SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 464-476, April 2024. Abstract. This manuscript presents an analytic solution to a generalization of the Wiener–Hopf equation in two variables and with three unknown functions. This equation arises in many applications, for example, when solving the discrete Helmholtz equation associated with scattering on a domain with perpendicular boundary. The traditional Wiener–Hopf method is suitable for problems involving boundary data on co-linear semi-infinite intervals, not for boundaries at an angle. This significant extension will enable the analytical solution to a new class of problems with more boundary configurations. Progress is made by defining an underlining manifold that links the two variables. This allows one to meromorphically continue the unknown functions on this manifold and formulate a jump condition. As a result the problem is fully solvable in terms of Cauchy-type integrals, which is surprising since this is not always possible for this type of functional equation.
{"title":"A Generalization of the Wiener–Hopf Method for an Equation in Two Variables with Three Unknown Functions","authors":"Anastasia V. Kisil","doi":"10.1137/23m1562445","DOIUrl":"https://doi.org/10.1137/23m1562445","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 464-476, April 2024. <br/> Abstract. This manuscript presents an analytic solution to a generalization of the Wiener–Hopf equation in two variables and with three unknown functions. This equation arises in many applications, for example, when solving the discrete Helmholtz equation associated with scattering on a domain with perpendicular boundary. The traditional Wiener–Hopf method is suitable for problems involving boundary data on co-linear semi-infinite intervals, not for boundaries at an angle. This significant extension will enable the analytical solution to a new class of problems with more boundary configurations. Progress is made by defining an underlining manifold that links the two variables. This allows one to meromorphically continue the unknown functions on this manifold and formulate a jump condition. As a result the problem is fully solvable in terms of Cauchy-type integrals, which is surprising since this is not always possible for this type of functional equation.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"24 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140172699","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}
SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 412-432, April 2024. Abstract. We discuss mapping properties of the parameter-to-state map of full waveform inversion and generalize the results of [M. Eller and A. Rieder, Inverse Problems, 37 (2021), 085011] from the acoustic to the viscoelastic wave equation. In particular, we establish injectivity of the Fréchet derivative of the parameter-to-state map for a semidiscrete seismic inverse problem in the viscoelastic regime. Here the finite-dimensional parameter space is restricted to functions having global support in the propagation medium (the nonlocal case) and that are locally linearly independent. As a consequence, we deduce local conditional well-posedness of this nonlinear inverse problem. Furthermore, we show that the tangential cone condition holds, which is an essential prerequisite in the convergence analysis of a variety of inversion algorithms for nonlinear ill-posed problems.
SIAM 应用数学杂志》第 84 卷第 2 期第 412-432 页，2024 年 4 月。 摘要我们讨论了全波形反演的参数到状态图的映射特性，并将 [M. Eller and A. Rieder, Inverse Problems, 37 (2021), 085011] 的结果从声波方程推广到粘弹性波方程。特别是，我们为粘弹性机制中的半离散地震逆问题建立了参数到状态图的弗雷谢特导数的注入性。这里的有限维参数空间仅限于在传播介质中具有全局支持（非局部情况）且局部线性独立的函数。因此，我们推导出了这一非线性逆问题的局部条件好求性。此外，我们还证明了切向锥条件的成立，这是对非线性问题的各种反演算法进行收敛分析的基本前提。
{"title":"Tangential Cone Condition for the Full Waveform Forward Operator in the Viscoelastic Regime: The Nonlocal Case","authors":"Matthias Eller, Roland Griesmaier, Andreas Rieder","doi":"10.1137/23m1551845","DOIUrl":"https://doi.org/10.1137/23m1551845","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 412-432, April 2024. <br/> Abstract. We discuss mapping properties of the parameter-to-state map of full waveform inversion and generalize the results of [M. Eller and A. Rieder, Inverse Problems, 37 (2021), 085011] from the acoustic to the viscoelastic wave equation. In particular, we establish injectivity of the Fréchet derivative of the parameter-to-state map for a semidiscrete seismic inverse problem in the viscoelastic regime. Here the finite-dimensional parameter space is restricted to functions having global support in the propagation medium (the nonlocal case) and that are locally linearly independent. As a consequence, we deduce local conditional well-posedness of this nonlinear inverse problem. Furthermore, we show that the tangential cone condition holds, which is an essential prerequisite in the convergence analysis of a variety of inversion algorithms for nonlinear ill-posed problems.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"70 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140128091","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}
SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 433-463, April 2024. Abstract. A weakly nonlinear stability analysis of isothermal Poiseuille flow in a fluid overlying porous domain is proposed and investigated in this article. The nonlinear interactions are studied by imposing finite amplitude disturbances to the classical model deliberated in Chang, Chen, and Straughan [J. Fluid Mech., 564 (2006), pp. 287–303]. The order parameter theory is used to ascertain the cubic Landau equation, and the regimes of instability for the bifurcations are determined henceforth. The well-established controlling parameters viz. the depth ratio [math] depth of fluid domain/depth of porous domain), the Beavers–Joseph constant [math], and the Darcy number [math] are inquired upon for the bifurcation phenomena. The imposed finite amplitude disturbances are viewed for bifurcations along the neutral stability curves and away from the critical point as a function of the wave number [math] and the Reynolds number [math]. The even-fluid-layer (porous) mode along the neutral stability curves correlates to the subcritical (supercritical) bifurcation phenomena. On perceiving the bifurcations as a function of [math] and [math] by moving away from the bifurcation/critical point, subcritical bifurcation is observed for increasing [math] and decreasing [math]. In contrast to only fluid flow through a channel, it is found that the inclusion of porous domain aids in the early appearance of subcritical bifurcation when [math]. A considerable difference between the computed skin friction coefficient for the base and the distorted state is observed for small (large) values of [math]. In addition, an intrinsic relation among the mode of instability, bifurcation phenomena, and secondary flow pattern is also observed.
{"title":"Finite Amplitude Analysis of Poiseuille Flow in Fluid Overlying Porous Domain","authors":"A. Aleria, A. Khan, P. Bera","doi":"10.1137/23m1575809","DOIUrl":"https://doi.org/10.1137/23m1575809","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 433-463, April 2024. <br/> Abstract. A weakly nonlinear stability analysis of isothermal Poiseuille flow in a fluid overlying porous domain is proposed and investigated in this article. The nonlinear interactions are studied by imposing finite amplitude disturbances to the classical model deliberated in Chang, Chen, and Straughan [J. Fluid Mech., 564 (2006), pp. 287–303]. The order parameter theory is used to ascertain the cubic Landau equation, and the regimes of instability for the bifurcations are determined henceforth. The well-established controlling parameters viz. the depth ratio [math] depth of fluid domain/depth of porous domain), the Beavers–Joseph constant [math], and the Darcy number [math] are inquired upon for the bifurcation phenomena. The imposed finite amplitude disturbances are viewed for bifurcations along the neutral stability curves and away from the critical point as a function of the wave number [math] and the Reynolds number [math]. The even-fluid-layer (porous) mode along the neutral stability curves correlates to the subcritical (supercritical) bifurcation phenomena. On perceiving the bifurcations as a function of [math] and [math] by moving away from the bifurcation/critical point, subcritical bifurcation is observed for increasing [math] and decreasing [math]. In contrast to only fluid flow through a channel, it is found that the inclusion of porous domain aids in the early appearance of subcritical bifurcation when [math]. A considerable difference between the computed skin friction coefficient for the base and the distorted state is observed for small (large) values of [math]. In addition, an intrinsic relation among the mode of instability, bifurcation phenomena, and secondary flow pattern is also observed.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"76 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140128113","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}
Omar Morandi, Nella Rotundo, Alfio Borzì, Luigi Barletti
SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 387-411, April 2024. Abstract. The Wigner quasi-density function allows a phase-space formulation of statistical quantum mechanics that is of fundamental importance in theoretical investigation and in applications. This work contributes to these tasks with the formulation and analysis of an optimal control problem for the Wigner equation, which describes the time evolution of the quasi-density function. For this purpose, two possible control mechanisms are considered, and, correspondingly, a detailed analysis in weighted Sobolev spaces for the controlled nonhomogeneous Wigner equation is presented. Further theoretical results are reported concerning existence of optimal controls and differentiability of the control-to-state map and of the ensemble cost functional, which allows the derivation of the optimality system that characterizes the optimal controls sought.
