Pub Date : 2026-01-14DOI: 10.1007/s00010-026-01263-z
Péter Tóth
In this paper we investigate the functional equation
$$begin{aligned} varphi left( frac{x+y}{2} right) left( psi _1(x) - psi _2(y) right) = 0 quad left( hbox { for all } x in I_1 hbox { and } y in I_2 right) end{aligned}$$
where ( I_1 , I_2 ) are open intervals of ( mathbb {R}), ( J = frac{1}{2} left( I_1 + I_2 right) ) moreover ( psi _1: I_1 rightarrow mathbb {R}), ( psi _2: I_2 rightarrow mathbb {R}) and ( varphi : J rightarrow mathbb {R}) are unknown functions. We describe the structure of the possible solutions assuming that ( varphi ) is measurable. In the case when ( varphi ) is a derivative, we give a complete characterization of the solutions. Furthermore, we present an example of a solution consisting of irregular Darboux functions. This provides the answer to an open problem proposed during the 59th International Symposium on Functional Equations.
本文研究了泛函方程$$begin{aligned} varphi left( frac{x+y}{2} right) left( psi _1(x) - psi _2(y) right) = 0 quad left( hbox { for all } x in I_1 hbox { and } y in I_2 right) end{aligned}$$，其中( I_1 , I_2 )为( mathbb {R})、( J = frac{1}{2} left( I_1 + I_2 right) )的开区间，( psi _1: I_1 rightarrow mathbb {R})、( psi _2: I_2 rightarrow mathbb {R})、( varphi : J rightarrow mathbb {R})为未知函数。假设( varphi )是可测量的，我们描述可能解的结构。当( varphi )为导数时，给出了解的完整表征。此外，我们还给出了一个由不规则达布函数组成的解的例子。这为第59届泛函方程国际研讨会上提出的一个开放性问题提供了答案。
{"title":"Measurable solutions of an alternative functional equation","authors":"Péter Tóth","doi":"10.1007/s00010-026-01263-z","DOIUrl":"10.1007/s00010-026-01263-z","url":null,"abstract":"<div><p>In this paper we investigate the functional equation </p><div><div><span>$$begin{aligned} varphi left( frac{x+y}{2} right) left( psi _1(x) - psi _2(y) right) = 0 quad left( hbox { for all } x in I_1 hbox { and } y in I_2 right) end{aligned}$$</span></div></div><p>where <span>( I_1 , I_2 )</span> are open intervals of <span>( mathbb {R})</span>, <span>( J = frac{1}{2} left( I_1 + I_2 right) )</span> moreover <span>( psi _1: I_1 rightarrow mathbb {R})</span>, <span>( psi _2: I_2 rightarrow mathbb {R})</span> and <span>( varphi : J rightarrow mathbb {R})</span> are unknown functions. We describe the structure of the possible solutions assuming that <span>( varphi )</span> is measurable. In the case when <span>( varphi )</span> is a derivative, we give a complete characterization of the solutions. Furthermore, we present an example of a solution consisting of irregular Darboux functions. This provides the answer to an open problem proposed during the <i>59th International Symposium on Functional Equations</i>.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-026-01263-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145982868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-14DOI: 10.1016/j.chaos.2026.117893
Lifen Yan, Mingfeng Wang, Dong Zhang, Haiyong Zhu
We predict the existence of petal-like vortex solitons (PVSs) carrying fractional angular momentum in a defocusing cubic nonlinear medium modulated by the cylindrical Bessel optical lattice. Linear stability analysis and direct simulations show that PVSs remain stable when the propagation constant exceeds a critical value but revert to a ring vortex soliton at a second threshold. Fundamental fractional vortex soliton possesses an opening gap in the intensity ring, and its power and angular momentum per photon vary continuously with the propagating constant. Higher-order PVS exhibits an odd number of petals, which results from a complex phase distribution containing an equal number of phase dislocations. Specifically, the stability domain of the soliton shrinks as the number of petals increases.
