Time-local existence of unique smooth solutions to the Navier-Stokes equations in the whole space with linearly growing initial data has been established, via smoothing properties of Ornstein-Uhlenbeck semigroup. It has also been shown that the solution is real-analytic in spatial variables around rotating flows. This note is devoted to prove the spatial analyticity for cases of straining flows and shear flows. It is estimated the size of radius of convergence of Taylor series, due to estimates for higher order derivatives and Cauchy-Hadamard theorem.
{"title":"A remark for spatial analyticity around straining flows","authors":"S. Hattori, O. Sawada","doi":"10.7153/DEA-2018-10-26","DOIUrl":"https://doi.org/10.7153/DEA-2018-10-26","url":null,"abstract":"Time-local existence of unique smooth solutions to the Navier-Stokes equations in the whole space with linearly growing initial data has been established, via smoothing properties of Ornstein-Uhlenbeck semigroup. It has also been shown that the solution is real-analytic in spatial variables around rotating flows. This note is devoted to prove the spatial analyticity for cases of straining flows and shear flows. It is estimated the size of radius of convergence of Taylor series, due to estimates for higher order derivatives and Cauchy-Hadamard theorem.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"54 1","pages":"387-395"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75095166","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}
This paper deals with study of the two-parametric quantum Hamiltonian systems. The main objective in our study is Lyapunov inequalities of the two-parametric quantum Hamiltonian systems. In this paper, we first define two-parametric quantum analogous of the Leibniz rule, Cauchy-Schwarz and Holder inequalities and consequently as theoretical part of our main results, by the use of new Leibniz rule and Cauchy-Schwarz inequality on the considered Hamiltonian systems we obtain corresponding Lyapunov inequalities. Applicability of the obtained Lyapunov inequalities is examined by presenting a disconjugacy and at the same time a nonexistence criterion for the related Hamiltonian systems.
{"title":"Lyapunov inequalities for two-parametric quantum Hamiltonian systems and their applications","authors":"Yousef Gholami","doi":"10.7153/DEA-2018-10-19","DOIUrl":"https://doi.org/10.7153/DEA-2018-10-19","url":null,"abstract":"This paper deals with study of the two-parametric quantum Hamiltonian systems. The main objective in our study is Lyapunov inequalities of the two-parametric quantum Hamiltonian systems. In this paper, we first define two-parametric quantum analogous of the Leibniz rule, Cauchy-Schwarz and Holder inequalities and consequently as theoretical part of our main results, by the use of new Leibniz rule and Cauchy-Schwarz inequality on the considered Hamiltonian systems we obtain corresponding Lyapunov inequalities. Applicability of the obtained Lyapunov inequalities is examined by presenting a disconjugacy and at the same time a nonexistence criterion for the related Hamiltonian systems.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"24 1","pages":"261-276"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88105250","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}
The problem on linear stability of stationary spatial flows of an inviscid compressible fluid entirely occupying a certain volume with quiescent solid impenetrable boundary in absence of external mass forces is studied. Applying the direct Lyapunov method, such flows are proved to be absolutely unstable under small three–dimensional (3D) perturbations. Constructive conditions for linear practical instability are obtained. The a priori exponential lower estimate for the growth of the considered perturbations in time is found.
{"title":"On Instability of Steady–State Three–Dimensional Flows of an Ideal Compressible Fluid","authors":"Y. Gubarev","doi":"10.12691/IJPDEA-5-1-2","DOIUrl":"https://doi.org/10.12691/IJPDEA-5-1-2","url":null,"abstract":"The problem on linear stability of stationary spatial flows of an inviscid compressible fluid entirely occupying a certain volume with quiescent solid impenetrable boundary in absence of external mass forces is studied. Applying the direct Lyapunov method, such flows are proved to be absolutely unstable under small three–dimensional (3D) perturbations. Constructive conditions for linear practical instability are obtained. The a priori exponential lower estimate for the growth of the considered perturbations in time is found.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"103 1","pages":"10-18"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80723230","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}
{"title":"Existence and multiplicity solutions for a nonlocal equation of Kirchhoff type","authors":"Lin Li, Jijiang Sun","doi":"10.7153/dea-2018-10-25","DOIUrl":"https://doi.org/10.7153/dea-2018-10-25","url":null,"abstract":"","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"138 1","pages":"369-386"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74551839","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 some existence and uniqueness results for periodic solutions for a class of fractional differential equations with the Caputo fractional derivative. The arguments are based upon the Banach contraction principle, and Schaefer’s fixed point theorem.
