Journal de Mathematiques Pures et Appliquees最新文献

{"title":"Strong c-concavity and stability in optimal transport","authors":"Anatole Gallouët ,&nbsp;Quentin Mérigot ,&nbsp;Boris Thibert","doi":"10.1016/j.matpur.2025.103773","DOIUrl":"10.1016/j.matpur.2025.103773","url":null,"abstract":"<div><div>The stability of solutions to optimal transport problems under variation of the measures is fundamental from a mathematical viewpoint: it is closely related to the convergence of numerical approaches to solve optimal transport problems and justifies many of the applications of optimal transport. In this article, we introduce the notion of strong <em>c</em>-concavity, and we show that it plays an important role for proving stability results in optimal transport for general cost functions <em>c</em>. We then introduce a differential criterion for proving that a function is strongly <em>c</em>-concave, under an hypothesis on the cost introduced originally by Ma-Trudinger-Wang for establishing regularity of optimal transport maps. Finally, we provide two examples where this stability result can be applied, for cost functions taking value +∞ on the sphere: the reflector problem and the Gaussian curvature measure prescription problem.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103773"},"PeriodicalIF":2.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144739250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Long time classical solutions of quasilinear Klein-Gordon equations with small weakly decaying initial data","authors":"Fei Hou ,&nbsp;Huicheng Yin","doi":"10.1016/j.matpur.2025.103803","DOIUrl":"10.1016/j.matpur.2025.103803","url":null,"abstract":"<div><div>It is well known that for the quasilinear Klein-Gordon equation with quadratic nonlinearity and sufficiently decaying small initial data, there exists a global smooth solution if the space dimensions <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>. When the initial data are of size <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span> in the Sobolev space, for the semilinear Klein-Gordon equation satisfying the null condition, the authors in the article (Delort and Fang, 2000 <span><span>[11]</span></span>) prove that the solution exists in time <span><math><mo>[</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>)</mo></math></span> with <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>≥</mo><mi>C</mi><msup><mrow><mi>e</mi></mrow><mrow><mi>C</mi><msup><mrow><mi>ε</mi></mrow><mrow><mo>−</mo><mi>μ</mi></mrow></msup></mrow></msup></math></span> (<span><math><mi>μ</mi><mo>=</mo><mn>1</mn></math></span> if <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>, <span><math><mi>μ</mi><mo>=</mo><mn>2</mn><mo>/</mo><mn>3</mn></math></span> if <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>). In the present paper, we will focus on the general quasilinear Klein-Gordon equation without the null condition and further show that the existence time of the solution can be improved to <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>=</mo><mo>+</mo><mo>∞</mo></math></span> if <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span> and <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>≥</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>C</mi><msup><mrow><mi>ε</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></msup></math></span> if <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>. In addition, for <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span> and any fixed number <span><math><mi>α</mi><mo>&gt;</mo><mn>0</mn></math></span>, if the weighted <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> norm of the initial data with the weight <span><math><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mo>|</mo><mi>x</mi><mo>|</mo><mo>)</mo></mrow><mrow><mi>α</mi></mrow></msup></math></span> is small, then the solution exists globally and scatters to a free solution. Our arguments are based on the introduction of a new good unknown, the Strichartz estimate, the weighted <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm estimate and the resonance analysis.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103803"},"PeriodicalIF":2.