The goal of this paper is to provide a rigorous justification of the asymptotic model proposed by Beneš et al. [Nonzero boundary condition for the unsteady micropolar pipe flow: well-posedness and asymptotics, Appl. Math. Comput. 427 (2022), Paper No. 127184, 22] for the time-dependent flow of a micropolar fluid in a thin cylindrical pipe. After proving the well-posedness of the governing initial-boundary value problem endowed with the dynamic boundary condition for the microrotation, we derive the suitable a priori estimates. Using this result, we evaluate the difference between the original solution and the asymptotic one in the corresponding functional norms. By doing that, we validate the usage of the proposed model and deduce the information about its order of accuracy.
{"title":"A remark on the nonsteady micropolar pipe flow with a dynamic boundary condition for the microrotation","authors":"Igor Pažanin, Borja Rukavina","doi":"10.1090/qam/1700","DOIUrl":"https://doi.org/10.1090/qam/1700","url":null,"abstract":"The goal of this paper is to provide a rigorous justification of the asymptotic model proposed by Beneš et al. [Nonzero boundary condition for the unsteady micropolar pipe flow: well-posedness and asymptotics, Appl. Math. Comput. 427 (2022), Paper No. 127184, 22] for the time-dependent flow of a micropolar fluid in a thin cylindrical pipe. After proving the well-posedness of the governing initial-boundary value problem endowed with the dynamic boundary condition for the microrotation, we derive the suitable a priori estimates. Using this result, we evaluate the difference between the original solution and the asymptotic one in the corresponding functional norms. By doing that, we validate the usage of the proposed model and deduce the information about its order of accuracy.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141814811","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 derive effective equations of a periodically heterogeneous Cosserat material encompassing intrinsic lengths modelling scale-size effects. The resultant homogenized material supports internal body torques and leads to an asymmetric effective stress providing a connection to the theory of odd elasticity. Furthermore, a link to the classical Cauchy stress is given. Moreover, the corresponding local problem exhibits asymmetry as well, due to the micropolar couple modulus inherited from the original microscopic Cosserat problem. We validate our results by conducting numerical simulations using the finite element method on circularly perforated square and rectangular unit cells, highlighting the impact, of not only volume fraction but also of internal body torques on effective coefficients. Additionally, we numerically quantify the “amount” that the body can torque internally.
{"title":"Scale-size dependent multi-continuum homogenization of complex bodies","authors":"Grigor Nika","doi":"10.1090/qam/1696","DOIUrl":"https://doi.org/10.1090/qam/1696","url":null,"abstract":"We derive effective equations of a periodically heterogeneous Cosserat material encompassing intrinsic lengths modelling scale-size effects. The resultant homogenized material supports internal body torques and leads to an asymmetric effective stress providing a connection to the theory of odd elasticity. Furthermore, a link to the classical Cauchy stress is given. Moreover, the corresponding local problem exhibits asymmetry as well, due to the micropolar couple modulus inherited from the original microscopic Cosserat problem. We validate our results by conducting numerical simulations using the finite element method on circularly perforated square and rectangular unit cells, highlighting the impact, of not only volume fraction but also of internal body torques on effective coefficients. Additionally, we numerically quantify the “amount” that the body can torque internally.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141371391","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 investigate the dynamics of a nonlinear model describing the motion of cells under the effect of porous-medium diffusion and transport and in the presence of nutrient and drug application. The momentum equation for the evolution of the velocity field is governed by Darcy’s law, while the evolution of the chemical attractant (nutrient or drug) is governed by a diffusion equation. The system evolves within a moving domain in R 3 mathbb {R}^3 accounting for the expansion or shrinkage of the tumor. The global existence of weak solutions is established with the aid of a regularized approximating scheme and an Arbitrary Lagrangian-Eulerian (ALE) mapping for the motion of the tumor. This work provides a variational framework suitable for both analysis and simulations.