{"title":"Periodic solutions for nonlinear fractional differential systems","authors":"S. Abbas, M. Benchohra, S. Bouriah, J. Nieto","doi":"10.7153/DEA-2018-10-21","DOIUrl":"https://doi.org/10.7153/DEA-2018-10-21","url":null,"abstract":"In this paper, we establish some existence and uniqueness results for periodic solutions for a class of fractional differential equations with the Caputo fractional derivative. The arguments are based upon the Banach contraction principle, and Schaefer’s fixed point theorem.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"52 1","pages":"299-316"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74869333","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}
B. Juárez-Campos, E. Kaikina, P. Naumkin, H. R. Paredes
We consider the question of global existence and asymptotics of small solutions of a certain pseudoparabolic equation in one dimension . This model is motivated by the wave equation for media with a strong spatial dispersion, which appear in the nonlinear theory of the quasi-stationary processes in the electric media. We develop the factorization technique to study the large time asymptotics of solutions.
{"title":"Factorization techniques for the nonlinear model of quasi-stationary processes in crystalline semiconductors","authors":"B. Juárez-Campos, E. Kaikina, P. Naumkin, H. R. Paredes","doi":"10.7153/DEA-2018-10-24","DOIUrl":"https://doi.org/10.7153/DEA-2018-10-24","url":null,"abstract":"We consider the question of global existence and asymptotics of small solutions of a certain pseudoparabolic equation in one dimension . This model is motivated by the wave equation for media with a strong spatial dispersion, which appear in the nonlinear theory of the quasi-stationary processes in the electric media. We develop the factorization technique to study the large time asymptotics of solutions.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"52 1","pages":"341-367"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90672289","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}
The box-counting dimension of graphs of oscillatory solutions to the Emden-Fowler equation is studied. The half-linear equation is also considered. Mathematics subject classification (2010): 34C10, 28A80.
{"title":"Box-counting dimension of oscillatory solutions to the Emden-Fowler equation","authors":"Takanao Kanemitsu, Satoshi Tanaka","doi":"10.7153/DEA-2018-10-17","DOIUrl":"https://doi.org/10.7153/DEA-2018-10-17","url":null,"abstract":"The box-counting dimension of graphs of oscillatory solutions to the Emden-Fowler equation is studied. The half-linear equation is also considered. Mathematics subject classification (2010): 34C10, 28A80.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"252 1","pages":"239-250"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77643463","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}
Nattapong Kamsrisuk, C. Promsakon, S. Ntouyas, J. Tariboon
In this paper we study existence and uniqueness of solutions for a boundary value problem for (p,q) -difference equations with nonlocal integral boundary conditions, by using classical fixed point theorems. Examples illustrating the main results are also presented. Mathematics subject classification (2010): 05A30, 39A13, 34A12.
{"title":"Nonlocal boundary value problems for (p, q)-difference equations","authors":"Nattapong Kamsrisuk, C. Promsakon, S. Ntouyas, J. Tariboon","doi":"10.7153/DEA-2018-10-11","DOIUrl":"https://doi.org/10.7153/DEA-2018-10-11","url":null,"abstract":"In this paper we study existence and uniqueness of solutions for a boundary value problem for (p,q) -difference equations with nonlocal integral boundary conditions, by using classical fixed point theorems. Examples illustrating the main results are also presented. Mathematics subject classification (2010): 05A30, 39A13, 34A12.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"71 1","pages":"183-195"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82122630","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 prove that the uniform exponential stability of time depended p -periodic system Ψ̇(t) = Π(t)Ψ(t), t ∈ R+, Ψ(t) ∈ Cn is equivalent to its Hyers–Ulam stability. As a tool, we consider the exact solution of the Cauchy problem { Θ̇(t) = Π(t)Θ(t)+ eiαtζ (t), t ∈ R+ Θ(0) = Θ0 as the approximate solution of Ψ̇(t) = Π(t)Ψ(t), t ∈ R+, Ψ(t)∈ Cn , where α is any real number, ζ (t) with ζ (0) = 0 , is a p -periodic bounded function on the Banach space S (R+,C) . More precisely we prove that the system Ψ̇(t) = Π(t)Ψ(t), t ∈ R+, Ψ(t) ∈ Cn is Hyers–Ulam stable if and only if it is exponentially stable. We argue that Hyers-Ulam stability concept is quite significant in realistic problems in numerical analysis and economics.