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145219207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Entropy estimates for uniform attractors of dissipative PDEs with non translation-compact external forces","authors":"Yangmin Xiong ,&nbsp;Anna Kostianko ,&nbsp;Chunyou Sun ,&nbsp;Sergey Zelik","doi":"10.1016/j.matpur.2025.103812","DOIUrl":"10.1016/j.matpur.2025.103812","url":null,"abstract":"<div><div>We study the Kolmogorov's entropy of uniform attractors for non-autonomous dissipative PDEs. The main attention is payed to the case where the external forces are not translation-compact. We present a new general scheme which allows us to give the upper bounds of this entropy for various classes of external forces through the entropy of proper projections of their hulls to the space of translation-compact functions. This result generalizes well known estimates of Vishik and Chepyzhov for the translation-compact case. The obtained results are applied to three model problems: sub-quintic 3D damped wave equation with Dirichlet boundary conditions, quintic 3D wave equation with periodic boundary conditions and 2D Navier-Stokes system in a bounded domain. The examples of finite-dimensional uniform attractors for some special external forces which are not translation-compact are also given.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103812"},"PeriodicalIF":2.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Global regularity for Einstein-Klein-Gordon system with U(1)×R isometry group, II","authors":"Haoyang Chen ,&nbsp;Yi Zhou","doi":"10.1016/j.matpur.2025.103813","DOIUrl":"10.1016/j.matpur.2025.103813","url":null,"abstract":"<div><div>This paper is devoted to the study of the global existence of smooth solutions to the 3+1 dimensional Einstein-Klein-Gordon systems with a <span><math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>×</mo><mi>R</mi></math></span> isometry group for a class of regular large Cauchy data. In our first paper <span><span>[14]</span></span>, we reduce these Einstein-Klein-Gordon equations to a 2+1 dimensional Einstein-wave-Klein-Gordon system. And we show that the first possible singularity can only occur at the axis. In this paper, we give a proof of the global regularity for this 2+1 dimensional system. The major difficulties arise from the lack of smallness at initial time <span><math><mi>t</mi><mo>=</mo><mn>0</mn></math></span> and near centre <span><math><mi>r</mi><mo>=</mo><mn>0</mn></math></span>. To overcome these obstacles, we firstly show the non-concentration of the energy near the first possible singularity, which transforms the large-data problem to a small energy problem. Then, near the first possible singularity, we derive spacetime energy estimates to prove global well-posedness for initial data with small energy.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103813"},"PeriodicalIF":2.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145361414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Controllability of a coupled wave system with a single control and different speeds","authors":"Pierre Lissy ,&nbsp;Jingrui Niu","doi":"10.1016/j.matpur.2025.103808","DOIUrl":"10.1016/j.matpur.2025.103808","url":null,"abstract":"<div><div>We consider an exact controllability problem in a smooth bounded domain Ω of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, <span><math><mi>d</mi><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, for a coupled wave system, with two different speeds and a single control acting on an open subset <em>ω</em> satisfying the Geometric Control Condition and acting on one speed only. Actions for the wave equations with the second speed are obtained through a coupling term. Firstly, we construct appropriate state spaces with compatibility conditions associated with the coupling structure. Secondly, in these well-prepared spaces, we prove that the coupled wave system is exactly controllable if and only if the coupling structure satisfies an operator Kalman rank condition.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103808"},"PeriodicalIF":2.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"From KP-I lump solution to travelling waves of Gross-Pitaevskii equation","authors":"Yong Liu ,&nbsp;Zhengping Wang ,&nbsp;Juncheng Wei ,&nbsp;Wen Yang","doi":"10.1016/j.matpur.2025.103801","DOIUrl":"10.1016/j.matpur.2025.103801","url":null,"abstract":"<div><div>Let <em>q</em> be a nondegenerate lump type solution to the KP-I (Kadomtsev-Petviashvili-I) equation<span><span><span><math><msubsup><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msubsup><mi>q</mi><mo>−</mo><mn>2</mn><msqrt><mrow><mn>2</mn></mrow></msqrt><msubsup><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mi>q</mi><mo>−</mo><mn>3</mn><msqrt><mrow><mn>2</mn></mrow></msqrt><msub><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>(</mo><msup><mrow><mo>(</mo><msub><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow></msub><mi>q</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>−</mo><mn>2</mn><msubsup><mrow><mo>∂</mo></mrow><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mi>q</mi><mo>=</mo><mn>0</mn><mo>.