Pub Date : 2022-06-01DOI: 10.4208/aam.oa-2020-0060
Pan Zhang, Mengmeng Liu, Fangying Song
. We investigate the thermal instability of a three-dimensional Rayleigh–B´enard (RB for short) problem without thermal diffusion in a bounded domain. First we construct unstable solutions in exponential growth modes for the linear RB problem. Then we derive energy estimates for the nonlinear solutions by a method of a prior energy estimates, and establish a Gronwall-type energy inequality for the nonlinear solutions. Finally, we estimate for the error of L 1 -norm between the both solutions of the linear and nonlinear problems, and prove the existence of escape times of nonlinear solutions. Thus we get the instability of nonlinear solutions under L 1 -norm.
{"title":"On Instability of the Rayleigh–Bénard Problem Without Thermal Diffusion in a Bounded Domain under $L^1$ -Norm","authors":"Pan Zhang, Mengmeng Liu, Fangying Song","doi":"10.4208/aam.oa-2020-0060","DOIUrl":"https://doi.org/10.4208/aam.oa-2020-0060","url":null,"abstract":". We investigate the thermal instability of a three-dimensional Rayleigh–B´enard (RB for short) problem without thermal diffusion in a bounded domain. First we construct unstable solutions in exponential growth modes for the linear RB problem. Then we derive energy estimates for the nonlinear solutions by a method of a prior energy estimates, and establish a Gronwall-type energy inequality for the nonlinear solutions. Finally, we estimate for the error of L 1 -norm between the both solutions of the linear and nonlinear problems, and prove the existence of escape times of nonlinear solutions. Thus we get the instability of nonlinear solutions under L 1 -norm.","PeriodicalId":58853,"journal":{"name":"应用数学年刊:英文版","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43190298","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 : 2021-11-27DOI: 10.4208/aam.oa-2022-0003
Z. Feng, Y. C. Li
Abstract. We introduce a population-age-time (PAT) model which describes the temporal evolution of the population distribution in age. The surprising result is that the qualitative nature of the population distribution dynamics is robust with respect to the birth rate and death rate distributions in age, and initial conditions. When the number of children born per woman is 2, the population distribution approaches an asymptotically steady state of a kink shape; thus the total population approaches a constant. When the number of children born per woman is greater than 2, the total population increases without bound; and when the number of children born per woman is less than 2, the total population decreases to zero.
{"title":"The PAT Model of Population Dynamics","authors":"Z. Feng, Y. C. Li","doi":"10.4208/aam.oa-2022-0003","DOIUrl":"https://doi.org/10.4208/aam.oa-2022-0003","url":null,"abstract":"Abstract. We introduce a population-age-time (PAT) model which describes the temporal evolution of the population distribution in age. The surprising result is that the qualitative nature of the population distribution dynamics is robust with respect to the birth rate and death rate distributions in age, and initial conditions. When the number of children born per woman is 2, the population distribution approaches an asymptotically steady state of a kink shape; thus the total population approaches a constant. When the number of children born per woman is greater than 2, the total population increases without bound; and when the number of children born per woman is less than 2, the total population decreases to zero.","PeriodicalId":58853,"journal":{"name":"应用数学年刊:英文版","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47308614","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 : 2021-08-18DOI: 10.4208/aam.oa-2021-0005
Tianling Jin, D. Kriventsov, Jingang Xiong
We study a Rayleigh-Faber-Krahn inequality for regional fractional Laplacian operators. In particular, we show that there exists a compactly supported nonnegative Sobolev function u0 that attains the infimum (which will be a positive real number) of the set {∫∫ {u>0}×{u>0} |u(x)− u(y)| |x− y| dxdy : u ∈ H̊(R), ∫ R u = 1, |{u > 0}| ≤ 1 } . Unlike the corresponding problem for the usual fractional Laplacian, where the domain of the integration is R × R, symmetrization techniques may not apply here. Our approach is instead based on the direct method and new a priori diameter estimates. We also present several remaining open questions concerning the regularity and shape of the minimizers, and the form of the Euler-Lagrange equations.
