Journal de Mathematiques Pures et Appliquees最新文献

{"title":"Non-uniqueness of parabolic solutions for advection-diffusion equation","authors":"Thérèse Moerschell , Massimo Sorella","doi":"10.1016/j.matpur.2025.103777","DOIUrl":"10.1016/j.matpur.2025.103777","url":null,"abstract":"<div><div>We present a novel example of a divergence–free velocity field <span><math><mi>b</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>;</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>)</mo></math></span> for <span><math><mi>p</mi><mo><</mo><mn>2</mn></math></span> arbitrary but fixed which leads to non-unique solutions of advection–diffusion in the class <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>x</mi></mrow><mrow><mo>∞</mo></mrow></msubsup><mo>∩</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msubsup><mrow><mi>H</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> while satisfying the local energy inequality. This result complements the known uniqueness result of bounded solutions for divergence-free and <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math></span> 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 <span><math><mi>b</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>;</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>)</mo></math></span>, for <span><math><mi>p</mi><mo><</mo><mn>2</mn></math></span> 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 <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span> divergence–free velocity fields <span><math><mi>b</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>;</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo><mo>)</mo></math></span> with <span><math><mi>p</mi><mo><</mo><mfrac><mrow><mn>2</mn><mi>d</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow></mfrac></math></span> could lead to non–unique solutions in the space <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mo>∞</mo></mrow></msubsup><msubsup><mrow><mi>L</mi></mrow><mrow><mi>x</mi></mrow><mrow><mfrac><mrow><mn>2</mn><mi>d</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></msubsup><mo>∩</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msubsup><mrow><mi>H</mi></mrow><mrow><","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103777"},"PeriodicalIF":2.3,"publicationDate":"2025-07-22","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":"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":"2025-07-21","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":"Local interpolation techniques for higher-order singular perturbations of non-convex functionals: Free-discontinuity problems","authors":"Margherita Solci","doi":"10.1016/j.matpur.2025.103776","DOIUrl":"10.1016/j.matpur.2025.103776","url":null,"abstract":"<div><div>We develop a general approach, using local interpolation inequalities, to non-convex integral functionals depending on the gradient with a singular perturbation by derivatives of order <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>. When applied to functionals giving rise to free-discontinuity energies, such methods permit to change boundary values for derivatives up to order <span><math><mi>k</mi><mo>−</mo><mn>1</mn></math></span> in problems defining density functions for the jump part, thus allowing to prove optimal-profile formulas, and to deduce compactness and lower bounds. As an application, we prove that for <em>k</em>-th order perturbations of energies depending on the gradient behaving as a constant at infinity, the jump energy density is a constant <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> times the <em>k</em>-th root of the jump size. The result is first proved for truncated quadratic energy densities and in the one-dimensional case, from which the general higher-dimensional case can be obtained by slicing techniques. A wide class of non-convex energies can be studied as an envelope of these particular ones. Finally, we remark that an approximation of the Mumford–Shah functional can be obtained by letting <em>k</em> tend to infinity. We also derive a new approximation of the Blake-Zisserman functional.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103776"},"PeriodicalIF":2.1,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144702622","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":"Parabolic Lusztig varieties and chromatic symmetric functions","authors":"Alex Abreu ,&nbsp;Antonio Nigro","doi":"10.1016/j.matpur.2025.103771","DOIUrl":"10.1016/j.matpur.2025.103771","url":null,"abstract":"<div><div>The characters of Kazhdan–Lusztig elements of the Hecke algebra over <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> (and in particular, the chromatic symmetric function of indifference graphs) are completely encoded in the (intersection) cohomology of Lusztig varieties. Considering the forgetful map to some partial flag variety, the decomposition theorem tells us that this cohomology splits as a sum of intersection cohomology groups with coefficients in some local systems of subvarieties of the partial flag variety. We prove that these local systems correspond to representations of subgroups of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. An explicit characterization of such representations would provide a recursive formula for the computation of such characters/chromatic symmetric functions, which could settle Haiman's conjecture about the positivity of the monomial characters of Kazhdan–Lusztig elements and Stanley–Stembridge conjecture about <em>e</em>-positivity of chromatic symmetric function of indifference graphs. We also find a connection between the character of certain homology groups of subvarieties of the partial flag varieties and the Grojnowski–Haiman hybrid basis of the Hecke algebra.