This work intends to prove that strong instabilities may appear for high order geometric optics expansions of weakly stable quasilinear hyperbolic boundary value problems, when the forcing boundary term is perturbed by a small amplitude oscillating function, with a transverse frequency. Since the boundary frequencies lie in the locus where the so-called Lopatinskii determinant is zero, the amplifications on the boundary give rise to a highly coupled system of equations for the profiles. A simplified model for this system is solved in an analytical framework using the Cauchy-Kovalevskaya theorem as well as a version of it ensuring analyticity in space and time for the solution. Then it is proven that, through resonances and amplification, a particular configuration for the phases may create an instability, in the sense that the small perturbation of the forcing term on the boundary interferes at the leading order in the asymptotic expansion of the solution. Finally we study the possibility for such a configuration of frequencies to happen for the isentropic Euler equations in space dimension three.
{"title":"Transverse instability of high frequency weakly stable quasilinear boundary value problems","authors":"Corentin Kilque","doi":"10.1090/qam/1637","DOIUrl":"https://doi.org/10.1090/qam/1637","url":null,"abstract":"This work intends to prove that strong instabilities may appear for high order geometric optics expansions of weakly stable quasilinear hyperbolic boundary value problems, when the forcing boundary term is perturbed by a small amplitude oscillating function, with a transverse frequency. Since the boundary frequencies lie in the locus where the so-called Lopatinskii determinant is zero, the amplifications on the boundary give rise to a highly coupled system of equations for the profiles. A simplified model for this system is solved in an analytical framework using the Cauchy-Kovalevskaya theorem as well as a version of it ensuring analyticity in space and time for the solution. Then it is proven that, through resonances and amplification, a particular configuration for the phases may create an instability, in the sense that the small perturbation of the forcing term on the boundary interferes at the leading order in the asymptotic expansion of the solution. Finally we study the possibility for such a configuration of frequencies to happen for the isentropic Euler equations in space dimension three.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44908045","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 article, we construct non-self-similar Riemann solutions for a two-dimensional quasilinear hyperbolic system of conservation laws which describes the fluid flow in a thin film for a perfectly soluble anti-surfactant solution. The initial Riemann data consists of two different constant states separated by a smooth curve in x − y x-y plane, so without using self-similarity transformation and dimension reduction, we establish solutions for five different cases. Further, we consider interaction of all possible nonlinear waves by taking initial discontinuity curve as a parabola to develop the structure of global entropy solutions explicitly.
{"title":"Two-dimensional non-self-similar Riemann solutions for a thin film model of a perfectly soluble anti-surfactant solution","authors":"R. Barthwal, T. Raja Sekhar","doi":"10.1090/qam/1625","DOIUrl":"https://doi.org/10.1090/qam/1625","url":null,"abstract":"In this article, we construct non-self-similar Riemann solutions for a two-dimensional quasilinear hyperbolic system of conservation laws which describes the fluid flow in a thin film for a perfectly soluble anti-surfactant solution. The initial Riemann data consists of two different constant states separated by a smooth curve in \u0000\u0000 \u0000 \u0000 x\u0000 −\u0000 y\u0000 \u0000 x-y\u0000 \u0000\u0000 plane, so without using self-similarity transformation and dimension reduction, we establish solutions for five different cases. Further, we consider interaction of all possible nonlinear waves by taking initial discontinuity curve as a parabola to develop the structure of global entropy solutions explicitly.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48021412","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 functional semilinear Rayleigh-Stokes equation involving fractional derivative. Our aim is to analyze some circumstances, in those the global solvability, and asymptotic behavior of solutions are addressed. By establishing a Halanay type inequality, we show the dissipativity and asymptotic stability of solutions to our problem. In addition, we prove the existence of a compact set of decay solutions by using local estimates and fixed point arguments.
{"title":"On stability for semilinear generalized Rayleigh-Stokes equation involving delays","authors":"Do Lan, P. Tuan","doi":"10.1090/qam/1624","DOIUrl":"https://doi.org/10.1090/qam/1624","url":null,"abstract":"We consider a functional semilinear Rayleigh-Stokes equation involving fractional derivative. Our aim is to analyze some circumstances, in those the global solvability, and asymptotic behavior of solutions are addressed. By establishing a Halanay type inequality, we show the dissipativity and asymptotic stability of solutions to our problem. In addition, we prove the existence of a compact set of decay solutions by using local estimates and fixed point arguments.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42594948","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 asymptotic emergent dynamics and the continuum limit for the Schrödinger-Lohe (SL) model and semi-discrete SL model. For the SL model, emergent dynamics has been mostly studied for systems with identical potentials in literature. In this paper, we further extend emergent dynamics and stability estimate for the SL model with nonidentical potentials. To achieve this, we use two-point correlation functions defined as an inner product between wave functions. For the semi-discrete SL model, we provide a global unique solvability and a sufficient framework for the smooth transition from the semi-discrete SL model to the SL model in any finite-time interval, as the mesh size tends to zero. Our convergence estimate depends on the uniform-in- h h Strichartz estimate and the uniform-stability of the SL models with respect to initial data.
