Journal of Differential Equations最新文献

{"title":"On the Liouville type theorems for the stationary Navier-Stokes equations in R3","authors":"Dongho Chae","doi":"10.1016/j.jde.2025.113597","DOIUrl":"10.1016/j.jde.2025.113597","url":null,"abstract":"<div><div>In this paper we prove Liouville type theorems for the stationary solution to the Navier-Stokes equations in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. Let <span><math><mo>(</mo><mi>u</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> a smooth stationary solution to the Navier-Stokes equations in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, satisfying <span><math><msub><mrow><mo>∫</mo></mrow><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msub><mo>|</mo><mi>∇</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>x</mi><mo>&lt;</mo><mo>+</mo><mo>∞</mo></math></span>, and <span><math><mi>Q</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>p</mi></math></span> is its head pressure, which vanishes at infinity. Then, we prove various Liouville type theorems, using the function <em>ξ</em> satisfying the Osgood criterion. These theorems improve some of previously known results. The proofs use the level set methods and the coarea formula.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113597"},"PeriodicalIF":2.4,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144571188","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":"Existence and stability of symmetric and asymmetric patterns for the half-Laplacian Gray-Scott system in one-dimensional domain","authors":"Min Gao ,&nbsp;Shanfa Lai ,&nbsp;Shuangquan Xie ,&nbsp;Wen Yang","doi":"10.1016/j.jde.2025.113588","DOIUrl":"10.1016/j.jde.2025.113588","url":null,"abstract":"<div><div>In this paper, we investigate the existence and stability of multiple spikes pattern to the fractional Gray-Scott model with periodic boundary conditions, where the fractional power is <span><math><mi>s</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. Employing the classical Lyapunov-Schmidt reduction method, we provide a rigorous proof for the existence of both symmetric multiple spikes and asymmetric two spikes solutions. Furthermore, we analyze the stability of these constructed solutions by studying the associated large and small eigenvalue problems. Our analysis crucially relies on the properties of the Green's function and its derivatives, as well as the study of two nonlocal eigenvalue problems, which play an important role in determining the stability characteristics of the solutions. Moreover, we find out the connection between the eigenvalues of the small eigenvalue problem and the spectral properties of a corresponding circulant matrix. Specially, using the properties of the polygamma function, we establish that all nonzero eigenvalues of this circulant matrix are negative regardless of the number of spikes.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113588"},"PeriodicalIF":2.4,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144557146","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":"Existence of very weak solutions to nonlinear elliptic equation with nonstandard growth and global weighted gradient estimates","authors":"Sun-Sig Byun ,&nbsp;Minkyu Lim","doi":"10.1016/j.jde.2025.113594","DOIUrl":"10.1016/j.jde.2025.113594","url":null,"abstract":"<div><div>We study a general class of quasilinear elliptic equations with nonstandard growth to prove the existence of a very weak solution to such a problem. A key ingredient in the proof is a priori global weighted gradient estimate of a very weak solution, where the right-hand side of the equation is the divergence of a vector-valued function with a low degree of integrability. To obtain this estimate, we adopt a notion of reverse Hölder class of Muckenhoupt weights. Another crucial part of the proof is a generalized weighted div-curl lemma in the setting of Orlicz spaces.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113594"},"PeriodicalIF":2.4,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144557147","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 the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system with Landau Potential","authors":"Li Fang ,&nbsp;Rui Nei ,&nbsp;Zhenhua Guo","doi":"10.1016/j.jde.2025.113585","DOIUrl":"10.1016/j.jde.2025.113585","url":null,"abstract":"<div><div>We study the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system in a bounded smooth domain in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, which describes the dynamics of nonhomogeneous incompressible two-phase viscous flows. We first give a blow-up criterion of local strong solution to the initial-boundary-value problem for the case of