Pub Date : 2025-09-11DOI: 10.1016/j.na.2025.113940
R. Dhanya , Jacques Giacomoni , Ritabrata Jana
In this article, we examine the Hölder regularity of solutions to equations involving a mixed local-nonlocal nonlinear nonhomogeneous operator with singular data, under the minimal assumption that . The regularity result is twofold: we establish interior gradient Hölder regularity for locally bounded data and boundary regularity for singular data. We prove both boundary Hölder and boundary gradient Hölder regularity depending on the degree of singularity. Additionally, we establish a strong comparison principle for this class of problems, which holds independent significance. As the applications of these qualitative results, we further study sublinear and subcritical perturbations of singular nonlinearity.
{"title":"Interior and boundary regularity of mixed local nonlocal problem with singular data and its applications","authors":"R. Dhanya , Jacques Giacomoni , Ritabrata Jana","doi":"10.1016/j.na.2025.113940","DOIUrl":"10.1016/j.na.2025.113940","url":null,"abstract":"<div><div>In this article, we examine the Hölder regularity of solutions to equations involving a mixed local-nonlocal nonlinear nonhomogeneous operator <span><math><mrow><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>+</mo><msubsup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow></mrow><mrow><mi>q</mi></mrow><mrow><mi>s</mi></mrow></msubsup></mrow></math></span> with singular data, under the minimal assumption that <span><math><mrow><mi>p</mi><mo>></mo><mi>s</mi><mi>q</mi></mrow></math></span>. The regularity result is twofold: we establish interior gradient Hölder regularity for locally bounded data and boundary regularity for singular data. We prove both boundary Hölder and boundary gradient Hölder regularity depending on the degree of singularity. Additionally, we establish a strong comparison principle for this class of problems, which holds independent significance. As the applications of these qualitative results, we further study sublinear and subcritical perturbations of singular nonlinearity.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113940"},"PeriodicalIF":1.3,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145048353","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}
Pub Date : 2025-09-09DOI: 10.1016/j.na.2025.113935
Qirui Peng
The goal of this paper is to construct non-trivial steady-state weak solutions of the three dimensional Electron Magnetohydrodynamics equations in the class of for some small . By exploiting the formulation of the stationary EMHD equations one can treat them as generalized Navier–Stokes equations with half Laplacian. Therefore with convex integration scheme we obtained such stationary weak solutions, which is not yet realizable in the case of classical 3D Navier–Stokes equations.
{"title":"Three dimensional stationary solutions of the Electron MHD equations","authors":"Qirui Peng","doi":"10.1016/j.na.2025.113935","DOIUrl":"10.1016/j.na.2025.113935","url":null,"abstract":"<div><div>The goal of this paper is to construct non-trivial steady-state weak solutions of the three dimensional Electron Magnetohydrodynamics equations in the class of <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> for some small <span><math><mrow><mi>s</mi><mo>></mo><mn>0</mn></mrow></math></span>. By exploiting the formulation of the stationary EMHD equations one can treat them as generalized Navier–Stokes equations with half Laplacian. Therefore with convex integration scheme we obtained such stationary weak solutions, which is not yet realizable in the case of classical 3D Navier–Stokes equations.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113935"},"PeriodicalIF":1.3,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145019143","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}
Pub Date : 2025-09-08DOI: 10.1016/j.na.2025.113941
Jiangfeng Han , Zhenhai Liu , Nikolaos S. Papageorgiou
We consider a Dirichlet problem driven by a double phase differential operator and a reaction which exhibits the combined effects of a parametric singular term and of an indefinite superlinear perturbation. The superlinearity condition on the perturbation is very general. Using variational tools, truncation and comparison techniques and critical groups, we prove an existence and multiplicity result which is global in the parameter (bifurcation-type result).
