Calculus of Variations and Partial Differential Equations最新文献

{"title":"On De Giorgi's conjecture of nonlocal approximations for free-discontinuity problems: The symmetric gradient case.","authors":"S Almi, E Davoli, A Kubin, E Tasso","doi":"10.1007/s00526-026-03338-w","DOIUrl":"10.1007/s00526-026-03338-w","url":null,"abstract":"<p><p>We prove that E. De Giorgi's conjecture for the nonlocal approximation of free-discontinuity problems extends to the case of functionals defined in terms of the symmetric gradient of the admissible field. After introducing a suitable class of continuous finite-difference approximants, we show the compactness of deformations with equibounded energies, as well as their Gamma-convergence. The compactness analysis is a crucial hurdle, which we overcome by generalizing a Fréchet-Kolmogorov approach previously introduced by two of the authors. A second essential difficulty is the identification of the limiting space of admissible deformations, since a control on the directional variations is, a priori, only available in average. A limiting representation in GSBD is eventually established via a novel characterization of this space.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 5","pages":"160"},"PeriodicalIF":2.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13070975/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147688520","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":"The global inverse fractional conductivity problem.","authors":"Giovanni Covi, Jesse Railo, Philipp Zimmermann","doi":"10.1007/s00526-026-03401-6","DOIUrl":"https://doi.org/10.1007/s00526-026-03401-6","url":null,"abstract":"<p><p>We prove <i>global</i> uniqueness for an inverse problem for the fractional conductivity equation on domains that are bounded in one direction. The conductivities are assumed to be nontrivial in the exterior of the domain and isotropic, while the data is given in the form of partial Dirichlet-to-Neumann (DN) maps measured in nondisjoint open subsets of the exterior. This can be seen as the fractional counterpart of the classical inverse conductivity problem. The proof is based on a unique continuation property (UCP) for the DN maps and a novel exterior determination method from the partial exterior DN maps, which is a nonlocal analog of the classical boundary determination method by Kohn and Vogelius. Exterior determination is achieved independently of the UCP and despite the nonlocality of the equation.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 9","pages":"258"},"PeriodicalIF":2.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13553603/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148918579","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":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Energy identity and no neck property for <ns0:math><ns0:mi>ε</ns0:mi></ns0:math> -harmonic and <ns0:math><ns0:mi>α</ns0:mi></ns0:math> -harmonic maps into homogeneous target manifolds.","authors":"Carolin Bayer, Andrew M Roberts","doi":"10.1007/s00526-025-03233-w","DOIUrl":"10.1007/s00526-025-03233-w","url":null,"abstract":"<p><p>In this paper we show the energy identity and the no-neck property for <math><mi>ε</mi></math> - and <math><mi>α</mi></math> -harmonic maps with homogeneous target manifolds. To prove this in the <math><mi>ε</mi></math> -harmonic case we introduce the idea of using an equivariant embedding of the homogeneous target manifold.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 2","pages":"57"},"PeriodicalIF":2.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12795924/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145970482","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":"Parabolic-elliptic and indirect-direct simplifications in chemotaxis systems driven by indirect signalling.","authors":"Le Trong Thanh Bui, Thi Kim Loan Huynh, Bao Quoc Tang, Bao-Ngoc Tran","doi":"10.1007/s00526-025-03247-4","DOIUrl":"10.1007/s00526-025-03247-4","url":null,"abstract":"&lt;p&gt;&lt;p&gt;Singular limits for the following indirect signalling chemotaxis system &lt;dispformula&gt; &lt;math&gt; &lt;mrow&gt; &lt;mtable&gt; &lt;mtr&gt; &lt;mtd&gt; &lt;mfenced&gt; &lt;mrow&gt; &lt;mtable&gt; &lt;mtr&gt; &lt;mtd&gt; &lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;∂&lt;/mi&gt; &lt;mi&gt;t&lt;/mi&gt;&lt;/msub&gt; &lt;mi&gt;n&lt;/mi&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;mi&gt;Δ&lt;/mi&gt; &lt;mi&gt;n&lt;/mi&gt; &lt;mo&gt;-&lt;/mo&gt; &lt;mi&gt;∇&lt;/mi&gt; &lt;mo&gt;·&lt;/mo&gt; &lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;n&lt;/mi&gt; &lt;mi&gt;∇&lt;/mi&gt; &lt;mi&gt;c&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;/mrow&gt; &lt;/mtd&gt; &lt;mtd&gt;&lt;mrow&gt;&lt;mtext&gt;in&lt;/mtext&gt; &lt;mspace&gt;&lt;/mspace&gt; &lt;mi&gt;Ω&lt;/mi&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;mi&gt;∞&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt; &lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt; &lt;/mtd&gt; &lt;/mtr&gt; &lt;mtr&gt; &lt;mtd&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;/mrow&gt; &lt;mi&gt;ε&lt;/mi&gt; &lt;msub&gt;&lt;mi&gt;∂&lt;/mi&gt; &lt;mi&gt;t&lt;/mi&gt;&lt;/msub&gt; &lt;mi&gt;c&lt;/mi&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;mi&gt;Δ&lt;/mi&gt; &lt;mi&gt;c&lt;/mi&gt; &lt;mo&gt;-&lt;/mo&gt; &lt;mi&gt;c&lt;/mi&gt; &lt;mo&gt;+&lt;/mo&gt; &lt;mi&gt;w&lt;/mi&gt;&lt;/mrow&gt; &lt;/mtd&gt; &lt;mtd&gt;&lt;mrow&gt;&lt;mtext&gt;in&lt;/mtext&gt; &lt;mspace&gt;&lt;/mspace&gt; &lt;mi&gt;Ω&lt;/mi&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;mi&gt;∞&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt; &lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt; &lt;/mtd&gt; &lt;/mtr&gt; &lt;mtr&gt; &lt;mtd&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;/mrow&gt; &lt;mi&gt;ε&lt;/mi&gt; &lt;msub&gt;&lt;mi&gt;∂&lt;/mi&gt; &lt;mi&gt;t&lt;/mi&gt;&lt;/msub&gt; &lt;mi&gt;w&lt;/mi&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;mi&gt;τ&lt;/mi&gt; &lt;mi&gt;Δ&lt;/mi&gt; &lt;mi&gt;w&lt;/mi&gt; &lt;mo&gt;-&lt;/mo&gt; &lt;mi&gt;w&lt;/mi&gt; &lt;mo&gt;+&lt;/mo&gt; &lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt; &lt;/mtd&gt; &lt;mtd&gt;&lt;mrow&gt;&lt;mtext&gt;in&lt;/mtext&gt; &lt;mspace&gt;&lt;/mspace&gt; &lt;mi&gt;Ω&lt;/mi&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;mi&gt;∞&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt; &lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt; &lt;/mtd&gt; &lt;/mtr&gt; &lt;mtr&gt; &lt;mtd&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;/mrow&gt; &lt;msub&gt;&lt;mi&gt;∂&lt;/mi&gt; &lt;mi&gt;ν&lt;/mi&gt;&lt;/msub&gt; &lt;mi&gt;n&lt;/mi&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;msub&gt;&lt;mi&gt;∂&lt;/mi&gt; &lt;mi&gt;ν&lt;/mi&gt;&lt;/msub&gt; &lt;mi&gt;c&lt;/mi&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;msub&gt;&lt;mi&gt;∂&lt;/mi&gt; &lt;mi&gt;ν&lt;/mi&gt;&lt;/msub&gt; &lt;mi&gt;w&lt;/mi&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;mn&gt;0&lt;/mn&gt; &lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt; &lt;/mtd&gt; &lt;mtd&gt;&lt;mrow&gt;&lt;mtext&gt;on&lt;/mtext&gt; &lt;mspace&gt;&lt;/mspace&gt; &lt;mi&gt;∂&lt;/mi&gt; &lt;mi&gt;Ω&lt;/mi&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;mi&gt;∞&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;/mtd&gt; &lt;/mtr&gt; &lt;/mtable&gt; &lt;/mrow&gt; &lt;/mfenced&gt; &lt;/mtd&gt; &lt;/mtr&gt; &lt;/mtable&gt; &lt;/mrow&gt; &lt;/math&gt; &lt;/dispformula&gt; are investigated. More precisely, we study parabolic-elliptic simplification, or PES, &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;ε&lt;/mi&gt; &lt;mo&gt;→&lt;/mo&gt; &lt;msup&gt;&lt;mn&gt;0&lt;/mn&gt; &lt;mo&gt;+&lt;/mo&gt;&lt;/msup&gt; &lt;/mrow&gt; &lt;/math&gt; with fixed &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;τ&lt;/mi&gt; &lt;mo&gt;&gt;&lt;/mo&gt; &lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt; &lt;/math&gt; up to the critical dimension &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt; &lt;/math&gt; , and indirect-direct simplification, or IDS, &lt;math&gt; &lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;ε&lt;/mi&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;mi&gt;τ&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;mo&gt;→&lt;/mo&gt; &lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;msup&gt;&lt;mn&gt;0&lt;/mn&gt; &lt;mo&gt;+&lt;/mo&gt;&lt;/msup&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;msup&gt;&lt;mn&gt;0&lt;/mn&gt; &lt;mo&gt;+&lt;/mo&gt;&lt;/msup&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;/mrow&gt; &lt;/math&gt; up to the critical dimension &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt; &lt;/math&gt; . These are relevant in