Journal of Differential Equations最新文献

{"title":"Bounded and periodic solutions of quasilinear parabolic equations in time-dependent domains","authors":"Mitsuhiro Nakao","doi":"10.1016/j.jde.2025.02.048","DOIUrl":"10.1016/j.jde.2025.02.048","url":null,"abstract":"<div><div>We show the existence and uniqueness of the bounded or periodic solution for the quasilinear parabolic equation of the form<span><span><span>(1.1)</span><span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>−</mo><mtext>div</mtext><mrow><mo>(</mo><mi>σ</mi><mo>(</mo><mo>|</mo><mi>∇</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>∇</mi><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mtext> in </mtext><mi>Q</mi><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mo>∞</mo><mo>)</mo></math></span></span></span> with the boundary condition <span><math><mi>u</mi><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mo>|</mo></mrow><mrow><mo>∂</mo><mi>Ω</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></msub><mo>=</mo><mn>0</mn></math></span>, where <span><math><mi>Ω</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> is a bounded domain in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> for each <span><math><mi>t</mi><mo>∈</mo><mi>R</mi></math></span> and <span><math><mi>Q</mi><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mo>∞</mo><mo>)</mo><mo>=</mo><msub><mrow><mo>∪</mo></mrow><mrow><mo>−</mo><mo>∞</mo><mo><</mo><mi>t</mi><mo><</mo><mo>∞</mo></mrow></msub><mi>Ω</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>×</mo><mo>{</mo><mi>t</mi><mo>}</mo></math></span>. Typical examples of <em>σ</em> are <span><math><mi>σ</mi><mo>(</mo><msup><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>=</mo><mo>|</mo><mi>v</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>m</mi></mrow></msup><mo>,</mo><mi>m</mi><mo>≥</mo><mn>0</mn><mo>,</mo><mi>σ</mi><mo>(</mo><msup><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>=</mo><mtext>log</mtext><mo>(</mo><mn>1</mn><mo>+</mo><msup><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> and <span><math><mi>σ</mi><mo>(</mo><msup><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>=</mo><mo>|</mo><mi>v</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>m</mi></mrow></msup><mo>/</mo><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msqrt><mo>,</mo><mi>m</mi><mo>≥</mo><mn>1</mn></math></span>. We derive a precise estimate for <span><math><msub><mrow><mi>sup</mi></mrow><mrow><mo>−</mo><mo>∞</mo><mo><</mo><mi>t</mi><mo><</mo><mo>∞</mo></mrow></msub><mo>⁡</mo><msub><mrow><mo>‖</mo><mi>∇</mi><mi>u</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>‖</mo></mrow><mrow><mi>Ω</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mo>∞</mo></mrow></msub></math></span> depending on <span><math><msub><mrow><mi>sup</mi></mrow><mrow><mo>−</mo><mo>∞</mo><mo><</mo><mi>t</mi><mo><</mo><mo>∞</mo></mrow></msub><mo>⁡</mo><msub><mrow><mo>‖</mo><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>‖</mo></mrow><mrow><mi>Ω</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mo>∞</mo></mrow></msub></math></span> and the movement of <span><math><mo>∂</mo><mi>Ω</mi><","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"436 ","pages":"Article 113177"},"PeriodicalIF":2.4,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143705433","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":"Effective medium theory for Van-der-Waals heterostructures","authors":"Xinlin Cao , Ahcene Ghandriche , Mourad Sini","doi":"10.1016/j.jde.2025.113260","DOIUrl":"10.1016/j.jde.2025.113260","url":null,"abstract":"<div><div>We derive the electromagnetic medium equivalent to a collection of all-dielectric nano-particles (enjoying high refractive indices) distributed locally non-periodically, precisely the medium is periodic with a unit cell composed of a cluster of multiple nano-particles, in a smooth domain Ω. Such distributions are used to model well-known structures in material sciences as the Van-der-Waals heterostructures. Since the nano-particles are all-dielectric, then the permittivity remains unchanged with respect to the background while the permeability is altered by this effective medium. This permeability is given in terms of three parameters. The first one is the polarization tensors of the used nano-particles. The second one is the averaged Magnetization matrix <span><math><mo>|</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mspace></mspace><msub><mrow><mo>∫</mo></mrow><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msub><munder><mi>∇</mi><mi>x</mi></munder><msub><mrow><mo>∫</mo></mrow><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msub><munder><mi>∇</mi><mi>y</mi></munder><msub><mrow><mi>Φ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>⋅</mo><msub><mrow><mi>I</mi></mrow><mrow><mn>3</mn></mrow></msub><mspace></mspace><mi>d</mi><mi>y</mi><mspace></mspace><mi>d</mi><mi>x</mi></math></span>, where <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mspace></mspace><mo>:</mo><mo>=</mo><mspace></mspace><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn><mspace></mspace><mi>π</mi><mspace></mspace><mo>|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo>|</mo></mrow></mfrac></math></span>, <span><math><msub><mrow><mi>I</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> is the identity matrix and <span><math><msub><mrow><mi>Ω</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is the unit cell. The third one is <span><math><mi>∇</mi><mi>∇</mi><msub><mrow><mi>Φ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>'s are locations of the local nano-particles distributed in the unit cell. This last tensor models the local strong interaction of the nano-particles. To our best knowledge, such tensors are new in both the mathematical and engineering oriented literature. This equivalent medium describes, in particular, the effective medium of 2 dimensional type Van-der-Waals heterostructures. These structures are 