{"title":"A singular double phase eigenvalue problem with a superlinear indefinite perturbation","authors":"Jiangfeng Han , Zhenhai Liu , Nikolaos S. Papageorgiou","doi":"10.1016/j.na.2025.113941","DOIUrl":"10.1016/j.na.2025.113941","url":null,"abstract":"<div><div>We consider a Dirichlet problem driven by a double phase differential operator and a reaction which exhibits the combined effects of a parametric singular term and of an indefinite superlinear perturbation. The superlinearity condition on the perturbation is very general. Using variational tools, truncation and comparison techniques and critical groups, we prove an existence and multiplicity result which is global in the parameter (bifurcation-type result).</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113941"},"PeriodicalIF":1.3,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145010445","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}
Pub Date : 2025-09-04DOI: 10.1016/j.na.2025.113921
Sekar Nugraheni , Paolo Giordano
The definition of a non-trivial space of generalized functions of a complex variable allowing to consider derivatives of continuous functions is a non-obvious task, e.g. because of Morera theorem, because distributional Cauchy–Riemann equations implies holomorphicity and of course because including Dirac delta seems incompatible with the identity theorem. Surprisingly, these results can be achieved if we consider a suitable non-Archimedean extension of the complex field, i.e. a ring where infinitesimal and infinite numbers return to be available. In this first paper, we set the definition of generalized holomorphic function and prove the extension of several classical theorems, such as Cauchy–Riemann equations, Goursat, Looman–Menchoff and Montel theorems, generalized complex differentiability implies smoothness, embedding of distributions, closure with respect to composition and hence non-linear operations on these generalized functions. The theory hence addresses several limitations of Colombeau theory of generalized holomorphic functions. The final aim of this series of papers is to prove the Cauchy–Kowalevski theorem including also distributional PDE or singular boundary conditions and nonlinear operations.
{"title":"Dirac delta as a generalized holomorphic function","authors":"Sekar Nugraheni , Paolo Giordano","doi":"10.1016/j.na.2025.113921","DOIUrl":"10.1016/j.na.2025.113921","url":null,"abstract":"<div><div>The definition of a non-trivial space of generalized functions of a complex variable allowing to consider derivatives of continuous functions is a non-obvious task, e.g. because of Morera theorem, because distributional Cauchy–Riemann equations implies holomorphicity and of course because including Dirac delta seems incompatible with the identity theorem. Surprisingly, these results can be achieved if we consider a suitable non-Archimedean extension of the complex field, i.e. a ring where infinitesimal and infinite numbers return to be available. In this first paper, we set the definition of generalized holomorphic function and prove the extension of several classical theorems, such as Cauchy–Riemann equations, Goursat, Looman–Menchoff and Montel theorems, generalized complex differentiability implies smoothness, embedding of distributions, closure with respect to composition and hence non-linear operations on these generalized functions. The theory hence addresses several limitations of Colombeau theory of generalized holomorphic functions. The final aim of this series of papers is to prove the Cauchy–Kowalevski theorem including also distributional PDE or singular boundary conditions and nonlinear operations.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113921"},"PeriodicalIF":1.3,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144988178","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}
Pub Date : 2025-09-03DOI: 10.1016/j.na.2025.113915
Stefano Biagi , Giovanni Cupini , Elvira Mascolo
We consider a class of energy integrals, associated to nonlinear and non-uniformly elliptic equations, with integrands satisfying anisotropic -growth conditions of the form for some exponents , and non-negative functions subject to suitable summability assumptions. We prove the local boundedness of scalar local quasi-minimizers of such integrals.