biological situations where the signalling process is on a much faster time scale compared to the species diffusion and all interactions. Showing singular limits in critical dimensions is challenging. To deal with the PES, we carefully combine the entropy function, an Adam-type inequality, the regularisation of slow evolution, and an energy equation method to obtain strong convergence in representative spaces. For the IDS, a bootstrap argument concerning the &lt;math&gt;&lt;msup&gt;&lt;mi&gt;L&lt;/mi&gt; &lt;mi&gt;p&lt;/mi&gt;&lt;/msup&gt; &lt;/math&gt; -energy function is devised, which allows us to obtain suitable uniform bounds for the singul","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 3","pages":"76"},"PeriodicalIF":2.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12913282/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146225681","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":"How close is too close for singular mean curvature flows?","authors":"J M Daniels-Holgate, Or Hershkovits","doi":"10.1007/s00526-026-03403-4","DOIUrl":"https://doi.org/10.1007/s00526-026-03403-4","url":null,"abstract":"<p><p>Suppose <math> <msub><mrow><mo>(</mo> <msubsup><mi>M</mi> <mi>t</mi> <mi>i</mi></msubsup> <mo>)</mo></mrow> <mrow><mi>t</mi> <mo>∈</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>)</mo></mrow> </msub> </math> , <math><mrow><mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn></mrow> </math> , are two mean curvature flows in <math> <msup><mrow><mi>R</mi></mrow> <mrow><mi>n</mi> <mo>+</mo> <mn>1</mn></mrow> </msup> </math> encountering a multiplicity one compact singularity at time <i>T</i>, in such a manner that for every <i>k</i>, the Hausdorff distance between the two flows, <math><msub><mi>d</mi> <mi>H</mi></msub> </math> , satisfies <math> <mrow><msub><mi>d</mi> <mi>H</mi></msub> <mrow><mo>(</mo> <msubsup><mi>M</mi> <mi>t</mi> <mn>1</mn></msubsup> <mo>,</mo> <msubsup><mi>M</mi> <mi>t</mi> <mn>2</mn></msubsup> <mo>)</mo></mrow> <mo>/</mo> <msup><mrow><mo>(</mo> <mi>T</mi> <mo>-</mo> <mi>t</mi> <mo>)</mo></mrow> <mi>k</mi></msup> <mo>→</mo> <mn>0</mn></mrow> </math> . We demonstrate that <math> <mrow><msubsup><mi>M</mi> <mi>t</mi> <mn>1</mn></msubsup> <mo>=</mo> <msubsup><mi>M</mi> <mi>t</mi> <mn>2</mn></msubsup> </mrow> </math> for every <i>t</i>. This generalizes a result of Martin-Hagemayer and Sesum, who proved the case where <math><msubsup><mi>M</mi> <mi>t</mi> <mn>1</mn></msubsup> </math> is itself a self-similarly shrinking flow.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 8","pages":"234"},"PeriodicalIF":2.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13380582/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148520116","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":"Explicit minimisers for anisotropic Riesz energies.","authors":"R L Frank, J Mateu, M G Mora, L Rondi, L Scardia, J Verdera","doi":"10.1007/s00526-025-03185-1","DOIUrl":"10.1007/s00526-025-03185-1","url":null,"abstract":"<p><p>In this paper we characterise the energy minimisers of a class of nonlocal interaction energies where the attraction is quadratic, and the repulsion is Riesz-like and anisotropic. In particular we show that, if the Fourier transform of the repulsive potential is positive, the minimiser is supported on a fully-dimensional ellipsoid, and its density is given by a Barenblatt-type profile. Our technique of proof is based on a Fourier representation of the potential of such measures, that extends a previous formula established by some of the authors in the Coulomb case.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 1","pages":"34"},"PeriodicalIF":2.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12701851/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145762323","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":"Comparison theorems on H-type sub-Riemannian manifolds.","authors":"Fabrice Baudoin, Erlend Grong, Luca Rizzi, Sylvie Vega-Molino","doi":"10.1007/s00526-025-02992-w","DOIUrl":"https://doi.org/10.1007/s00526-025-02992-w","url":null,"abstract":"<p><p>On H-type sub-Riemannian manifolds we establish sub-Hessian and sub-Laplacian comparison theorems which are uniform for a family of approximating Riemannian metrics converging to the sub-Riemannian one. We also prove a sharp sub-Riemannian Bonnet-Myers theorem that extends to this general setting results previously proved on contact and quaternionic contact manifolds.