我们研究了一个非线性模型的动力学，该模型描述了在多孔介质扩散和传输作用下以及在施用营养物和药物的情况下细胞的运动。速度场演变的动量方程受达西定律支配，而化学吸引剂（营养物或药物）的演变受扩散方程支配。系统在 R 3 mathbb {R}^3 的移动域内演化，考虑到肿瘤的扩张或收缩。借助正则化近似方案和肿瘤运动的任意拉格朗日-欧勒（ALE）映射，确定了弱解的全局存在性。这项工作提供了一个既适合分析又适合模拟的变分框架。
{"title":"On a nonlinear diffussive model for the evolution of cells within a moving domain","authors":"Tessa Thorsen, K. Trivisa","doi":"10.1090/qam/1690","DOIUrl":"https://doi.org/10.1090/qam/1690","url":null,"abstract":"We investigate the dynamics of a nonlinear model describing the motion of cells under the effect of porous-medium diffusion and transport and in the presence of nutrient and drug application. The momentum equation for the evolution of the velocity field is governed by Darcy’s law, while the evolution of the chemical attractant (nutrient or drug) is governed by a diffusion equation. The system evolves within a moving domain in \u0000\u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 3\u0000 \u0000 mathbb {R}^3\u0000 \u0000\u0000 accounting for the expansion or shrinkage of the tumor. The global existence of weak solutions is established with the aid of a regularized approximating scheme and an Arbitrary Lagrangian-Eulerian (ALE) mapping for the motion of the tumor. This work provides a variational framework suitable for both analysis and simulations.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141114144","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}
Thin polycrystalline solid state films, which are used in many technological applications, can exhibit various phenomena, such as wetting, dewetting, and hole formation. We focus on a model system containing two contacting grains which surround a hole. For simplicity, the system is assumed to be axisymmetric, to be supported by a planar substrate and to be bounded within an inert semi-infinite cylinder. We assume that the exterior surfaces of the grains evolve by surface diffusion and the grain boundary between the adjacent grains evolve by motion by mean curvature. Boundary conditions are imposed following W.W. Mullins, 1958. Parametric formulas are derived for the steady states, which contain two nodoids describing the exterior surfaces, which are coupled to a catenoid which describes the grain boundary. At steady state, the physical parameters of the system may be prescribed via two angles, β beta , the angle between the exterior surface and the grain boundary, and θ c theta _c , the contact angle between the exterior surface and the substrate; additionally, there are two dimensionless geometric parameters which must satisfy certain constraints. We prove that if β ∈ ( π / 2 , π ) beta in (pi /2, pi ) and θ c = π theta _c=pi , then there exists a continuum of steady states. Numerical calculations indicate that steady state profiles can exhibit physical features, such as hillock formation; a fuller numerical study of the steady states and their properties recently appeared in Zigelman and Novick-Cohen [J. Appl. Phys. 134 (2023), 135302], which relies on the formulas and results derived here.
{"title":"Coupled surface diffusion and mean curvature motion: An axisymmetric system with two grains and a hole","authors":"Katrine Golubkov, A. Novick-Cohen, Yotam Vaknin","doi":"10.1090/qam/1691","DOIUrl":"https://doi.org/10.1090/qam/1691","url":null,"abstract":"Thin polycrystalline solid state films, which are used in many technological applications, can exhibit various phenomena, such as wetting, dewetting, and hole formation. We focus on a model system containing two contacting grains which surround a hole. For simplicity, the system is assumed to be axisymmetric, to be supported by a planar substrate and to be bounded within an inert semi-infinite cylinder. We assume that the exterior surfaces of the grains evolve by surface diffusion and the grain boundary between the adjacent grains evolve by motion by mean curvature. Boundary conditions are imposed following W.W. Mullins, 1958. Parametric formulas are derived for the steady states, which contain two nodoids describing the exterior surfaces, which are coupled to a catenoid which describes the grain boundary. At steady state, the physical parameters of the system may be prescribed via two angles, \u0000\u0000 \u0000 β\u0000 beta\u0000 \u0000\u0000, the angle between the exterior surface and the grain boundary, and \u0000\u0000 \u0000 \u0000 θ\u0000 c\u0000 \u0000 theta _c\u0000 \u0000\u0000, the contact angle between the exterior surface and the substrate; additionally, there are two dimensionless geometric parameters which must satisfy certain constraints. We prove that if \u0000\u0000 \u0000 \u0000 β\u0000 ∈\u0000 (\u0000 π\u0000 \u0000 /\u0000 \u0000 2\u0000 ,\u0000 π\u0000 )\u0000 \u0000 beta in (pi /2, pi )\u0000 \u0000\u0000 and \u0000\u0000 \u0000 \u0000 \u0000 θ\u0000 c\u0000 \u0000 =\u0000 π\u0000 \u0000 theta _c=pi\u0000 \u0000\u0000, then there exists a continuum of steady states. Numerical calculations indicate that steady state profiles can exhibit physical features, such as hillock formation; a fuller numerical study of the steady states and their properties recently appeared in Zigelman and Novick-Cohen [J. Appl. Phys. 134 (2023), 135302], which relies on the formulas and results derived here.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140793128","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}
The Maxey-Riley-Gatignol (MRG) equation, which describes the dynamics of an inertial particle in nonuniform and unsteady flow, is an integro-differential equation with a memory term and its solution lacks a well-defined Taylor series at t = 0 t=0 . In particulate flows, one often seeks trajectories of millions of particles simultaneously, and the numerical solution to the MRG equation for each particle becomes prohibitively expensive due to its ever-rising memory costs. In this paper, we present an explicit numerical integrator for the MRG equation that inherits the benefits of standard time-integrators, namely a constant memory storage cost, a linear growth of operational effort with simulation time, and the ability to restart a simulation with the final state as the new initial condition. The integrator is based on a Markovian embedding of the MRG equation. The integrator and the embedding are consequences of a spectral representation of the solution to the linear MRG equation. We exploit these to extend the work of Cox and Matthews [J. Comput. Phys. 176 (2002), 430–455] and derive Runge-Kutta type iterative schemes of differing orders for the MRG equation. Our approach may be generalized to a large class of systems with memory effects.
{"title":"Explicit integrators for nonlocal equations: The case of the Maxey-Riley-Gatignol equation","authors":"Divya Jaganathan, Rama Govindarajan, V. Vasan","doi":"10.1090/qam/1693","DOIUrl":"https://doi.org/10.1090/qam/1693","url":null,"abstract":"The Maxey-Riley-Gatignol (MRG) equation, which describes the dynamics of an inertial particle in nonuniform and unsteady flow, is an integro-differential equation with a memory term and its solution lacks a well-defined Taylor series at \u0000\u0000 \u0000 \u0000 t\u0000 =\u0000 0\u0000 \u0000 t=0\u0000 \u0000\u0000. In particulate flows, one often seeks trajectories of millions of particles simultaneously, and the numerical solution to the MRG equation for each particle becomes prohibitively expensive due to its ever-rising memory costs. In this paper, we present an explicit numerical integrator for the MRG equation that inherits the benefits of standard time-integrators, namely a constant memory storage cost, a linear growth of operational effort with simulation time, and the ability to restart a simulation with the final state as the new initial condition. The integrator is based on a Markovian embedding of the MRG equation. The integrator and the embedding are consequences of a spectral representation of the solution to the linear MRG equation. We exploit these to extend the work of Cox and Matthews [J. Comput. Phys. 176 (2002), 430–455] and derive Runge-Kutta type iterative schemes of differing orders for the MRG equation. Our approach may be generalized to a large class of systems with memory effects.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140382522","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}
The paper proves existence of stationary solutions to the Boltzmann equation in a bounded set of R 2 mathbb {R}^2 for given indata, hard forces and truncation in the collision kernel for small velocities and close to parallel colliding velocities. It does not use any averaging in velocity lemma. Instead, it is based on stability techniques employing the Kolmogorov-Riesz-Fréchet theorem, from the discrete velocity stationary case, where the averaging in velocity lemmas are not valid.
