Journal of Differential Equations最新文献

{"title":"Stability and large-time behavior for Euler-like equations","authors":"Jiahong Wu ,&nbsp;Xiaojing Xu ,&nbsp;Yueyuan Zhong ,&nbsp;Ning Zhu","doi":"10.1016/j.jde.2025.113578","DOIUrl":"10.1016/j.jde.2025.113578","url":null,"abstract":"<div><div>This paper intends to understand the long-time existence and stability of solutions to an Euler-like equation. An Euler-like equation is the 2D incompressible Euler equation with an extra singular integral operator (SIO) type term. In contrast to the 2D Euler equation, the vorticity to the 2D Euler-like equation is not known to be bounded due to the unboundedness of the SIO on the space <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>. As a consequence, classical Yudovich theory fails on the Euler-like equation. The global existence, regularity and stability problems on the Euler-like equation are generally open. This paper makes progress on an Euler-like equation arising in the study of several fluids. We establish a long-time existence and stability result. When the Sobolev size of the initial data is of order <em>ε</em>, the solution is shown to live on a time interval of the size <span><math><mn>1</mn><mo>/</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. When the initial data is restricted to a class with special symmetry, we obtain the global existence and nonlinear stability.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"444 ","pages":"Article 113578"},"PeriodicalIF":2.4,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491342","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":"Regularity of the 3D stochastic viscous Primitive Equations","authors":"Zhao Dong ,&nbsp;Hao Xiong ,&nbsp;Guoli Zhou","doi":"10.1016/j.jde.2025.113579","DOIUrl":"10.1016/j.jde.2025.113579","url":null,"abstract":"<div><div>Utilizing the method of hydrostatic decomposition, we obtain the smoothness property and uniform <em>a</em> <span><math><mi>p</mi><mi>r</mi><mi>i</mi><mi>o</mi><mi>r</mi><mi>i</mi></math></span> estimates for the strong solution to 3D stochastic Primitive Equations (PEs) of large-scale ocean and atmosphere dynamics with non-periodic condition. Consequently, we derive the existence of invariant measures and the smoothness of random attractor.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113579"},"PeriodicalIF":2.4,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144480526","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 stability of traveling waves for Nagumo equations with degenerate diffusion","authors":"Tianyuan Xu ,&nbsp;Shanming Ji ,&nbsp;Ming Mei ,&nbsp;Jingxue Yin","doi":"10.1016/j.jde.2025.113587","DOIUrl":"10.1016/j.jde.2025.113587","url":null,"abstract":"<div><div>This paper is concerned with the global nonlinear stability with possibly large perturbations of the unique sharp / smooth traveling waves for the degenerate diffusion equations with Nagumo (bistable) reaction. Two technical issues arise in this study. One is the shortage of weak regularity of sharp traveling waves, the other difficulty is the non-absorbing initial-perturbation around the smooth traveling waves at the far field <span><math><mi>x</mi><mo>=</mo><mo>+</mo><mo>∞</mo></math></span>. For the sharp traveling wave case, we technically construct weak sub- and super-solutions with semi-compact supports via translation and scaling of the unique sharp traveling wave to characterize the motion of the steep moving edges and avoid the weak regularity of the solution near the steep edges. For the smooth traveling wave case, we artfully combine both the translation and scaling type sub- and super-solutions and the translation and superposition type sub- and super-solutions in a systematical manner.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113587"},"PeriodicalIF":2.4,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144490352","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":"Isolated singularities of solutions of linear and semilinear elliptic equations with singular drifts","authors":"Hyunseok Kim","doi":"10.1016/j.jde.2025.113574","DOIUrl":"10.1016/j.jde.2025.113574","url":null,"abstract":"<div><div>We study isolated singularities of solutions of linear and semilinear elliptic equations in divergence form with singular drifts. First, extending a classical result for isolated singularities of harmonic functions, we establish a removable isolated singularity theorem for linear equations with drifts <strong>b</strong> in <span><math><msup><mrow><mi>L</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>R</mi></mrow></msub><mtext>; </mtext><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> for some <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≥</mo><mi>n</mi></math></span>, where <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> is the dimension. Then this theorem is applied to prove removability theorems for isolated singularities of solutions of some semilinear equations with drifts in <span><math><msup><mrow><mi>L</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>R</mi></mrow></msub><mtext>; </mtext><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span>. One novelty of our results is that the critical case <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mi>n</mi></math></span> is allowed for removable singularity theorems for both linear and semilinear equations. Moreover, our methods of proofs rely only on interior <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>q</mi></mrow></msup></math></span>-estimates for solutions on annuli and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup></math></span>-estimates for their traces on spheres but not pointwise estimates like the maximum principle, which can be thus applied to linear and nonlinear systems.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"444 ","pages":"Article 113574"},"PeriodicalIF":2.4,"publicationDate":"2025-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144471820","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":"Kato-Ponce inequality for fractional nonlocal parabolic operators","authors":"Meng Qu ,&nbsp;Xinfeng Wu","doi":"10.1016/j.jde.2025.113572","DOIUrl":"10.1016/j.jde.2025.113572","url":null,"abstract":"<div><div>We establish Kato-Ponce inequality (or fractional Leibniz rule) for fractional nonlocal parabolic operators <span><math><msup><mrow><mo>(</mo><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mo>−</mo><msub><mrow><mo>△</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>s</mi><mo>/</mo><mn>2</mn></mrow></msup></math></span> and <span><math><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mo>−</mo><msub><mrow><mo>△</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>s</mi><mo>/</mo><mn>2</mn></mrow></msup></math></span> of arbitrary order <span><math><mi>s</mi><mo>&gt;</mo><mn>0</mn></math></span> for a full range of Lebesgue indices including the endpoints, and determine the sharp range of <em>s</em>. We also prove a sharp Kato-Ponce commutator inequality for <span><math><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mo>−</mo><msub><mrow><mo>△</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>s</mi><mo>/</mo><mn>2</mn></mrow></msup></math></span>. To achieve these results, we not only adapt the methods of Bourgain-Li <span><span>[11]</span></span>, Grafakos-Oh <span><span>[25]</span></span> and Oh-Wu <span><span>[50]</span></span> to the present parabolic setting, but build up sharp decay estimates for higher-order hyper-singular integrals of Nogin-Rubin <span><span>[48]</span></span> and Stinga-Torrea <span><span>[54]</span></span>, which are crucial for us to derive the sharp ranges of <em>s</em>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"443 ","pages":"Article 113572"},"PeriodicalIF":2.4,"publicationDate":"2025-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144470604","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":"Transport equations for Osgood velocity fields","authors":"U.S. Fjordholm,&nbsp;O. Mæhlen","doi":"10.1016/j.jde.2025.113566","DOIUrl":"10.1016/j.jde.2025.113566","url":null,"abstract":"<div><div>We consider the transport equation with a velocity field satisfying the Osgood condition. The weak formulation is not meaningful in the usual Lebesgue sense, meaning that the usual DiPerna–Lions treatment of the problem is not applicable (in particular, the divergence of the velocity might be unbounded). Instead, we use Riemann–Stieltjes integration to interpret the weak formulation, leading to a well-posedness theory in regimes not covered by existing works. The most general results are for the one-dimensional problem, with generalisations to multiple dimensions in the particular case of log-Lipschitz velocities.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"443 ","pages":"Article 113566"},"PeriodicalIF":2.4,"publicationDate":"2025-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144470603","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":"Low regularity results for degenerate Poisson problems","authors":"Marta Calanchi ,&nbsp;Massimo Grossi","doi":"10.1016/j.jde.2025.113567","DOIUrl":"10.1016/j.jde.2025.113567","url":null,"abstract":"<div><div>In this paper we study the Poisson problem,<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><mrow><mi>div</mi></mrow><mo>(</mo><msup><mrow><mi>d</mi></mrow><mrow><mi>β</mi></mrow></msup><mi>∇</mi><mi>u</mi><mo>)</mo><mo>=</mo><mi>f</mi></mtd><mtd><mrow><mi>in</mi></mrow><mspace></mspace><mi>Ω</mi></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mn>0</mn></mtd><mtd><mrow><mi>on</mi></mrow><mspace></mspace><mo>∂</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, <span><math><mi>N</mi><mo>≥</mo><mn>2</mn></math></span> is a smooth bounded domain, <em>f</em> is a continuous function, <span><math><mi>β</mi><mo>&lt;</mo><mn>1</mn></math></span>, and <span><math><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>d</mi><mi>i</mi><mi>s</mi><mi>t</mi><mo>(</mo><mi>x</mi><mo>,</mo><mo>∂</mo><mi>Ω</mi><mo>)</mo></math></span>. We describe the behaviour of <em>u</em> near ∂Ω and discuss some of its regularity properties.