Pub Date : 2022-01-01DOI: 10.4310/dpde.2022.v19.n2.a1
A. Esfahani, S. Levandosky
{"title":"Traveling waves of a generalized nonlinear Beam equation","authors":"A. Esfahani, S. Levandosky","doi":"10.4310/dpde.2022.v19.n2.a1","DOIUrl":"https://doi.org/10.4310/dpde.2022.v19.n2.a1","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70427191","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 : 2022-01-01DOI: 10.4310/dpde.2022.v19.n4.a4
Y. Guan, Michal Feckan, Jinrong Wang
{"title":"Constant vorticity atmospheric Ekman flows in the modified $beta$-plane approximation","authors":"Y. Guan, Michal Feckan, Jinrong Wang","doi":"10.4310/dpde.2022.v19.n4.a4","DOIUrl":"https://doi.org/10.4310/dpde.2022.v19.n4.a4","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70426895","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 : 2022-01-01DOI: 10.4310/dpde.2022.v19.n2.a3
J. H. Kang, Timothy Robertson
{"title":"An elliptic nonlinear system of multiple functions with application","authors":"J. H. Kang, Timothy Robertson","doi":"10.4310/dpde.2022.v19.n2.a3","DOIUrl":"https://doi.org/10.4310/dpde.2022.v19.n2.a3","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70427202","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 : 2022-01-01DOI: 10.4310/dpde.2022.v19.n1.a2
L. Ferreira, J. E. Pérez-López
{"title":"On the well-posedness of the incompressible Euler equations in a larger space of Besov–Morrey type","authors":"L. Ferreira, J. E. Pérez-López","doi":"10.4310/dpde.2022.v19.n1.a2","DOIUrl":"https://doi.org/10.4310/dpde.2022.v19.n1.a2","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70427141","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 : 2022-01-01DOI: 10.4310/dpde.2022.v19.n4.a2
Dongbing Zha, Minghui Sun
{"title":"Asymptotic behavior of global solutions to some multidimensional quasilinear hyperbolic systems","authors":"Dongbing Zha, Minghui Sun","doi":"10.4310/dpde.2022.v19.n4.a2","DOIUrl":"https://doi.org/10.4310/dpde.2022.v19.n4.a2","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70427332","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 : 2021-12-02DOI: 10.4310/dpde.2021.v18.n4.a4
Huiying Fan, Meng Wang
In this paper, we are concentrated on demonstrating the Liouville type theorem for the stationary Magnetohydrodynamic equations in mixednorm Lebesgue spaces and weighted mixed-norm Lebesgue spaces. In particular, we show that, under some sufficient conditions in (weighted) mixed-norm Lebesgue spaces, the solution of stationary MHDs are identically zero. Precisely, we investigate solutions of MHDs that may decay to zero in different rates as $lvert x rvert to infty$ in different directions. In un-mixed norm case, the result recovers available results. With some additional geometric assumptions on the supports of solutions in weighted mixed-norm Lebesgue spaces, this work also provides several other important Liouville type theorems of solutions in weighted mixed-norm Lebesgue spaces.
本文主要讨论了混合范数Lebesgue空间和加权混合范数Lebesgue空间中平稳磁流体动力学方程的Liouville型定理。特别地，我们证明了在(加权)混合范数Lebesgue空间中的一些充分条件下，平稳mhd的解是同零的。确切地说，我们研究了可能在不同方向上以$lvert x rvert to infty$的不同速率衰减到零的mhd的解。在非混合范数情况下，结果恢复了可用结果。通过对加权混合范数Lebesgue空间中解的支撑的一些附加几何假设，本文还给出了加权混合范数Lebesgue空间中解的几个重要的Liouville型定理。
{"title":"The Liouville type theorem for the stationary magnetohydrodynamic equations in weighted mixed-norm Lebesgue spaces","authors":"Huiying Fan, Meng Wang","doi":"10.4310/dpde.2021.v18.n4.a4","DOIUrl":"https://doi.org/10.4310/dpde.2021.v18.n4.a4","url":null,"abstract":"In this paper, we are concentrated on demonstrating the Liouville type theorem for the stationary Magnetohydrodynamic equations in mixednorm Lebesgue spaces and weighted mixed-norm Lebesgue spaces. In particular, we show that, under some sufficient conditions in (weighted) mixed-norm Lebesgue spaces, the solution of stationary MHDs are identically zero. Precisely, we investigate solutions of MHDs that may decay to zero in different rates as $lvert x rvert to infty$ in different directions. In un-mixed norm case, the result recovers available results. With some additional geometric assumptions on the supports of solutions in weighted mixed-norm Lebesgue spaces, this work also provides several other important Liouville type theorems of solutions in weighted mixed-norm Lebesgue spaces.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138539415","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 : 2021-05-23DOI: 10.4310/dpde.2022.v19.n2.a2
Xinyue Cheng, Dong Li, Jiao Xu, Dongbing Zha
We consider the two-dimensional quasilinear wave equations with standard nullform type quadratic nonlinearities. We prove global wellposedness without using the Lorentz boost vector fields.
