Pub Date : 2025-12-16DOI: 10.1016/j.na.2025.114038
Vitor Gusson , Claudio Pessoa , Lucas Queiroz
In this work, we study isochronous centers on center manifolds of three-dimensional systems of differential equations. We describe in detail an algorithm that provides the necessary conditions for a Hopf point, when it is a center, to be isochronous, without initially requiring the system to be restricted to an explicit expression of the center manifold or a finite power series expansion of this expression. In addition, we determine necessary and sufficient conditions for a center on the center manifold of certain classes of quadratic three-dimensional systems of differential equations to be isochronous. The algorithm has an algebraic nature, inspired by a counterpart developed for the two-dimensional case, and offers lower computational cost in obtaining the isochronicity conditions.
{"title":"Isochronous centers on center manifolds in R3","authors":"Vitor Gusson , Claudio Pessoa , Lucas Queiroz","doi":"10.1016/j.na.2025.114038","DOIUrl":"10.1016/j.na.2025.114038","url":null,"abstract":"<div><div>In this work, we study isochronous centers on center manifolds of three-dimensional systems of differential equations. We describe in detail an algorithm that provides the necessary conditions for a Hopf point, when it is a center, to be isochronous, without initially requiring the system to be restricted to an explicit expression of the center manifold or a finite power series expansion of this expression. In addition, we determine necessary and sufficient conditions for a center on the center manifold of certain classes of quadratic three-dimensional systems of differential equations to be isochronous. The algorithm has an algebraic nature, inspired by a counterpart developed for the two-dimensional case, and offers lower computational cost in obtaining the isochronicity conditions.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"266 ","pages":"Article 114038"},"PeriodicalIF":1.3,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145797905","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 : 2025-12-16DOI: 10.1016/j.na.2025.114036
Sitao Zhang
Based on the concept of mixed LYZ ellipsoid and the technique of isotropic embedding, a sharp reverse affine isoperimetric inequality is established in this paper. This inequality is a generalization of a weak version of the Mahler conjecture obtained by Lutwak, Yang, and Zhang [31].
{"title":"A sharp volume inequality for mixed LYZ ellipsoids","authors":"Sitao Zhang","doi":"10.1016/j.na.2025.114036","DOIUrl":"10.1016/j.na.2025.114036","url":null,"abstract":"<div><div>Based on the concept of mixed LYZ ellipsoid and the technique of isotropic embedding, a sharp reverse affine isoperimetric inequality is established in this paper. This inequality is a generalization of a weak version of the Mahler conjecture obtained by Lutwak, Yang, and Zhang [31].</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"266 ","pages":"Article 114036"},"PeriodicalIF":1.3,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798491","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 : 2025-12-09DOI: 10.1016/j.na.2025.114028
Animesh Biswas , Mikil D. Foss , Petronela Radu
In this paper we introduce and study the concept of nonlocal ordered curvature. In the classical (differential) setting, the problem was introduced by Li and Nirenberg in [1, 2] where they conjectured (and proved in some cases) that if a bounded smooth surface has its mean curvature ordered in a particular direction, then the surface must be symmetric with respect to some hyperplane orthogonal to that direction. The conjecture was finally settled by Li et al in 2021 [3]. Here we study the counterpart problem in the nonlocal setting, where the nonlocal mean curvature of a set Ω, at any point x on its boundary, is defined as and the kernel function J is radially symmetric, non-increasing, integrable and compactly supported. Using a generalization of Alexandrov’s moving plane method, we prove a similar result in the nonlocal setting.
