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Long time behavior of solutions of an electroconvection model in $${mathbb {R}}^2$$ 电对流模型解在 $${mathbb {R}}^2$ 中的长时间行为
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2024-02-10 DOI: 10.1007/s00028-024-00944-z
{"title":"Long time behavior of solutions of an electroconvection model in $${mathbb {R}}^2$$","authors":"","doi":"10.1007/s00028-024-00944-z","DOIUrl":"https://doi.org/10.1007/s00028-024-00944-z","url":null,"abstract":"<h3>Abstract</h3> <p>We consider a two dimensional electroconvection model which consists of a nonlinear and nonlocal system coupling the evolutions of a charge distribution and a fluid. We show that the solutions decay in time in <span> <span>(L^2({{mathbb {R}}}^2))</span> </span> at the same sharp rate as the linear uncoupled system. This is achieved by proving that the difference between the nonlinear and linear evolution decays at a faster rate than the linear evolution. In order to prove the sharp <span> <span>(L^2)</span> </span> decay we establish bounds for decay in <span> <span>(H^2({{mathbb {R}}}^2))</span> </span> and a logarithmic growth in time of a quadratic moment of the charge density.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"66 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757334","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}
引用次数: 0
Existence of a weak solution and blow-up of strong solutions for a two-component Fornberg–Whitham system 双分量福恩贝格-惠瑟姆系统弱解的存在和强解的膨胀
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2024-02-10 DOI: 10.1007/s00028-023-00941-8
Zhihao Bai, Yang Wang, Long Wei
{"title":"Existence of a weak solution and blow-up of strong solutions for a two-component Fornberg–Whitham system","authors":"Zhihao Bai, Yang Wang, Long Wei","doi":"10.1007/s00028-023-00941-8","DOIUrl":"https://doi.org/10.1007/s00028-023-00941-8","url":null,"abstract":"<p>In this paper, we investigate the existence of a weak solution and blow-up of strong solutions to a two-component Fornberg–Whitham system. Due to the absence of some useful conservation laws, we establish the existence of a weak solution to the system in lower order Sobolev spaces <span>(H^{s}times H^{s-1})</span> (<span>(sin (1,3/2])</span>) via a modified pseudo-parabolic regularization method. And then, a blow-up scenario for strong solutions to this system is shown. By the analysis of Riccati-type inequalities recently, we present some sufficient conditions on the initial data that lead to the blow-up for corresponding strong solutions to the system.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"45 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757335","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}
引用次数: 0
Well-posedness and asynchronous exponential growth of an age-weighted structured fish population model with diffusion in $$L^1$$ 在 $$L^1$$ 中具有扩散作用的年龄加权结构化鱼类种群模型的良好拟合和非同步指数增长
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2024-02-10 DOI: 10.1007/s00028-023-00942-7
Samir Boujijane, Said Boulite, Mohamed Halloumi, Lahcen Maniar, Abdelaziz Rhandi
{"title":"Well-posedness and asynchronous exponential growth of an age-weighted structured fish population model with diffusion in $$L^1$$","authors":"Samir Boujijane, Said Boulite, Mohamed Halloumi, Lahcen Maniar, Abdelaziz Rhandi","doi":"10.1007/s00028-023-00942-7","DOIUrl":"https://doi.org/10.1007/s00028-023-00942-7","url":null,"abstract":"<p>In the present paper, we address the asymptotic behavior of a fish population system structured in age and weight, while also incorporating spatial effects. Initially, we develop an abstract perturbation result concerning the essential spectral radius, employing the regular systems approach. Following that, we present the model in the form of a perturbed boundary problem, which involves unbounded operators on the boundary. Using time-invariant regular techniques, we construct the corresponding semigroup solution. Then, we designate an operator characteristic equation of the primary system via the radius of a bounded linear operator defined on the boundary space. Moreover, we provide a characterization of the uniform exponential stability and the asynchronous exponential growth property (AEG) by localizing the essential radius and proving the irreducibility of the perturbed semigroup. Finally, we precise the projection that emerged from the (AEG) property; this depends on the developed characteristic equation.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"66 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757516","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}
引用次数: 0
On a thermodynamically consistent model for magnetoviscoelastic fluids in 3D 关于三维磁致弹性流体的热力学一致模型
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2024-02-10 DOI: 10.1007/s00028-023-00938-3
Hengrong Du, Yuanzhen Shao, Gieri Simonett