{"title":"Petal-like vortex solitons with fractional angular momentum in Bessel optical lattices","authors":"Lifen Yan, Mingfeng Wang, Dong Zhang, Haiyong Zhu","doi":"10.1016/j.chaos.2026.117893","DOIUrl":"10.1016/j.chaos.2026.117893","url":null,"abstract":"<div><div>We predict the existence of petal-like vortex solitons (PVSs) carrying fractional angular momentum in a defocusing cubic nonlinear medium modulated by the cylindrical Bessel optical lattice. Linear stability analysis and direct simulations show that PVSs remain stable when the propagation constant exceeds a critical value but revert to a ring vortex soliton at a second threshold. Fundamental fractional vortex soliton possesses an opening gap in the intensity ring, and its power and angular momentum per photon vary continuously with the propagating constant. Higher-order PVS exhibits an odd number of petals, which results from a complex phase distribution containing an equal number of phase dislocations. Specifically, the stability domain of the soliton shrinks as the number of petals increases.</div></div>","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"206 ","pages":"Article 117893"},"PeriodicalIF":5.6,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145962059","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}
Pub Date : 2026-01-14DOI: 10.1016/j.chaos.2026.117877
Yifeng Zhang , Wei Zhang , Yufei Zhang
Low-altitude aircrafts are subject to complex aerodynamic and environmental disturbances, which place stringent demands on lightweight structures with reliable dynamic stability. Functionally graded graphene-reinforced composites (FG-GRCs), owing to their high specific strength and tunable properties, are promising candidates for load-bearing and vibration-sensitive components in such vehicles. This study investigates the complex nonlinear dynamics, particularly multi-pulse chaotic vibrations, of cantilever laminated curved plate subjected simultaneous transverse and in-plane parametric excitations. A nonlinear dynamic model incorporating geometric curvature and material gradation effects is established and validated by finite element analysis, revealing a critical 1:1 internal resonance between specific modes under particular geometric configurations. The extended Melnikov method is applied to theoretically predict the Shilnikov-type multi-pulse chaotic motions. Bi-parameter threshold surfaces derived from the Melnikov function are proposed and quantitatively validated against two-dimensional maximum Lyapunov exponent (MLE) diagrams, exhibiting good agreement in identifying chaotic regions. Extensive numerical simulations, including bifurcation diagrams, phase portraits, Poincaré maps, and time histories, confirm the presence of complex dynamics such as periodic motions, chaotic vibrations, and their transitions. The results provide crucial insights into the parameter domains that may trigger hazardous chaotic responses in FG-GRC curved plates, supporting the safe and reliable design of next-generation low-altitude aircraft structures such as wings, rotor arms, and fuselage skins.
{"title":"Nonlinear chaotic dynamics of functionally graded graphene composite curved plates for the wings of low-altitude aircrafts","authors":"Yifeng Zhang , Wei Zhang , Yufei Zhang","doi":"10.1016/j.chaos.2026.117877","DOIUrl":"10.1016/j.chaos.2026.117877","url":null,"abstract":"<div><div>Low-altitude aircrafts are subject to complex aerodynamic and environmental disturbances, which place stringent demands on lightweight structures with reliable dynamic stability. Functionally graded graphene-reinforced composites (FG-GRCs), owing to their high specific strength and tunable properties, are promising candidates for load-bearing and vibration-sensitive components in such vehicles. This study investigates the complex nonlinear dynamics, particularly multi-pulse chaotic vibrations, of cantilever laminated curved plate subjected simultaneous transverse and in-plane parametric excitations. A nonlinear dynamic model incorporating geometric curvature and material gradation effects is established and validated