{"title":"An Optimal Control Problem for the Wigner Equation","authors":"Omar Morandi, Nella Rotundo, Alfio Borzì, Luigi Barletti","doi":"10.1137/22m1515033","DOIUrl":"https://doi.org/10.1137/22m1515033","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 387-411, April 2024. <br/> Abstract. The Wigner quasi-density function allows a phase-space formulation of statistical quantum mechanics that is of fundamental importance in theoretical investigation and in applications. This work contributes to these tasks with the formulation and analysis of an optimal control problem for the Wigner equation, which describes the time evolution of the quasi-density function. For this purpose, two possible control mechanisms are considered, and, correspondingly, a detailed analysis in weighted Sobolev spaces for the controlled nonhomogeneous Wigner equation is presented. Further theoretical results are reported concerning existence of optimal controls and differentiability of the control-to-state map and of the ensemble cost functional, which allows the derivation of the optimality system that characterizes the optimal controls sought.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"30 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140070622","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}
SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 362-386, April 2024. Abstract. A lubrication model can be used to describe the dynamics of a weakly volatile viscous fluid layer on a hydrophobic substrate. Thin layers of the fluid are unstable to perturbations and break up into slowly evolving interacting droplets. A reduced-order dynamical system is derived from the lubrication model based on the nearest-neighbor droplet interactions in the weak condensation limit. Dynamics for periodic arrays of identical drops and pairwise droplet interactions are investigated, providing insights into the coarsening dynamics of a large droplet system. Weak condensation is shown to be a singular perturbation, fundamentally changing the long-time coarsening dynamics for the droplets and the overall mass of the fluid in two additional regimes of long-time dynamics.
{"title":"Coarsening of Thin Films with Weak Condensation","authors":"Hangjie Ji, Thomas P. Witelski","doi":"10.1137/23m1559336","DOIUrl":"https://doi.org/10.1137/23m1559336","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 362-386, April 2024. <br/> Abstract. A lubrication model can be used to describe the dynamics of a weakly volatile viscous fluid layer on a hydrophobic substrate. Thin layers of the fluid are unstable to perturbations and break up into slowly evolving interacting droplets. A reduced-order dynamical system is derived from the lubrication model based on the nearest-neighbor droplet interactions in the weak condensation limit. Dynamics for periodic arrays of identical drops and pairwise droplet interactions are investigated, providing insights into the coarsening dynamics of a large droplet system. Weak condensation is shown to be a singular perturbation, fundamentally changing the long-time coarsening dynamics for the droplets and the overall mass of the fluid in two additional regimes of long-time dynamics.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"31 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140043897","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}
SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 338-361, April 2024. Abstract. In this paper, a study is carried out on the spatiotemporal dynamics of a Lengyel–Epstein model describing the chlorite-iodine-malonic-acid (CIMA) reaction with time delay and the Neumann boundary condition in a two-dimensional region. The existences for Turing, Hopf, Turing–Turing, Turing–Hopf, and Bogdanov–Takens bifurcations are derived by analyzing the dispersion relation between eigenvalues and wave numbers. In particular, to study the dynamics around a Turing–Hopf bifurcation singularity, the amplitude equations near a codimension-two bifurcation point are derived by employing the weakly nonlinear analysis method. Different spatiotemporal patterns for the system in parameter space are classified and various patterns identified, including spatially homogeneous periodic solutions, mixed mode, coexistence mode, bistable phenomenon, square, hexagon, black eye, two-phase oscillating staggered hexagon lattice, and other complex spatiotemporal patterns. The theoretical predictions are verified by numerical simulations showing an excellent agreement with many reported experiment results not only in chemistry but also in physics and biology. Results presented in this article reveal the mechanism of generating the spatiotemporal patterns of the CIMA reaction.
{"title":"Spatiotemporal Patterns in a Lengyel–Epstein Model Near a Turing–Hopf Singular Point","authors":"Shuangrui Zhao, Pei Yu, Hongbin Wang","doi":"10.1137/23m1552668","DOIUrl":"https://doi.org/10.1137/23m1552668","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 338-361, April 2024. <br/> Abstract. In this paper, a study is carried out on the spatiotemporal dynamics of a Lengyel–Epstein model describing the chlorite-iodine-malonic-acid (CIMA) reaction with time delay and the Neumann boundary condition in a two-dimensional region. The existences for Turing, Hopf, Turing–Turing, Turing–Hopf, and Bogdanov–Takens bifurcations are derived by analyzing the dispersion relation between eigenvalues and wave numbers. In particular, to study the dynamics around a Turing–Hopf bifurcation singularity, the amplitude equations near a codimension-two bifurcation point are derived by employing the weakly nonlinear analysis method. Different spatiotemporal patterns for the system in parameter space are classified and various patterns identified, including spatially homogeneous periodic solutions, mixed mode, coexistence mode, bistable phenomenon, square, hexagon, black eye, two-phase oscillating staggered hexagon lattice, and other complex spatiotemporal patterns. The theoretical predictions are verified by numerical simulations showing an excellent agreement with many reported experiment results not only in chemistry but also in physics and biology. Results presented in this article reveal the mechanism of generating the spatiotemporal patterns of the CIMA reaction.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"26 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140017601","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}
SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 317-337, April 2024. Abstract. A method for computing the Riesz [math]-capacity, [math], of a general set [math] is given. The method is based on simulations of isotropic [math]-stable motion paths in [math]-dimensions. The familiar walk-on-spheres method, often utilized for simulating Brownian motion, is modified to a novel walk-in-and-out-of-balls method adapted for modeling the stable path process on the exterior of regions “probed” by this type of generalized random walk. It accounts for the propensity of this class of random walk to jump through boundaries because of the path discontinuity. This method allows for the computationally efficient simulation of hitting locations of stable paths launched from the exterior of probed sets. Reliable methods of computing capacity from these locations are given, along with non-standard confidence intervals. Illustrative calculations are performed for representative types of sets [math], where both [math] and [math] are varied.
