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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
Global well-posedness of the incompressible Hall-MHD system in critical spaces 临界空间中不可压缩霍尔-MHD 系统的全局拟合性
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2024-01-11 DOI: 10.1007/s00028-023-00933-8
Mikihiro Fujii
{"title":"Global well-posedness of the incompressible Hall-MHD system in critical spaces","authors":"Mikihiro Fujii","doi":"10.1007/s00028-023-00933-8","DOIUrl":"https://doi.org/10.1007/s00028-023-00933-8","url":null,"abstract":"<p>In this paper, we consider the initial value problem of the incompressible Hall-MHD system and prove the global well-posedness in the scaling critical class <span>({dot{B}}_{p,infty }^{-1+frac{3}{p}}(mathbb {R}^3)times ({dot{B}}_{p,infty }^{-1+frac{3}{p}}(mathbb {R}^3) cap L^{infty }(mathbb {R}^3)))</span> for <span>(3&lt; p &lt; infty )</span>. Moreover, we also refine the smallness conditions and show that our global well-posedness holds for initial data whose <span>({dot{B}}_{p,infty }^{-1+frac{3}{p}}(mathbb {R}^3))</span>-norm is large, provided that some weaker norm is sufficiently small.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"47 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139421392","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
Rate of convergence for reaction–diffusion equations with nonlinear Neumann boundary conditions and $${mathcal {C}}^1$$ variation of the domain 具有非线性诺伊曼边界条件和 $${mathcal {C}}^1$ 域变化的反应扩散方程的收敛速率
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2024-01-11 DOI: 10.1007/s00028-023-00934-7
Marcone C. Pereira, Leonardo Pires
{"title":"Rate of convergence for reaction–diffusion equations with nonlinear Neumann boundary conditions and $${mathcal {C}}^1$$ variation of the domain","authors":"Marcone C. Pereira, Leonardo Pires","doi":"10.1007/s00028-023-00934-7","DOIUrl":"https://doi.org/10.1007/s00028-023-00934-7","url":null,"abstract":"<p>In this paper, we propose the compact convergence approach to deal with the continuity of attractors of some reaction–diffusion equations under smooth perturbations of the domain subject to nonlinear Neumann boundary conditions. We define a family of invertible linear operators to compare the dynamics of perturbed and unperturbed problems in the same phase space. All continuity arising from small smooth perturbations will be estimated by a rate of convergence given by the domain variation in a <span>({mathcal {C}}^1)</span> topology.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"41 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139461785","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 the Weierstraß form of infinite-dimensional differential algebraic equations. 论无穷维微分代数方程的 Weierstraß 形式。
IF 1.1 3区 数学
Journal of Evolution Equations Pub Date : 2024-01-01 Epub Date: 2024-09-02 DOI: 10.1007/s00028-024-01003-3
Mehmet Erbay, Birgit Jacob, Kirsten Morris
{"title":"On the Weierstraß form of infinite-dimensional differential algebraic equations.","authors":"Mehmet Erbay, Birgit Jacob, Kirsten Morris","doi":"10.1007/s00028-024-01003-3","DOIUrl":"https://doi.org/10.1007/s00028-024-01003-3","url":null,"abstract":"<p><p>The solvability for infinite-dimensional differential algebraic equations possessing a resolvent index and a Weierstraß form is studied. In particular, the concept of integrated semigroups is used to determine a subset on which solutions exist and are unique. This information is later used for a important class of systems, namely, port-Hamiltonian differential algebraic equations.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"24 4","pages":"73"},"PeriodicalIF":1.1,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11369006/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142134393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Gradient profile for the reconnection of vortex lines with the boundary in type-II superconductors II 型超导体中涡旋线与边界重新连接的梯度分布图
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2023-12-18 DOI: 10.1007/s00028-023-00932-9
Yi C. Huang, Hatem Zaag
{"title":"Gradient profile for the reconnection of vortex lines with the boundary in type-II superconductors","authors":"Yi C. Huang, Hatem Zaag","doi":"10.1007/s00028-023-00932-9","DOIUrl":"https://doi.org/10.1007/s00028-023-00932-9","url":null,"abstract":"<p>In a recent work, Duong, Ghoul and Zaag determined the gradient profile for blowup solutions of standard semilinear heat equation with power nonlinearities in the (supposed to be) generic case. Their method refines the constructive techniques introduced by Bricmont and Kupiainen and further developed by Merle and Zaag. In this paper, we extend their refinement to the problem about the reconnection of vortex lines with the boundary in a type-II superconductor under planar approximation, a physical model derived by Chapman, Hunton and Ockendon featuring the finite time quenching for the nonlinear heat equation </p><span>$$begin{aligned} frac{partial h}{partial t}=frac{partial ^2 h}{partial x^2}+e^{-h}-frac{1}{h^beta },quad beta &gt;0 end{aligned}$$</span><p>subject to initial boundary value conditions </p><span>$$begin{aligned} h(cdot ,0)=h_0&gt;0,quad h(pm 1,t)=1. end{aligned}$$</span><p>We derive the intermediate extinction profile with refined asymptotics, and with extinction time <i>T</i> and extinction point 0, the gradient profile behaves as <span>(xrightarrow 0)</span> like </p><span>$$begin{aligned} lim _{trightarrow T},(nabla h)(x,t)quad sim quad frac{1}{sqrt{2beta }}frac{x}{|x|}frac{1}{sqrt{|log |x||}} left[ frac{(beta +1)^2}{8beta }frac{|x|^2}{|log |x||}right] ^{frac{1}{beta +1}-frac{1}{2}}, end{aligned}$$</span><p>agreeing with the gradient of the extinction profile previously derived by Merle and Zaag. Our result holds with general boundary conditions and in higher dimensions.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"33 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138743641","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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