{"title":"On a thermodynamically consistent model for magnetoviscoelastic fluids in 3D","authors":"Hengrong Du, Yuanzhen Shao, Gieri Simonett","doi":"10.1007/s00028-023-00938-3","DOIUrl":"https://doi.org/10.1007/s00028-023-00938-3","url":null,"abstract":"<p>We introduce a system of equations that models a non-isothermal magnetoviscoelastic fluid. We show that the model is thermodynamically consistent, and that the critical points of the entropy functional with prescribed energy correspond exactly with the equilibria of the system. The system is investigated in the framework of quasilinear parabolic systems and shown to be locally well-posed in an <span>(L_p)</span>-setting. Furthermore, we prove that constant equilibria are normally stable. In particular, we show that solutions that start close to a constant equilibrium exist globally and converge exponentially fast to a (possibly different) constant equilibrium. Finally, we establish that the negative entropy serves as a strict Lyapunov functional and we then show that every solution that is eventually bounded in the topology of the natural state space exists globally and converges to the set of equilibria.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"5 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757306","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}
引用次数: 0
Finite- and infinite-time cluster formation for alignment dynamics on the real line 实线上排列动力学的有限时间和无限时间集群形成
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2024-02-10 DOI: 10.1007/s00028-023-00939-2
Trevor M. Leslie, Changhui Tan
{"title":"Finite- and infinite-time cluster formation for alignment dynamics on the real line","authors":"Trevor M. Leslie, Changhui Tan","doi":"10.1007/s00028-023-00939-2","DOIUrl":"https://doi.org/10.1007/s00028-023-00939-2","url":null,"abstract":"<p>We show that the locations where finite- and infinite-time clustering occur for the 1D Euler-alignment system can be determined using only the initial data. Our present work provides the first results on the structure of the finite-time singularity set and asymptotic clusters associated with a weak solution. In many cases, the eventual size of the cluster can be read off directly from the flux associated with a scalar balance law formulation of the system.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"53 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757330","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}
引用次数: 0
Stability and optimal decay for the 3D magnetohydrodynamic equations with only horizontal dissipation 仅有水平耗散的三维磁流体动力学方程的稳定性和最优衰减
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2024-02-10 DOI: 10.1007/s00028-023-00940-9
Haifeng Shang, Jiahong Wu, Qian Zhang
{"title":"Stability and optimal decay for the 3D magnetohydrodynamic equations with only horizontal dissipation","authors":"Haifeng Shang, Jiahong Wu, Qian Zhang","doi":"10.1007/s00028-023-00940-9","DOIUrl":"https://doi.org/10.1007/s00028-023-00940-9","url":null,"abstract":"<p>This paper develops an effective approach to establishing the optimal decay estimates on solutions of the 3D anisotropic magnetohydrodynamic (MHD) equations with only horizontal dissipation. As our first step, we prove the global existence and stability of solutions to the aforementioned MHD system emanating from any initial data with small <span>(H^1)</span>-norm. Due to the lack of dissipation in the vertical direction, the large-time behavior does not follow from the classical approaches. The analysis of the nonlinear terms are much more difficult than in the case of full dissipation. In particular, we need to represent the MHD equations in an integral form, exploit cancellations and other properties such as the incompressibility in order to control terms involving vertical derivatives.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"53 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757354","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}
引用次数: 0
Pointwise space-time estimates of two-phase fluid model in dimension three 三维两相流体模型的点时空估算
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2024-02-10 DOI: 10.1007/s00028-024-00943-0
Zhigang Wu, Wenyue Zhou
{"title":"Pointwise space-time estimates of two-phase fluid model in dimension three","authors":"Zhigang Wu, Wenyue Zhou","doi":"10.1007/s00028-024-00943-0","DOIUrl":"https://doi.org/10.1007/s00028-024-00943-0","url":null,"abstract":"<p>We studied the pointwise space-time behavior of the classical solution to the Cauchy problem of two-phase fluid model derived by Choi (SIAM J Math Anal 48:3090–3122, 2016) when the initial data is sufficiently small and regular. This model is the compressible damped Euler system coupled with the compressible Naiver–Stokes system via a drag force. As we know, Liu and Wang (Commun Math Phys 196:145–173, 1998) verified that the solution of the compressible Naiver–Stokes system obeys the generalized Huygens’ principle, while Wang and Yang (J Differ Equ 173:410–450, 2001) verified the solution of the compressible Euler system does not obey the generalized Huygens’ principle due to the damped mechanism. In this paper, we proved that both of two densities and two momentums for the two-phase fluid model obey the generalized Huygens’ principle as that in Liu and Wang (Commun Math Phys 196:145–173, 1998). The main contribution is to overcome the difficulty of the non-conservation arising from the damped mechanism of the system. As a byproduct, we also extended <span>(L^2)</span>-estimate in Wu et al. (SIAM J Math Anal 52(6):5748–5774, 2020) to <span>(L^p)</span>-estimate with <span>(p&gt;1)</span>.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"45 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757511","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}
