We study emergent behaviors of thermomechanical Cucker-Smale (TCS) ensemble confined in a harmonic potential field. In the absence of external force field, emergent dynamics of TCS particles has been extensively studied recently under various frameworks formulated in terms of initial configuration, system parameters and network topologies. Moreover, the TCS model does not exhibit rotating motions in the absence of an external force field. In this paper, we show the emergence of periodically rotating one-point cluster for the TCS model in a harmonic potential field using elementary energy estimates and continuity argument. We also provide several numerical simulations and compare them with analytical results.
本文研究了受谐波势场约束的热机械cucker - small (TCS)系综的涌现行为。在没有外力作用的情况下，TCS粒子在初始构型、系统参数和网络拓扑结构等不同框架下的涌现动力学得到了广泛的研究。此外，在没有外力的情况下，TCS模型不表现出旋转运动。本文利用初等能量估计和连续性论证，证明了调和势场中TCS模型周期性旋转的一点簇的出现。我们还提供了几个数值模拟，并与分析结果进行了比较。
{"title":"Emergence of a periodically rotating one-point cluster in a thermodynamic Cucker-Smale ensemble","authors":"H. Cho, Linglong Du, Seung‐Yeal Ha","doi":"10.1090/qam/1602","DOIUrl":"https://doi.org/10.1090/qam/1602","url":null,"abstract":"We study emergent behaviors of thermomechanical Cucker-Smale (TCS) ensemble confined in a harmonic potential field. In the absence of external force field, emergent dynamics of TCS particles has been extensively studied recently under various frameworks formulated in terms of initial configuration, system parameters and network topologies. Moreover, the TCS model does not exhibit rotating motions in the absence of an external force field. In this paper, we show the emergence of periodically rotating one-point cluster for the TCS model in a harmonic potential field using elementary energy estimates and continuity argument. We also provide several numerical simulations and compare them with analytical results.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48436989","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 the inertial spin model which consists of two variables: velocity as a mechanical observable and spin as an internal variable. In this paper, we slightly modified the original inertial spin model where the spin in the dynamics of the velocity is replaced by the average of spins. Moreover, by introducing two external control functions (rotation control and alignment control), we show the emergence of velocity and spin alignments mainly depends on these control functions. Finally, we perform numerical simulations that support and complement our theoretical results.
{"title":"Complete solvability of the inertial spin model with an averaged spin","authors":"Hyungjin Huh, Dohyun Kim","doi":"10.1090/qam/1601","DOIUrl":"https://doi.org/10.1090/qam/1601","url":null,"abstract":"We study the inertial spin model which consists of two variables: velocity as a mechanical observable and spin as an internal variable. In this paper, we slightly modified the original inertial spin model where the spin in the dynamics of the velocity is replaced by the average of spins. Moreover, by introducing two external control functions (rotation control and alignment control), we show the emergence of velocity and spin alignments mainly depends on these control functions. Finally, we perform numerical simulations that support and complement our theoretical results.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42538147","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 the degenerate semi-linear Schrödinger and Korteweg-de Vries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which turn out to have compact support, compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDEs and for appropriate variational problems as well. We provide a complete spectral characterization of such waves, for all values of p p . Namely, we show that all waves are spectrally stable for 2 > p ≤ 8 2>pleq 8 , while a single mode instability occurs for p > 8 p>8 . This extends previous work of Germain, Harrop-Griffiths and Marzuola [Quart. Appl. Math. 78 (2020), pp. 1–32] who have previously established orbital stability for some specific waves, in the range p > 8 p>8 .
{"title":"On the stability of the compacton waves for the degenerate KdV and NLS models","authors":"S. Hakkaev, A. Ramadan, A. Stefanov","doi":"10.1090/qam/1616","DOIUrl":"https://doi.org/10.1090/qam/1616","url":null,"abstract":"In this paper, we consider the degenerate semi-linear Schrödinger and Korteweg-de Vries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which turn out to have compact support, compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDEs and for appropriate variational problems as well. We provide a complete spectral characterization of such waves, for all values of \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000. Namely, we show that all waves are spectrally stable for \u0000\u0000 \u0000 \u0000 2\u0000 >\u0000 p\u0000 ≤\u0000 8\u0000 \u0000 2>pleq 8\u0000 \u0000\u0000, while a single mode instability occurs for \u0000\u0000 \u0000 \u0000 p\u0000 >\u0000 8\u0000 \u0000 p>8\u0000 \u0000\u0000. This extends previous work of Germain, Harrop-Griffiths and Marzuola [Quart. Appl. Math. 78 (2020), pp. 1–32] who have previously established orbital stability for some specific waves, in the range \u0000\u0000 \u0000 \u0000 p\u0000 >\u0000 8\u0000 \u0000 p>8\u0000 \u0000\u0000.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49063822","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}
Susanna V. Haziot, V. M. Hur, W. Strauss, J. Toland, Erik Wahl'en, Samuel Walsh, Miles H. Wheeler
This survey covers the mathematical theory of steady water waves with an emphasis on topics that are at the forefront of current research. These areas include: variational characterizations of traveling water waves; analytical and numerical studies of periodic waves with critical layers that may overhang; existence, nonexistence, and qualitative theory of solitary waves and fronts; traveling waves with localized vorticity or density stratification; and waves in three dimensions.