{"title":"Local boundedness for solutions of a class of non-uniformly elliptic anisotropic problems","authors":"Stefano Biagi , Giovanni Cupini , Elvira Mascolo","doi":"10.1016/j.na.2025.113915","DOIUrl":"10.1016/j.na.2025.113915","url":null,"abstract":"<div><div>We consider a class of energy integrals, associated to nonlinear and non-uniformly elliptic equations, with integrands <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>ξ</mi><mo>)</mo></mrow></mrow></math></span> satisfying anisotropic <span><math><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>q</mi></mrow></math></span>-growth conditions of the form <span><math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msup><mo>≤</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>ξ</mi><mo>)</mo></mrow><mo>≤</mo><mi>μ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mfenced><mrow><msup><mrow><mrow><mo>|</mo><mi>ξ</mi><mo>|</mo></mrow></mrow><mrow><mi>q</mi></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>γ</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></math></span> for some exponents <span><math><mrow><mi>γ</mi><mo>≥</mo><mi>q</mi><mo>≥</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><mn>1</mn></mrow></math></span>, and non-negative functions <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>μ</mi></mrow></math></span> subject to suitable summability assumptions. We prove the local boundedness of scalar local quasi-minimizers of such integrals.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113915"},"PeriodicalIF":1.3,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144988180","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}
Pub Date : 2025-09-03DOI: 10.1016/j.na.2025.113937
Niklas Knobel
This article considers the ideal 2D magnetohydrodynamic equations in a infinite periodic channel close to a combination of an affine shear flow, called Couette flow, and a constant magnetic field. This incorporates important physical effects, including mixing and coupling of velocity and magnetic field. We establish the existence and stability of the velocity and magnetic field for Gevrey-class perturbations of size , valid up to times . Additionally, the vorticity and current grow as and there is no inviscid damping of the velocity and magnetic field. This is similar to the above threshold case for the Navier–Stokes (Jacob Bedrossian et al., 2022) where growth in ‘streaks’ leads to time scales of . In particular, for the ideal MHD equations, our article suggests that for a wide range of initial data, the scenario “induction by shear vorticity and current growth vorticity and current breakdown” leads to instability and possible turbulences.
本文考虑了无限周期通道中的理想二维磁流体动力学方程，该通道接近仿射剪切流（称为Couette流）和恒定磁场的组合。这包含了重要的物理效应，包括速度和磁场的混合和耦合。我们建立了大小为i的gevrey类扰动的速度和磁场的存在性和稳定性，有效到t ~ i−1次。此外，涡度和电流以O(t)增长，并且速度和磁场没有无粘阻尼。这类似于上述三维Navier-Stokes的阈值情况（Jacob Bedrossian et al., 2022），其中“条纹”的增长导致时间尺度为t ~ ε−1。特别是，对于理想的MHD方程，我们的文章表明，对于大范围的初始数据，“剪切诱导⇒涡度和电流增长⇒涡度和电流击穿”的情况会导致不稳定和可能的湍流。
{"title":"Ideal magnetohydrodynamics around couette flow: Long time stability and vorticity–current instability","authors":"Niklas Knobel","doi":"10.1016/j.na.2025.113937","DOIUrl":"10.1016/j.na.2025.113937","url":null,"abstract":"<div><div>This article considers the ideal 2D magnetohydrodynamic equations in a infinite periodic channel close to a combination of an affine shear flow, called Couette flow, and a constant magnetic field. This incorporates important physical effects, including mixing and coupling of velocity and magnetic field. We establish the existence and stability of the velocity and magnetic field for Gevrey-class perturbations of size <span><math><mi>ɛ</mi></math></span>, valid up to times <span><math><mrow><mi>t</mi><mo>∼</mo><msup><mrow><mi>ɛ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span>. Additionally, the vorticity and current grow as <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> and there is no inviscid damping of the velocity and magnetic field. This is similar to the above threshold case for the <span><math><mrow><mn>3</mn><mi>D</mi></mrow></math></span> Navier–Stokes (Jacob Bedrossian et al., 2022) where growth in ‘streaks’ leads to time scales of <span><math><mrow><mi>t</mi><mo>∼</mo><msup><mrow><mi>ɛ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span>. In particular, for the ideal MHD equations, our article suggests that for a wide range of initial data, the scenario “induction by shear <span><math><mo>⇒</mo></math></span> vorticity and current growth <span><math><mo>⇒</mo></math></span> vorticity and current breakdown” leads to instability and possible turbulences.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113937"},"PeriodicalIF":1.3,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144932827","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}
Pub Date : 2025-09-03DOI: 10.1016/j.na.2025.113939
Tongkeun Chang, Kyungkeun Kang
Local behavior near the boundary is analyzed for solutions of the Stokes and Navier–Stokes equations in the half space with localized non-smooth boundary data. We construct solutions to the Stokes equations whose velocity fields are unbounded near the boundary away from the support of boundary data, although the velocity and its gradient of solutions are locally square integrable. This is an improvement compared to known results in the sense that the velocity field itself is unbounded, since previously constructed solutions were bounded near the boundary, although their normal derivatives are singular. We also establish singular solutions and their derivatives that do not belong to near the boundary for . For such examples, the corresponding pressures turn out not to be locally integrable. A similar construction, via a perturbation argument, is available to the Navier–Stokes equations near the boundary as well.
