Journal of Differential Equations最新文献

{"title":"Global Lipschitz estimates for fully non-linear singular perturbation problems with non-homogeneous degeneracy","authors":"Elzon C. Bezerra Júnior,&nbsp;João Vitor da Silva","doi":"10.1016/j.jde.2025.02.020","DOIUrl":"10.1016/j.jde.2025.02.020","url":null,"abstract":"<div><div>This manuscript investigates the global regularity of singularly perturbed unbalanced models with variable exponents. In this context, we aim to find a non-negative function <span><math><msup><mrow><mi>u</mi></mrow><mrow><mi>ϵ</mi></mrow></msup></math></span> that satisfies the following equation for each fixed <span><math><mi>ϵ</mi><mo>&gt;</mo><mn>0</mn></math></span><span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mi>H</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>∇</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>ϵ</mi></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>[</mo><mi>Δ</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>ϵ</mi></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>b</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>⋅</mo><mi>∇</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>ϵ</mi></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>]</mo><mspace></mspace></mtd><mtd><mo>=</mo><mspace></mspace></mtd><mtd><msub><mrow><mi>ζ</mi></mrow><mrow><mi>ϵ</mi></mrow></msub><mo>(</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>ϵ</mi></mrow></msup><mo>)</mo><mo>+</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>ϵ</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mspace></mspace></mtd><mtd><mtext>in</mtext><mspace></mspace></mtd><mtd><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><msup><mrow><mi>u</mi></mrow><mrow><mi>ϵ</mi></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mspace></mspace></mtd><mtd><mo>=</mo><mspace></mspace></mtd><mtd><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mspace></mspace></mtd><mtd><mtext>on</mtext><mspace></mspace></mtd><mtd><mo>∂</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where Ω is a bounded regular domain in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, and <span><math><mi>H</mi></math></span> represents a function exhibiting a variable degeneracy signature. Additionally, <span><math><msub><mrow><mi>ζ</mi></mrow><mrow><mi>ϵ</mi></mrow></msub></math></span> approaches the Dirac measure <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> as <em>ϵ</em> tends to zero, and <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>ϵ</mi></mrow></msub></math></span> remains bounded away from zero and infinity. We aim to establish global gradient bounds that are unaffected by the parameter <em>ϵ</em>. Specifically, this family of solutions converges uniformly to a Lipschitz limiting profile associated with a one-phase Bernoulli-type free boundary problem.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"428 ","pages":"Pages 623-653"},"PeriodicalIF":2.4,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143422436","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 weak solutions for a volume-filling model of cell invasion into extracellular matrix","authors":"Rebecca M. Crossley ,&nbsp;Jan-Frederik Pietschmann ,&nbsp;Markus Schmidtchen","doi":"10.1016/j.jde.2025.02.023","DOIUrl":"10.1016/j.jde.2025.02.023","url":null,"abstract":"<div><div>We study the existence of weak solutions for a model of cell invasion into the extracellular matrix (ECM), consisting of a non-linear partial differential equation (PDE) for cell density coupled with an ordinary differential equation (ODE) for ECM density. The model includes cross-species density-dependent diffusion and proliferation terms, capturing the role of the ECM in supporting cells during invasion and preventing growth via volume-filling effects. The occurrence of cross-diffusion terms is a common theme in the system of interacting species with excluded-volume interactions. Additionally, ECM degradation by cells is included. We present an existence result for weak solutions, exploiting the partial gradient flow structure to overcome the non-regularising nature of the ODE. Furthermore, we present simulations that illustrate travelling wave solutions and investigate asymptotic behaviour as the ECM degradation rate tends to infinity.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"428 ","pages":"Pages 721-746"},"PeriodicalIF":2.4,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143422435","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":"Abstract left-definite theory: A model operator approach, examples, fractional Sobolev spaces, and interpolation theory","authors":"Christoph Fischbacher,&nbsp;Fritz Gesztesy,&nbsp;Paul Hagelstein ,&nbsp;Lance L. Littlejohn","doi":"10.1016/j.jde.2025.02.013","DOIUrl":"10.1016/j.jde.2025.02.013","url":null,"abstract":"<div><div>We use a model operator approach and the spectral theorem for self-adjoint operators in a Hilbert space to derive the basic results of abstract left-definite theory in a straightforward manner. The theory is amply illustrated with a variety of concrete examples employing scales of Hilbert spaces, fractional Sobolev spaces, and domains of (strictly) positive fractional powers of operators, employing interpolation theory.