{"title":"Nonlocal ordered mean curvature with integrable kernels","authors":"Animesh Biswas , Mikil D. Foss , Petronela Radu","doi":"10.1016/j.na.2025.114028","DOIUrl":"10.1016/j.na.2025.114028","url":null,"abstract":"<div><div>In this paper we introduce and study the concept of nonlocal ordered curvature. In the classical (differential) setting, the problem was introduced by Li and Nirenberg in [1, 2] where they conjectured (and proved in some cases) that if a bounded smooth surface has its mean curvature ordered in a particular direction, then the surface must be symmetric with respect to some hyperplane orthogonal to that direction. The conjecture was finally settled by Li et al in 2021 [3]. Here we study the counterpart problem in the nonlocal setting, where the nonlocal mean curvature of a set Ω, at any point <em>x</em> on its boundary, is defined as <span><math><mrow><msubsup><mi>H</mi><mstyle><mi>Ω</mi></mstyle><mi>J</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∫</mo><msup><mstyle><mi>Ω</mi></mstyle><mi>c</mi></msup></msub><mi>J</mi><mrow><mo>(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo>)</mo></mrow><mi>d</mi><mi>y</mi><mo>−</mo><msub><mo>∫</mo><mstyle><mi>Ω</mi></mstyle></msub><mi>J</mi><mrow><mo>(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo>)</mo></mrow><mi>d</mi><mi>y</mi></mrow></math></span> and the kernel function <em>J</em> is radially symmetric, non-increasing, integrable and compactly supported. Using a generalization of Alexandrov’s moving plane method, we prove a similar result in the nonlocal setting.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"266 ","pages":"Article 114028"},"PeriodicalIF":1.3,"publicationDate":"2025-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145749108","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 : 2025-12-05DOI: 10.1016/j.na.2025.114031
Hao Chen , Yongkai Liao , Ling Wan
We study global-in-time spherically symmetric solutions for a viscous, compressible, heat-conducting ionized gas in a n-dimensional unbounded exterior domain with large initial data, where n ≥ 2 is the space dimension. The properties of ionized gases, combined with the unboundedness of the exterior domain, make it challenging to estimate the first-order spatial derivatives of the bulk velocity and the absolute temperature. For a class of constant non-vacuum equilibrium states, we obtain the uniform-in-time bounds on the dissipative estimates for both the bulk velocity and the absolute temperature. Based on such estimates, we establish the global existence and asymptotic behavior of spherically symmetric solutions to the viscous and heat-conducting ionized gas in unbounded exterior domains with large initial data. The key point lies in deducing the lower and upper bounds on the specific volume and the temperature.
{"title":"Global existence and large-time behavior of spherically symmetric solutions for a viscous heat-conducting ionized gas in exterior domains","authors":"Hao Chen , Yongkai Liao , Ling Wan","doi":"10.1016/j.na.2025.114031","DOIUrl":"10.1016/j.na.2025.114031","url":null,"abstract":"<div><div>We study global-in-time spherically symmetric solutions for a viscous, compressible, heat-conducting ionized gas in a <em>n</em>-dimensional unbounded exterior domain with large initial data, where <em>n</em> ≥ 2 is the space dimension. The properties of ionized gases, combined with the unboundedness of the exterior domain, make it challenging to estimate the first-order spatial derivatives of the bulk velocity and the absolute temperature. For a class of constant non-vacuum equilibrium states, we obtain the uniform-in-time bounds on the dissipative estimates for both the bulk velocity and the absolute temperature. Based on such estimates, we establish the global existence and asymptotic behavior of spherically symmetric solutions to the viscous and heat-conducting ionized gas in unbounded exterior domains with large initial data. The key point lies in deducing the lower and upper bounds on the specific volume and the temperature.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"266 ","pages":"Article 114031"},"PeriodicalIF":1.3,"publicationDate":"2025-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145694789","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 : 2025-12-04DOI: 10.1016/j.na.2025.114029
Francesco Nobili, Ivan Yuri Violo
We study the generalized existence of extremizers for the sharp p-Sobolev inequality on noncompact Riemannian manifolds in connection with nonnegative curvature and Euclidean volume growth assumptions. Assuming a nonnegative Ricci curvature lower bound, we show that almost extremal functions are close in gradient norm to radial Euclidean bubbles. In the case of nonnegative sectional curvature lower bounds, we additionally deduce that vanishing is the only possible behavior, in the sense that almost extremal functions are almost zero globally. Our arguments rely on nonsmooth concentration compactness methods and Mosco-convergence results for the Cheeger energy on noncompact varying spaces, generalized to every exponent p ∈ (1, ∞).
