Pub Date : 2020-05-31DOI: 10.1017/9781108839808.011
A. Nandakumaran, P. S. Datti
{"title":"Wave Equation in Higher Dimensions","authors":"A. Nandakumaran, P. S. Datti","doi":"10.1017/9781108839808.011","DOIUrl":"https://doi.org/10.1017/9781108839808.011","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84264611","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-31DOI: 10.1017/9781108839808.013
{"title":"A Peep into Weak Derivatives, Sobolev Spaces and Weak Formulation","authors":"","doi":"10.1017/9781108839808.013","DOIUrl":"https://doi.org/10.1017/9781108839808.013","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79205234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-31DOI: 10.1017/9781108839808.008
A. Nandakumaran, P. S. Datti
{"title":"Laplace and Poisson Equations","authors":"A. Nandakumaran, P. S. Datti","doi":"10.1017/9781108839808.008","DOIUrl":"https://doi.org/10.1017/9781108839808.008","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87931741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-03-12DOI: 10.4310/DPDE.2021.v18.n4.a3
Pham Truong Xuan
This paper we study the viscous simplified Bardina equations on two-dimensional closed manifolds $M$ imbedded in $mathbb{R}^3$. First we will show that the existence and uniqueness of the weak solutions. Then the existence of a maximal attractor is proved and the upper bound for the global Hausdorff and fractal dimensions of the attractor is obtained. The applications to the two-dimensional sphere ${S}^2$ and the square torus ${T}^2$ will be treated. Finally, we prove the existence of the inertial manifolds in the case ${S}^2$.
{"title":"The simplified Bardina equation on two-dimensional closed manifolds","authors":"Pham Truong Xuan","doi":"10.4310/DPDE.2021.v18.n4.a3","DOIUrl":"https://doi.org/10.4310/DPDE.2021.v18.n4.a3","url":null,"abstract":"This paper we study the viscous simplified Bardina equations on two-dimensional closed manifolds $M$ imbedded in $mathbb{R}^3$. First we will show that the existence and uniqueness of the weak solutions. Then the existence of a maximal attractor is proved and the upper bound for the global Hausdorff and fractal dimensions of the attractor is obtained. The applications to the two-dimensional sphere ${S}^2$ and the square torus ${T}^2$ will be treated. Finally, we prove the existence of the inertial manifolds in the case ${S}^2$.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45364439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.4310/dpde.2020.v17.n3.a1
B. Guo, Yunrui Zheng
{"title":"A steady model on Navier–Stokes equations with a free surface","authors":"B. Guo, Yunrui Zheng","doi":"10.4310/dpde.2020.v17.n3.a1","DOIUrl":"https://doi.org/10.4310/dpde.2020.v17.n3.a1","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70426525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.4310/dpde.2020.v17.n2.a2
Yanjuan Yang, Xun Wang
{"title":"Exact and explicit internal water waves at arbitrary latitude with underlying currents","authors":"Yanjuan Yang, Xun Wang","doi":"10.4310/dpde.2020.v17.n2.a2","DOIUrl":"https://doi.org/10.4310/dpde.2020.v17.n2.a2","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70426467","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.4310/dpde.2020.v17.n2.a4
W. Wang, H. Yue
{"title":"Almost sure existence of global weak solutions to the Boussinesq equations","authors":"W. Wang, H. Yue","doi":"10.4310/dpde.2020.v17.n2.a4","DOIUrl":"https://doi.org/10.4310/dpde.2020.v17.n2.a4","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70426515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.4310/dpde.2020.v17.n1.a3
Jishan Fan, Lulu Jing, G. Nakamura, T. Tang
{"title":"Global solutions of the 3D compressible MHD system in a bounded domain","authors":"Jishan Fan, Lulu Jing, G. Nakamura, T. Tang","doi":"10.4310/dpde.2020.v17.n1.a3","DOIUrl":"https://doi.org/10.4310/dpde.2020.v17.n1.a3","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70426404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.4310/dpde.2020.v17.n1.a4
Qixiang Yang
{"title":"Symmetric and uniform analytic solutions in phase space for Navier–Stokes equations","authors":"Qixiang Yang","doi":"10.4310/dpde.2020.v17.n1.a4","DOIUrl":"https://doi.org/10.4310/dpde.2020.v17.n1.a4","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70426455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.4310/dpde.2020.v17.n3.a3
Z. Ye
. In this paper, we consider the n -dimensional regularized incompressible Boussinesq equations with a Leray-regularization through a smooth- ing kernel of order α in the quadratic term and a β -fractional Laplacian in the velocity equation. We prove the global regularity of the solution to the n dimensional logarithmically supercritical Boussinesq equations with zero diffu- sion. As a direct corollary, we obtain the global regularity result for the regularized Boussinesq equations with zero diffusion in the critical case α + β = 12 + n 4 . Therefore, our results settle the global regularity case previously mentioned in the literatures.
{"title":"Global regularity of the regularized Boussinesq equations with zero diffusion","authors":"Z. Ye","doi":"10.4310/dpde.2020.v17.n3.a3","DOIUrl":"https://doi.org/10.4310/dpde.2020.v17.n3.a3","url":null,"abstract":". In this paper, we consider the n -dimensional regularized incompressible Boussinesq equations with a Leray-regularization through a smooth- ing kernel of order α in the quadratic term and a β -fractional Laplacian in the velocity equation. We prove the global regularity of the solution to the n dimensional logarithmically supercritical Boussinesq equations with zero diffu- sion. As a direct corollary, we obtain the global regularity result for the regularized Boussinesq equations with zero diffusion in the critical case α + β = 12 + n 4 . Therefore, our results settle the global regularity case previously mentioned in the literatures.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70426533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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