</div><div>In particular, we explicitly describe the domains of positive powers of the harmonic oscillator operator in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo><mspace></mspace><mo>(</mo></math></span>and hence that of the Hermite operator in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>;</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msup><mi>d</mi><mi>x</mi><mo>)</mo><mo>)</mo></math></span> in terms of fractional Sobolev spaces, certain commutation techniques, and positive powers of (the absolute value of) the operator of multiplication by the independent variable in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"428 ","pages":"Pages 422-482"},"PeriodicalIF":2.4,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143422438","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 the uniqueness and non-uniqueness of the steady planar Navier-Stokes equations in an exterior domain","authors":"Zhengguang Guo ,&nbsp;Wendong Wang","doi":"10.1016/j.jde.2025.02.031","DOIUrl":"10.1016/j.jde.2025.02.031","url":null,"abstract":"<div><div>In this paper we investigate the uniqueness of solutions of the steady planar Navier-Stokes equations with different boundary conditions in the exterior domain. We prove the uniqueness of the solution under the enhanced Navier boundary conditions for a class of incompressible flow with constant vorticity. Meanwhile, some counterexamples are given to show that the uniqueness of the solution fails under the Navier boundary conditions. For the general incompressible flow with Dirichlet boundary condition, we establish various sufficient conditions to guarantee the uniqueness of the solution.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"428 ","pages":"Pages 483-510"},"PeriodicalIF":2.4,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143422439","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 the global well-posedness of interface dynamics for gravity Stokes flow","authors":"Francisco Gancedo ,&nbsp;Rafael Granero-Belinchón ,&nbsp;Elena Salguero","doi":"10.1016/j.jde.2025.02.032","DOIUrl":"10.1016/j.jde.2025.02.032","url":null,"abstract":"<div><div>In this paper, we establish the global-in-time well-posedness for an arbitrary <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>γ</mi></mrow></msup></math></span>, <span><math><mn>0</mn><mo>&lt;</mo><mi>γ</mi><mo>&lt;</mo><mn>1</mn></math></span>, initial internal periodic wave for the free boundary gravity Stokes system in two dimensions. This classical well-posedness result is complemented by a weak solvability result in the case of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>γ</mi></mrow></msup></math></span> or Lipschitz interfaces. In particular, we show new cancellations that prevent the so-called two-dimensional Stokes paradox, despite the polynomial growth of the Stokeslet in this horizontally periodic setting. The bounds obtained in this work are exponential in time, which are in strong agreement with the growth of the solutions obtained in <span><span>[22]</span></span>. Additionally, these new cancellations are used to establish global-in-time well-posedness for the Stokes-transport system with initial densities in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> for <span><math><mn>2</mn><mo>&lt;</mo><mi>p</mi><mo>&lt;</mo><mo>∞</mo></math></span>. Furthermore, we also propose and analyze several one-dimensional models that capture different aspects of the full internal wave problem for the gravity Stokes system, showing that all of these models exhibit finite-time singularities. This fact evidences the fine structure of the nonlinearity in the full system, which allows the free boundary problem to be globally well-posed, while simplified versions blow-up in finite time.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"428 ","pages":"Pages 654-687"},"PeriodicalIF":2.4,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143422305","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 suitable weak solutions to an anisotropic electrokinetic flow model","authors":"Dietmar Hömberg ,&nbsp;Robert Lasarzik ,&nbsp;Luisa Plato","doi":"10.1016/j.jde.2025.02.018","DOIUrl":"10.1016/j.jde.2025.02.018","url":null,"abstract":"<div><div>In this article we present a system of coupled non-linear PDEs modeling an anisotropic electrokinetic flow. We show the existence of suitable weak solutions in three spatial dimensions, that is weak solutions which fulfill an energy inequality, via a regularized system. The flow is modeled by a Navier–Stokes–Nernst–Planck–Poisson system and the anisotropy is introduced via space dependent diffusion matrices in the Nernst–Planck and Poisson equations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"428 ","pages":"Pages 511-584"},"PeriodicalIF":2.4,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143422427","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 dynamical interpretation of the