by finite element analysis, revealing a critical 1:1 internal resonance between specific modes under particular geometric configurations. The extended Melnikov method is applied to theoretically predict the Shilnikov-type multi-pulse chaotic motions. Bi-parameter threshold surfaces derived from the Melnikov function are proposed and quantitatively validated against two-dimensional maximum Lyapunov exponent (MLE) diagrams, exhibiting good agreement in identifying chaotic regions. Extensive numerical simulations, including bifurcation diagrams, phase portraits, Poincaré maps, and time histories, confirm the presence of complex dynamics such as periodic motions, chaotic vibrations, and their transitions. The results provide crucial insights into the parameter domains that may trigger hazardous chaotic responses in FG-GRC curved plates, supporting the safe and reliable design of next-generation low-altitude aircraft structures such as wings, rotor arms, and fuselage skins.</div></div>","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"206 ","pages":"Article 117877"},"PeriodicalIF":5.6,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976349","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}
Pub Date : 2026-01-14DOI: 10.1016/j.jmaa.2026.130417
Juan Pablo Alcon Apaza
<div><div>We establish local boundedness and local Hölder continuity of weak solutions to the following prototype problem:<span><span><span><math><mo>−</mo><mi>div</mi><mspace></mspace><mrow><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mn>2</mn><mi>β</mi></mrow></msup><mo>|</mo><mi>∇</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>∇</mi><mi>u</mi><mo>)</mo></mrow><mo>+</mo><msubsup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>Ω</mi><mo>,</mo><mi>p</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mo>⋅</mo><mo>)</mo><mo>,</mo><mi>β</mi></mrow><mrow><mi>s</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mo>⋅</mo><mo>)</mo></mrow></msubsup><mi>u</mi><mo>=</mo><mn>0</mn><mspace></mspace><mtext> in </mtext><mspace></mspace><mi>Ω</mi><mo>,</mo></math></span></span></span> where <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, is a bounded domain. The nonlocal operator is defined by<span><span><span><math><msubsup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>Ω</mi><mo>,</mo><mi>p</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mo>⋅</mo><mo>)</mo><mo>,</mo><mi>β</mi></mrow><mrow><mi>s</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mo>⋅</mo><mo>)</mo></mrow></msubsup><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mrow><mi>P</mi><mo>.</mo><mi>V</mi><mi>.</mi></mrow><mspace></mspace><munder><mo>∫</mo><mrow><mi>Ω</mi></mrow></munder><mfrac><mrow><mo>|</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>u</mi><mo>(</mo><mi>y</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>−</mo><mn>2</mn></mrow></msup><mo>(</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>u</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>)</mo></mrow><mrow><mo>|</mo><mi>x</mi><mo>−</mo><mi>y</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>n</mi><mo>+</mo><mi>s</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></msup></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>β</mi></mrow></msup><mo>|</mo><mi>y</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>β</mi></mrow></msup></mrow></mfrac><mspace></mspace><mi>d</mi><mi>y</mi><mo>.</mo></math></span></span></span> Here, <span><math><mi>p</mi><mo>:</mo><mi>Ω</mi><mo>×</mo><mi>Ω</mi><mo>→</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> and <span><math><mi>s</mi><mo>:</mo><mi>Ω</mi><mo>×</mo><mi>Ω</mi><mo>→</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> are measurable functions, <span><math><mi>q</mi><mo>:</mo><mo>=</mo><mi>ess</mi><mspace></mspace><msub><mrow><mi>sup</mi></mrow><mrow><mi>Ω</mi><mo>×</mo><mi>Ω</mi></mrow></msub><mo></mo><mspace></mspace><mi>p</mi></math></span>, and <span><math><mn>0</mn><mo>≤</mo><mn>2</mn>