3 dimensional which are built as superposition of identical (2D)-sheets each supporting locally non-periodic distributions of nano-particles. An explicit form of thi","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"433 ","pages":"Article 113260"},"PeriodicalIF":2.4,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704189","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":"Nonlinear Rayleigh-Taylor instability in compressible viscoelastic fluids with an upper free boundary","authors":"Caifeng Liu ,&nbsp;Wanwan Zhang","doi":"10.1016/j.jde.2025.113248","DOIUrl":"10.1016/j.jde.2025.113248","url":null,"abstract":"<div><div>In this paper, we study the nonlinear Rayleigh-Taylor instability in the compressible viscoelastic fluid governed by the gravity-driven Oldroyd-B model in a finitely deep and horizontally periodic moving domain. Under the instability condition <span><math><mi>κ</mi><mo>&lt;</mo><msub><mrow><mi>κ</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span>, we first prove that there exist growing mode solutions to the linearized equations around a viscoelastic equilibrium <span><math><mo>(</mo><mover><mrow><mi>ρ</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>,</mo><mn>0</mn><mo>,</mo><mover><mrow><mi>u</mi></mrow><mrow><mo>¯</mo></mrow></mover><mi>I</mi><mo>)</mo></math></span> for a smooth increasing density profile <span><math><mover><mrow><mi>ρ</mi></mrow><mrow><mo>¯</mo></mrow></mover></math></span>. Based on the finding of growing modes, we then show the nonlinear Rayleigh-Taylor instability of the above profile by constructing appropriate initial data for the nonlinear perturbation problem departing from the equilibrium and conducting some instability bootstrap arguments.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"432 ","pages":"Article 113248"},"PeriodicalIF":2.4,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704854","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":"Multiple solutions to an elliptic problem driven by a singular nonlinearity","authors":"Debajyoti Choudhuri ,&nbsp;Ratan Kr. Giri ,&nbsp;K. Saoudi","doi":"10.1016/j.jde.2025.113263","DOIUrl":"10.1016/j.jde.2025.113263","url":null,"abstract":"<div><div>We will prove the existence of multiple solutions to an elliptic problem driven by a singular nonlinearity. Further we studied the singular problem and analyzed the regularity of the solutions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"436 ","pages":"Article 113263"},"PeriodicalIF":2.4,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143705434","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":"Fractional mean field equations: Theory and application on finite graphs","authors":"Yang Liu","doi":"10.1016/j.jde.2025.113264","DOIUrl":"10.1016/j.jde.2025.113264","url":null,"abstract":"<div><div>In this paper, the author introduces a nonlocal perspective by incorporating the fractional Laplacian <span><math><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup></math></span>, and considers the fractional mean field equation on a finite graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span>, say<span><span><span><math><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup><mi>u</mi><mo>=</mo><mi>ρ</mi><mrow><mo>(</mo><mfrac><mrow><mi>h</mi><msup><mrow><mi>e</mi></mrow><mrow><mi>u</mi></mrow></msup></mrow><mrow><msub><mrow><mo>∫</mo></mrow><mrow><mi>V</mi></mrow></msub><mi>h</mi><msup><mrow><mi>e</mi></mrow><mrow><mi>u</mi></mrow></msup><mi>d</mi><mi>μ</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow></mfrac><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mo>∀</mo><mspace></mspace><mi>x</mi><mo>∈</mo><mi>V</mi><mo>,</mo></math></span></span></span> where <span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, <span><math><mi>ρ</mi><mo>∈</mo><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mn>0</mn><mo>)</mo><mo>∪</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></math></span> are some fixed parameters, <em>h</em> denotes a given real value function on <em>V</em>. Based on the sign of the prescribed function <em>h</em>, using various methods such as variational method, topological degree and two mean field type heat flows, the author obtains the existence of solutions for the above problem in three cases respectively. These results extend the relevant research of Lin-Yang (Calc. Var., 2021), Sun-Wang (Adv. Math., 2022) and Liu-Zhang (J. Math. Anal. Appl., 2023) in the case of <span><math><mi>s</mi><mo>=</mo><mn>1</mn></math></span>, and potentially broaden the understanding and application of fractional operators in discrete mathematical structures, emphasizing connections to both continuous and discrete theories.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"436 ","pages":"Article 113264"},"PeriodicalIF":2.4,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143705435","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":"On long time behavior of the solutions of the coupled Zakharov-Kuznetsov system","authors":"Qihe Niang,&nbsp;Deqin Zhou","doi":"10.1016/j.jde.2025.113256","DOIUrl":"10.1016/j.jde.2025.113256","url":null,"abstract":"<div><div>We consider the long time asymptotic behavior of the solutions of the coupled Zakharov-Kuznetsov system. Compared with the classical Zakharov-Kuznetsov equation, the linear coupling term coefficients and the convection term coefficients will not affect its well-posedness but will affect the local energy decay property. Two local energy decay results of the coupled Zakharov-Kuznetsov system are obtained when the initial data are in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>×</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> and <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>×</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>, respectively. The local decay results indicate the impossibility of the breather solutions for some coupled Zakharov-Kuznetsov system in a growing domain depending on time variable but not contains the soliton region.