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Communications in Mathematical Analysis and Applications最新文献

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Weak Solutions to the Equations of Stationary Compressible Flows in Active Liquid Crystals 有源液晶中静态可压缩流动方程的弱解
Communications in Mathematical Analysis and Applications Pub Date : 2022-04-30 DOI: 10.4208/cmaa.2022-0021
Zhilei Liang, A. Majumdar, Dehua Wang, Yixuan Wang
{"title":"Weak Solutions to the Equations of Stationary Compressible Flows in Active Liquid Crystals","authors":"Zhilei Liang, A. Majumdar, Dehua Wang, Yixuan Wang","doi":"10.4208/cmaa.2022-0021","DOIUrl":"https://doi.org/10.4208/cmaa.2022-0021","url":null,"abstract":"The equations of stationary compressible flows of active liquid crystals are considered in a bounded three-dimensional domain. The system consists of the stationary Navier-Stokes equations coupled with the equation of Q-tensors and the equation of the active particles. The existence of weak solutions to the stationary problem is established through a two-level approximation scheme, compactness estimates and weak convergence arguments. Novel techniques are developed to overcome the difficulties due to the lower regularity of stationary solutions, a Moser-type iteration is used to deal with the strong coupling of active particles and fluids, and some weighted estimates on the energy functions are achieved so that the weak solutions can be constructed for all values of the adiabatic exponent $gamma>1$.","PeriodicalId":371957,"journal":{"name":"Communications in Mathematical Analysis and Applications","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123253620","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: 0
3D Hyperbolic Navier-Stokes Equations in a Thin Strip: Global Well-Posedness and Hydrostatic Limit in Gevrey Space 薄带中的三维双曲Navier-Stokes方程:Gevrey空间中的全局适定性和流体静力极限
Communications in Mathematical Analysis and Applications Pub Date : 2022-04-20 DOI: 10.4208/cmaa.2022-0007
Wei-Xi Li, Tong Yang
{"title":"3D Hyperbolic Navier-Stokes Equations in a Thin Strip: Global Well-Posedness and Hydrostatic Limit in Gevrey Space","authors":"Wei-Xi Li, Tong Yang","doi":"10.4208/cmaa.2022-0007","DOIUrl":"https://doi.org/10.4208/cmaa.2022-0007","url":null,"abstract":"We consider the hyperbolic version of three-dimensional anisotropic Naver-Stokes equations in a thin strip and its hydrostatic limit that is a hyperbolic Prandtl type equations. We prove the global-in-time existence and uniqueness for the two systems and the hydrostatic limit when the initial data belong to the Gevrey function space with index 2. The proof is based on a direct energy method by observing the damping e ff ect in the systems.","PeriodicalId":371957,"journal":{"name":"Communications in Mathematical Analysis and Applications","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132197736","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
Vector Fields of Cancellation for the Prandtl Operators Prandtl算子的消去向量场
Communications in Mathematical Analysis and Applications Pub Date : 2022-01-25 DOI: 10.4208/cmaa.2022-0004
Tong Yang
{"title":"Vector Fields of Cancellation for the Prandtl Operators","authors":"Tong Yang","doi":"10.4208/cmaa.2022-0004","DOIUrl":"https://doi.org/10.4208/cmaa.2022-0004","url":null,"abstract":"Abstract. It has been a fascinating topic in the study of boundary layer theory about the well-posedness of Prandtl equation that was derived in 1904. Recently, new ideas about cancellation to overcome the loss of tangential derivatives were obtained so that Prandtl equation can be shown to be well-posed in Sobolev spaces to avoid the use of Crocco transformation as in the classical work of Oleinik. This short note aims to show that the cancellation mechanism is in fact related to some intrinsic directional derivative that can be used to recover the tangential derivative under some structural assumption on the fluid near the boundary.","PeriodicalId":371957,"journal":{"name":"Communications in Mathematical Analysis and Applications","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114102054","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
On Smooth Solutions to the Thermostated Boltzmann Equation with Deformation 带变形的热稳态玻尔兹曼方程的光滑解
Communications in Mathematical Analysis and Applications Pub Date : 2021-12-22 DOI: 10.4208/cmaa.2021-0004
Renjun Duan, Shuangqiang Liu
{"title":"On Smooth Solutions to the Thermostated Boltzmann Equation with Deformation","authors":"Renjun Duan, Shuangqiang Liu","doi":"10.4208/cmaa.2021-0004","DOIUrl":"https://doi.org/10.4208/cmaa.2021-0004","url":null,"abstract":"This paper concerns a kinetic model of the thermostated Boltzmann equation with a linear deformation force described by a constant matrix. The collision kernel under consideration includes both the Maxwell molecule and general hard potentials with angular cutoff. We construct the smooth steady solutions via a perturbation approach when the deformation strength is sufficiently small. The steady solution is a spatially homogeneous non Maxwellian state and may have the polynomial tail at large velocities. Moreover, we also establish the long time asymptotics toward steady states for the Cauchy problem on the corresponding spatially inhomogeneous equation in torus, which in turn gives the non-negativity of steady solutions.","PeriodicalId":371957,"journal":{"name":"Communications in Mathematical Analysis and Applications","volume":"144 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116442236","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
Convergence Toward a Periodically Rotating One-Point Cluster in the Kinetic Thermodynamic Cucker-Smale Model 动态热力学cucker - small模型中周期旋转单点簇的收敛性
Communications in Mathematical Analysis and Applications Pub Date : 1900-01-01 DOI: 10.4208/cmaa.2021-0002
Linglong Du, Seung‐Yeal Ha
{"title":"Convergence Toward a Periodically Rotating One-Point Cluster in the Kinetic Thermodynamic Cucker-Smale Model","authors":"Linglong Du, Seung‐Yeal Ha","doi":"10.4208/cmaa.2021-0002","DOIUrl":"https://doi.org/10.4208/cmaa.2021-0002","url":null,"abstract":"","PeriodicalId":371957,"journal":{"name":"Communications in Mathematical Analysis and Applications","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125145615","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: 0
The Dirichlet Problem for Stochastic Degenerate Parabolic-Hyperbolic Equations 随机退化抛物-双曲型方程的Dirichlet问题
Communications in Mathematical Analysis and Applications Pub Date : 1900-01-01 DOI: 10.4208/cmaa.2021-0001
H. Frid, Yachun Li
{"title":"The Dirichlet Problem for Stochastic Degenerate Parabolic-Hyperbolic Equations","authors":"H. Frid, Yachun Li","doi":"10.4208/cmaa.2021-0001","DOIUrl":"https://doi.org/10.4208/cmaa.2021-0001","url":null,"abstract":"We consider the Dirichlet problem for a quasilinear degenerate parabolic stochastic partial differential equation with multiplicative noise and nonhomogeneous Dirichlet boundary condition. We introduce the definition of kinetic solution for this problem and prove existence and uniqueness of solutions. For the uniqueness of kinetic solutions we prove a new version of the doubling of variables method and use it to deduce a comparison principle between solutions. The proof requires a delicate analysis of the boundary values of the solutions for which we develop some techniques that enable the usage of the existence of normal weak traces for divergence measure fields in this stochastic setting. The existence of solutions, as usual, is obtained through a two-level approximation scheme consisting of nondegenerate regularizations of the equations which we show to be consistent with our definition of solutions. In particular, the regularity conditions that give meaning to the boundary values of the solutions are shown to be inherited by limits of nondegenerate parabolic approximations provided by the vanishing viscosity method.","PeriodicalId":371957,"journal":{"name":"Communications in Mathematical Analysis and Applications","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125533519","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: 4
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