几乎相同的数据大小限制，我们获得了与（《非线性偏分方程分析讲座2013 237–290Int.Press》）中发现的弱解相匹配的弱解的存在。Press），当这里强加的假设和（in Lectures on the Analysis of非线性偏微分方程2013 237–290 Int.Press）中假设的假设都满足时；此外，我们还证明了这种解决方案是独一无二的。对于稍强的大小限制，我们获得了Leray问题的解和相应的含时问题的全局温和解的存在性和唯一性，同时表明我们构造的平稳解和含时解都比弱解强一点。我们建立结果的设置使我们能够通过证明我们的时间相关解在时间上呈指数收敛来达到讨论的高潮，因此特别是我们的平稳解是渐近稳定的。我们还讨论了允许在某些域上增加数据大小的附加功能，包括广义意义上的薄数据。
{"title":"Existence, uniqueness, and asymptotic stability results for the 3-D steady and unsteady Navier–Stokes equations on multi-connected domains with inhomogeneous boundary conditions","authors":"J. Avrin","doi":"10.3233/asy-221816","DOIUrl":"https://doi.org/10.3233/asy-221816","url":null,"abstract":"We consider both stationary and time-dependent solutions of the 3-D Navier–Stokes equations (NSE) on a multi-connected bounded domain Ω ⊂ R 3 with inhomogeneous boundary values on ∂ Ω = Γ; here Γ is a union of disjoint surfaces Γ 0 , Γ 1 , … , Γ l . Our starting point is Leray’s classic problem, which is to find a weak solution u ∈ H 1 ( Ω ) of the stationary problem assuming that on the boundary u = β ∈ H 1 / 2 ( Γ ). The general flux condition ∑ j = 0 l ∫ Γ j β · n d S = 0 must be satisfied due to compatibility considerations. Early results on this problem including the initial results in (J. Math. Pures Appl. 12 (1933) 1–82) assumed the more restrictive flux condition ∫ Γ j β · n d S = 0 for each j = 1 , … , l. More recent results, of which those in (An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. II 1994 Springer–Verlag) and (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) are particularly representative, assume only the general flux condition in exchange for size restrictions on the data. In this paper we also assume only the general flux condition throughout, and for virtually the same size restrictions on the data as in (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) we obtain the existence of a weak solution that matches that found in (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) when the assumptions imposed here and those assumed in (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) are both met; additionally we demonstrate that this solution is unique. For slightly stronger size restrictions we obtain the existence and uniqueness of solutions of both Leray’s problem and global mild solutions of the corresponding time-dependent problem, while showing that both the stationary and time-dependent solutions we construct are a bit stronger than weak solutions. The settings in which we establish our results allow us to culminate our discussion by showing that our time-dependent solutions converge to each other exponentially in time, so that in particular our stationary solutions are asymptotically stable. We also discuss additional features which allow for data of increased size on certain domains, including those which are thin in a generalized sense.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45166456","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 present a null controllability result for a thermoelastic Rayleigh system. Instead of working directly with the control system, we obtain the controlled system as the modulus of elasticity in shear tends to infinity in the corresponding thermoelastic Mindlin–Timoshenko system. Our results follow the seminal book of Lagnese and Lions (Rech. Math. Appl. 6(1988)) where the controllability of a Kirkhhoff model is proposed as the limit of a controlled Mindlin–Timoshenko one. We use estimates for some eigenvalues of the beam model that were obtained in (SIAM J. Control Optim. 47 (2008) 1909–1938) and the recent paper of Komornik and Tenenbaum (Evolution Equations and Control Theory 4(3) (2015) 297–314) where explicit estimates for systems with real and complex eigenvalues are proposed.