我们建立了以下原型问题弱解的局部有界性和局部Hölder连续性：−div(|x|−2β|∇u|q−2∇u)+（−Δ）Ω，p(⋅,⋅),βs(⋅,⋅)u=0在Ω中，其中Ω∧Rn， n≥2是一个有界域。外地操作符被定义为(−Δ)Ω,p(⋅⋅),β年代(⋅⋅)u (x): = pv∫Ω| u (x)−u (y) | p (x, y)−2 (u (x)−u (y)) x−y | | n + s (x, y) p (x, y)⋅1 | x |βy | |βdy。在这里,p:Ω×Ω→(∞)和年代:Ω×Ω→(0,1)是可测函数,问:= esssupΩ×Ωp,和0≤2β& lt; n。我们的方法是解析的，依赖于De Giorgi-Nash-Moser理论对具有可变指数和权重的混合局部-非局部框架的适应。
{"title":"Local Hölder regularity for quasilinear elliptic equations with mixed local-nonlocal operators, variable exponents, and weights","authors":"Juan Pablo Alcon Apaza","doi":"10.1016/j.jmaa.2026.130417","DOIUrl":"10.1016/j.jmaa.2026.130417","url":null,"abstract":"<div><div>We establish local boundedness and local Hölder continuity of weak solutions to the following prototype problem:<span><span><span><math><mo>−</mo><mi>div</mi><mspace></mspace><mrow><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mn>2</mn><mi>β</mi></mrow></msup><mo>|</mo><mi>∇</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>∇</mi><mi>u</mi><mo>)</mo></mrow><mo>+</mo><msubsup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>Ω</mi><mo>,</mo><mi>p</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mo>⋅</mo><mo>)</mo><mo>,</mo><mi>β</mi></mrow><mrow><mi>s</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mo>⋅</mo><mo>)</mo></mrow></msubsup><mi>u</mi><mo>=</mo><mn>0</mn><mspace></mspace><mtext> in </mtext><mspace></mspace><mi>Ω</mi><mo>,</mo></math></span></span></span> where <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, is a bounded domain. The nonlocal operator is defined by<span><span><span><math><msubsup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>Ω</mi><mo>,</mo><mi>p</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mo>⋅</mo><mo>)</mo><mo>,</mo><mi>β</mi></mrow><mrow><mi>s</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mo>⋅</mo><mo>)</mo></mrow></msubsup><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mrow><mi>P</mi><mo>.</mo><mi>V</mi><mi>.</mi></mrow><mspace></mspace><munder><mo>∫</mo><mrow><mi>Ω</mi></mrow></munder><mfrac><mrow><mo>|</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>u</mi><mo>(</mo><mi>y</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>−</mo><mn>2</mn></mrow></msup><mo>(</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>u</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>)</mo></mrow><mrow><mo>|</mo><mi>x</mi><mo>−</mo><mi>y</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>n</mi><mo>+</mo><mi>s</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></msup></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>β</mi></mrow></msup><mo>|</mo><mi>y</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>β</mi></mrow></msup></mrow></mfrac><mspace></mspace><mi>d</mi><mi>y</mi><mo>.</mo></math></span></span></span> Here, <span><math><mi>p</mi><mo>:</mo><mi>Ω</mi><mo>×</mo><mi>Ω</mi><mo>→</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> and <span><math><mi>s</mi><mo>:</mo><mi>Ω</mi><mo>×</mo><mi>Ω</mi><mo>→</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> are measurable functions, <span><math><mi>q</mi><mo>:</mo><mo>=</mo><mi>ess</mi><mspace></mspace><msub><mrow><mi>sup</mi></mrow><mrow><mi>Ω</mi><mo>×</mo><mi>Ω</mi></mrow></msub><mo></mo><mspace></mspace><mi>p</mi></math></span>, and <span><math><mn>0</mn><mo>≤</mo><mn>2</mn>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130417"},"PeriodicalIF":1.2,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-14DOI: 10.1016/j.na.2026.114057
Mostafa Meliani
We study the local existence of solutions to the Navier–Stokes–Fourier-magnetohydrodynamics (NSF-MHD) system describing the motion of a compressible, viscous, electrically and heat conducting fluid in the Lp–Lq class with inhomogeneous boundary conditions. The open system is allowed to receive incoming matter from the outside through (part of) the boundary which we refer to as an inflow boundary. This setup brings about a difficulty in estimating the regularity of the density ϱ which we remedy by assuming appropriate hypotheses on the velocity field, domain boundary and on the boundary and initial data of ϱ. The main result ensures the local well-posedness of the full NSF-MHD system which is shown through a linearization combined with a Banach fixed-point theorem.