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"203 ","pages":"Article 103771"},"PeriodicalIF":2.1,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144696692","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":"Moduli spaces of parabolic bundles over P1 with five marked points","authors":"Zhi Hu ,&nbsp;Pengfei Huang ,&nbsp;Runhong Zong","doi":"10.1016/j.matpur.2025.103775","DOIUrl":"10.1016/j.matpur.2025.103775","url":null,"abstract":"<div><div>This paper considers the moduli spaces/stacks of parabolic bundles (parabolic logarithmic flat bundles and parabolic logarithmic Higgs bundles with given spectrum) of rank 2 and degree 1 over <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> with five marked points. The foliation and stratification structures on these moduli spaces/stacks are investigated. In particular, we confirm Simpson's conjecture for the moduli space of parabolic logarithmic flat bundles with certain non-special weight system.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103775"},"PeriodicalIF":2.3,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144739251","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":"Hidden asymptotics for the weak solutions of the strongly stratified Boussinesq system without rotation","authors":"Frédéric Charve","doi":"10.1016/j.matpur.2025.103750","DOIUrl":"10.1016/j.matpur.2025.103750","url":null,"abstract":"<div><div>The asymptotics of the strongly stratified Boussinesq system when the Froude number goes to zero have been previously investigated, but the resulting limit system surprisingly did not depend on the thermal diffusivity <span><math><msup><mrow><mi>ν</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>. In this article we obtain richer asymptotics (depending on <span><math><msup><mrow><mi>ν</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>) for more general ill-prepared initial data.</div><div>As for the rotating fluids system, the only way to reach this limit consists in finding suitable non-conventional initial data: here, to a function classically depending on the full space variable, we add a second one only depending on the vertical coordinate.</div><div>Thanks to a refined study of the structure of the limit system and to new adapted Strichartz estimates, we obtain convergence in the context of weak Leray-type solutions providing explicit convergence rates when possible. In the usually simpler case <span><math><mi>ν</mi><mo>=</mo><msup><mrow><mi>ν</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> we are able to improve the Strichartz estimates and the convergence rates. The last part of the appendix is devoted to the proof of a new and crucial dispersion estimate, as classical methods fail.</div><div>Finally, our theorems can also be rewritten as a global existence result and asymptotic expansion for the classical Boussinesq system near an explicit stationary solution and for large non-conventional vertically stratified initial data.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"202 ","pages":"Article 103750"},"PeriodicalIF":2.1,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313661","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":"Positive temperature in nonlinear thermoviscoelasticity and the derivation of linearized models","authors":"Rufat Badal ,&nbsp;Manuel Friedrich ,&nbsp;Martin Kružík ,&nbsp;Lennart Machill","doi":"10.1016/j.matpur.2025.103751","DOIUrl":"10.1016/j.matpur.2025.103751","url":null,"abstract":"<div><div>According to the Nernst theorem or, equivalently, the third law of thermodynamics, the absolute zero temperature is not attainable. Starting with an initial positive temperature, we show that there exist solutions to a Kelvin-Voigt model for quasi-static nonlinear thermoviscoelasticity at a finite-strain setting <span><span>[45]</span></span>, obeying an exponential-in-time lower bound on the temperature. Afterwards, we focus on the case of deformations near the identity and temperatures near a critical positive temperature, and we show that weak solutions of the nonlinear system converge in a suitable sense to solutions of a system in linearized thermoviscoelasticity. Our result extends the recent linearization result in <span><span>[4]</span></span>, as it allows the critical temperature to be positive.