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"443 ","pages":"Article 113567"},"PeriodicalIF":2.4,"publicationDate":"2025-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144470605","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":"Positive harmonically bounded solutions for semi-linear equations","authors":"Wolfhard Hansen ,&nbsp;Krzysztof Bogdan","doi":"10.1016/j.jde.2025.113544","DOIUrl":"10.1016/j.jde.2025.113544","url":null,"abstract":"<div><div>For open sets <em>U</em> in some space <em>X</em>, we are interested in positive solutions to semi-linear equations <span><math><mi>L</mi><mi>u</mi><mo>=</mo><mi>φ</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mi>u</mi><mo>)</mo><mi>μ</mi></math></span> on <em>U</em>. Here <em>L</em> may be an elliptic or parabolic operator of second order (generator of a diffusion process) or an integro-differential operator (generator of a jump process), <em>μ</em> is a positive measure on <em>U</em> and <em>φ</em> is an arbitrary measurable real function on <span><math><mi>U</mi><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> such that the functions <span><math><mi>t</mi><mo>↦</mo><mi>φ</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span>, <span><math><mi>x</mi><mo>∈</mo><mi>U</mi></math></span>, are continuous, increasing and vanish at <span><math><mi>t</mi><mo>=</mo><mn>0</mn></math></span>.</div><div>More precisely, given a measurable function <span><math><mi>h</mi><mo>≥</mo><mn>0</mn></math></span> on <em>X</em> which is <em>L</em>-harmonic on <em>U</em>, that is, continuous real on <em>U</em> with <span><math><mi>L</mi><mi>h</mi><mo>=</mo><mn>0</mn></math></span> on <em>U</em>, we give necessary and sufficient conditions for the existence of positive solutions <em>u</em> such that <span><math><mi>u</mi><mo>=</mo><mi>h</mi></math></span> on <span><math><mi>X</mi><mo>∖</mo><mi>U</mi></math></span> and <em>u</em> has the same “boundary behavior” as <em>h</em> on <em>U</em> (Problem 1) or, alternatively, <span><math><mi>u</mi><mo>≤</mo><mi>h</mi></math></span> on <em>U</em>, but <span><math><mi>u</mi><mo>≢</mo><mn>0</mn></math></span> on <em>U</em> (Problem 2).</div><div>We show that these problems are equivalent to problems of the existence of positive solutions to certain integral equations <span><math><mi>u</mi><mo>+</mo><mi>K</mi><mi>φ</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mi>u</mi><mo>)</mo><mo>=</mo><mi>g</mi></math></span> on <em>U</em>, <em>K</em> being a potential kernel. We solve them in the general setting of balayage spaces <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>W</mi><mo>)</mo></math></span> which, in probabilistic terms, corresponds to the setting of transient Hunt processes with strong Feller resolvent.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"443 ","pages":"Article 113544"},"PeriodicalIF":2.4,"publicationDate":"2025-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144470602","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 nonlocal Neumann problem","authors":"Leonhard Frerick,&nbsp;Christian Vollmann,&nbsp;Michael Vu","doi":"10.1016/j.jde.2025.113553","DOIUrl":"10.1016/j.jde.2025.113553","url":null,"abstract":"<div><div>The classical local Neumann problem is well studied and solutions of this problem lie, in general, in a Sobolev space. In this work, we focus on <em>nonlocal</em> Neumann problems with measurable, nonnegative kernels, whose solutions require less regularity assumptions. For kernels of this kind we formulate and study the weak formulation of the nonlocal Neumann problem and we investigate a nonlocal counterpart of the Sobolev space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> as well as a resulting nonlocal trace space. We further establish, mainly for symmetric kernels, various existence results for the weak solution of the Neumann problem and we discuss related necessary conditions. Both, homogeneous and nonhomogeneous Neumann boundary conditions are considered.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"443 ","pages":"Article 113553"},"PeriodicalIF":2.4,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144338921","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":"Very weak solutions of the Dirichlet problem for 2-Hessian equation","authors":"Tongtong Li ,&nbsp;Guohuan Qiu","doi":"10.1016/j.jde.2025.113577","DOIUrl":"10.1016/j.jde.2025.113577","url":null,"abstract":"<div><div>For any <em>α</em> small, we construct infinitely many <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span> very weak solutions to the 2-Hessian equation with prescribed boundary value. The proof relies on the convex integration method and cut-off technique.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"444 ","pages":"Article 113577"},"PeriodicalIF":2.4,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144364901","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}