{"title":"Singular velocity of the Stokes and Navier–Stokes equations near boundary in the half-space","authors":"Tongkeun Chang, Kyungkeun Kang","doi":"10.1016/j.na.2025.113939","DOIUrl":"10.1016/j.na.2025.113939","url":null,"abstract":"<div><div>Local behavior near the boundary is analyzed for solutions of the Stokes and Navier–Stokes equations in the half space with localized non-smooth boundary data. We construct solutions to the Stokes equations whose velocity fields are unbounded near the boundary away from the support of boundary data, although the velocity and its gradient of solutions are locally square integrable. This is an improvement compared to known results in the sense that the velocity field itself is unbounded, since previously constructed solutions were bounded near the boundary, although their normal derivatives are singular. We also establish singular solutions and their derivatives that do not belong to <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>loc</mi></mrow><mrow><mi>q</mi></mrow></msubsup></math></span> near the boundary for <span><math><mrow><mi>q</mi><mo>></mo><mn>1</mn></mrow></math></span>. For such examples, the corresponding pressures turn out not to be locally integrable. A similar construction, via a perturbation argument, is available to the Navier–Stokes equations near the boundary as well.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113939"},"PeriodicalIF":1.3,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144932826","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}
Pub Date : 2025-09-03DOI: 10.1016/j.na.2025.113936
Pêdra D.S. Andrade , Julio C. Correa
In this article, we study a class of fully nonlinear double-divergence systems with free boundaries associated with a minimization problem. The variational structure of the Hessian-dependent functional plays a fundamental role in proving the existence of the minimizers and then the existence of the solutions for the system. In addition, we establish improvements in integrability for the equation in the double-divergence form. Consequently, we improve the regularity for the fully nonlinear equation in Sobolev and Hölder spaces.
{"title":"Two-phase free boundary problems for a class of fully nonlinear double-divergence systems","authors":"Pêdra D.S. Andrade , Julio C. Correa","doi":"10.1016/j.na.2025.113936","DOIUrl":"10.1016/j.na.2025.113936","url":null,"abstract":"<div><div>In this article, we study a class of fully nonlinear double-divergence systems with free boundaries associated with a minimization problem. The variational structure of the Hessian-dependent functional plays a fundamental role in proving the existence of the minimizers and then the existence of the solutions for the system. In addition, we establish improvements in integrability for the equation in the double-divergence form. Consequently, we improve the regularity for the fully nonlinear equation in Sobolev and Hölder spaces.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113936"},"PeriodicalIF":1.3,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144988179","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}
-type foliations are studied in the framework of sub-Riemannian geometry with bracket generating distribution defined as the bundle transversal to the fibers. Equipping with the Bott connection we consider the scalar horizontal curvature as well as a new local invariant induced from the vertical distribution. We extend recent results on the small-time asymptotics of the sub-Riemannian heat kernel on quaternion-contact (qc-)manifolds due to A. Laaroussi and we express the second heat invariant in sub-Riemannian geometry as a linear combination of and . The use of an analog to normal coordinates in Riemannian geometry that are well-adapted to the geometric structure of -type foliations allows us to consider the pull-back of Korányi balls to . We explicitly obtain the first three terms in the asymptotic expansion of their Popp volume for small radii. Finally, we address the question of when is locally isometric as a sub-Riemannian manifold to its -type tangent group.