{"title":"On a Rayleigh-Faber-Krahn Inequality for the Regional Fractional Laplacian","authors":"Tianling Jin, D. Kriventsov, Jingang Xiong","doi":"10.4208/aam.oa-2021-0005","DOIUrl":"https://doi.org/10.4208/aam.oa-2021-0005","url":null,"abstract":"We study a Rayleigh-Faber-Krahn inequality for regional fractional Laplacian operators. In particular, we show that there exists a compactly supported nonnegative Sobolev function u0 that attains the infimum (which will be a positive real number) of the set {∫∫ {u>0}×{u>0} |u(x)− u(y)| |x− y| dxdy : u ∈ H̊(R), ∫ R u = 1, |{u > 0}| ≤ 1 } . Unlike the corresponding problem for the usual fractional Laplacian, where the domain of the integration is R × R, symmetrization techniques may not apply here. Our approach is instead based on the direct method and new a priori diameter estimates. We also present several remaining open questions concerning the regularity and shape of the minimizers, and the form of the Euler-Lagrange equations.","PeriodicalId":58853,"journal":{"name":"应用数学年刊:英文版","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48266928","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 : 2021-08-11DOI: 10.4208/aam.OA-2022-0006
A. Biswas, K. Brown, V. Martinez
. This paper is dedicated to the expansion of the framework of general interpolant observables introduced by Azouani, Olson, and Titi for continuous data assimilation of nonlinear partial differential equations. The main feature of this expanded framework is its mesh-free aspect, which allows the observational data itself to dictate the subdivision of the domain via partition of unity in the spirit of the so-called Partition of Unity Method by Babuska and Melenk. As an application of this framework, we consider a nudging-based scheme for data assimilation applied to the context of the two-dimensional Navier-Stokes equations as a paradigmatic example and establish convergence to the reference solution in all higher-order Sobolev topologies in a periodic, mean-free setting. The convergence analysis also makes use of absorbing ball bounds in higher-order Sobolev norms, for which explicit bounds appear to be available in the literature only up to H 2 ; such bounds are additionally proved for all integer levels of Sobolev regularity above H 2 .
{"title":"Mesh-Free Interpolant Observables for Continuous Data Assimilation","authors":"A. Biswas, K. Brown, V. Martinez","doi":"10.4208/aam.OA-2022-0006","DOIUrl":"https://doi.org/10.4208/aam.OA-2022-0006","url":null,"abstract":". This paper is dedicated to the expansion of the framework of general interpolant observables introduced by Azouani, Olson, and Titi for continuous data assimilation of nonlinear partial differential equations. The main feature of this expanded framework is its mesh-free aspect, which allows the observational data itself to dictate the subdivision of the domain via partition of unity in the spirit of the so-called Partition of Unity Method by Babuska and Melenk. As an application of this framework, we consider a nudging-based scheme for data assimilation applied to the context of the two-dimensional Navier-Stokes equations as a paradigmatic example and establish convergence to the reference solution in all higher-order Sobolev topologies in a periodic, mean-free setting. The convergence analysis also makes use of absorbing ball bounds in higher-order Sobolev norms, for which explicit bounds appear to be available in the literature only up to H 2 ; such bounds are additionally proved for all integer levels of Sobolev regularity above H 2 .","PeriodicalId":58853,"journal":{"name":"应用数学年刊:英文版","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46341046","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 : 2021-06-01DOI: 10.4208/aam.oa-2021-0011
global sci
{"title":"On the Cahn-Hilliard-Brinkman Equations in $mathbb{R}^4$: Global Well-Posedness","authors":"global sci","doi":"10.4208/aam.oa-2021-0011","DOIUrl":"https://doi.org/10.4208/aam.oa-2021-0011","url":null,"abstract":"","PeriodicalId":58853,"journal":{"name":"应用数学年刊:英文版","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42869328","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 : 2021-06-01DOI: 10.4208/aam.oa-2021-0003
global DongLi
Many physical problems such as Allen-Cahn flows have natural maximum principles which yield strong point-wise control of the physical solutions in terms of the boundary data, the initial conditions and the operator coefficients. Sharp/strict maximum principles insomuch of fundamental importance for the continuous problem often do not persist under numerical discretization. A lot of past research concentrates on designing fine numerical schemes which preserves the sharp maximum principles especially for nonlinear problems. However these sharp principles not only sometimes introduce unwanted stringent conditions on the numerical schemes but also completely leaves many powerful frequency-based methods unattended and rarely analyzed directly in the sharp maximum norm topology. A prominent example is the spectral methods in the family of weighted residual methods. In this work we introduce and develop a new framework of almost sharp maximum principles which allow the numerical solutions to deviate from the sharp bound by a controllable discretization error: we call them effective maximum principles. We showcase the analysis for the classical Fourier spectral methods including Fourier Galerkin and Fourier collocation in space with forward Euler in time or second order Strang splitting. The model equations include the Allen-Cahn equations with double well potential, the Burgers equation and the Navier-Stokes equations. We give a comprehensive proof of the effective maximum principles under very general parametric conditions.
