Journal of Differential Equations最新文献

{"title":"Periodic normal forms for bifurcations of limit cycles in DDEs","authors":"Bram Lentjes ,&nbsp;Len Spek ,&nbsp;Maikel M. Bosschaert ,&nbsp;Yuri A. Kuznetsov","doi":"10.1016/j.jde.2025.01.064","DOIUrl":"10.1016/j.jde.2025.01.064","url":null,"abstract":"<div><div>A recent work <span><span>[1]</span></span> by the authors on the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic cycle in delay differential equations motivates the derivation of normal forms. In this paper, we prove the existence of a special coordinate system on the center manifold that allows us to describe the local dynamics on the center manifold near the cycle in terms of periodic normal forms. To construct the linear part of this coordinate system, we prove the existence of time periodic smooth Jordan chains for the original and adjoint system. Moreover, we establish duality and spectral relations between both systems by using tools from the theory of delay and Volterra integral equations, dual perturbation theory, duality theory, and evolution semigroups.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"423 ","pages":"Pages 631-694"},"PeriodicalIF":2.4,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143158367","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimization of the Steklov-Lamé eigenvalues with respect to the domain","authors":"Pedro R.S. Antunes ,&nbsp;Beniamin Bogosel","doi":"10.1016/j.jde.2025.01.043","DOIUrl":"10.1016/j.jde.2025.01.043","url":null,"abstract":"<div><div>This work deals with theoretical and numerical aspects related to the behavior of the Steklov-Lamé eigenvalues on variable domains. After establishing the eigenstructure for the disk, we prove that for a certain class of Lamé parameters, the disk maximizes the first non-zero eigenvalue under area or perimeter constraints in dimension two. Upper bounds for these eigenvalues can be found in terms of the scalar Steklov eigenvalues, involving various geometric quantities. We prove that the Steklov-Lamé eigenvalues are upper semicontinuous for the complementary Hausdorff convergence of <em>ε</em>-cone domains and, as a consequence, there exist shapes maximizing these eigenvalues under convexity and volume constraints. A numerical method based on fundamental solutions is proposed for computing the Steklov-Lamé eigenvalues, allowing to study numerically the shapes maximizing the first ten non-zero eigenvalues.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"426 ","pages":"Pages 1-35"},"PeriodicalIF":2.4,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143095489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Global well-posedness for a quasilinear chemotaxis system with decaying diffusivity and consumption of a chemoattractant","authors":"Juan Yang ,&nbsp;Chunyou Sun","doi":"10.1016/j.jde.2025.01.062","DOIUrl":"10.1016/j.jde.2025.01.062","url":null,"abstract":"&lt;div&gt;&lt;div&gt;The Neumann initial-boundary value problem for the quasilinear parabolic-parabolic chemotaxis model:&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; is considered in smoothly bounded domains &lt;span&gt;&lt;math&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. The diffusivity sensitivity &lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and chemotaxis sensitivity &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; satisfy&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mtext&gt;and&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mtext&gt; for any&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for all &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;)&lt;/m","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"426 ","pages":"Pages 36-71"},"PeriodicalIF":2.4,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143095488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The incompressible Navier-Stokes-Fourier-Poisson limit of the Vlasov-Poisson-Boltzmann-Fermi-Dirac equation","authors":"Bowen Han","doi":"10.1016/j.jde.2025.01.048","DOIUrl":"10.1016/j.jde.2025.01.048","url":null,"abstract":"<div><div>For the Vlasov-Poisson-Boltzmann-Fermi-Dirac (VPBFD) system in the scaling under which the moments of fluctuations formally converge to the incompressible Navier-Stokes-Fourier-Poisson (NSFP) system, we prove the uniform estimates with respect to the Knudsen number <em>ε</em> for the fluctuations. As a consequence, the existence of global-in-time classical solutions of VPBFD with all <span><math><mi>ε</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> is established in whole space under small initial data, and the convergence to incompressible NSFP as <em>ε</em> goes to 0 is rigorously justified.