论文证明了在 R 2 mathbb {R}^2 的有界集合中，对于给定的 indata、硬力和碰撞内核中的截断，小速度和接近平行碰撞速度的玻尔兹曼方程静止解的存在性。它不使用任何速度平均法。取而代之的是，它基于离散速度静止情况下的稳定性技术，采用了柯尔莫哥洛夫-里兹-弗雷谢定理，在离散速度静止情况下，速度平均定理是无效的。
{"title":"Stationary solutions to the Boltzmann equation in the plane","authors":"L. Arkeryd, A. Nouri","doi":"10.1090/qam/1692","DOIUrl":"https://doi.org/10.1090/qam/1692","url":null,"abstract":"The paper proves existence of stationary solutions to the Boltzmann equation in a bounded set of \u0000\u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 2\u0000 \u0000 mathbb {R}^2\u0000 \u0000\u0000 for given indata, hard forces and truncation in the collision kernel for small velocities and close to parallel colliding velocities. It does not use any averaging in velocity lemma. Instead, it is based on stability techniques employing the Kolmogorov-Riesz-Fréchet theorem, from the discrete velocity stationary case, where the averaging in velocity lemmas are not valid.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140384701","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}
The relativistic Vlasov-Maxwell-Landau (r-VML) system and the relativistic Landau (r-LAN) equation are fundamental models that describe the dynamics of an electron gas. In this paper, we introduce a novel weighted energy method and establish the validity of the Hilbert expansion for the one-species r-VML system and r-LAN equation. As the Knudsen number shrinks to zero, we rigorously demonstrate the relativistic Euler-Maxwell limit and relativistic Euler limit, respectively. This successfully resolves the long-standing open problem regarding the hydrodynamic limits of Landau-type equations.
{"title":"Hilbert expansion for Coulomb collisional kinetic models","authors":"Zhimeng Ouyang, Lei Wu, Qinghua Xiao","doi":"10.1090/qam/1689","DOIUrl":"https://doi.org/10.1090/qam/1689","url":null,"abstract":"The relativistic Vlasov-Maxwell-Landau (r-VML) system and the relativistic Landau (r-LAN) equation are fundamental models that describe the dynamics of an electron gas. In this paper, we introduce a novel weighted energy method and establish the validity of the Hilbert expansion for the one-species r-VML system and r-LAN equation. As the Knudsen number shrinks to zero, we rigorously demonstrate the relativistic Euler-Maxwell limit and relativistic Euler limit, respectively. This successfully resolves the long-standing open problem regarding the hydrodynamic limits of Landau-type equations.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140249623","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":"Preface for the second special issue in honor of Bob Pego","authors":"Govind Menon","doi":"10.1090/qam/1683","DOIUrl":"https://doi.org/10.1090/qam/1683","url":null,"abstract":"","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140442776","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 paper serves as a short corrigendum, describing an error in the derivation of the linearized equation. The main theorems remain correct, but they are based upon an incorrect linearization of the Birkhoff-Rott equation and hence are not directly applicable to the original physical problem.
{"title":"Corrigendum to “Non-linear singularity formation for circular vortex sheets”","authors":"Ryan Murray, Galen Wilcox","doi":"10.1090/qam/1688","DOIUrl":"https://doi.org/10.1090/qam/1688","url":null,"abstract":"This paper serves as a short corrigendum, describing an error in the derivation of the linearized equation. The main theorems remain correct, but they are based upon an incorrect linearization of the Birkhoff-Rott equation and hence are not directly applicable to the original physical problem.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139801577","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 paper serves as a short corrigendum, describing an error in the derivation of the linearized equation. The main theorems remain correct, but they are based upon an incorrect linearization of the Birkhoff-Rott equation and hence are not directly applicable to the original physical problem.
{"title":"Corrigendum to “Non-linear singularity formation for circular vortex sheets”","authors":"Ryan Murray, Galen Wilcox","doi":"10.1090/qam/1688","DOIUrl":"https://doi.org/10.1090/qam/1688","url":null,"abstract":"This paper serves as a short corrigendum, describing an error in the derivation of the linearized equation. The main theorems remain correct, but they are based upon an incorrect linearization of the Birkhoff-Rott equation and hence are not directly applicable to the original physical problem.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139861304","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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