{"title":"Global wellposedness for 2D quasilinear wave without Lorentz","authors":"Xinyue Cheng, Dong Li, Jiao Xu, Dongbing Zha","doi":"10.4310/dpde.2022.v19.n2.a2","DOIUrl":"https://doi.org/10.4310/dpde.2022.v19.n2.a2","url":null,"abstract":"We consider the two-dimensional quasilinear wave equations with standard nullform type quadratic nonlinearities. We prove global wellposedness without using the Lorentz boost vector fields.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46257929","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 : 2021-03-06DOI: 10.4310/DPDE.2021.V18.N2.A4
Chunqiu Li, Desheng Li, Jintao Wang
In this paper we present some local dynamic bifurcation results in terms of invariant sets of nonlinear evolution equations. We show that if the trivial solution is an isolated invariant set of the system at the critical value $lambda=lambda_0$, then either there exists a one-sided neighborhood $I^-$ of $lambda_0$ such that for each $lambdain I^-$, the system bifurcates from the trivial solution to an isolated nonempty compact invariant set $K_lambda$ with $0notin K_lambda$, or there is a one-sided neighborhood $I^+$ of $lambda_0$ such that the system undergoes an attractor bifurcation for $lambdain I^+$ from $(0,lambda_0)$. Then we give a modified version of the attractor bifurcation theorem. Finally, we consider the classical Swift-Hohenberg equation and illustrate how to apply our results to a concrete evolution equation.
{"title":"A remark on attractor bifurcation","authors":"Chunqiu Li, Desheng Li, Jintao Wang","doi":"10.4310/DPDE.2021.V18.N2.A4","DOIUrl":"https://doi.org/10.4310/DPDE.2021.V18.N2.A4","url":null,"abstract":"In this paper we present some local dynamic bifurcation results in terms of invariant sets of nonlinear evolution equations. We show that if the trivial solution is an isolated invariant set of the system at the critical value $lambda=lambda_0$, then either there exists a one-sided neighborhood $I^-$ of $lambda_0$ such that for each $lambdain I^-$, the system bifurcates from the trivial solution to an isolated nonempty compact invariant set $K_lambda$ with $0notin K_lambda$, or there is a one-sided neighborhood $I^+$ of $lambda_0$ such that the system undergoes an attractor bifurcation for $lambdain I^+$ from $(0,lambda_0)$. Then we give a modified version of the attractor bifurcation theorem. Finally, we consider the classical Swift-Hohenberg equation and illustrate how to apply our results to a concrete evolution equation.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46671604","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 : 2021-02-04DOI: 10.4310/dpde.2022.v19.n3.a2
Dixi Wang, Cheng Yu, Xinhua Zhao
In this paper, we consider the inviscid limit of inhomogeneous incompressible Navier-Stokes equations under the weak Kolmogorov hypothesis in R. In particular, we first deduce the Kolmogorov-type hypothesis in R, which yields the uniform bounds of α-order fractional derivatives of √ ρμu in Lx for some α > 0, independent of the viscosity. The uniform bounds can provide strong convergence of √ ρμu in L space. This shows that the inviscid limit is a weak solution to the corresponding Euler equations.
{"title":"Inviscid limit of the inhomogeneous incompressible Navier–Stokes equations under the weak Kolmogorov hypothesis in $mathbb{R}^3$","authors":"Dixi Wang, Cheng Yu, Xinhua Zhao","doi":"10.4310/dpde.2022.v19.n3.a2","DOIUrl":"https://doi.org/10.4310/dpde.2022.v19.n3.a2","url":null,"abstract":"In this paper, we consider the inviscid limit of inhomogeneous incompressible Navier-Stokes equations under the weak Kolmogorov hypothesis in R. In particular, we first deduce the Kolmogorov-type hypothesis in R, which yields the uniform bounds of α-order fractional derivatives of √ ρμu in Lx for some α > 0, independent of the viscosity. The uniform bounds can provide strong convergence of √ ρμu in L space. This shows that the inviscid limit is a weak solution to the corresponding Euler equations.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43294829","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 : 2021-01-04DOI: 10.4310/dpde.2022.v19.n2.a4
R. Frier, Shuanglin Shao
In this paper, we study the extremal problem for the Strichartz inequality for the Schrödinger equation on R. We show that the solutions to the associated Euler-Lagrange equation are exponentially decaying in the Fourier space and thus can be extended to be complex analytic. Consequently we provide a new proof to the characterization of the extremal functions: the only extremals are Gaussian functions, which was investigated previously by Foschi [7] and Hundertmark-Zharnitsky [11].
{"title":"A remark on the Strichartz inequality in one dimension","authors":"R. Frier, Shuanglin Shao","doi":"10.4310/dpde.2022.v19.n2.a4","DOIUrl":"https://doi.org/10.4310/dpde.2022.v19.n2.a4","url":null,"abstract":"In this paper, we study the extremal problem for the Strichartz inequality for the Schrödinger equation on R. We show that the solutions to the associated Euler-Lagrange equation are exponentially decaying in the Fourier space and thus can be extended to be complex analytic. Consequently we provide a new proof to the characterization of the extremal functions: the only extremals are Gaussian functions, which was investigated previously by Foschi [7] and Hundertmark-Zharnitsky [11].","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43330944","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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