</mo></math></span></span></span> We show the existence of travelling wave solutions with the form <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>−</mo><mi>c</mi><mi>t</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span>, for the GP (Gross-Pitaevskii) equation<span><span><span><math><mi>i</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>Ψ</mi><mo>+</mo><mi>Δ</mi><mi>Ψ</mi><mo>+</mo><mo>(</mo><mn>1</mn><mo>−</mo><mo>|</mo><mi>Ψ</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>Ψ</mi><mo>=</mo><mn>0</mn><mspace></mspace><mspace></mspace><mspace></mspace><mtext>in</mtext><mspace></mspace><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo></math></span></span></span> with travelling speed <span><math><mi>c</mi><mo>=</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>−</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, and <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>=</mo><mn>1</mn><mo>+</mo><mi>i</mi><mi>ε</mi><mi>q</mi><mo>+</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>. This proves the existence of finite energy solutions in the so-called Jones-Roberts program within the transonic regime <span><math><mi>c</mi><mo>∈</mo><mo>(</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>−</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>)</mo></math></span>. The main ingredients in our proof are detailed point-wise estimates for the Green functions associated to a family of fourth order hypoelliptic operators. In view of the classification of lump type solutions of the KP-I equation, our proof also indicates that for fixed small <em>ε</em>, there should exist a sequence of travelling wave solutions to GP equation, with energy tends to infinity.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103801"},"PeriodicalIF":2.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145265977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the magnetic Dirichlet to Neumann operator on the exterior of the disk – Diamagnetism, weak-magnetic field limit and flux effects","authors":"Bernard Helffer,&nbsp;François Nicoleau","doi":"10.1016/j.matpur.2025.103799","DOIUrl":"10.1016/j.matpur.2025.103799","url":null,"abstract":"<div><div>In this paper, we analyze the magnetic Dirichlet-to-Neumann operator (D-to-N map) <span><math><mover><mrow><mi>Λ</mi></mrow><mrow><mo>ˇ</mo></mrow></mover><mo>(</mo><mi>b</mi><mo>,</mo><mi>ν</mi><mo>)</mo></math></span> on the exterior of the disk with respect to a magnetic potential <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>b</mi><mo>,</mo><mi>ν</mi></mrow></msub><mo>=</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>b</mi></mrow></msup><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>ν</mi></mrow></msub></math></span> where, for <span><math><mi>b</mi><mo>∈</mo><mi>R</mi></math></span> and <span><math><mi>ν</mi><mo>∈</mo><mi>R</mi></math></span>, <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>b</mi></mrow></msup><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>b</mi><mspace></mspace><mo>(</mo><mo>−</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>ν</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> is the Aharonov-Bohm potential centered at the origin of flux <span><math><mn>2</mn><mi>π</mi><mi>ν</mi></math></span>. First, we show that the limit of <span><math><mover><mrow><mi>Λ</mi></mrow><mrow><mo>ˇ</mo></mrow></mover><mo>(</mo><mi>b</mi><mo>,</mo><mi>ν</mi><mo>)</mo></math></span> as <span><math><mi>b</mi><mo>→</mo><mn>0</mn></math></span> is equal to the D-to-N map <span><math><mover><mrow><mi>Λ</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><mi>ν</mi><mo>)</mo></math></span> on the interior of the disk associated with the potential <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>ν</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span>. Secondly, we study the ground state energy of the D-to-N map <span><math><mover><mrow><mi>Λ</mi></mrow><mrow><mo>ˇ</mo></mrow></mover><mo>(</mo><mi>b</mi><mo>,</mo><mi>ν</mi><mo>)</mo></math></span> and show that the strong diamagnetism property holds. Finally we slightly extend to the exterior case the asymptotic results as <span><math><mi>b</mi><mo>→</mo><mo>∞</mo></math></span> obtained in the interior case for general domains.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103799"},"PeriodicalIF":2.