connection map of an attractor-repeller decomposition","authors":"J.J. Sánchez-Gabites","doi":"10.1016/j.jde.2025.02.042","DOIUrl":"10.1016/j.jde.2025.02.042","url":null,"abstract":"<div><div>In Conley index theory one may study an invariant set <em>S</em> by decomposing it into an attractor <em>A</em>, a repeller <em>R</em>, and the orbits connecting the two. The Conley indices of <em>S</em>, <em>A</em> and <em>R</em> fit into an exact sequence where a certain connection homomorphism Γ plays an important role. In this paper we provide a dynamical interpretation of this map. Roughly, <em>R</em> “emits” an element of its Conley index as a “wavefront”, part of which intersects the connecting orbits in <em>S</em>. This subset of the wavefront evolves towards <em>A</em> and is then “received” by it to produce an element in its Conley index.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"428 ","pages":"Pages 688-720"},"PeriodicalIF":2.4,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143422306","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":"Ground states of a coupled pseudo-relativistic Hartree system: Existence and concentration behavior","authors":"Huiting He ,&nbsp;Chungen Liu ,&nbsp;Jiabin Zuo","doi":"10.1016/j.jde.2025.02.019","DOIUrl":"10.1016/j.jde.2025.02.019","url":null,"abstract":"<div><div>This paper is concerned with the ground states of a coupled pseudo-relativistic Hartree system in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> with trapping potentials, where the intraspecies and the interspecies interaction are both attractive. By investigating an associated constraint minimization problem, the existence and non-existence of ground states are classified completely. Under certain conditions on the trapping potentials, we present a precise analysis on the concentration behavior of the minimizers as the coupling coefficient goes to a critical value, where the minimizers blow up and the maximum point sequence concentrates at the global minima of the associated trapping potentials. We also identify an optimal blowing up rate under polynomial potentials by establishing some delicate estimates of energy functionals.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"428 ","pages":"Pages 585-622"},"PeriodicalIF":2.4,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143422440","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":"Some existence and regularity results for a non-local transport-diffusion equation with fractional derivatives in time and space","authors":"Diego Chamorro ,&nbsp;Miguel Yangari","doi":"10.1016/j.jde.2025.02.027","DOIUrl":"10.1016/j.jde.2025.02.027","url":null,"abstract":"<div><div>We study the existence of global weak solutions of a nonlinear transport-diffusion equation with a fractional derivative in the time variable and, under some extra hypotheses, we also study some regularity properties for this type of solutions. In the system considered here, the diffusion operator is given by a fractional Laplacian and the nonlinear drift is assumed to be divergence free and it is assumed to satisfy some general stability and boundedness properties in Lebesgue spaces. In order to obtain global solutions, we first introduce an hyperviscosity perturbation and we perform a fixed-point argument to obtain a solution of the perturbed equation. Then, by using suitable a priori information, given by an energy inequality, we can extend the solutions and we finally obtain global weak solutions of the original problem.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"428 ","pages":"Pages 389-421"},"PeriodicalIF":2.4,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143403387","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":"Stabilization effect of temperature on three-dimensional inviscid compressible fluid","authors":"Tao Liang ,&nbsp;Yongsheng Li ,&nbsp;Xiaoping Zhai","doi":"10.1016/j.jde.2025.02.026","DOIUrl":"10.1016/j.jde.2025.02.026","url":null,"abstract":"<div><div>This paper solves the stability problem for a three-dimensional inviscid non-isentropic compressible fluid with radial symmetrical data in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span>. We establish the stability for the nonlinear system and derive precise large-time behavior of the solutions. The result presented in this paper reveals a remarkable phenomenon for the inviscid non-isentropic compressible fluids. That is, the temperature actually smooths and stabilizes the irrotational flows. If the temperature were not present, the fluid is governed by the 3D compressible Euler equations and its stability remains open. It is the coupling and interaction between the temperature and the velocity in the inviscid system that makes the stability problem studied here possible. Mathematically the system can be reduced to degenerate and damped wave equations that fuel the stabilization.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"428 ","pages":"Pages 348-388"},"PeriodicalIF":2.4,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143403386","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}