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"64 5","pages":"143"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12053226/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143954637","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":"Continuity up to the boundary for minimizers of the one-phase Bernoulli problem.","authors":"Xavier Fernández-Real, Florian Gruen","doi":"10.1007/s00526-025-03040-3","DOIUrl":"10.1007/s00526-025-03040-3","url":null,"abstract":"<p><p>We prove new boundary regularity results for minimizers to the one-phase Alt-Caffarelli functional (also known as Bernoulli free boundary problem) in the case of continuous and Hölder-continuous boundary data. As an application, we use them to extend recent generic uniqueness and regularity results to families of continuous functions.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"64 5","pages":"166"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12116618/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144180638","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":"Steady bubbles and drops in inviscid fluids.","authors":"David Meyer, Lukas Niebel, Christian Seis","doi":"10.1007/s00526-025-03144-w","DOIUrl":"10.1007/s00526-025-03144-w","url":null,"abstract":"<p><p>We construct steady non-spherical bubbles and drops, which are traveling wave solutions to the axisymmetric two-phase Euler equations with surface tension, whose inner phase is a bounded connected domain. The solutions have a uniform vorticity distribution in this inner phase and they have a vortex sheet on its surface. Our construction relies on a perturbative approach around an explicit spherical solution, given by Hill's vortex enclosed by a spherical vortex sheet. The construction is sensitive to the Weber numbers describing the flow. At critical Weber numbers, we perform a bifurcation analysis utilizing the Crandall-Rabinowitz theorem in Sobolev spaces on the 2-sphere. Away from these critical numbers, our construction relies on the implicit function theorem. Our results imply that the model containing surface tension is richer than the ordinary one-phase Euler equations, in the sense that for the latter, Hill's spherical vortex is unique (modulo translations) among all axisymmetric simply connected uniform vortices of a given circulation.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"64 9","pages":"299"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12528203/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145328313","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":"Globally stable blowup profile for supercritical wave maps in all dimensions.","authors":"Irfan Glogić","doi":"10.1007/s00526-024-02901-7","DOIUrl":"https://doi.org/10.1007/s00526-024-02901-7","url":null,"abstract":"<p><p>We consider wave maps from the <math><mrow><mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>d</mi> <mo>)</mo></mrow> </math> -dimensional Minkowski space into the <i>d</i>-sphere. It is known from the work of Bizoń and Biernat (Commun Math Phys 338(3): 1443-1450, 2015) that in the energy-supercritical case, i.e., for <math><mrow><mi>d</mi> <mo>≥</mo> <mn>3</mn></mrow> </math> , this model admits a closed-form corotational self-similar blowup solution. We show that this blowup profile is globally nonlinearly stable for all <math><mrow><mi>d</mi> <mo>≥</mo> <mn>3</mn></mrow> </math> , thereby verifying a perturbative version of the conjecture posed in Bizoń and Biernat (Commun Math Phys 338(3): 1443-1450, 2015) about the generic large data blowup behavior for this model. To accomplish this, we develop a novel stability analysis approach based on similarity variables posed on the whole space <math> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </math> . As a result, we draw a general road map for studying spatially global stability of self-similar blowup profiles for nonlinear wave equations in the radial case for arbitrary dimension <math><mrow><mi>d</mi> <mo>≥</mo> <mn>3</mn></mrow> </math> .</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"64 2","pages":"46"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11703941/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142945062","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}