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Global existence of strong solutions to the planar compressible magnetohydrodynamic equations with large initial data in unbounded domains 无界区域大初始数据平面可压缩磁流体动力学方程强解的整体存在性
arXiv: Analysis of PDEs Pub Date : 2020-09-18 DOI: 10.4310/cms.2021.v19.n6.a9
Boqiang Lu, Xiaoding Shi, C. Xiong
{"title":"Global existence of strong solutions to the planar compressible magnetohydrodynamic equations with large initial data in unbounded domains","authors":"Boqiang Lu, Xiaoding Shi, C. Xiong","doi":"10.4310/cms.2021.v19.n6.a9","DOIUrl":"https://doi.org/10.4310/cms.2021.v19.n6.a9","url":null,"abstract":"In one-dimensional unbounded domains, we consider the equations of a planar compressible magnetohydrodynamic (MHD) flow with constant viscosity and heat conductivity. More precisely, we prove the global existence of strong solutions to the MHD equations with large initial data satisfying the same conditions as those of Kazhikhov's theory in bounded domains (Kazhikhov 1987 Boundary Value Problems for Equations of Mathematical Physics (Krasnoyarsk)). In particular, our result generalizes the Kazhikhov's theory for the initial boundary value problem in bounded domains to the unbounded case.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82587016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Geometric series expansion of the Neumann–Poincaré operator: Application to composite materials neumann - poincarcarr算子的几何级数展开:在复合材料中的应用
arXiv: Analysis of PDEs Pub Date : 2020-09-16 DOI: 10.1017/S0956792521000127
E. Cherkaev, Minwoo Kim, Mikyoung Lim
{"title":"Geometric series expansion of the Neumann–Poincaré operator: Application to composite materials","authors":"E. Cherkaev, Minwoo Kim, Mikyoung Lim","doi":"10.1017/S0956792521000127","DOIUrl":"https://doi.org/10.1017/S0956792521000127","url":null,"abstract":"The Neumann-Poincare operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the Neumann-Poincare operator was developed in two dimensions based on geometric function theory. In this paper, we investigate geometric properties of composite materials by using this series expansion. In particular, we obtain explicit formulas for the polarization tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the Neumann--Poincare operator has a monotonic behavior with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain by using the properties of the polarization tensor corresponding to the domain.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74733257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Parabolic and elliptic equations with singular or degenerate coefficients: The Dirichlet problem 具有奇异或退化系数的抛物型和椭圆型方程:狄利克雷问题
arXiv: Analysis of PDEs Pub Date : 2020-09-16 DOI: 10.1090/TRAN/8397
Hongjie Dong, T. Phan
{"title":"Parabolic and elliptic equations with singular or degenerate coefficients: The Dirichlet problem","authors":"Hongjie Dong, T. Phan","doi":"10.1090/TRAN/8397","DOIUrl":"https://doi.org/10.1090/TRAN/8397","url":null,"abstract":"We consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space $mathbb{R}^d_+$, where the coefficients are the product of $x_d^alpha, alpha in (-infty, 1),$ and a bounded uniformly elliptic matrix of coefficients. Thus, the coefficients are singular or degenerate near the boundary ${x_d =0}$ and they may not locally integrable. The novelty of the work is that we find proper weights under which the existence, uniqueness, and regularity of solutions in Sobolev spaces are established. These results appear to be the first of their kind and are new even if the coefficients are constant. They are also readily extended to systems of equations.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85873678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 18
A limiting absorption principle for Helmholtz systems and time-harmonic isotropic Maxwell’s equations 亥姆霍兹系统的极限吸收原理及时谐各向同性麦克斯韦方程组
arXiv: Analysis of PDEs Pub Date : 2020-09-10 DOI: 10.5445/IR/1000124275
Lucrezia Cossetti, Rainer Mandel
{"title":"A limiting absorption principle for Helmholtz systems and time-harmonic isotropic Maxwell’s equations","authors":"Lucrezia Cossetti, Rainer Mandel","doi":"10.5445/IR/1000124275","DOIUrl":"https://doi.org/10.5445/IR/1000124275","url":null,"abstract":"In this work we investigate the $L^p-L^q$-mapping properties of the resolvent associated with the time-harmonic isotropic Maxwell operator. As spectral parameters close to the spectrum are also covered by our analysis, we obtain an $L^p-L^q$-type Limiting Absorption Principle for this operator. Our analysis relies on new results for Helmholtz systems with zero order non-Hermitian perturbations. Moreover, we provide an improved version of the Limiting Absorption Principle for Hermitian (self-adjoint) Helmholtz systems.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79730857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
Stability of the boundary layer expansion for the 3D plane parallel MHD flow 三维平面平行MHD流动边界层扩展的稳定性
arXiv: Analysis of PDEs Pub Date : 2020-09-10 DOI: 10.1063/5.0031449
S. Ding, Zhilin Lin, Dongjuan Niu
