{"title":"Strict separation and numerical approximation for a non–local Cahn–Hilliard equation with single–well potential","authors":"Abramo Agosti, Elisabetta Rocca, Luca Scarpa","doi":"10.3934/dcdss.2023213","DOIUrl":"https://doi.org/10.3934/dcdss.2023213","url":null,"abstract":"","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"22 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140026153","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}
Gregorio Gerardo Spinelli, Jana Gericke, Kannan Masilamani, Harald Günther Klimach
{"title":"Key ingredients for wall-modeled LES with the Lattice Boltzmann method: Systematic comparison of collision schemes, SGS models, and wall functions on simulation accuracy and efficiency for turbulent channel flow","authors":"Gregorio Gerardo Spinelli, Jana Gericke, Kannan Masilamani, Harald Günther Klimach","doi":"10.3934/dcdss.2023212","DOIUrl":"https://doi.org/10.3934/dcdss.2023212","url":null,"abstract":"","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"23 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025997","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":"Abstract action spaces and their topological and dynamic properties","authors":"Riccarda Rossi, Giuseppe Savaré","doi":"10.3934/dcdss.2023208","DOIUrl":"https://doi.org/10.3934/dcdss.2023208","url":null,"abstract":"","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"42 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025995","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 mobility-SAV approach for a Cahn-Hilliard equation with degenerate mobilities","authors":"Elie Bretin, Luca Calatroni, Simon Masnou","doi":"10.3934/dcdss.2023177","DOIUrl":"https://doi.org/10.3934/dcdss.2023177","url":null,"abstract":"","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"49 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140026325","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 consider a nonlinear system of PDEs coupling the viscous Cahn-Hilliard-Oono equation with dynamic boundary conditions enjoying a similar structure on the boundary. After proving well-posedness of the corresponding initial boundary value problem, we study an associated optimal control problem related to a tracking-type cost functional, proving first-order necessary conditions of optimality. The controls enter the system in the form of a distributed and a boundary source. We can account for general potentials in the bulk and in the boundary part under the common assumption that the boundary potential is dominant with respect to the bulk one. For example, the regular quartic potential as well as the logarithmic potential can be considered in our analysis.
{"title":"Well-posedness and optimal control for a viscous Cahn–Hilliard–Oono system with dynamic boundary conditions","authors":"G. Gilardi, E. Rocca, A. Signori","doi":"10.3934/dcdss.2023127","DOIUrl":"https://doi.org/10.3934/dcdss.2023127","url":null,"abstract":"In this paper we consider a nonlinear system of PDEs coupling the viscous Cahn-Hilliard-Oono equation with dynamic boundary conditions enjoying a similar structure on the boundary. After proving well-posedness of the corresponding initial boundary value problem, we study an associated optimal control problem related to a tracking-type cost functional, proving first-order necessary conditions of optimality. The controls enter the system in the form of a distributed and a boundary source. We can account for general potentials in the bulk and in the boundary part under the common assumption that the boundary potential is dominant with respect to the bulk one. For example, the regular quartic potential as well as the logarithmic potential can be considered in our analysis.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"32 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84527366","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 address a distributed control problem for a system of partial differential equations describing the evolution of a tumor that takes the biological mechanism of chemotaxis into account. The system describing the evolution is obtained as a nontrivial combination of a Cahn-Hilliard type system accounting for the segregation between tumor cells and healthy cells, with a Keller-Segel type equation accounting for the evolution of a nutrient species and modeling the chemotaxis phenomenon. First, we develop a robust mathematical background that allows us to analyze an associated optimal control problem. This analysis forced us to select a source term of logistic type in the nutrient equation and to restrict the analysis to the case of two space dimensions. Then, the existence of an optimal control and first-order necessary conditions for optimality are established.
{"title":"Nutrient control for a viscous Cahn–Hilliard–Keller–Segel model with logistic source describing tumor growth","authors":"G. Gilardi, A. Signori, J. Sprekels","doi":"10.3934/dcdss.2023123","DOIUrl":"https://doi.org/10.3934/dcdss.2023123","url":null,"abstract":"In this paper, we address a distributed control problem for a system of partial differential equations describing the evolution of a tumor that takes the biological mechanism of chemotaxis into account. The system describing the evolution is obtained as a nontrivial combination of a Cahn-Hilliard type system accounting for the segregation between tumor cells and healthy cells, with a Keller-Segel type equation accounting for the evolution of a nutrient species and modeling the chemotaxis phenomenon. First, we develop a robust mathematical background that allows us to analyze an associated optimal control problem. This analysis forced us to select a source term of logistic type in the nutrient equation and to restrict the analysis to the case of two space dimensions. Then, the existence of an optimal control and first-order necessary conditions for optimality are established.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"103 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79454287","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 consider a coupled system, known as Kobayashi--Warren--Carter system, abbreviated as the KWC system. KWC system consists of an Allen--Cahn type equation and a singular diffusion equation, and it was proposed by [Kobayashi et al, Phys. D, 140, 141--150 (2000)] as a possible mathematical model of grain boundary motion. The focus of this work is on the dynamic boundary condition imposed in our KWC system, and the mathematical interest is in a conflicting situation between: the continuity of the transmission condition included in the dynamic boundary condition; and the discontinuity encouraged by the singular diffusion equation. On this basis, we will prove the Main Theorem concerned with the existence of solution to our KWC system with energy-dissipation. Additionally, as a sub-result, we will prove a key-lemma that is to give a certain mathematical interpretation for the conflicting situation.