{"title":"Generalized existence of extremizers for the sharp p-Sobolev inequality on Riemannian manifolds with nonnegative curvature","authors":"Francesco Nobili, Ivan Yuri Violo","doi":"10.1016/j.na.2025.114029","DOIUrl":"10.1016/j.na.2025.114029","url":null,"abstract":"<div><div>We study the generalized existence of extremizers for the sharp <em>p</em>-Sobolev inequality on noncompact Riemannian manifolds in connection with nonnegative curvature and Euclidean volume growth assumptions. Assuming a nonnegative Ricci curvature lower bound, we show that almost extremal functions are close in gradient norm to radial Euclidean bubbles. In the case of nonnegative sectional curvature lower bounds, we additionally deduce that vanishing is the only possible behavior, in the sense that almost extremal functions are almost zero globally. Our arguments rely on nonsmooth concentration compactness methods and Mosco-convergence results for the Cheeger energy on noncompact varying spaces, generalized to every exponent <em>p</em> ∈ (1, ∞).</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"266 ","pages":"Article 114029"},"PeriodicalIF":1.3,"publicationDate":"2025-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145694793","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 : 2025-12-03DOI: 10.1016/j.na.2025.114019
Ilia Naumkin
We investigate a stochastic nonlinear Schrödinger equation (NSE) posed on a multidimensional hyperoctant, where randomness enters both the domain interior and the boundary. The model incorporates additive interior noise and time-dependent stochastic Dirichlet boundary conditions, making it a prototypical system for analyzing the interplay between bulk stochasticity and boundary-driven randomness. We establish well-posedness by first developing a linear deterministic framework using Laplace and Riemann-Hilbert transform techniques, adapted to nonhomogeneous boundary data. The stochastic structure is then rigorously handled via fixed-point arguments in suitable function spaces, accounting for both cylindrical Wiener processes and stochastic convolutions. The solution is represented explicitly in terms of deterministic and stochastic Green operators, with an additional boundary evolution term that captures the diffusion of noise from the boundary into the domain. Our results provide conditions for existence, uniqueness, and regularity of mild solutions, and highlight how boundary noise can influence solution behavior through resonance, instability, or enhanced dispersion effects. This work contributes to the mathematical understanding of boundary-sensitive stochastic dispersive systems and lays a foundation for future analysis of noise-induced phenomena in high-dimensional domains.
{"title":"Stochastic forcing in nonlinear dispersive systems: Interior–boundary noise interactions on hyperoctants","authors":"Ilia Naumkin","doi":"10.1016/j.na.2025.114019","DOIUrl":"10.1016/j.na.2025.114019","url":null,"abstract":"<div><div>We investigate a stochastic nonlinear Schrödinger equation (NSE) posed on a multidimensional hyperoctant, where randomness enters both the domain interior and the boundary. The model incorporates additive interior noise and time-dependent stochastic Dirichlet boundary conditions, making it a prototypical system for analyzing the interplay between bulk stochasticity and boundary-driven randomness. We establish well-posedness by first developing a linear deterministic framework using Laplace and Riemann-Hilbert transform techniques, adapted to nonhomogeneous boundary data. The stochastic structure is then rigorously handled via fixed-point arguments in suitable function spaces, accounting for both cylindrical Wiener processes and stochastic convolutions. The solution is represented explicitly in terms of deterministic and stochastic Green operators, with an additional boundary evolution term that captures the diffusion of noise from the boundary into the domain. Our results provide conditions for existence, uniqueness, and regularity of mild solutions, and highlight how boundary noise can influence solution behavior through resonance, instability, or enhanced dispersion effects. This work contributes to the mathematical understanding of boundary-sensitive stochastic dispersive systems and lays a foundation for future analysis of noise-induced phenomena in high-dimensional domains.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"266 ","pages":"Article 114019"},"PeriodicalIF":1.3,"publicationDate":"2025-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145659107","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 : 2025-11-30DOI: 10.1016/j.na.2025.114027
André de Laire, Erwan Le Quiniou
We study a defocusing quasilinear Schrödinger equation with nonzero conditions at infinity in dimension one. This quasilinear model corresponds to a weakly nonlocal approximation of the nonlocal Gross–Pitaevskii equation, and can also be derived by considering the effects of surface tension in superfluids. When the quasilinear term is neglected, the resulting equation is the classical Gross–Pitaevskii equation, which possesses a well-known stable branch of subsonic traveling waves solution, given by dark solitons.
Our goal is to investigate how the quasilinear term and the intensity-dependent dispersion affect the traveling-wave solutions. We provide a complete classification of finite energy traveling waves of the equation, in terms of two parameters: the speed and the strength of the quasilinear term. This classification leads to the existence of dark and antidark solitons, as well as more exotic localized solutions like dark cuspons, compactons, and composite waves, even for supersonic speeds. Depending on the parameters, these types of solutions can coexist, showing that finite energy solutions are not unique. Furthermore, we prove that some of these dark solitons can be obtained as minimizers of the energy, at fixed momentum, and that they are orbitally stable.