研究了Schrödinger-Lohe (SL)模型和半离散SL模型的渐近涌现动力学和连续极限。对于SL模型，文献中大多研究具有相同势的系统的涌现动力学。在本文中，我们进一步推广了具有非相同电位的SL模型的涌现动力学和稳定性估计。为了实现这一点，我们使用两点相关函数定义为波函数之间的内积。对于半离散SL模型，我们提供了一个全局唯一的可解性和一个充分的框架，使得半离散SL模型在任何有限时间间隔内，当网格大小趋于零时，可以平滑地从半离散SL模型过渡到SL模型。我们的收敛估计依赖于均匀- h - h strihartz估计和SL模型相对于初始数据的均匀稳定性。
{"title":"Two-point correlation function and its applications to the Schrödinger-Lohe type models","authors":"Seung‐Yeal Ha, Gyuyoung Hwang, Dohyun Kim","doi":"10.1090/qam/1623","DOIUrl":"https://doi.org/10.1090/qam/1623","url":null,"abstract":"We study the asymptotic emergent dynamics and the continuum limit for the Schrödinger-Lohe (SL) model and semi-discrete SL model. For the SL model, emergent dynamics has been mostly studied for systems with identical potentials in literature. In this paper, we further extend emergent dynamics and stability estimate for the SL model with nonidentical potentials. To achieve this, we use two-point correlation functions defined as an inner product between wave functions. For the semi-discrete SL model, we provide a global unique solvability and a sufficient framework for the smooth transition from the semi-discrete SL model to the SL model in any finite-time interval, as the mesh size tends to zero. Our convergence estimate depends on the uniform-in-\u0000\u0000 \u0000 h\u0000 h\u0000 \u0000\u0000 Strichartz estimate and the uniform-stability of the SL models with respect to initial data.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41582760","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 a reaction-diffusion-convection problem with non-linear drift posed in a domain with periodically arranged obstacles. The non-linearity in the drift is linked to the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. Because of the imposed large drift scaling, this non-linearity is expected to explode in the limit of a vanishing scaling parameter. As main working techniques, we employ two-scale formal homogenization asymptotics with drift to derive the corresponding upscaled model equations as well as the structure of the effective transport tensors. Finally, we use Schauder’s fixed point theorem as well as monotonicity arguments to study the weak solvability of the upscaled model posed in an unbounded domain. This study wants to contribute with theoretical understanding needed when designing thin composite materials that are resistant to high velocity impacts.
{"title":"Upscaling of a reaction-diffusion-convection problem with exploding non-linear drift","authors":"Vishnu Raveendran, E. Cirillo, A. Muntean","doi":"10.1090/qam/1622","DOIUrl":"https://doi.org/10.1090/qam/1622","url":null,"abstract":"We study a reaction-diffusion-convection problem with non-linear drift posed in a domain with periodically arranged obstacles. The non-linearity in the drift is linked to the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. Because of the imposed large drift scaling, this non-linearity is expected to explode in the limit of a vanishing scaling parameter. As main working techniques, we employ two-scale formal homogenization asymptotics with drift to derive the corresponding upscaled model equations as well as the structure of the effective transport tensors. Finally, we use Schauder’s fixed point theorem as well as monotonicity arguments to study the weak solvability of the upscaled model posed in an unbounded domain. This study wants to contribute with theoretical understanding needed when designing thin composite materials that are resistant to high velocity impacts.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43745816","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":"Error estimates for pressure-driven Hele-Shaw flow","authors":"J. Fabricius, Salvador Manjate, P. Wall","doi":"10.1090/qam/1619","DOIUrl":"https://doi.org/10.1090/qam/1619","url":null,"abstract":"","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41618691","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 review very briefly the main mathematical structures and results in some important areas of Quantum Mechanics involving PDEs and formulate open problems.
我们简要回顾了量子力学中涉及偏微分方程的一些重要领域的主要数学结构和结果，并提出了开放问题。
{"title":"Differential equations of quantum mechanics","authors":"I. Sigal","doi":"10.1090/qam/1611","DOIUrl":"https://doi.org/10.1090/qam/1611","url":null,"abstract":"We review very briefly the main mathematical structures and results in some important areas of Quantum Mechanics involving PDEs and formulate open problems.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48567205","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 concerns the two-dimensional free surface cavity flow past an inclined plate in a finite depth. To close the cavity, a Riabouchinsky model is considered. The fluid is assumed to be inviscid and incompressible and the flow to be steady and irrotational. When the gravity and surface tension are negligible, an exact free streamline solution is derived. We solve the cavity flow problem by using two numerical methods. These methods allow us to compute solutions including the effects of gravity and surface tension. The first method is the series truncation and the second is the boundary integral equations, based on Cauchy integral formula. Numerical solutions are found for different values of the angle of inclination γ gamma and for various values of the Weber number, the Froude number. Good agreement between the two numerical schemes and the exact solution provides a check on the numerical methods.