{"title":"Lp–Lq existence for the open compressible MHD system","authors":"Mostafa Meliani","doi":"10.1016/j.na.2026.114057","DOIUrl":"10.1016/j.na.2026.114057","url":null,"abstract":"<div><div>We study the local existence of solutions to the Navier–Stokes–Fourier-magnetohydrodynamics (NSF-MHD) system describing the motion of a compressible, viscous, electrically and heat conducting fluid in the <em>L<sup>p</sup></em>–<em>L<sup>q</sup></em> class with inhomogeneous boundary conditions. The open system is allowed to receive incoming matter from the outside through (part of) the boundary which we refer to as an inflow boundary. This setup brings about a difficulty in estimating the regularity of the density ϱ which we remedy by assuming appropriate hypotheses on the velocity field, domain boundary and on the boundary and initial data of ϱ. The main result ensures the local well-posedness of the full NSF-MHD system which is shown through a linearization combined with a Banach fixed-point theorem.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114057"},"PeriodicalIF":1.3,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979449","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 : 2026-01-14DOI: 10.1016/j.jmaa.2026.130421
K. Benmoussa , J. Deteix , D. Yakoubi
In this paper, we study the three-dimensional velocity–vorticity–helicity (VVH) formulation modeling the flow of incompressible Newtonian fluids. We present an analysis of the formulation that encompasses the existence and regularity of solutions, providing a rigorous functional framework in which the VVH formulation is shown to be equivalent to the more traditional velocity–pressure formulation.
{"title":"On the well-posedness of the unsteady velocity-vorticity-helicity formulation of the Navier–Stokes equations","authors":"K. Benmoussa , J. Deteix , D. Yakoubi","doi":"10.1016/j.jmaa.2026.130421","DOIUrl":"10.1016/j.jmaa.2026.130421","url":null,"abstract":"<div><div>In this paper, we study the three-dimensional velocity–vorticity–helicity (VVH) formulation modeling the flow of incompressible Newtonian fluids. We present an analysis of the formulation that encompasses the existence and regularity of solutions, providing a rigorous functional framework in which the VVH formulation is shown to be equivalent to the more traditional velocity–pressure formulation.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"559 1","pages":"Article 130421"},"PeriodicalIF":1.2,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-14DOI: 10.1007/s11538-025-01576-1
Lukas Eigentler, Mattia Sensi
Periodic travelling waves (PTWs) are a common solution type of models describing spatio-temporal patterns in biology and ecology. Particularly in ecology, pattern formation is regarded as a resilience mechanism and an ecosystem's ability to change its pattern wavelength is seen as a tool to adapt to environmental change. PTW solutions of corresponding mathematical models also possess this ability and typically undergo a cascade of wavelength changes in response to a gradual change in a bifurcation parameter. Extensive analysis has been conducted to develop a predictive understanding of parameter thresholds leading to wavelength changes. By contrast, theory on what determines PTW wavelength selection during a wavelength change is currently lacking and most conjectures stem from limited observations of specific simulations, or apply to special cases only. In this unsolved problems article, we first provide a review of how linear stability analysis and Busse balloon theory are used to predict parameter values at which PTW wavelength changes occur. On the topic of wavelength selection, we review the special case of PTWs in - systems, often used to predict wavelengths of predator-prey dynamics in the wake of an invasion front. For more general systems, we highlight that the Busse balloon theory that is so successful in determining parameter values of wavelength changes is unlikely able to provide information on PTW wavelength selection. Finally, we present new numerical trends of PTW wavelength selection during PTW-to-PTW transitions that highlight that some stable wavelengths are more frequently selected than others, and that cascades of wavelength changes can also result in extinction events despite bistability of the extinction state with PTWs. We conclude with a tentative list of potential approaches to unravel a deeper understanding of this topic. Combined, we aim to stimulate new approaches to gain more insights into the unsolved problem of PTW wavelength selection during PTW-to-PTW transitions.