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145219206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Isotrivial Lagrangian fibrations of compact hyper-Kähler manifolds","authors":"Yoon-Joo Kim ,&nbsp;Radu Laza ,&nbsp;Olivier Martin","doi":"10.1016/j.matpur.2025.103810","DOIUrl":"10.1016/j.matpur.2025.103810","url":null,"abstract":"<div><div>This article initiates the study of isotrivial Lagrangian fibrations of compact hyper-Kähler manifolds. We present four foundational results that extend well-known facts about isotrivial elliptic fibrations of K3 surfaces. First, we prove that smooth fibers of an isotrivial Lagrangian fibration are isogenous to a power of an elliptic curve. Second, we exhibit a dichotomy between two types of isotrivial Lagrangian fibrations, which we call A and B. Third, we give a classification result for type A isotrivial Lagrangian fibrations. Namely, if a type A isotrivial Lagrangian fibration admits a rational section, then it is birational to one of two straightforward examples of isotrivial fibrations of hyper-Kähler manifolds of <span><math><msup><mrow><mtext>K3</mtext></mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></msup></math></span>-type and <span><math><msub><mrow><mtext>Kum</mtext></mrow><mrow><mi>n</mi></mrow></msub></math></span>-type. Finally, we prove that a genericity assumption on the smooth fiber of an isotrivial Lagrangian fibration without multiple fibers ensures that the fibration is of type A.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103810"},"PeriodicalIF":2.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145415606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-uniqueness of parabolic solutions for advection-diffusion equation","authors":"Thérèse Moerschell ,&nbsp;Massimo Sorella","doi":"10.1016/j.matpur.2025.103777","DOIUrl":"10.1016/j.matpur.2025.103777","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We present a novel example of a divergence–free velocity field &lt;span&gt;&lt;math&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;T&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;/math&gt;&lt;/span&gt; for &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; arbitrary but fixed which leads to non-unique solutions of advection–diffusion in the class &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; while satisfying the local energy inequality. This result complements the known uniqueness result of bounded solutions for divergence-free and &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; integrable velocity fields. Additionally, we also prove the necessity of time integrability of the velocity field for the uniqueness result. More precisely, we construct another divergence–free velocity field &lt;span&gt;&lt;math&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;T&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;/math&gt;&lt;/span&gt;, for &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; fixed, but arbitrary, with non–unique aforementioned solutions. Our contribution closes the gap between the regime of uniqueness and non-uniqueness in this context. Previously, it was shown with the convex integration technique that for &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; divergence–free velocity fields &lt;span&gt;&lt;math&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; could lead to non–unique solutions in the space &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103777"},"PeriodicalIF":2.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144738741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Global axisymmetric solution to the 3D incompressible anisotropic Navier–Stokes equations","authors":"Hui Chen ,&nbsp;Zijin Li ,&nbsp;Ping Zhang","doi":"10.1016/j.matpur.2025.103807","DOIUrl":"10.1016/j.matpur.2025.103807","url":null,"abstract":"<div><div>In this paper, we prove the global existence and uniqueness of axisymmetric solution to the 3D incompressible anisotropic Navier–Stokes equations in a cylindrical domain with Navier boundary condition provided that the swirl component of the initial velocity is sufficiently small. The main idea of the proof is to perform energy estimates for the pair <span><math><mo>(</mo><mi>J</mi><mo>,</mo><msup><mrow><mi>Ω</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>)</mo></math></span>, where <figure><img></figure> and <figure><img></figure> is a corrector of <figure><img></figure>. In order to close the energy estimates, we introduced the derivative-reduction technique and new elliptic estimates of the pressure function, which are established to overcome difficulties arising from the lower-order terms in the Navier boundary condition. We also consider the global regularity of the axisymmetric solution to the Navier–Stokes equations with full viscosity subject to the total-slip Navier boundary condition. Several new inequalities are established to address the challenges posed by the weak horizontal diffusion of the swirl component.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103807"},"PeriodicalIF":2.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145265978","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}