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"426 ","pages":"Pages 104-185"},"PeriodicalIF":2.4,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143095492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Hamilton-Jacobi approach to neural field equations","authors":"Wen Tao,&nbsp;Wan-Tong Li,&nbsp;Jian-Wen Sun","doi":"10.1016/j.jde.2025.01.065","DOIUrl":"10.1016/j.jde.2025.01.065","url":null,"abstract":"<div><div>This paper explores the long time/large space dynamics of the neural field equation with an exponentially decaying initial data. By establishing a Harnack type inequality, we derive the Hamilton-Jacobi equation corresponding to the neural field equation due to the elegant theory developed by Freidlin [<em>Ann. Probab.</em> (1985)], Evans and Souganidis [<em>Indiana Univ. Math. J.</em> (1989)]. In addition, we obtain the exact formula for the motion of the interface by constructing the explicit viscosity solutions for the underlying Hamilton-Jacobi equation. It is then shown that the propagation speed of the interface is determined by the decay rate of the initial value. As an intriguing implication, we find that the propagation speed of interface is related to the speed of traveling waves. Finally, we study the spreading speed of the corresponding Cauchy problem. To the best of our knowledge, it is the first time that the Hamilton-Jacobi approach is used in the study of dynamics of neural field equations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"422 ","pages":"Pages 659-695"},"PeriodicalIF":2.4,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143174123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Strong solutions to the three-dimensional two-phase magnetohydrodynamic equations","authors":"Tian Jing ,&nbsp;Dehua Wang","doi":"10.1016/j.jde.2025.01.017","DOIUrl":"10.1016/j.jde.2025.01.017","url":null,"abstract":"<div><div>In this paper, we study the existence of strong solutions to the two-phase magnetohydrodynamic equations in a bounded domain <span><math><mi>Ω</mi><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. The fluids are incompressible, viscous, and resistive. The surface tension is considered. The equations are reformulated using the Hanzawa transformation, which turns the free interface into a fixed one for a short time. The study of the new equations is then divided into the principal part and the nonlinear part. Due to the effect of the magnetic field and the complexity of the transformation in generic bounded domains, the Fréchet derivatives of nonlinearities have to be carefully estimated. The equations can then be solved using the fixed-point argument by finding a contraction mapping, which follows the estimates of the nonlinear part.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"425 ","pages":"Pages 661-727"},"PeriodicalIF":2.4,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094062","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Global well-posedness and stability of classical solutions to the pressureless Navier-Stokes system in 3D","authors":"Boling Guo ,&nbsp;Houzhi Tang ,&nbsp;Bin Zhao","doi":"10.1016/j.jde.2025.01.074","DOIUrl":"10.1016/j.jde.2025.01.074","url":null,"abstract":"<div><div>In this paper, we consider the Cauchy problem of the pressureless Navier-Stokes system which can be derived by taking the high Mach number limit for compressible Navier-Stokes equations. Due to the absence of pressure, the classical perturbation theory developed by Matsumura and Nishida (1980) <span><span>[34]</span></span> is not applicable to this model. The major difficulty lies in lack of the uniform boundedness of density. To solve this problem, we employ the spectral analysis and energy method to obtain the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> time-decay rates of velocity under the additional assumption that the initial velocity is small in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> norm. Then combining the time-decay rates of velocity yields the uniform boundedness of density. Furthermore, it is proved that the velocity of pressureless flow decays to the motionless state at an optimal time-decay rate of <span><math><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>t</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></math></span> in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"422 ","pages":"Pages 696-716"},"PeriodicalIF":2.4,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143174124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Limit cycle bifurcations for a homoclinic loop with a nilpotent equilibrium","authors":"Yanqin Xiong ,&nbsp;Xingwu Chen","doi":"10.1016/j.jde.2025.01.056","DOIUrl":"10.1016/j.jde.2025.01.056","url":null,"abstract":"<div><div>In this paper we investigate the limit cycle bifurcation problem of system <span><math><mover><mrow><mi>x</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>=</mo><mi>y</mi><mo>,</mo><mspace></mspace><mover><mrow><mi>y</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>=</mo><mo>−</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> under polynomial perturbations of degree <em>n</em>, where <span><math><mi>m</mi><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> and this unperturbed system possesses a homoclinic loop (denoted by <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>) connecting a nilpotent equilibrium of order <span><math><mo>[</mo><mi>m</mi><mo>/</mo><mn>2</mn><mo>]</mo></math></span> at the origin. By exploiting the inherent structure of the unperturbed system, we provide a general expression for the first-order Melnikov function in a neighborhood of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> as well as formulas of all its coefficients. Moreover, the independence of these coefficients is utilized to explore limit cycle bifurcations near <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> by determining the maximum possible number of simple zeros of the first-order Melnikov function in the vicinity of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>, which gives the number of limit cycles depending on general <em>m</em> and <em>n</em>. Following our main results, the growth rate of the maximal number of limit cycles near a homoclinic loop is at least <span><math><mi>n</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>4</mn></math></span> for near-Hamiltonian systems, and at least 2<em>n</em> for Liénard systems.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"426 ","pages":"Pages 72-103"},"PeriodicalIF":2.4,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143095490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-unique ergodicity for the 2D stochastic Navier-Stokes equations with derivative of space-time white noise","authors":"Huaxiang Lü,&nbsp;Xiangchan Zhu","doi":"10.1016/j.jde.2025.01.041","DOIUrl":"10.1016/j.jde.2025.01.041","url":null,"abstract":"<div><div>We prove existence of infinitely many stationary solutions as well as ergodic stationary solutions for the stochastic Navier-Stokes equations on <span><math><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span><span><span><span><math><mrow><mi>d</mi><mi>u</mi><mo>+</mo><mrow><mi>div</mi></mrow><mo>(</mo><mi>u</mi><mo>⊗</mo><mi>u</mi><mo>)</mo><mi>d</mi><mi>t</mi><mo>+</mo><mi>∇</mi><mi>p</mi><mi>d</mi><mi>t</mi><mo>=</mo><mi>Δ</mi><mi>u</mi><mi>d</mi><mi>t</mi><mo>+</mo><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>a</mi><mo>/</mo><mn>2</mn></mrow></msup><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mrow><mi>div</mi></mrow><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo></mrow></math></span></span></span> driven by the derivative of space-time white noise, where <span><math><mi>a</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></math></span>. In this setting, the solutions are not function valued and probabilistic renormalization is required to give a meaning to the equations. Finally, we show that the stationary distributions are not Gaussian distribution <span><math><mi>N</mi><mo>(</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>a</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span>. The proof relies on a time-dependent decomposition and a stochastic version of the convex integration method which provides uniform moment bounds in some function spaces.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"425 ","pages":"Pages 383-433"},"PeriodicalIF":2.4,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143102807","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stability for the 3D magneto-micropolar fluids with only velocity dissipation near a background magnetic field","authors":"Xiaoping Zhai ,&nbsp;Jiahong Wu ,&nbsp;Fuyi Xu","doi":"10.1016/j.jde.2025.01.047","DOIUrl":"10.1016/j.jde.2025.01.047","url":null,"abstract":"<div><div>Many biological fluids such as blood contain ions that can interact with an applied magnetic field, and can thus be treated as electrically conducting fluids. In many recent studies related to bioengineering such as the development of magnetic tracers and medical applications, the magneto-micropolar systems have been used to model the dynamics of these biofluids. A natural question related to these studies is the influence of a static background magnetic field on the stability and long-time behavior of the biofluids. This paper intends to present a rigorous theory in terms of the magneto-micropolar system with only velocity dissipation but without magnetic dissipation and angular dissipation. The spatial domain is taken to be a periodic box and the background magnetic field is assumed to satisfy a Diophantine condition. This Diophantine condition is satisfied by almost every vector field. We establish the asymptotic stability of any perturbation and its precise long-time behavior. This result reflects the stabilizing effect of the background magnetic field. Without it, it is almost impossible to analyze this 3D nonlinear magneto-micropolar system. The mathematical objects underlying the smoothing and stabilizing effect are the wave structures hidden in this magneto-micropolar system.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"425 ","pages":"Pages 596-626"},"PeriodicalIF":2.4,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}