在本文中，我们考虑一个耦合系统，称为Kobayashi- Warren- Carter系统，简称为KWC系统。KWC系统由Allen—Cahn型方程和奇异扩散方程组成，由[Kobayashi et al .， Phys]提出。作为晶界运动可能的数学模型[D]， 140, 141—150(2000)。本工作的重点是在我们的KWC系统中施加的动态边界条件，而数学上的兴趣处于一种冲突的局面:动态边界条件中包含的传输条件的连续性;以及由奇异扩散方程引起的不连续。在此基础上，我们将证明具有能量耗散的KWC系统解的存在性的主要定理。此外，作为子结果，我们将证明一个关键引理，即对冲突情况给出一定的数学解释。
{"title":"Kobayashi-Warren-Carter system of singular type under dynamic boundary condition","authors":"Ryota Nakayashiki, K. Shirakawa","doi":"10.3934/dcdss.2023162","DOIUrl":"https://doi.org/10.3934/dcdss.2023162","url":null,"abstract":"In this paper, we consider a coupled system, known as Kobayashi--Warren--Carter system, abbreviated as the KWC system. KWC system consists of an Allen--Cahn type equation and a singular diffusion equation, and it was proposed by [Kobayashi et al, Phys. D, 140, 141--150 (2000)] as a possible mathematical model of grain boundary motion. The focus of this work is on the dynamic boundary condition imposed in our KWC system, and the mathematical interest is in a conflicting situation between: the continuity of the transmission condition included in the dynamic boundary condition; and the discontinuity encouraged by the singular diffusion equation. On this basis, we will prove the Main Theorem concerned with the existence of solution to our KWC system with energy-dissipation. Additionally, as a sub-result, we will prove a key-lemma that is to give a certain mathematical interpretation for the conflicting situation.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"78 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74509825","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 show that the classical strong maximum principle, concerning positive supersolutions of linear elliptic equations vanishing on the boundary of the domain $Omega $ can be extended, under suitable conditions, to the case in which the forcing term $f(x)$ is changing sign. In addition, in the case of solutions, the normal derivative on the boundary may also vanish on the boundary (definition of flat solutions). This leads to examples in which the unique continuation property fails. As a first application, we show the existence of positive solutions for a sublinear semilinear elliptic problem of indefinite sign. A second application, concerning the positivity of solutions of the linear heat equation, for some large values of time, with forcing and/or initial datum changing sign is also given.
{"title":"Beyond the classical strong maximum principle: Forcing changing sign near the boundary and flat solutions","authors":"Jes'us Ildefonso D'iaz, J. Hern'andez","doi":"10.3934/dcdss.2023151","DOIUrl":"https://doi.org/10.3934/dcdss.2023151","url":null,"abstract":"We show that the classical strong maximum principle, concerning positive supersolutions of linear elliptic equations vanishing on the boundary of the domain $Omega $ can be extended, under suitable conditions, to the case in which the forcing term $f(x)$ is changing sign. In addition, in the case of solutions, the normal derivative on the boundary may also vanish on the boundary (definition of flat solutions). This leads to examples in which the unique continuation property fails. As a first application, we show the existence of positive solutions for a sublinear semilinear elliptic problem of indefinite sign. A second application, concerning the positivity of solutions of the linear heat equation, for some large values of time, with forcing and/or initial datum changing sign is also given.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"104 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79185232","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}
Well-posedness in $L_infty$ of the nonlocal Gray-Scott model is studied for integrable kernels, along with the stability of the semi-trivial spatially homogeneous steady state. In addition, it is shown that the solutions to the nonlocal Gray-Scott system converge to those to the classical Gray-Scott system in the diffusive limit.
{"title":"A nonlocal Gray-Scott model: Well-posedness and diffusive limit","authors":"Philippe Laurencçot, Christoph Walker","doi":"10.3934/dcdss.2023158","DOIUrl":"https://doi.org/10.3934/dcdss.2023158","url":null,"abstract":"Well-posedness in $L_infty$ of the nonlocal Gray-Scott model is studied for integrable kernels, along with the stability of the semi-trivial spatially homogeneous steady state. In addition, it is shown that the solutions to the nonlocal Gray-Scott system converge to those to the classical Gray-Scott system in the diffusive limit.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"126 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87635170","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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