{"title":"The influence of surface tension and gravity on cavitating flow past an inclined plate in a channel","authors":"A. Laiadi","doi":"10.1090/qam/1617","DOIUrl":"https://doi.org/10.1090/qam/1617","url":null,"abstract":"This paper concerns the two-dimensional free surface cavity flow past an inclined plate in a finite depth. To close the cavity, a Riabouchinsky model is considered. The fluid is assumed to be inviscid and incompressible and the flow to be steady and irrotational. When the gravity and surface tension are negligible, an exact free streamline solution is derived. We solve the cavity flow problem by using two numerical methods. These methods allow us to compute solutions including the effects of gravity and surface tension. The first method is the series truncation and the second is the boundary integral equations, based on Cauchy integral formula. Numerical solutions are found for different values of the angle of inclination \u0000\u0000 \u0000 γ\u0000 gamma\u0000 \u0000\u0000 and for various values of the Weber number, the Froude number. Good agreement between the two numerical schemes and the exact solution provides a check on the numerical methods.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47592856","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}
Hyunjin Ahn, Seung‐Yeal Ha, Doheon Kim, F. Schlöder, Woojoo Shim
We study the mean-field limit of the Cucker-Smale (C-S) model for flocking on complete smooth Riemannian manifolds. For this, we first formally derive the kinetic manifold C-S model on manifolds using the BBGKY hierarchy and derive several a priori estimates on emergent dynamics. Then, we present a rigorous mean-field limit from the particle model to the corresponding kinetic model by using the generalized particle-in-cell method. As a byproduct of our rigorous mean-field limit estimate, we also establish a global existence of a measure-valued solution for the derived kinetic model. Compared to the corresponding results on R d mathbb {R}^d , our procedure requires additional assumption on holonomy and proper a priori bound on the derivative of parallel transports. As a concrete example, we verify that hyperbolic space H d mathbb {H}^d satisfies our proposed standing assumptions.
研究了完全光滑黎曼流形上簇的cucker - small (C-S)模型的平均场极限。为此，我们首先使用BBGKY层次正式推导了流形上的动力学流形C-S模型，并推导了对紧急动力学的几个先验估计。然后，利用广义胞内粒子法给出了从粒子模型到相应动力学模型的严格平均场极限。作为我们严格的平均场极限估计的副产品，我们还建立了导出的动力学模型的测量值解的全局存在性。与R d mathbb {R}^d上的相应结果相比，我们的方法需要额外的完整假设和平行输运导数的固有先验界。作为一个具体的例子，我们验证了双曲空间H d mathbb {H}^d满足我们提出的常值假设。
{"title":"The mean-field limit of the Cucker-Smale model on complete Riemannian manifolds","authors":"Hyunjin Ahn, Seung‐Yeal Ha, Doheon Kim, F. Schlöder, Woojoo Shim","doi":"10.1090/qam/1613","DOIUrl":"https://doi.org/10.1090/qam/1613","url":null,"abstract":"We study the mean-field limit of the Cucker-Smale (C-S) model for flocking on complete smooth Riemannian manifolds. For this, we first formally derive the kinetic manifold C-S model on manifolds using the BBGKY hierarchy and derive several a priori estimates on emergent dynamics. Then, we present a rigorous mean-field limit from the particle model to the corresponding kinetic model by using the generalized particle-in-cell method. As a byproduct of our rigorous mean-field limit estimate, we also establish a global existence of a measure-valued solution for the derived kinetic model. Compared to the corresponding results on \u0000\u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 d\u0000 \u0000 mathbb {R}^d\u0000 \u0000\u0000, our procedure requires additional assumption on holonomy and proper a priori bound on the derivative of parallel transports. As a concrete example, we verify that hyperbolic space \u0000\u0000 \u0000 \u0000 \u0000 H\u0000 \u0000 d\u0000 \u0000 mathbb {H}^d\u0000 \u0000\u0000 satisfies our proposed standing assumptions.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45905864","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 identifies a non-(or /iso-)thermal variant of Ruggeri’s 1983 formulation of viscous heat-conductive fluid dynamics as a hyperbolic system of balance laws and shows that both the original model and this variant have (a) time-asymptotically stable equilibria and (b) principal parts deriving from a protopotential: a single scalar function that induces the temporospatial flux as an appropriate part of its Hessian.
{"title":"Time-asymptotic stability for first-order symmetric hyperbolic systems of balance laws in dissipative compressible fluid dynamics","authors":"H. Freistühler","doi":"10.1090/qam/1620","DOIUrl":"https://doi.org/10.1090/qam/1620","url":null,"abstract":"This paper identifies a non-(or /iso-)thermal variant of Ruggeri’s 1983 formulation of viscous heat-conductive fluid dynamics as a hyperbolic system of balance laws and shows that both the original model and this variant have (a) time-asymptotically stable equilibria and (b) principal parts deriving from a protopotential: a single scalar function that induces the temporospatial flux as an appropriate part of its Hessian.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45557377","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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