{"title":"Exotic traveling waves for a quasilinear Schrödinger equation with nonzero background","authors":"André de Laire, Erwan Le Quiniou","doi":"10.1016/j.na.2025.114027","DOIUrl":"10.1016/j.na.2025.114027","url":null,"abstract":"<div><div>We study a defocusing quasilinear Schrödinger equation with nonzero conditions at infinity in dimension one. This quasilinear model corresponds to a weakly nonlocal approximation of the nonlocal Gross–Pitaevskii equation, and can also be derived by considering the effects of surface tension in superfluids. When the quasilinear term is neglected, the resulting equation is the classical Gross–Pitaevskii equation, which possesses a well-known stable branch of subsonic traveling waves solution, given by dark solitons.</div><div>Our goal is to investigate how the quasilinear term and the intensity-dependent dispersion affect the traveling-wave solutions. We provide a complete classification of finite energy traveling waves of the equation, in terms of two parameters: the speed and the strength of the quasilinear term. This classification leads to the existence of dark and antidark solitons, as well as more exotic localized solutions like dark cuspons, compactons, and composite waves, even for supersonic speeds. Depending on the parameters, these types of solutions can coexist, showing that finite energy solutions are not unique. Furthermore, we prove that some of these dark solitons can be obtained as minimizers of the energy, at fixed momentum, and that they are orbitally stable.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"265 ","pages":"Article 114027"},"PeriodicalIF":1.3,"publicationDate":"2025-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685496","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 : 2025-11-24DOI: 10.1016/j.na.2025.114015
Bowen Zheng , Tohru Ozawa
<div><div>This paper is dedicated to the blow-up solution for the divergence Schrödinger equations with inhomogeneous nonlinearity (dINLS for short) <span><span><span><math><mrow><mi>i</mi><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>+</mo><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mi>b</mi></mrow></msup><mo>∇</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mi>c</mi></mrow></msup><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi></mrow></msup><mi>u</mi><mo>,</mo><mspace></mspace><mspace></mspace><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mn>2</mn><mo>−</mo><mi>n</mi><mo><</mo><mi>b</mi><mo><</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><mi>c</mi><mo>></mo><mi>b</mi><mo>−</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mi>p</mi><mo>−</mo><mn>2</mn><mi>c</mi><mo><</mo><mrow><mo>(</mo><mn>2</mn><mo>−</mo><mi>b</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>p</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>. First, for radial blow-up solutions in <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>b</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup></math></span>, we prove an upper bound on the blow-up rate for the intercritical dNLS. Moreover, an <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm concentration in the mass-critical case is also obtained by giving a compact lemma. Next, we turn to the non-radial case. By establishing two types of Gagliardo–Nirenberg inequalities, we show the existence of finite time blow-up solutions in <span><math><mrow><msup><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>̇</mo></mrow></mover></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></msup><mo>∩</mo><msubsup><mrow><mover><mrow><mi>W</mi></mrow><mrow><mo>̇</mo></mrow></mover></mrow><mrow><mi>b</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup></mrow></math></span>, where <span><math><mrow><msup><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>̇</mo></mrow></mover></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mfrac><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span> and <span><math><mrow><msubsup><mrow><mover><mrow><mi>W</mi></mrow><mrow><mo>̇</mo></mrow></mover></mrow><mrow><mi>b</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup
{"title":"The blow-up dynamics for the divergence Schrödinger equations with inhomogeneous nonlinearity","authors":"Bowen Zheng , Tohru Ozawa","doi":"10.1016/j.na.2025.114015","DOIUrl":"10.1016/j.na.2025.114015","url":null,"abstract":"<div><div>This paper is dedicated to the blow-up solution for the divergence Schrödinger equations with inhomogeneous nonlinearity (dINLS for short) <span><span><span><math><mrow><mi>i</mi><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>+</mo><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mi>b</mi></mrow></msup><mo>∇</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mi>c</mi></mrow></msup><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi></mrow></msup><mi>u</mi><mo>,</mo><mspace></mspace><mspace></mspace><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mn>2</mn><mo>−</mo><mi>n</mi><mo><</mo><mi>b</mi><mo><</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><mi>c</mi><mo>></mo><mi>b</mi><mo>−</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mi>p</mi><mo>−</mo><mn>2</mn><mi>c</mi><mo><</mo><mrow><mo>(</mo><mn>2</mn><mo>−</mo><mi>b</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>p</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>. First, for radial blow-up solutions in <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>b</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup></math></span>, we prove