本文给出了热弹性瑞利系统的零可控性结果。而不是直接与控制系统工作，我们得到控制系统的剪切弹性模量趋于无穷大，在相应的热弹性Mindlin-Timoshenko系统。我们的研究结果遵循了影响深远的《拉格内斯与狮子》一书。数学。其中Kirkhhoff模型的可控性被提出为受控Mindlin-Timoshenko模型的极限。我们使用了在(SIAM J. Control Optim. 47(2008) 1909-1938)和Komornik和Tenenbaum最近的论文(进化方程和控制理论4(3)(2015)297-314)中获得的梁模型的一些特征值的估计，其中提出了对具有实特征值和复数特征值的系统的显式估计。
{"title":"Controllability of a thermoelastic system","authors":"F. D. Araruna, A. Mercado, Luz de Teresa","doi":"10.3233/asy-221815","DOIUrl":"https://doi.org/10.3233/asy-221815","url":null,"abstract":"In this paper we present a null controllability result for a thermoelastic Rayleigh system. Instead of working directly with the control system, we obtain the controlled system as the modulus of elasticity in shear tends to infinity in the corresponding thermoelastic Mindlin–Timoshenko system. Our results follow the seminal book of Lagnese and Lions (Rech. Math. Appl. 6(1988)) where the controllability of a Kirkhhoff model is proposed as the limit of a controlled Mindlin–Timoshenko one. We use estimates for some eigenvalues of the beam model that were obtained in (SIAM J. Control Optim. 47 (2008) 1909–1938) and the recent paper of Komornik and Tenenbaum (Evolution Equations and Control Theory 4(3) (2015) 297–314) where explicit estimates for systems with real and complex eigenvalues are proposed.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48834737","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 Dirichlet problem for the Poisson equation in a periodically perforated domain. The geometry of the domain is controlled by two parameters: a real number ϵ > 0, proportional to the radius of the holes, and a map ϕ, which models the shape of the holes. So, if g denotes the Dirichlet boundary datum and f the Poisson datum, we have a solution for each quadruple ( ϵ , ϕ , g , f ). Our aim is to study how the solution depends on ( ϵ , ϕ , g , f ), especially when ϵ is very small and the holes narrow to points. In contrast with previous works, we do not introduce the assumption that f has zero integral on the fundamental periodicity cell. This brings in a certain singular behavior for ϵ close to 0. We show that, when the dimension n of the ambient space is greater than or equal to 3, a suitable restriction of the solution can be represented with an analytic map of the quadruple ( ϵ , ϕ , g , f ) multiplied by the factor 1 / ϵ n − 2 . In case of dimension n = 2, we have to add log ϵ times the integral of f / 2 π.
{"title":"Singular behavior for a multi-parameter periodic Dirichlet problem","authors":"M. D. Riva, Paolo Luzzini, P. Musolino","doi":"10.3233/asy-231831","DOIUrl":"https://doi.org/10.3233/asy-231831","url":null,"abstract":"We consider a Dirichlet problem for the Poisson equation in a periodically perforated domain. The geometry of the domain is controlled by two parameters: a real number ϵ > 0, proportional to the radius of the holes, and a map ϕ, which models the shape of the holes. So, if g denotes the Dirichlet boundary datum and f the Poisson datum, we have a solution for each quadruple ( ϵ , ϕ , g , f ). Our aim is to study how the solution depends on ( ϵ , ϕ , g , f ), especially when ϵ is very small and the holes narrow to points. In contrast with previous works, we do not introduce the assumption that f has zero integral on the fundamental periodicity cell. This brings in a certain singular behavior for ϵ close to 0. We show that, when the dimension n of the ambient space is greater than or equal to 3, a suitable restriction of the solution can be represented with an analytic map of the quadruple ( ϵ , ϕ , g , f ) multiplied by the factor 1 / ϵ n − 2 . In case of dimension n = 2, we have to add log ϵ times the integral of f / 2 π.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48621729","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 work, we investigate stochastic fractional diffusion equations with Caputo–Fabrizio fractional derivatives and multiplicative noise, involving finite and infinite delays. Initially, the existence and uniqueness of mild solution in the spaces C p ( [ − a , b ] ; L q ( Ω , H ˙ r ) ) ) and C δ ( ( − ∞ , b ] ; L q ( Ω , H ˙ r ) ) ) are established. Next, besides investigating the regularity properties, we show the continuity of mild solutions with respect to the initial functions and the order of the fractional derivative for both cases of delay separately.
{"title":"Stochastic fractional diffusion equations containing finite and infinite delays with multiplicative noise","authors":"N. Tuan, T. Caraballo, Tran Ngoc Thach","doi":"10.3233/asy-221811","DOIUrl":"https://doi.org/10.3233/asy-221811","url":null,"abstract":"In this work, we investigate stochastic fractional diffusion equations with Caputo–Fabrizio fractional derivatives and multiplicative noise, involving finite and infinite delays. Initially, the existence and uniqueness of mild solution in the spaces C p ( [ − a , b ] ; L q ( Ω , H ˙ r ) ) ) and C δ ( ( − ∞ , b ] ; L q ( Ω , H ˙ r ) ) ) are established. Next, besides investigating the regularity properties, we show the continuity of mild solutions with respect to the initial functions and the order of the fractional derivative for both cases of delay separately.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42088395","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}
Using variational methods, we establish the existence of infinitely many solutions to an elliptic problem driven by a Choquard term and a singular nonlinearity. We further show that if the problem has a positive solution, then it is bounded a.e. in the domain Ω and is Hölder continuous.