initial density away from zero. After establishing some key a priori with the help of the Landau Potential, we obtain the global existence and the decay-in-time of strong solution, provided that the initial datum <span><math><msub><mrow><mo>‖</mo><mi>∇</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></msub><mo>+</mo><msub><mrow><mo>‖</mo><mi>∇</mi><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></msub><mo>+</mo><msub><mrow><mo>‖</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></msub></math></span> is suitably small. Precisely, we provide a systematic analysis of the Navier-Stokes-Cahn-Hilliard system through detailed a priori estimates, covering blow-up criterion, global existence and decay behavior of strong solutions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113585"},"PeriodicalIF":2.4,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144534438","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":"Periodic solutions for the coupled wave equations with concave-convex nonlinearities","authors":"Jianhua Liu,&nbsp;Jiayu Deng,&nbsp;Shuguan Ji","doi":"10.1016/j.jde.2025.113584","DOIUrl":"10.1016/j.jde.2025.113584","url":null,"abstract":"<div><div>In this work, we show that the coupled wave equations possess infinitely many time-periodic solutions. The nonlinearities are the combination of a convex term and a concave term, and we mainly focus on the effect of the concave term on the solution structure of the periodic-Dirichlet wave equations. The proof is based on the detailed analysis for the spectrum structure of the corresponding linear operator, and we obtain the existence and multiplicity of time-periodic solutions to our problem by critical point theory and approximation arguments.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113584"},"PeriodicalIF":2.4,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144534437","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":"Large-space and large-time asymptotics of the Camassa-Holm soliton gas","authors":"Xianguo Geng ,&nbsp;Dedi Yan ,&nbsp;Minxin Jia","doi":"10.1016/j.jde.2025.113581","DOIUrl":"10.1016/j.jde.2025.113581","url":null,"abstract":"<div><div>We investigate the large-space and large-time asymptotic behaviors of the soliton gas associated with the Camassa-Holm equation. Utilizing the nonlinear steepest descent method, we demonstrate that the soliton gas is slowly approaching an elliptic function with constant coefficients for <span><math><mi>y</mi><mo>→</mo><mo>−</mo><mo>∞</mo></math></span>. In the regime <span><math><mi>t</mi><mo>→</mo><mo>+</mo><mo>∞</mo></math></span>, we establish a global large-time asymptotic description of the Camassa-Holm soliton gas. The half-plane <span><math><mo>{</mo><mo>(</mo><mi>y</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>:</mo><mo>−</mo><mo>∞</mo><mo>&lt;</mo><mi>y</mi><mo>&lt;</mo><mo>+</mo><mo>∞</mo><mo>,</mo><mi>t</mi><mo>&gt;</mo><mn>0</mn><mo>}</mo></math></span> is divided into sharply separated regions with different asymptotics. To facilitate the large-time asymptotic analysis, we construct a series of <em>g</em>-functions. To reconstruct the solution, we control the behavior of <em>g</em>-functions when <span><math><mi>k</mi><mo>→</mo><mfrac><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> and <span><math><mi>k</mi><mo>→</mo><mo>∞</mo></math></span>. The relevant asymptotic sector exists only if the <em>g</em>-function exists, we categorize the values of <span><math><msub><mrow><mi>η</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>η</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> into two distinct cases and rigorously prove the existence of these <em>g</em>-functions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"444 ","pages":"Article 113581"},"PeriodicalIF":2.4,"publicationDate":"2025-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144524290","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":"A type of anisotropic flows and dual Orlicz Christoffel-Minkowski type equations","authors":"Shanwei Ding,&nbsp;Guanghan Li","doi":"10.1016/j.jde.2025.113586","DOIUrl":"10.1016/j.jde.2025.113586","url":null,"abstract":"<div><div>In this paper, we study the long-time existence and asymptotic behavior for a class of anisotropic non-homogeneous curvature flows without global forcing terms. By the stationary solutions of such anisotropic flows, we obtain existence results for a class of dual Orlicz Christoffel-Minkowski type problems, which is equivalent to solve the PDE <span><math><mi>G</mi><mo>(</mo><mi>x</mi><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>,</mo><mi>D</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>)</mo><mi>F</mi><mo>(</mo><mo>[</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>K</mi></mrow></msub><mi>I</mi><mo>]</mo><mo>)</mo><mo>=</mo><mn>1</mn></math></span> on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> for