{"title":"Stability of the boundary layer expansion for the 3D plane parallel MHD flow","authors":"S. Ding, Zhilin Lin, Dongjuan Niu","doi":"10.1063/5.0031449","DOIUrl":"https://doi.org/10.1063/5.0031449","url":null,"abstract":"In this paper, we establish the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear plane parallel flows of viscous incompressible magnetohydrodynamic (MHD) flow with no-slip boundary condition of velocity and perfectly conducting wall for magnetic fields. The convergence is shown under various Sobolev norms, including the physically important space-time uniform norm $L^infty(H^1)$. In addition, the similar convergence results are also obtained under the case with uniform magnetic fields. This implies the stabilizing effects of magnetic fields. Besides, the higher-order expansion is also considered.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74988814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Sharp estimates for solutions to elliptic problems with mixed boundary conditions 混合边界条件下椭圆型问题解的尖锐估计
arXiv: Analysis of PDEs Pub Date : 2020-09-09 DOI: 10.1016/j.matpur.2020.12.003
A. Alvino, F. Chiacchio, C. Nitsch, C. Trombetti
{"title":"Sharp estimates for solutions to elliptic problems with mixed boundary conditions","authors":"A. Alvino, F. Chiacchio, C. Nitsch, C. Trombetti","doi":"10.1016/j.matpur.2020.12.003","DOIUrl":"https://doi.org/10.1016/j.matpur.2020.12.003","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90161090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 18
Nonexistence of variational minimizers related to a quasilinear singular problem in metric measure spaces 度量度量空间中拟线性奇异问题的变分极小值的不存在性
arXiv: Analysis of PDEs Pub Date : 2020-09-08 DOI: 10.1090/proc/15417
Prashanta Garain, J. Kinnunen
{"title":"Nonexistence of variational minimizers related to a quasilinear singular problem in metric measure spaces","authors":"Prashanta Garain, J. Kinnunen","doi":"10.1090/proc/15417","DOIUrl":"https://doi.org/10.1090/proc/15417","url":null,"abstract":"In this article we consider a variational problem related to a quasilinear singular problem and obtain a nonexistence result in a metric measure space with a doubling measure and a Poincare inequality. Our method is purely variational and to the best of our knowledge, this is the first work concerning singular problems in a general metric setting.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74079613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
Grassmannian reduction of cucker-smale systems and dynamical opinion games 小库克系统的格拉斯曼约简与动态意见博弈
arXiv: Analysis of PDEs Pub Date : 2020-09-08 DOI: 10.3934/dcds.2021095
Daniel Lear, David N. Reynolds, R. Shvydkoy
{"title":"Grassmannian reduction of cucker-smale systems and dynamical opinion games","authors":"Daniel Lear, David N. Reynolds, R. Shvydkoy","doi":"10.3934/dcds.2021095","DOIUrl":"https://doi.org/10.3934/dcds.2021095","url":null,"abstract":"In this note we study a new class of alignment models with self-propulsion and Rayleigh-type friction forces, which describes the collective behavior of agents with individual characteristic parameters. We describe the long time dynamics via a new method which allows to reduce analysis from the multidimensional system to a simpler family of two-dimensional systems parametrized by a proper Grassmannian. With this method we demonstrate exponential alignment for a large (and sharp) class of initial velocity configurations confined to a sector of opening less than $pi$. \u0000In the case when characteristic parameters remain frozen, the system governs dynamics of opinions for a set of players with constant convictions. Viewed as a dynamical non-cooperative game, the system is shown to possess a unique stable Nash equilibrium, which represents a settlement of opinions most agreeable to all agents. Such an agreement is furthermore shown to be a global attractor for any set of initial opinions.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74844605","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
Weighted Spectral Cluster Bounds and a Sharp Multiplier Theorem for Ultraspherical Grushin Operators 超球面Grushin算子的加权谱聚类界和Sharp乘子定理
arXiv: Analysis of PDEs Pub Date : 2020-09-07 DOI: 10.1093/imrn/rnab007
Valentina Casarino, P. Ciatti, Alessio Martini
{"title":"Weighted Spectral Cluster Bounds and a Sharp Multiplier Theorem for Ultraspherical Grushin Operators","authors":"Valentina Casarino, P. Ciatti, Alessio Martini","doi":"10.1093/imrn/rnab007","DOIUrl":"https://doi.org/10.1093/imrn/rnab007","url":null,"abstract":"We study degenerate elliptic operators of Grushin type on the $d$-dimensional sphere, which are singular on a $k$-dimensional sphere for some $k < d$. For these operators we prove a spectral multiplier theorem of Mihlin-Hormander type, which is optimal whenever $2k leq d$, and a corresponding Bochner-Riesz summability result. The proof hinges on suitable weighted spectral cluster bounds, which in turn depend on precise estimates for ultraspherical polynomials.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72559704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
Final state problem for nonlinear Schr"{o}dinger equations with time-decaying harmonic oscillators 具有时间衰减谐振子的非线性Schr {o}dinger方程的终态问题
arXiv: Analysis of PDEs Pub Date : 2020-09-03 DOI: 10.1016/j.jmaa.2021.125292
Masaki Kawamoto
{"title":"Final state problem for nonlinear Schr\"{o}dinger equations with time-decaying harmonic oscillators","authors":"Masaki Kawamoto","doi":"10.1016/j.jmaa.2021.125292","DOIUrl":"https://doi.org/10.1016/j.jmaa.2021.125292","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83779503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
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