an upper bound on the blow-up rate for the intercritical dNLS. Moreover, an <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm concentration in the mass-critical case is also obtained by giving a compact lemma. Next, we turn to the non-radial case. By establishing two types of Gagliardo–Nirenberg inequalities, we show the existence of finite time blow-up solutions in <span><math><mrow><msup><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>̇</mo></mrow></mover></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></msup><mo>∩</mo><msubsup><mrow><mover><mrow><mi>W</mi></mrow><mrow><mo>̇</mo></mrow></mover></mrow><mrow><mi>b</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup></mrow></math></span>, where <span><math><mrow><msup><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>̇</mo></mrow></mover></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mfrac><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span> and <span><math><mrow><msubsup><mrow><mover><mrow><mi>W</mi></mrow><mrow><mo>̇</mo></mrow></mover></mrow><mrow><mi>b</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"265 ","pages":"Article 114015"},"PeriodicalIF":1.3,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618595","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 : 2025-11-20DOI: 10.1016/j.na.2025.114018
Irene De Blasi
This work presents some results regarding three-dimensional billiards having a non-constant potential of Keplerian type inside a regular domain . Two models will be analysed: in the first one, only an inner Keplerian potential is present, and every time the particle encounters the boundary of is reflected back by keeping constant its tangential component to , while the normal one changes its sign. The second model is a refractive billiard, where the inner Keplerian potential is coupled with a harmonic outer one; in this case, the interaction with results in a generalised refraction Snell’s law. In both cases, the analysis of a particular type of straight equilibrium trajectories, called homothetic, is carried on, and their presence is linked to the topological chaoticity of the dynamics for large inner energies.
{"title":"Keplerian billiards in three dimensions: Stability of equilibrium orbits and conditions for chaos","authors":"Irene De Blasi","doi":"10.1016/j.na.2025.114018","DOIUrl":"10.1016/j.na.2025.114018","url":null,"abstract":"<div><div>This work presents some results regarding three-dimensional billiards having a non-constant potential of Keplerian type inside a regular domain <span><math><mrow><mi>D</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></math></span>. Two models will be analysed: in the first one, only an inner Keplerian potential is present, and every time the particle encounters the boundary of <span><math><mi>D</mi></math></span> is reflected back by keeping constant its tangential component to <span><math><mrow><mi>∂</mi><mi>D</mi></mrow></math></span>, while the normal one changes its sign. The second model is a refractive billiard, where the inner Keplerian potential is coupled with a harmonic outer one; in this case, the interaction with <span><math><mrow><mi>∂</mi><mi>D</mi></mrow></math></span> results in a generalised refraction Snell’s law. In both cases, the analysis of a particular type of straight equilibrium trajectories, called <em>homothetic</em>, is carried on, and their presence is linked to the topological chaoticity of the dynamics for large inner energies.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"265 ","pages":"Article 114018"},"PeriodicalIF":1.3,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145571025","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 : 2025-11-19DOI: 10.1016/j.na.2025.114016
R.D. Ayissi , G. Deugoué , J. Ngandjou Zangue , T. Tachim Medjo
In this paper, we study a feedback optimal control problem for the stochastic nonlocal Cahn–Hilliard–Navier–Stokes model in a two-dimensional bounded domain. The model consists of the stochastic Navier–Stokes equations for the velocity, coupled with a nonlocal Cahn-Hilliard system for the order (phase) parameter. We prove the existence of an optimal feedback control for the stochastic nonlocal Cahn–Hilliard-Navier-Stokes system. Moreover using the Galerkin approximation, we show that the optimal cost can be approximated by a sequence of finite dimensional optimal costs.
{"title":"On the existence of optimal and ɛ-optimal controls for the stochastic 2D nonlocal Cahn–Hilliard–Navier–Stokes system","authors":"R.D. Ayissi , G. Deugoué , J. Ngandjou Zangue , T. Tachim Medjo","doi":"10.1016/j.na.2025.114016","DOIUrl":"10.1016/j.na.2025.114016","url":null,"abstract":"<div><div>In this paper, we study a feedback optimal control problem for the stochastic nonlocal Cahn–Hilliard–Navier–Stokes model in a two-dimensional bounded domain. The model consists of the stochastic Navier–Stokes equations for the velocity, coupled with a nonlocal Cahn-Hilliard system for the order (phase) parameter. We prove the existence of an optimal feedback control for the stochastic nonlocal Cahn–Hilliard-Navier-Stokes system. Moreover using the Galerkin approximation, we show that the optimal cost can be approximated by a sequence of finite dimensional optimal costs.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"265 ","pages":"Article 114016"},"PeriodicalIF":1.3,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145537288","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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