a convex body <em>K</em>, where <em>D</em> is the covariant derivative with respect to the standard metric on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> and <em>I</em> is the unit matrix of order <em>n</em>. This result covers many previous known solutions to <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> dual Minkowski problem, <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> dual Christoffel-Minkowski problem, and dual Orlicz Minkowski problem etc. Meanwhile, the variational formula of some modified quermassintegrals and the corresponding prescribed area measure problem (Orlicz Christoffel-Minkowski type problem) are considered, and inequalities involving modified quermassintegrals are also derived. As corollary, this also provides a sufficient condition for the existence to the general prescribed curvature equation <span><math><msub><mrow><mi>F</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mi>κ</mi><mo>)</mo><mo>=</mo><mi>G</mi><mo>(</mo><mi>ν</mi><mo>,</mo><mi>X</mi><mo>)</mo></math></span> raised in <span><span>[20]</span></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113586"},"PeriodicalIF":2.4,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144517454","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":"Stable determination of the first order perturbation of the biharmonic operator from partial data","authors":"Boya Liu ,&nbsp;Salem Selim","doi":"10.1016/j.jde.2025.113575","DOIUrl":"10.1016/j.jde.2025.113575","url":null,"abstract":"<div><div>We consider an inverse boundary value problem for the biharmonic operator with the first order perturbation in a bounded domain of dimension three or higher. Assuming that the first and the zeroth order perturbations are known in a neighborhood of the boundary, we establish log-type stability estimates for these perturbations from a partial Dirichlet-to-Neumann map. Specifically, measurements are taken only on arbitrarily small open subsets of the boundary.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113575"},"PeriodicalIF":2.4,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144517456","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":"Existence of periodic probability measure solutions to Fokker-Planck equations for SDEs with random switching","authors":"Dan Li ,&nbsp;Yong Li","doi":"10.1016/j.jde.2025.113583","DOIUrl":"10.1016/j.jde.2025.113583","url":null,"abstract":"<div><div>This paper is concerned with the existence of periodic probability measure solutions to the Fokker-Planck equation for a periodic stochastic differential equation driving by a continuous-time, discrete-state jump process. The jump rates of this jump process can also be time-periodic and dependent on the state variables of the system. We prove the existence and smoothness of principal eigenfunctions for a cooperative weakly coupled periodic-parabolic system of partial differential equations (PDEs), in which the boundary operator is time-dependent and its zero-order coefficients may be negative. In addition, the results on Hölder estimates and Harnack inequality for a cooperative weakly coupled parabolic PDE system are extended. Based on these results, we establish a Lyapunov function criterion for the existence of periodic probability measure solutions to the Fokker-Planck equation in both non-degenerate and degenerate cases.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113583"},"PeriodicalIF":2.4,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144517452","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":"Relative sequence entropy for amenable group actions","authors":"Chunlin Liu ,&nbsp;Changlin Wang ,&nbsp;Weisheng Wu","doi":"10.1016/j.jde.2025.113582","DOIUrl":"10.1016/j.jde.2025.113582","url":null,"abstract":"<div><div>We introduce the concept of relative sequence entropy for amenable group actions and explore the interplay between relative sequence entropy and Kronecker algebras for amenable group actions, and rigid algebras for abelian group actions. Our investigation extends to the application of relative sequence entropy in various mixing concepts within both measure-theoretic and topological systems. Additionally, we refine the notion of relative sequence entropy by introducing the concept of relative sequence entropy pairs for amenable group actions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113582"},"PeriodicalIF":2.4,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144517455","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}