{"title":"On elliptic problems with Choquard term and singular nonlinearity","authors":"D. Choudhuri, Dušan D. Repovš, K. Saoudi","doi":"10.3233/ASY-221812","DOIUrl":"https://doi.org/10.3233/ASY-221812","url":null,"abstract":"Using variational methods, we establish the existence of infinitely many solutions to an elliptic problem driven by a Choquard term and a singular nonlinearity. We further show that if the problem has a positive solution, then it is bounded a.e. in the domain Ω and is Hölder continuous.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46894450","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 deal with the initial value fractional damped wave equation on G, a compact Lie group, with power-type nonlinearity. The aim of this manuscript is twofold. First, using the Fourier analysis on compact Lie groups, we prove a local in-time existence result in the energy space for the fractional damped wave equation on G. Moreover, a finite time blow-up result is established under certain conditions on the initial data. In the next part of the paper, we consider fractional wave equation with lower order terms, i.e., damping and mass with the same power type nonlinearity on compact Lie groups, and prove the global in-time existence of small data solutions in the energy evolution space.
{"title":"Nonlinear fractional damped wave equation on compact Lie groups","authors":"Aparajita Dasgupta, Vishvesh Kumar, Shyam Swarup Mondal","doi":"10.3233/asy-231842","DOIUrl":"https://doi.org/10.3233/asy-231842","url":null,"abstract":"In this paper, we deal with the initial value fractional damped wave equation on G, a compact Lie group, with power-type nonlinearity. The aim of this manuscript is twofold. First, using the Fourier analysis on compact Lie groups, we prove a local in-time existence result in the energy space for the fractional damped wave equation on G. Moreover, a finite time blow-up result is established under certain conditions on the initial data. In the next part of the paper, we consider fractional wave equation with lower order terms, i.e., damping and mass with the same power type nonlinearity on compact Lie groups, and prove the global in-time existence of small data solutions in the energy evolution space.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44679698","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 derive uniform local energy decay results for wave equations with a short-range potential in an exterior domain. In this study, we considered this problem within the framework of non-compactly supported initial data, unlike previously reported studies. The essential parts of analysis are both L 2 -estimates of the solution itself and the weighted energy estimates. Only a multiplier method is used, and we do not rely on any resolvent estimates.
{"title":"A note on local energy decay results for wave equations with a potential","authors":"R. Ikehata","doi":"10.3233/asy-231835","DOIUrl":"https://doi.org/10.3233/asy-231835","url":null,"abstract":"In this paper, we derive uniform local energy decay results for wave equations with a short-range potential in an exterior domain. In this study, we considered this problem within the framework of non-compactly supported initial data, unlike previously reported studies. The essential parts of analysis are both L 2 -estimates of the solution itself and the weighted energy estimates. Only a multiplier method is used, and we do not rely on any resolvent estimates.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43801409","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 introduce new diffuse approximations of the Willmore functional and the Willmore flow. They are based on a corresponding approximation of the perimeter that has been studied by Amstutz-van Goethem [Interfaces Free Bound. 14 (2012)]. We identify the candidate for the Γ-convergence, prove the Γ-limsup statement and justify the convergence to the Willmore flow by an asymptotic expansion. Furthermore, we present numerical simulations that are based on the new approximation.
{"title":"“Gradient-free” diffuse approximations of the Willmore functional and Willmore flow","authors":"Nils Dabrock, Sascha Knuttel, M. Roger","doi":"10.3233/ASY-221810","DOIUrl":"https://doi.org/10.3233/ASY-221810","url":null,"abstract":"We introduce new diffuse approximations of the Willmore functional and the Willmore flow. They are based on a corresponding approximation of the perimeter that has been studied by Amstutz-van Goethem [Interfaces Free Bound. 14 (2012)]. We identify the candidate for the Γ-convergence, prove the Γ-limsup statement and justify the convergence to the Willmore flow by an asymptotic expansion. Furthermore, we present numerical simulations that are based on the new approximation.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48417407","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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