Calculus of Variations and Partial Differential Equations最新文献

Long-time behaviour and bifurcation analysis of a two-species aggregation-diffusion system on the torus. 环面上两种聚集-扩散系统的长时间行为和分岔分析。

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2026-01-01 Epub Date: 2025-12-06 DOI: 10.1007/s00526-025-03132-0

José A Carrillo, Yurij Salmaniw

{"title":"Long-time behaviour and bifurcation analysis of a two-species aggregation-diffusion system on the torus.","authors":"José A Carrillo, Yurij Salmaniw","doi":"10.1007/s00526-025-03132-0","DOIUrl":"10.1007/s00526-025-03132-0","url":null,"abstract":"We investigate stationary states, including their existence and stability, in a class of nonlocal aggregation-diffusion equations with linear diffusion and symmetric nonlocal interactions. For the scalar case, we extend previous results by showing that key model features, such as existence, regularity, bifurcation structure, and stability exchange, continue to hold under a mere bounded variation hypothesis. For the corresponding two-species system, we carry out a fully rigorous bifurcation analysis using the bifurcation theory of Crandall & Rabinowitz. This framework allows us to classify all solution branches from homogeneous states, with particular attention given to those arising from the self-interaction strength and the cross-interaction strength, as well as the stability of the branch at a point of critical stability. The analysis relies on an equivalent classification of solutions through fixed points of a nonlinear map, followed by a careful derivation of Fréchet derivatives up to third order. An interesting application to cell-cell adhesion arises from our analysis, yielding stable segregation patterns that appear at the onset of cell sorting in a modelling regime where all interactions are purely attractive.","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 1","pages":"19"},"PeriodicalIF":2.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12681474/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145707398","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}

引用次数: 0

Best constants in subelliptic fractional Sobolev and Gagliardo-Nirenberg inequalities and ground states on stratified Lie groups. 亚椭圆分数Sobolev和Gagliardo-Nirenberg不等式的最佳常数和分层李群上的基态。

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2026-01-01 Epub Date: 2025-12-13 DOI: 10.1007/s00526-025-03191-3

Sekhar Ghosh, Vishvesh Kumar, Michael Ruzhansky

{"title":"Best constants in subelliptic fractional Sobolev and Gagliardo-Nirenberg inequalities and ground states on stratified Lie groups.","authors":"Sekhar Ghosh, Vishvesh Kumar, Michael Ruzhansky","doi":"10.1007/s00526-025-03191-3","DOIUrl":"10.1007/s00526-025-03191-3","url":null,"abstract":"In this paper, we establish the sharp fractional subelliptic Sobolev inequalities and Gagliardo-Nirenberg inequalities on stratified Lie groups. The best constants are given in terms of a ground state solution of a fractional subelliptic equation involving the fractional p-sublaplacian ( <math><mrow><mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi></mrow> </math> ) on stratified Lie groups. We also prove the existence of ground state (least energy) solutions to nonlinear subelliptic fractional Schrödinger equation on stratified Lie groups. Different from the proofs of analogous results in the setting of classical Sobolev spaces on Euclidean spaces given by Weinstein (Comm. Math. Phys. 87(4):576-676, 1982/1983) using the rearrangement inequality which is not available in stratified Lie groups, we apply a subelliptic version of vanishing lemma due to Lions extended in the setting of stratified Lie groups combining it with the compact embedding theorem for subelliptic fractional Sobolev spaces obtained in our previous paper (Math. Ann. 388(4):4201-4249, 2024). We also present subelliptic fractional logarithmic Sobolev inequalities with explicit constants on stratified Lie groups. The main results are new for <math><mrow><mi>p</mi> <mo>=</mo> <mn>2</mn></mrow> </math> even in the context of the Heisenberg group.","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 1","pages":"28"},"PeriodicalIF":2.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12701863/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145762226","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}

引用次数: 0

Average-field approximation for dilute almost-bosonic anyons. 稀几乎玻色子的平均场近似。

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2026-01-01 Epub Date: 2026-09-12 DOI: 10.1007/s00526-026-03428-9