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"202 ","pages":"Article 103751"},"PeriodicalIF":2.1,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322454","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":"Convergence of free boundaries in the incompressible limit of tumor growth models","authors":"Jiajun Tong ,&nbsp;Yuming Paul Zhang","doi":"10.1016/j.matpur.2025.103752","DOIUrl":"10.1016/j.matpur.2025.103752","url":null,"abstract":"<div><div>We investigate the general Porous Medium Equations with drift and source terms that model tumor growth. Incompressible limit of such models has been well-studied in the literature, where convergence of the density and pressure variables are established, while it remains unclear whether the free boundaries of the solutions exhibit convergence as well. In this paper, we provide an affirmative result by showing that the free boundaries converge in the Hausdorff distance in the incompressible limit. To achieve this, we quantify the relation between the free boundary motion and spatial average of the pressure, and establish a uniform-in-<em>m</em> strict expansion property of the pressure supports. As a corollary, we derive upper bounds for the Hausdorff dimensions of the free boundaries and show that the limiting free boundary has finite <span><math><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-dimensional Hausdorff measure.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"203 ","pages":"Article 103752"},"PeriodicalIF":2.1,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144320782","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":"Formation and construction of a shock wave for 1-D n × n strictly hyperbolic conservation laws with small smooth initial data","authors":"Min Ding ,&nbsp;Huicheng Yin","doi":"10.1016/j.matpur.2025.103754","DOIUrl":"10.1016/j.matpur.2025.103754","url":null,"abstract":"<div><div>Under the genuinely nonlinear assumption for 1-D <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> strictly hyperbolic conservation laws, we investigate the geometric blowup of smooth solutions and the development of singularities when the small initial data fulfill the generic nondegenerate condition. At first, near the unique blowup point we give a precise description on the space-time blowup rate of the smooth solution and meanwhile derive the cusp singularity structure of characteristic envelope. These results are established through extending the smooth solution of the completely nonlinear blowup system across the blowup time. Subsequently, by utilizing a new form on the resulting 1-D strictly hyperbolic system with <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> good components and one bad component, together with the choice of an efficient iterative scheme and some involved analyses, a weak entropy shock wave starting from the blowup point is constructed. As a byproduct, our result can be applied to the shock formation and construction for the 2-D supersonic steady compressible full Euler equations (<span><math><mn>4</mn><mo>×</mo><mn>4</mn></math></span> system), 1-D MHD equations (<span><math><mn>5</mn><mo>×</mo><mn>5</mn></math></span> system), 1-D elastic wave equations (<span><math><mn>6</mn><mo>×</mo><mn>6</mn></math></span> system) and 1-D full ideal compressible MHD equations (<span><math><mn>7</mn><mo>×</mo><mn>7</mn></math></span> system).</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103754"},"PeriodicalIF":2.1,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321251","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 small-time bilinear control of a nonlinear heat equation: Global approximate controllability and exact controllability to trajectories","authors":"Alessandro Duca ,&nbsp;Eugenio Pozzoli ,&nbsp;Cristina Urbani","doi":"10.1016/j.matpur.2025.103758","DOIUrl":"10.1016/j.matpur.2025.103758","url":null,"abstract":"<div><div>In this work we analyse the small-time reachability properties of a nonlinear parabolic equation, by means of a bilinear control, posed on a torus of arbitrary dimension <em>d</em>. Under a saturation hypothesis on the control operators, we show the small-time approximate controllability between states sharing the same sign. Moreover, in the one-dimensional case <span><math><mi>d</mi><mo>=</mo><mn>1</mn></math></span>, we combine this property with a local exact controllability result, and prove the small-time exact controllability of any positive states towards the ground state of the evolution operator.</div><div>Dans ce travail, nous analysons les propriétés d'accessibilité en temps court d'une équation parabolique non linéaire, à l'aide d'un contrôle bilinéaire, posée sur un tore de dimension arbitraire <em>d</em>. Sous une hypothèse de saturation sur les opérateurs de contrôle, nous montrons la contrôlabilité approchée en temps court entre les états qui ont le même signe. De plus, dans le cas unidimensionnel <span><math><mi>d</mi><mo>=</mo><mn>1</mn></math></span>, nous combinons cette propriété avec un résultat de contrôlabilité locale exacte, et prouvons la contrôlabilité exacte en temps court de tout état positif vers l'état fondamental de l'opérateur d'évolution.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"203 ","pages":"Article 103758"},"PeriodicalIF":2.1,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144320781","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}