{"title":"Local invariants and geometry of the sub-Laplacian on H-type foliations","authors":"Wolfram Bauer , Irina Markina , Abdellah Laaroussi , Sylvie Vega-Molino","doi":"10.1016/j.na.2025.113934","DOIUrl":"10.1016/j.na.2025.113934","url":null,"abstract":"<div><div><span><math><mi>H</mi></math></span>-type foliations <span><math><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>H</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>)</mo></mrow></math></span> are studied in the framework of sub-Riemannian geometry with bracket generating distribution defined as the bundle transversal to the fibers. Equipping <span><math><mi>M</mi></math></span> with the Bott connection we consider the scalar horizontal curvature <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>H</mi></mrow></msub></math></span> as well as a new local invariant <span><math><msub><mrow><mi>τ</mi></mrow><mrow><mi>V</mi></mrow></msub></math></span> induced from the vertical distribution. We extend recent results on the small-time asymptotics of the sub-Riemannian heat kernel on quaternion-contact (qc-)manifolds due to A. Laaroussi and we express the second heat invariant in sub-Riemannian geometry as a linear combination of <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>H</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>τ</mi></mrow><mrow><mi>V</mi></mrow></msub></math></span>. The use of an analog to normal coordinates in Riemannian geometry that are well-adapted to the geometric structure of <span><math><mi>H</mi></math></span>-type foliations allows us to consider the pull-back of Korányi balls to <span><math><mi>M</mi></math></span>. We explicitly obtain the first three terms in the asymptotic expansion of their Popp volume for small radii. Finally, we address the question of when <span><math><mi>M</mi></math></span> is locally isometric as a sub-Riemannian manifold to its <span><math><mi>H</mi></math></span>-type tangent group.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113934"},"PeriodicalIF":1.3,"publicationDate":"2025-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144920377","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}
Pub Date : 2025-08-26DOI: 10.1016/j.na.2025.113930
Thomas Hamori , Changhui Tan
We present a new family of second-order traffic flow models, extending the Aw–Rascle–Zhang (ARZ) model to incorporate nonlocal interactions. Our model includes a specific nonlocal Arrhenius-type look-ahead slowdown factor. We establish both local and global well-posedness theories for these nonlocal ARZ models.
In contrast to the local ARZ model, where generic smooth initial data typically lead to finite-time shock formation, we show that our nonlocal ARZ model exhibits global regularity for a class of smooth subcritical initial data. Our result highlights the potential of nonlocal interactions to mitigate shock formations in second-order traffic flow models.
Our analytical approach relies on investigating phase plane dynamics. We introduce a novel comparison principle based on a mediant inequality to effectively handle the nonlocal information inherent in our model.
{"title":"On the Aw–Rascle–Zhang traffic models with nonlocal look-ahead interactions","authors":"Thomas Hamori , Changhui Tan","doi":"10.1016/j.na.2025.113930","DOIUrl":"10.1016/j.na.2025.113930","url":null,"abstract":"<div><div>We present a new family of second-order traffic flow models, extending the Aw–Rascle–Zhang (ARZ) model to incorporate nonlocal interactions. Our model includes a specific nonlocal Arrhenius-type look-ahead slowdown factor. We establish both local and global well-posedness theories for these nonlocal ARZ models.</div><div>In contrast to the local ARZ model, where generic smooth initial data typically lead to finite-time shock formation, we show that our nonlocal ARZ model exhibits global regularity for a class of smooth subcritical initial data. Our result highlights the potential of nonlocal interactions to mitigate shock formations in second-order traffic flow models.</div><div>Our analytical approach relies on investigating phase plane dynamics. We introduce a novel comparison principle based on a mediant inequality to effectively handle the nonlocal information inherent in our model.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113930"},"PeriodicalIF":1.3,"publicationDate":"2025-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144895313","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}
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