{"title":"Computation of Riesz [math]-Capacity [math] of General Sets in [math] Using Stable Random Walks","authors":"John P. Nolan, Debra J. Audus, Jack F. Douglas","doi":"10.1137/23m1568077","DOIUrl":"https://doi.org/10.1137/23m1568077","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 317-337, April 2024. <br/> Abstract. A method for computing the Riesz [math]-capacity, [math], of a general set [math] is given. The method is based on simulations of isotropic [math]-stable motion paths in [math]-dimensions. The familiar walk-on-spheres method, often utilized for simulating Brownian motion, is modified to a novel walk-in-and-out-of-balls method adapted for modeling the stable path process on the exterior of regions “probed” by this type of generalized random walk. It accounts for the propensity of this class of random walk to jump through boundaries because of the path discontinuity. This method allows for the computationally efficient simulation of hitting locations of stable paths launched from the exterior of probed sets. Reliable methods of computing capacity from these locations are given, along with non-standard confidence intervals. Illustrative calculations are performed for representative types of sets [math], where both [math] and [math] are varied.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"40 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140007537","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}
SIAM Journal on Applied Mathematics, Volume 84, Issue 1, Page 285-315, February 2024. Abstract. We analyze and quantify the amount of heat generated by a nanoparticle, injected in a background medium, while excited by incident electromagnetic waves. These nanoparticles are dispersive with electric permittivity following the Lorentz model. The purpose is to determine the quantity of heat generated extremely close to the nanoparticle (at a distance proportional to the radius of the nanoparticle). This study extends our previous results, derived in the 2D TM and TE regimes, to the full Maxwell system. We show that by exciting the medium with incident frequencies close to the plasmonic or Dielectric resonant frequencies, we can generate any desired amount of heat close to the injected nanoparticle while the amount of heat decreases away from it. These results offer a wide range of potential applications in the areas of photo-thermal therapy, drug delivery, and material science, to cite a few. To do so, we employ time-domain integral equations and asymptotic analysis techniques to study the corresponding mathematical model for heat generation. This model is given by the heat equation where the body source term comes from the modulus of the electric field generated by the used incident electromagnetic field. Therefore, we first analyze the dominant term of this electric field by studying the full Maxwell scattering problem in the presence of plasmonic or all-dielectric nanoparticles. As a second step, we analyze the propagation of this dominant electric field in the estimation of the heat potential. For both the electromagnetic and parabolic models, the presence of the nanoparticles is translated into the appearance of large scales in the contrasts for the heat-conductivity (for the parabolic model) and the permittivity (for the full Maxwell system) between the nanoparticle and its surroundings.