{"title":"Traveling water waves — the ebb and flow of two centuries","authors":"Susanna V. Haziot, V. M. Hur, W. Strauss, J. Toland, Erik Wahl'en, Samuel Walsh, Miles H. Wheeler","doi":"10.1090/qam/1614","DOIUrl":"https://doi.org/10.1090/qam/1614","url":null,"abstract":"This survey covers the mathematical theory of steady water waves with an emphasis on topics that are at the forefront of current research. These areas include: variational characterizations of traveling water waves; analytical and numerical studies of periodic waves with critical layers that may overhang; existence, nonexistence, and qualitative theory of solitary waves and fronts; traveling waves with localized vorticity or density stratification; and waves in three dimensions.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46799026","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 the large time behaviour of the solution of a linear dispersive PDEs posed on a finite interval, when the prescribed boundary conditions are time periodic. We use the approach pioneered in [7] for nonlinear integrable PDEs. and then applied to linear problems on the half-line in [6], to characterise necessary conditions for the solution of such a problem to be periodic, at least in an asymptotic sense. We then fully describe the periodicity properties of the solution in three important illustrative examples, recovering known results for the second-order cases and establishing new ones for the third order one.
{"title":"Time-periodic linear boundary value problems on a finite interval","authors":"A. S. Fokas, B. Pelloni, D. Smith","doi":"10.1090/qam/1615","DOIUrl":"https://doi.org/10.1090/qam/1615","url":null,"abstract":"We study the large time behaviour of the solution of a linear dispersive PDEs posed on a finite interval, when the prescribed boundary conditions are time periodic. We use the approach pioneered in [7] for nonlinear integrable PDEs. and then applied to linear problems on the half-line in [6], to characterise necessary conditions for the solution of such a problem to be periodic, at least in an asymptotic sense. We then fully describe the periodicity properties of the solution in three important illustrative examples, recovering known results for the second-order cases and establishing new ones for the third order one.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41434493","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 formal methodology for developing variational principles corresponding to a given nonlinear PDE system is discussed. The scheme is demonstrated in the context of the incompressible Navier-Stokes equations, systems of first-order conservation laws, and systems of Hamilton-Jacobi equations.
{"title":"Variational principles for nonlinear PDE systems via duality","authors":"A. Acharya","doi":"10.1090/qam/1631","DOIUrl":"https://doi.org/10.1090/qam/1631","url":null,"abstract":"A formal methodology for developing variational principles corresponding to a given nonlinear PDE system is discussed. The scheme is demonstrated in the context of the incompressible Navier-Stokes equations, systems of first-order conservation laws, and systems of Hamilton-Jacobi equations.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47828446","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}
Vishnu Raveendran, E. Cirillo, I. Bonis, A. Muntean
We study the question of periodic homogenization of a variably scaled reaction-diffusion problem with non-linear drift posed for a domain crossed by a flat composite thin layer. The structure of the non-linearity in the drift was obtained in earlier works as hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) for a population of interacting particles crossing a domain with obstacle. Using energy-type estimates as well as concepts like thin-layer convergence and two-scale convergence, we derive the homogenized evolution equation and the corresponding effective model parameters for a regularized problem. Special attention is paid to the derivation of the effective transmission conditions across the separating limit interface in essentially two different situations: (i) finitely thin layer and (ii) infinitely thin layer. This study should be seen as a preliminary step needed for the investigation of averaging fast non-linear drifts across material interfaces—a topic with direct applications in the design of thin composite materials meant to be impenetrable to high-velocity impacts.
{"title":"Scaling effects on the periodic homogenization of a reaction-diffusion-convection problem posed in homogeneous domains connected by a thin composite layer","authors":"Vishnu Raveendran, E. Cirillo, I. Bonis, A. Muntean","doi":"10.1090/qam/1607","DOIUrl":"https://doi.org/10.1090/qam/1607","url":null,"abstract":"We study the question of periodic homogenization of a variably scaled reaction-diffusion problem with non-linear drift posed for a domain crossed by a flat composite thin layer. The structure of the non-linearity in the drift was obtained in earlier works as hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) for a population of interacting particles crossing a domain with obstacle.\u0000\u0000Using energy-type estimates as well as concepts like thin-layer convergence and two-scale convergence, we derive the homogenized evolution equation and the corresponding effective model parameters for a regularized problem. Special attention is paid to the derivation of the effective transmission conditions across the separating limit interface in essentially two different situations: (i) finitely thin layer and (ii) infinitely thin layer.\u0000\u0000This study should be seen as a preliminary step needed for the investigation of averaging fast non-linear drifts across material interfaces—a topic with direct applications in the design of thin composite materials meant to be impenetrable to high-velocity impacts.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44295435","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":"Models of bacteria swimming in a nematic liquid crystal","authors":"Mochong Duan, N. Walkington","doi":"10.1090/QAM/1598","DOIUrl":"https://doi.org/10.1090/QAM/1598","url":null,"abstract":"","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45818155","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":"The Cauchy problem for a modified Euler-Poisson system in one dimension","authors":"Long Wei","doi":"10.1090/QAM/1597","DOIUrl":"https://doi.org/10.1090/QAM/1597","url":null,"abstract":"","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":"1"},"PeriodicalIF":0.8,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46343816","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":"Reversal permanent charge and concentrations in ionic flows via Poisson-Nernst-Planck models","authors":"Hamid Mofidi","doi":"10.1090/QAM/1593","DOIUrl":"https://doi.org/10.1090/QAM/1593","url":null,"abstract":"","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":"1"},"PeriodicalIF":0.8,"publicationDate":"2021-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48512080","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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