François L A Visconti

{"title":"Average-field approximation for dilute almost-bosonic anyons.","authors":"François L A Visconti","doi":"10.1007/s00526-026-03428-9","DOIUrl":"10.1007/s00526-026-03428-9","url":null,"abstract":"We study the ground state of a system of N two-dimensional trapped almost-bosonic anyons. This setup can equivalently be viewed as bosons interacting through long-range magnetic potentials generated by magnetic charges carried by each particle. These magnetic charges are assumed to be smeared over discs of radius R-a model known as extended anyons. To recover the point-like anyons perspective, we consider the joint limit <math><mrow><mi>R</mi> <mo>→</mo> <mn>0</mn></mrow> </math> as <math><mrow><mi>N</mi> <mo>→</mo> <mi>∞</mi></mrow> </math> . We rigorously justify the average-field approximation for any radius R that decays polynomially, and even for certain exponentially decaying R. The average-field approximation asserts that the particles behave like independent, identical bosons interacting through a self-consistent magnetic field. Our result significantly improves upon the best-known estimates from [1, 2], and it in particular covers radii that are much smaller than the mean interparticle distance. The proof strategy builds on a recent work on two-dimensional attractive Bose gases by Junge and the author [3].","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 9","pages":"260"},"PeriodicalIF":2.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13570908/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148963067","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}

引用次数: 0

Adiabatic Klein-Gordon dynamics for the renormalized Nelson model. 重归一化Nelson模型的绝热Klein-Gordon动力学。

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2026-01-01 Epub Date: 2026-10-12 DOI: 10.1007/s00526-026-03436-9

Morris Brooks, David Mitrouskas

{"title":"Adiabatic Klein-Gordon dynamics for the renormalized Nelson model.","authors":"Morris Brooks, David Mitrouskas","doi":"10.1007/s00526-026-03436-9","DOIUrl":"10.1007/s00526-026-03436-9","url":null,"abstract":"We study the renormalized Nelson model in a semiclassical regime where the field becomes classical while the particle remains quantum. The degree of classicality is measured by a small parameter <math><mrow><mi>ε</mi> <mo>≪</mo> <mn>1</mn></mrow> </math> . In this scaling the particle evolves on microscopic times, whereas the field exhibits nontrivial dynamics only on the macroscopic scale <math><mrow><mi>t</mi> <mo>=</mo> <mi>O</mi> <mo>(</mo> <msup><mi>ε</mi> <mrow><mo>-</mo> <mn>2</mn></mrow> </msup> <mo>)</mo></mrow> </math> . The natural semiclassical model is the coupled Schrödinger-Klein-Gordon (SKG) system, which encodes the time-scale separation through an explicit <math><mi>ε</mi></math> -dependence. Based on this scale separation in SKG, we apply the adiabatic principle to derive a new PDE for the classical field, the <math><mi>ε</mi></math> -free adiabatic Klein-Gordon (aKG) equation, where the field is driven by the instantaneous ground state of the particle. Our main result is a norm approximation of the Nelson dynamics by the aKG solution corrected by a quasi-free fluctuation dynamics around the classical field, generated by a renormalized Bogoliubov-Nelson Hamiltonian. As a corollary, we obtain convergence of the reduced one-body densities for both subsystems, where the fluctuation correction vanishes, thereby justifying aKG as a semiclassical Born-Oppenheimer type approximation of the renormalized Nelson model.","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 10","pages":"270"},"PeriodicalIF":2.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13654888/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149164857","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}

引用次数: 0

Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. 具有一般狄利克雷边界条件的Fokker-Planck方程的变分结构。

IF 2 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2026-01-01 Epub Date: 2025-12-06 DOI: 10.1007/s00526-025-03193-1