引用次数: 0
Persistence and asymptotic analysis of solutions of nonlinear wave equations 非线性波方程解的持久性和渐近分析
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2024-01-17 DOI: 10.1007/s00028-023-00937-4
Igor Leite Freire
{"title":"Persistence and asymptotic analysis of solutions of nonlinear wave equations","authors":"Igor Leite Freire","doi":"10.1007/s00028-023-00937-4","DOIUrl":"https://doi.org/10.1007/s00028-023-00937-4","url":null,"abstract":"<p>We consider persistence properties of solutions for a generalised wave equation including vibration in elastic rods and shallow water models, such as the BBM, the Dai’s, the Camassa–Holm, and the Dullin–Gottwald–Holm equations, as well as some recent shallow water equations with Coriolis effect. We establish unique continuation results and exhibit asymptotic profiles for the solutions of the general class considered. From these results we prove the non-existence of non-trivial spatially compactly supported solutions for the equation. As an aftermath, we study the equations earlier mentioned in light of our results for the general class.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"1 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139481617","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}
引用次数: 0
Orbital instability of periodic waves for scalar viscous balance laws 标量粘性平衡定律周期波的轨道不稳定性
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2024-01-17 DOI: 10.1007/s00028-023-00936-5
Enrique Álvarez, Jaime Angulo Pava, Ramón G. Plaza
{"title":"Orbital instability of periodic waves for scalar viscous balance laws","authors":"Enrique Álvarez, Jaime Angulo Pava, Ramón G. Plaza","doi":"10.1007/s00028-023-00936-5","DOIUrl":"https://doi.org/10.1007/s00028-023-00936-5","url":null,"abstract":"<p>The purpose of this paper is to prove that, for a large class of nonlinear evolution equations known as scalar viscous balance laws, the spectral (linear) instability condition of periodic traveling wave solutions implies their orbital (nonlinear) instability in appropriate periodic Sobolev spaces. The analysis is based on the well-posedness theory, the smoothness of the data-solution map, and an abstract result of instability of equilibria under nonlinear iterations. The resulting instability criterion is applied to two families of periodic waves. The first family consists of small amplitude waves with finite fundamental period which emerge from a local Hopf bifurcation around a critical value of the velocity. The second family comprises arbitrarily large period waves which arise from a homoclinic (global) bifurcation and tend to a limiting traveling pulse when their fundamental period tends to infinity. In the case of both families, the criterion is applied to conclude their orbital instability under the flow of the nonlinear viscous balance law in periodic Sobolev spaces with same period as the fundamental period of the wave.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"29 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139481668","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}
引用次数: 0
A Kalman condition for the controllability of a coupled system of Stokes equations 斯托克斯方程耦合系统可控性的卡尔曼条件
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2024-01-11 DOI: 10.1007/s00028-023-00935-6
Takéo Takahashi, Luz de Teresa, Yingying Wu-Zhang
{"title":"A Kalman condition for the controllability of a coupled system of Stokes equations","authors":"Takéo Takahashi, Luz de Teresa, Yingying Wu-Zhang","doi":"10.1007/s00028-023-00935-6","DOIUrl":"https://doi.org/10.1007/s00028-023-00935-6","url":null,"abstract":"<p>We consider the controllability of a class of systems of <i>n</i> Stokes equations, coupled through terms of order zero and controlled by <i>m</i> distributed controls. Our main result states that such a system is null-controllable if and only if a Kalman type condition is satisfied. This generalizes the case of finite-dimensional systems and the case of systems of coupled linear heat equations. The proof of the main result relies on the use of the Kalman operator introduced in [1] and on a Carleman estimate for a cascade type system of Stokes equations. Using a fixed-point argument, we also obtain that if the Kalman condition is verified, then the corresponding system of Navier–Stokes equations is locally null-controllable.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"13 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139421434","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}
引用次数: 0
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