{"title":"Effective Maximum Principles for Spectral Methods","authors":"global DongLi","doi":"10.4208/aam.oa-2021-0003","DOIUrl":"https://doi.org/10.4208/aam.oa-2021-0003","url":null,"abstract":"Many physical problems such as Allen-Cahn flows have natural maximum principles which yield strong point-wise control of the physical solutions in terms of the boundary data, the initial conditions and the operator coefficients. Sharp/strict maximum principles insomuch of fundamental importance for the continuous problem often do not persist under numerical discretization. A lot of past research concentrates on designing fine numerical schemes which preserves the sharp maximum principles especially for nonlinear problems. However these sharp principles not only sometimes introduce unwanted stringent conditions on the numerical schemes but also completely leaves many powerful frequency-based methods unattended and rarely analyzed directly in the sharp maximum norm topology. A prominent example is the spectral methods in the family of weighted residual methods. In this work we introduce and develop a new framework of almost sharp maximum principles which allow the numerical solutions to deviate from the sharp bound by a controllable discretization error: we call them effective maximum principles. We showcase the analysis for the classical Fourier spectral methods including Fourier Galerkin and Fourier collocation in space with forward Euler in time or second order Strang splitting. The model equations include the Allen-Cahn equations with double well potential, the Burgers equation and the Navier-Stokes equations. We give a comprehensive proof of the effective maximum principles under very general parametric conditions.","PeriodicalId":58853,"journal":{"name":"应用数学年刊:英文版","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47199002","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 : 2021-06-01DOI: 10.4208/AAM.OA-2020-0004
Zefu Feng
. In this paper, we study long-time dynamics and diffusion limit of large-data solutions to a system of balance laws arising from a chemotaxis model with logarithmic sensitivity and nonlinear production/degradation rate. Utilizing energy methods, we show that under time-dependent Dirichlet boundary conditions, long-time dynamics of solutions are driven by their boundary data, and there is no restriction on the magnitude of initial energy. Moreover, the zero chemical diffusivity limit is established under zero Dirichlet boundary conditions, which has not been observed in previous studies on related models
{"title":"Initial and Boundary Value Problem for a System of Balance Laws from Chemotaxis: Global Dynamics and Diffusivity Limit","authors":"Zefu Feng","doi":"10.4208/AAM.OA-2020-0004","DOIUrl":"https://doi.org/10.4208/AAM.OA-2020-0004","url":null,"abstract":". In this paper, we study long-time dynamics and diffusion limit of large-data solutions to a system of balance laws arising from a chemotaxis model with logarithmic sensitivity and nonlinear production/degradation rate. Utilizing energy methods, we show that under time-dependent Dirichlet boundary conditions, long-time dynamics of solutions are driven by their boundary data, and there is no restriction on the magnitude of initial energy. Moreover, the zero chemical diffusivity limit is established under zero Dirichlet boundary conditions, which has not been observed in previous studies on related models","PeriodicalId":58853,"journal":{"name":"应用数学年刊:英文版","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46014022","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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