Filippo Quattrocchi

{"title":"Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions.","authors":"Filippo Quattrocchi","doi":"10.1007/s00526-025-03193-1","DOIUrl":"https://doi.org/10.1007/s00526-025-03193-1","url":null,"abstract":"We prove the convergence of a modified Jordan-Kinderlehrer-Otto scheme to a solution to the Fokker-Planck equation in <math><mrow><mi>Ω</mi> <mo>⋐</mo> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </mrow> </math> with general-strictly positive and temporally constant-Dirichlet boundary conditions. We work under mild assumptions on the domain, the drift, and the initial datum. In the special case where <math><mi>Ω</mi></math> is an interval in <math> <msup><mrow><mi>R</mi></mrow> <mn>1</mn></msup> </math> , we prove that such a solution is a gradient flow-curve of maximal slope-within a suitable space of measures, endowed with a modified Wasserstein distance. Our discrete scheme and modified distance draw inspiration from contributions by A. Figalli and N. Gigli [J. Math. Pures Appl. 94, (2010), pp. 107-130], and J. Morales [J. Math. Pures Appl. 112, (2018), pp. 41-88] on an optimal-transport approach to evolution equations with Dirichlet boundary conditions. Similarly to these works, we allow the mass to flow from/to the boundary <math><mrow><mi>∂</mi> <mi>Ω</mi></mrow> </math> throughout the evolution. However, our leading idea is to also keep track of the mass at the boundary by working with measures defined on the whole closure <math><mover><mi>Ω</mi> <mo>¯</mo></mover> </math> . The driving functional is a modification of the classical relative entropy that also makes use of the information at the boundary. As an intermediate result, when <math><mi>Ω</mi></math> is an interval in <math> <msup><mrow><mi>R</mi></mrow> <mn>1</mn></msup> </math> , we find a formula for the descending slope of this geodesically nonconvex functional.","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 1","pages":"23"},"PeriodicalIF":2.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12681467/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145707396","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}

引用次数: 0

Existence of nontrival n-harmonic maps via min-max methods. 非平凡n调和映射的存在性。

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2026-01-01 Epub Date: 2026-07-03 DOI: 10.1007/s00526-026-03399-x

Dorian Martino, Katarzyna Mazowiecka, Armin Schikorra

引用次数: 0

Free boundary minimal Möbius bands in toroids. 环面中的自由边界极小Möbius带。

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2026-01-01 Epub Date: 2026-06-23 DOI: 10.1007/s00526-026-03382-6

Mario B Schulz

引用次数: 0

Thermo-elastodynamics of nonlinearly viscous solids. 非线性粘性固体的热弹性动力学。

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2026-01-01 Epub Date: 2026-02-28 DOI: 10.1007/s00526-026-03282-9

Stefano Almi, Rufat Badal, Manuel Friedrich, Sebastian Schwarzacher

{"title":"Thermo-elastodynamics of nonlinearly viscous solids.","authors":"Stefano Almi, Rufat Badal, Manuel Friedrich, Sebastian Schwarzacher","doi":"10.1007/s00526-026-03282-9","DOIUrl":"10.1007/s00526-026-03282-9","url":null,"abstract":"In this paper, we study the thermo-elastodynamics of nonlinearly viscous solids in the Kelvin-Voigt rheology where both the elastic and the viscous stress tensors comply with the frame-indifference principle. The system features a force balance including inertia in the frame of nonsimple materials and a heat-transfer equation which is governed by the Fourier law in the deformed configuration. Combining a staggered minimizing movement scheme for quasi-static thermoviscoelasticity [2, 37] with a variational approach to hyperbolic PDEs developed in [5], our main result consists in establishing the existence of weak solutions in the dynamic case. This is first achieved by including an additional higher-order regularization for the dissipation. Afterwards, this regularization can be removed by passing to a weaker formulation of the heat-transfer equation which complies with a total energy balance. The latter description hinges on regularity theory for the fourth order p-Laplacian which induces regularity estimates of the deformation beyond the standard estimates available from energy bounds. Besides being crucial for the proof, these extra regularity properties might be of independent interest and seem to be new in the setting of nonlinear viscoelasticity, also in the static or quasi-static case.","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 4","pages":"111"},"PeriodicalIF":2.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12950105/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147343738","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}

引用次数: 0

JKO schemes with general transport costs. JKO计划及一般运输费用。

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2026-01-01 Epub Date: 2026-02-17 DOI: 10.1007/s00526-026-03256-x

Cale Rankin, Ting-Kam Leonard Wong

引用次数: 0

Surgery and positive Bakry-Émery Ricci curvature. 手术和阳性的Bakry-Émery Ricci曲度。

IF 2 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2026-01-01 Epub Date: 2026-01-12 DOI: 10.1007/s00526-025-03211-2

Philipp Reiser, Francesca Tripaldi

引用次数: 0