{"title":"Uniform exponential stability in the sense of Hyers and Ulam for periodic time varying linear systems","authors":"Bakht Zada","doi":"10.7153/DEA-2018-10-15","DOIUrl":"https://doi.org/10.7153/DEA-2018-10-15","url":null,"abstract":"We prove that the uniform exponential stability of time depended p -periodic system Ψ̇(t) = Π(t)Ψ(t), t ∈ R+, Ψ(t) ∈ Cn is equivalent to its Hyers–Ulam stability. As a tool, we consider the exact solution of the Cauchy problem { Θ̇(t) = Π(t)Θ(t)+ eiαtζ (t), t ∈ R+ Θ(0) = Θ0 as the approximate solution of Ψ̇(t) = Π(t)Ψ(t), t ∈ R+, Ψ(t)∈ Cn , where α is any real number, ζ (t) with ζ (0) = 0 , is a p -periodic bounded function on the Banach space S (R+,C) . More precisely we prove that the system Ψ̇(t) = Π(t)Ψ(t), t ∈ R+, Ψ(t) ∈ Cn is Hyers–Ulam stable if and only if it is exponentially stable. We argue that Hyers-Ulam stability concept is quite significant in realistic problems in numerical analysis and economics.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"17 1","pages":"227-234"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87179674","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}
Professor Robert JANIN was a former teacher in Poitiers (1988-1998); he died in October 2016. His speciality was Optimization Theory and Theory of Control. He obtained his thesis in 1974, at the University of PARIS VI, under the supervision of Professor Pallu de la Barrière. Then, he was hired as a researcher IRIA (nowadays INRIA) where he stayed there from 1967 to 1972 and then was appointed as a researcher at C.N.R.S. (National Center for Scientific Research in France) from 1972 to 1979. During that period, he spent two years (1976-1978) in Africa, in Togo more precisely. He applied to a full professorship position at University CUAG (Centre Universitaire AntillesGuyane) in 1979, where he spent nine years before pursuing his career
Robert JANIN教授曾在普瓦捷任教(1988-1998);他于2016年10月去世。他的专业是最优化理论和控制理论。1974年，他在巴黎第六大学，在帕卢·德拉·巴瑞特教授的指导下获得了论文。1967年至1972年，他被聘为法国国家科学研究中心(IRIA)研究员，1972年至1979年被聘为法国国家科学研究中心(cnr.s.)研究员。在此期间，他在非洲呆了两年(1976-1978)，更准确地说是在多哥。1979年，他申请了安的列斯-圭亚那大学中心(Centre Universitaire antilles - guyane)的正教授职位，在那里待了9年才开始他的职业生涯
{"title":"Dedicated to memory of Professor ROBERT JANIN","authors":"J. Rakotoson","doi":"10.7153/dea-2018-10-01","DOIUrl":"https://doi.org/10.7153/dea-2018-10-01","url":null,"abstract":"Professor Robert JANIN was a former teacher in Poitiers (1988-1998); he died in October 2016. His speciality was Optimization Theory and Theory of Control. He obtained his thesis in 1974, at the University of PARIS VI, under the supervision of Professor Pallu de la Barrière. Then, he was hired as a researcher IRIA (nowadays INRIA) where he stayed there from 1967 to 1972 and then was appointed as a researcher at C.N.R.S. (National Center for Scientific Research in France) from 1972 to 1979. During that period, he spent two years (1976-1978) in Africa, in Togo more precisely. He applied to a full professorship position at University CUAG (Centre Universitaire AntillesGuyane) in 1979, where he spent nine years before pursuing his career","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"38 1","pages":"1-2"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84224659","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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