{"title":"Heat Generation Using Lorentzian Nanoparticles. The Full Maxwell System","authors":"Arpan Mukherjee, Mourad Sini","doi":"10.1137/23m1547597","DOIUrl":"https://doi.org/10.1137/23m1547597","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 1, Page 285-315, February 2024. <br/> Abstract. We analyze and quantify the amount of heat generated by a nanoparticle, injected in a background medium, while excited by incident electromagnetic waves. These nanoparticles are dispersive with electric permittivity following the Lorentz model. The purpose is to determine the quantity of heat generated extremely close to the nanoparticle (at a distance proportional to the radius of the nanoparticle). This study extends our previous results, derived in the 2D TM and TE regimes, to the full Maxwell system. We show that by exciting the medium with incident frequencies close to the plasmonic or Dielectric resonant frequencies, we can generate any desired amount of heat close to the injected nanoparticle while the amount of heat decreases away from it. These results offer a wide range of potential applications in the areas of photo-thermal therapy, drug delivery, and material science, to cite a few. To do so, we employ time-domain integral equations and asymptotic analysis techniques to study the corresponding mathematical model for heat generation. This model is given by the heat equation where the body source term comes from the modulus of the electric field generated by the used incident electromagnetic field. Therefore, we first analyze the dominant term of this electric field by studying the full Maxwell scattering problem in the presence of plasmonic or all-dielectric nanoparticles. As a second step, we analyze the propagation of this dominant electric field in the estimation of the heat potential. For both the electromagnetic and parabolic models, the presence of the nanoparticles is translated into the appearance of large scales in the contrasts for the heat-conductivity (for the parabolic model) and the permittivity (for the full Maxwell system) between the nanoparticle and its surroundings.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"65 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139909629","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}
SIAM Journal on Applied Mathematics, Volume 84, Issue 1, Page 262-284, February 2024. Abstract. In this paper, we develop a general mathematical framework for perfect and approximate hydrodynamic cloaking and shielding of electro-osmotic flow, which is governed by a coupled PDE system via the field-effect electro-osmosis. We first establish the representation formula of the solution of the coupled system using the layer potential techniques. Based on the Fourier series, the perfect hydrodynamic cloaking and shielding conditions are derived for the control region with the cross-sectional shape being an annulus or a confocal ellipses. Then we further propose an optimization scheme for the design of approximate cloaks and shields within general geometries. The well-posedness of the optimization problem is proved. In particular, the conditions that can ensure the occurrence of approximate cloaks and shields for general geometries are also established. Our theoretical findings are validated and supplemented by a variety of numerical results. The results in this paper also provide a mathematical foundation for more complex hydrodynamic cloaking and shielding.
{"title":"A Mathematical Theory of Microscale Hydrodynamic Cloaking and Shielding by Electro-Osmosis","authors":"Hongyu Liu, Zhi-Qiang Miao, Guang-Hui Zheng","doi":"10.1137/23m1554837","DOIUrl":"https://doi.org/10.1137/23m1554837","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 1, Page 262-284, February 2024. <br/> Abstract. In this paper, we develop a general mathematical framework for perfect and approximate hydrodynamic cloaking and shielding of electro-osmotic flow, which is governed by a coupled PDE system via the field-effect electro-osmosis. We first establish the representation formula of the solution of the coupled system using the layer potential techniques. Based on the Fourier series, the perfect hydrodynamic cloaking and shielding conditions are derived for the control region with the cross-sectional shape being an annulus or a confocal ellipses. Then we further propose an optimization scheme for the design of approximate cloaks and shields within general geometries. The well-posedness of the optimization problem is proved. In particular, the conditions that can ensure the occurrence of approximate cloaks and shields for general geometries are also established. Our theoretical findings are validated and supplemented by a variety of numerical results. The results in this paper also provide a mathematical foundation for more complex hydrodynamic cloaking and shielding.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"16 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139771008","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}
Jeffrey Galkowski, Pierre Marchand, Jian Wang, Maciej Zworski
SIAM Journal on Applied Mathematics, Volume 84, Issue 1, Page 246-261, February 2024. Abstract. The scattering phase, defined as [math] where [math] is the (unitary) scattering matrix, is the analogue of the counting function for eigenvalues when dealing with exterior domains and is closely related to Kreĭn’s spectral shift function. We revisit classical results on asymptotics of the scattering phase and point out that it is never monotone in the case of strong trapping of waves. Perhaps more importantly, we provide the first numerical calculations of scattering phases for nonradial scatterers. They show that the asymptotic Weyl law is accurate even at low frequencies and reveal effects of trapping such as lack of monotonicity. This is achieved by using the recent high level multiphysics finite element software FreeFEM.
{"title":"The Scattering Phase: Seen at Last","authors":"Jeffrey Galkowski, Pierre Marchand, Jian Wang, Maciej Zworski","doi":"10.1137/23m1547147","DOIUrl":"https://doi.org/10.1137/23m1547147","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 1, Page 246-261, February 2024. <br/> Abstract. The scattering phase, defined as [math] where [math] is the (unitary) scattering matrix, is the analogue of the counting function for eigenvalues when dealing with exterior domains and is closely related to Kreĭn’s spectral shift function. We revisit classical results on asymptotics of the scattering phase and point out that it is never monotone in the case of strong trapping of waves. Perhaps more importantly, we provide the first numerical calculations of scattering phases for nonradial scatterers. They show that the asymptotic Weyl law is accurate even at low frequencies and reveal effects of trapping such as lack of monotonicity. This is achieved by using the recent high level multiphysics finite element software FreeFEM.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"39 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139771047","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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