{"title":"Wavelength Selection for Periodic Travelling Waves: An Unsolved Problem.","authors":"Lukas Eigentler, Mattia Sensi","doi":"10.1007/s11538-025-01576-1","DOIUrl":"10.1007/s11538-025-01576-1","url":null,"abstract":"<p><p>Periodic travelling waves (PTWs) are a common solution type of models describing spatio-temporal patterns in biology and ecology. Particularly in ecology, pattern formation is regarded as a resilience mechanism and an ecosystem's ability to change its pattern wavelength is seen as a tool to adapt to environmental change. PTW solutions of corresponding mathematical models also possess this ability and typically undergo a cascade of wavelength changes in response to a gradual change in a bifurcation parameter. Extensive analysis has been conducted to develop a predictive understanding of parameter thresholds leading to wavelength changes. By contrast, theory on what determines PTW wavelength selection during a wavelength change is currently lacking and most conjectures stem from limited observations of specific simulations, or apply to special cases only. In this unsolved problems article, we first provide a review of how linear stability analysis and Busse balloon theory are used to predict parameter values at which PTW wavelength changes occur. On the topic of wavelength selection, we review the special case of PTWs in <math><mi>λ</mi></math> - <math><mi>ω</mi></math> systems, often used to predict wavelengths of predator-prey dynamics in the wake of an invasion front. For more general systems, we highlight that the Busse balloon theory that is so successful in determining parameter values of wavelength changes is unlikely able to provide information on PTW wavelength selection. Finally, we present new numerical trends of PTW wavelength selection during PTW-to-PTW transitions that highlight that some stable wavelengths are more frequently selected than others, and that cascades of wavelength changes can also result in extinction events despite bistability of the extinction state with PTWs. We conclude with a tentative list of potential approaches to unravel a deeper understanding of this topic. Combined, we aim to stimulate new approaches to gain more insights into the unsolved problem of PTW wavelength selection during PTW-to-PTW transitions.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 2","pages":"22"},"PeriodicalIF":2.2,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12804330/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145965226","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-14DOI: 10.1016/j.cnsns.2026.109664
Zenghui Hu , Jiamin Liu , Yiming Wang , Lijie You
The almost surely exponential stability (ES a.s.) is investigated for nonlinear systems subject to the effects of stochastic noises and random impulses. The randomness of impulsive effects is considered in two aspects: impulsive intensity and density. In detail, the impulsive instants and jumps are respectively impelled by a renewal process and a Markov chain. By analyzing the coupling effect among stochastic noise, randomly impulsive intensity and density, novel criteria of ES a.s. are obtained by employing Lyapunov-based approach. The proposed results not only capture the positive effect of stochastic noise, but also remove the non-zero property of solution required in existing works. As an application, the synchronization problem of master-slave stochastic chaotic systems is solved by applying the randomly impulsive control method. Two examples are provided to illustrate the effectiveness and application of the proposed results.
{"title":"Almost surely exponential stability of nonlinear stochastic systems with random impulses and its application in chaos synchronization","authors":"Zenghui Hu , Jiamin Liu , Yiming Wang , Lijie You","doi":"10.1016/j.cnsns.2026.109664","DOIUrl":"10.1016/j.cnsns.2026.109664","url":null,"abstract":"<div><div>The almost surely exponential stability (ES a.s.) is investigated for nonlinear systems subject to the effects of stochastic noises and random impulses. The randomness of impulsive effects is considered in two aspects: impulsive intensity and density. In detail, the impulsive instants and jumps are respectively impelled by a renewal process and a Markov chain. By analyzing the coupling effect among stochastic noise, randomly impulsive intensity and density, novel criteria of ES a.s. are obtained by employing Lyapunov-based approach. The proposed results not only capture the positive effect of stochastic noise, but also remove the non-zero property of solution required in existing works. As an application, the synchronization problem of master-slave stochastic chaotic systems is solved by applying the randomly impulsive control method. Two examples are provided to illustrate the effectiveness and application of the proposed results.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"157 ","pages":"Article 109664"},"PeriodicalIF":3.8,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145962060","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 : 2026-01-14DOI: 10.1016/j.camwa.2025.12.031
Haihong Zhou , Huan Zhang , Xiaomin Pan
In this paper, we extend the selection of auxiliary variables by proposing a hyperbolic tangent scalar auxiliary variable (tanh-SAV) approach for solving gradient flows. The proposed tanh-SAV schemes introduce an auxiliary variable based on the hyperbolic tangent function, providing a well-defined formulation that enables the construction of decoupled, linear, and efficient numerical schemes. We demonstrate the construction of first-order, second-order, and higher-order unconditionally energy-stable schemes, utilizing either the Crank–Nicolson method or a k-step backward differentiation formula for time discretization. Only one constant-coefficient equation needs to be solved per time step, and we prove that all resulting tanh-SAV schemes are uniquely solvable at each time level. Furthermore, the theoretical analysis demonstrates the discrete energy stability of the proposed numerical schemes and proves the positivity property of the auxiliary variable. For the tanh-SAV/BDFk schemes () combined with a Fourier pseudo-spectral spatial discretization, we further establish fully discrete optimal-order error estimates. In addition, we provide numerical simulations of one- and two-dimensional Cahn–Hilliard, Allen–Cahn, and phase-field crystal models. The results demonstrate that, consistent with the theoretical analysis, the proposed schemes preserve the positivity of the auxiliary variable, maintain excellent stability, and achieve the desired temporal accuracy.
