Pub Date : 2020-09-18DOI: 10.4310/cms.2021.v19.n6.a9
Boqiang Lu, Xiaoding Shi, C. Xiong
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.
{"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":"24 1","pages":""},"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}
Pub Date : 2020-09-16DOI: 10.1017/S0956792521000127
E. Cherkaev, Minwoo Kim, Mikyoung Lim
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.
{"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":"94 1","pages":""},"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}
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.
考虑了上半空间$mathbb{R}^d_+$上的一类椭圆型和抛物型方程的Dirichlet问题，其中系数是$x_d^alpha, alpha in (-infty, 1),$与有界一致椭圆型系数矩阵的乘积。因此，系数在边界${x_d =0}$附近是奇异的或退化的，它们可能不是局部可积的。该工作的新颖之处在于我们找到了适当的权值，在此权值下建立了Sobolev空间中解的存在性、唯一性和正则性。这些结果似乎是同类中的第一个，即使系数是恒定的，也是新的。它们也很容易推广到方程组中。
{"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":"89 1","pages":""},"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}
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.
{"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":"22 1","pages":""},"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}
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.
{"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":"98 1","pages":""},"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}
Pub Date : 2020-09-09DOI: 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":"41 1","pages":""},"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}
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.
{"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":"42 1","pages":""},"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}
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 $pi$. In 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.
{"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":"15 1","pages":""},"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}
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.
{"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":"59 5","pages":""},"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}
Pub Date : 2020-09-03DOI: 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":"91 1","pages":""},"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}
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