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"432 ","pages":"Article 113256"},"PeriodicalIF":2.4,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143705532","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":"Dynamics and integrability of polynomial vector fields on the n-dimensional sphere","authors":"Supriyo Jana,&nbsp;Soumen Sarkar","doi":"10.1016/j.jde.2025.113253","DOIUrl":"10.1016/j.jde.2025.113253","url":null,"abstract":"<div><div>In this paper, we characterize arbitrary polynomial vector fields on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. We establish a necessary and sufficient condition for a degree one vector field on the odd-dimensional sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> to be Hamiltonian. Additionally, we classify polynomial vector fields on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> up to degree two that possess an invariant great <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-sphere. We present a class of completely integrable vector fields on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. We found a sharp bound for the number of invariant meridian hyperplanes for a polynomial vector field on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Furthermore, we compute the sharp bound for the number of invariant parallel hyperplanes for any polynomial vector field on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. Finally, we study homogeneous polynomial vector fields on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, providing a characterization of their invariant <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-spheres.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"435 ","pages":"Article 113253"},"PeriodicalIF":2.4,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143705668","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 of non-resistive quantum MHD system","authors":"Sinan Wang,&nbsp;Jianfeng Zhou","doi":"10.1016/j.jde.2025.113255","DOIUrl":"10.1016/j.jde.2025.113255","url":null,"abstract":"<div><div>We are concerned with the global well-posedness of viscous non-resistive compressible quantum magnetohydrodynamic (QMHD) system in Lagrangian coordinates. By using a two-tier energy method, we study an initial-boundary value problem of compressible QMHD system in an infinite flat layer. We prove the global existence, uniqueness and decay estimate of smooth solution to the system around a suitably small uniform magnetic field which is non-parallel to the layer.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"436 ","pages":"Article 113255"},"PeriodicalIF":2.4,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143705541","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 analysis for two coupled second order evolution equations","authors":"Jianghao Hao ,&nbsp;Zhaobin Kuang ,&nbsp;Zhuangyi Liu ,&nbsp;Jiongmin Yong","doi":"10.1016/j.jde.2025.113246","DOIUrl":"10.1016/j.jde.2025.113246","url":null,"abstract":"<div><div>In this paper, we provide a stability analysis for the following abstract system of two coupled second order evolution equations<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>=</mo><mo>−</mo><mi>a</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>γ</mi></mrow></msup><mi>u</mi><mo>+</mo><mi>b</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msup><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>=</mo><mo>−</mo><mi>A</mi><mi>y</mi><mo>−</mo><mi>b</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>−</mo><mi>k</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>β</mi></mrow></msup><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mspace></mspace><mi>y</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>.</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <em>A</em> is a self-adjoint, positive definite operator on a complex Hilbert space <em>H</em>, and parameters <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mi>γ</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>]</mo><mo>×</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> with <span><math><mi>γ</mi><mo>∈</mo><mo>[</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>2</mn><mo>]</mo></math></span>. We are able to completely divide the parameter region into subsets where the semigroup associated with the system is (i) exponentially stable, (ii) polynomially stable of optimal order, and (iii) merely strong stable.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"432 ","pages":"Article 113246"},"PeriodicalIF":2.4,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704853","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":"Local exact controllability to the trajectories for the two-dimensional magnetohydrodynamic system with controls acting only on the velocity field","authors":"Qiang Tao ,&nbsp;Zheng-an Yao ,&nbsp;Xuan Yin","doi":"10.1016/j.jde.2025.113237","DOIUrl":"10.1016/j.jde.2025.113237","url":null,"abstract":"<div><div>In this paper, we study the local exact controllability to the trajectories for the two-dimensional incompressible magnetohydrodynamic system on a bounded domain with no-slip boundary condition on the velocity field and the perfect insulating condition on the magnetic field. The controls are distributed in an arbitrarily small nonempty open subset and act only on the velocity field. In this situation, the divergence free condition for the magnetic field can be inherited from the initial value. With this condition, we transform the magnetohydrodynamic system into a coupled system between the Navier-Stokes equations and a scalar equation. Our proof relies on a new Carleman inequality for two kinds of boundary conditions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"433 ","pages":"Article 113237"},"PeriodicalIF":2.4,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704190","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}