{"title":"Unconditionally energy-stable and accurate schemes based on hyperbolic tangent scalar auxiliary variable approach for gradient flows","authors":"Haihong Zhou , Huan Zhang , Xiaomin Pan","doi":"10.1016/j.camwa.2025.12.031","DOIUrl":"10.1016/j.camwa.2025.12.031","url":null,"abstract":"<div><div>In this paper, we extend the selection of auxiliary variables by proposing a hyperbolic tangent scalar auxiliary variable (tanh-SAV) approach for solving gradient flows. The proposed tanh-SAV schemes introduce an auxiliary variable based on the hyperbolic tangent function, providing a well-defined formulation that enables the construction of decoupled, linear, and efficient numerical schemes. We demonstrate the construction of first-order, second-order, and higher-order unconditionally energy-stable schemes, utilizing either the Crank–Nicolson method or a <em>k</em>-step backward differentiation formula for time discretization. Only one constant-coefficient equation needs to be solved per time step, and we prove that all resulting tanh-SAV schemes are uniquely solvable at each time level. Furthermore, the theoretical analysis demonstrates the discrete energy stability of the proposed numerical schemes and proves the positivity property of the auxiliary variable. For the tanh-SAV/BDF<em>k</em> schemes (<span><math><mrow><mi>k</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn></mrow></math></span>) combined with a Fourier pseudo-spectral spatial discretization, we further establish fully discrete optimal-order error estimates. In addition, we provide numerical simulations of one- and two-dimensional Cahn–Hilliard, Allen–Cahn, and phase-field crystal models. The results demonstrate that, consistent with the theoretical analysis, the proposed schemes preserve the positivity of the auxiliary variable, maintain excellent stability, and achieve the desired temporal accuracy.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"204 ","pages":"Pages 263-282"},"PeriodicalIF":2.5,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977942","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 : 2026-01-14DOI: 10.1016/j.amc.2026.129950
Liangcai Mei , Yingchao Zhang , Boying Wu , Yingzhen Lin
In this article, a spatial concept of direct sum space based on the discontinuous reproducing kernel method (RKM for short) is proposed for impulsive differential equations, and a reproducing kernel numerical solution method is constructed. Based on the piecewise smoothness of the solution, a discontinuous reproducing kernel is constructed, and a direct sum space is constructed in vector form to represent the structure of the equation system. Furthermore, the simplified RKM is used to solve the operator equation, and an approximate solution in series form is obtained. Finally, the regularity analysis and uniform convergence analysis are carried out, and numerical experiments verify the second-order convergence and stability of the algorithm.
{"title":"A simplified discontinuous reproducing kernel method for impulsive differential equations","authors":"Liangcai Mei , Yingchao Zhang , Boying Wu , Yingzhen Lin","doi":"10.1016/j.amc.2026.129950","DOIUrl":"10.1016/j.amc.2026.129950","url":null,"abstract":"<div><div>In this article, a spatial concept of direct sum space based on the discontinuous reproducing kernel method (RKM for short) is proposed for impulsive differential equations, and a reproducing kernel numerical solution method is constructed. Based on the piecewise smoothness of the solution, a discontinuous reproducing kernel is constructed, and a direct sum space is constructed in vector form to represent the structure of the equation system. Furthermore, the simplified RKM is used to solve the operator equation, and an approximate solution in series form is obtained. Finally, the regularity analysis and uniform convergence analysis are carried out, and numerical experiments verify the second-order convergence and stability of the algorithm.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"520 ","pages":"Article 129950"},"PeriodicalIF":3.4,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980862","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号