Pub Date : 2024-05-23DOI: 10.1088/1361-6544/ad4adf
J F Carreño-Diaz and E I Kaikina
We consider the initial-boundary value problem for the Ginzburg–Landau equation with fractional Laplacian on a upper-right quarter plane where , and is a fractional Laplacian defined as We study the main questions of the theory of IBV- problems for nonlocal equations: the existence and uniqueness of a solution, the asymptotic behavior of the solution for large time and the influence of initial and boundary data on the basic properties of the solution. We generalize the concept of the well-posedness of IBV- problem in the based Sobolev spaces to the case of a multidimensional domain. We also give optimal relations between the orders of the Sobolev spaces to which the initial and boundary data belong. The lower order compatibility conditions between initial and boundary data are also discussed.
{"title":"Ginzburg–Landau equation with fractional Laplacian on a upper- right quarter plane","authors":"J F Carreño-Diaz and E I Kaikina","doi":"10.1088/1361-6544/ad4adf","DOIUrl":"https://doi.org/10.1088/1361-6544/ad4adf","url":null,"abstract":"We consider the initial-boundary value problem for the Ginzburg–Landau equation with fractional Laplacian on a upper-right quarter plane where , and is a fractional Laplacian defined as We study the main questions of the theory of IBV- problems for nonlocal equations: the existence and uniqueness of a solution, the asymptotic behavior of the solution for large time and the influence of initial and boundary data on the basic properties of the solution. We generalize the concept of the well-posedness of IBV- problem in the based Sobolev spaces to the case of a multidimensional domain. We also give optimal relations between the orders of the Sobolev spaces to which the initial and boundary data belong. The lower order compatibility conditions between initial and boundary data are also discussed.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"22 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148306","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 : 2024-05-21DOI: 10.1088/1361-6544/ad4948
Xiao-Chuan Liu and Fábio Armando Tal
We present a dichotomy for surface homeomorphisms in the isotopy class of the identity. We show that, in the absence of a degenerate fixed point set, either there exists a uniform bound on the diameter of orbits of non-wandering points for the lifted dynamics in the universal covering space, or the map has non-contractible periodic orbits. We then use this new tool to characterize the dynamics of area preserving homeomorphisms of the torus without non-contractible periodic orbits, showing that if the fixed point set is non-degenerate, then either the lifted dynamics is uniformly bounded, or it has a single strong irrational dynamical direction.
{"title":"On non-contractible periodic orbits and bounded deviations","authors":"Xiao-Chuan Liu and Fábio Armando Tal","doi":"10.1088/1361-6544/ad4948","DOIUrl":"https://doi.org/10.1088/1361-6544/ad4948","url":null,"abstract":"We present a dichotomy for surface homeomorphisms in the isotopy class of the identity. We show that, in the absence of a degenerate fixed point set, either there exists a uniform bound on the diameter of orbits of non-wandering points for the lifted dynamics in the universal covering space, or the map has non-contractible periodic orbits. We then use this new tool to characterize the dynamics of area preserving homeomorphisms of the torus without non-contractible periodic orbits, showing that if the fixed point set is non-degenerate, then either the lifted dynamics is uniformly bounded, or it has a single strong irrational dynamical direction.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"46 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148289","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 : 2024-05-20DOI: 10.1088/1361-6544/ad48f3
Cinzia Bisi, Jonathan D Hauenstein and Tuyen Trung Truong
We study a family of birational maps of smooth affine quadric 3-folds, over the complex numbers, of the form constant, which seems to have some (among many others) interesting/unexpected characters: (a) they are cohomologically hyperbolic, (b) their second dynamical degree is an algebraic number but not an algebraic integer, and (c) the logarithmic growth of their periodic points is strictly smaller than their algebraic entropy. These maps are restrictions of a polynomial map on preserving each of the quadrics. The study in this paper is a mixture of rigorous and experimental ones, where for the experimental study we rely on Bertini which is a reliable and fast software for expensive numerical calculations in complex algebraic geometry.
{"title":"Some interesting birational morphisms of smooth affine quadric 3-folds *","authors":"Cinzia Bisi, Jonathan D Hauenstein and Tuyen Trung Truong","doi":"10.1088/1361-6544/ad48f3","DOIUrl":"https://doi.org/10.1088/1361-6544/ad48f3","url":null,"abstract":"We study a family of birational maps of smooth affine quadric 3-folds, over the complex numbers, of the form constant, which seems to have some (among many others) interesting/unexpected characters: (a) they are cohomologically hyperbolic, (b) their second dynamical degree is an algebraic number but not an algebraic integer, and (c) the logarithmic growth of their periodic points is strictly smaller than their algebraic entropy. These maps are restrictions of a polynomial map on preserving each of the quadrics. The study in this paper is a mixture of rigorous and experimental ones, where for the experimental study we rely on Bertini which is a reliable and fast software for expensive numerical calculations in complex algebraic geometry.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"68 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148286","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 : 2024-05-19DOI: 10.1088/1361-6544/ad3f66
Timothée Crin-Barat, Ling-Yun Shou and Jin Tan
We derive a novel two-phase flow system in porous media as a relaxation limit of compressible multi-fluid systems. Considering a one-velocity Baer–Nunziato system with friction forces, we first justify its pressure-relaxation limit toward a Kapila model in a uniform manner with respect to the time-relaxation parameter associated with the friction forces. Then, we show that the diffusely rescaled solutions of the damped Kapila system converge to the solutions of the new two-phase porous media system as the time-relaxation parameter tends to zero. In addition, we also prove the convergence of the Baer–Nunziato system to the same two-phase porous media system as both relaxation parameters tend to zero. For each relaxation limit, we exhibit sharp rates of convergence in a critical regularity setting. Our proof is based on an elaborate low-frequency and high-frequency analysis via the Littlewood–Paley decomposition and includes three main ingredients: a refined spectral analysis of the linearized problem to determine the frequency threshold explicitly in terms of the time-relaxation parameter, the introduction of an effective flux in the low-frequency region to overcome the loss of parameters due to the overdamping phenomenon, and renormalized energy estimates in the high-frequency region to cancel higher-order nonlinear terms. To justify the convergence rates, we discover several auxiliary unknowns allowing us to recover crucial bounds.
{"title":"Quantitative derivation of a two-phase porous media system from the one-velocity Baer–Nunziato and Kapila systems","authors":"Timothée Crin-Barat, Ling-Yun Shou and Jin Tan","doi":"10.1088/1361-6544/ad3f66","DOIUrl":"https://doi.org/10.1088/1361-6544/ad3f66","url":null,"abstract":"We derive a novel two-phase flow system in porous media as a relaxation limit of compressible multi-fluid systems. Considering a one-velocity Baer–Nunziato system with friction forces, we first justify its pressure-relaxation limit toward a Kapila model in a uniform manner with respect to the time-relaxation parameter associated with the friction forces. Then, we show that the diffusely rescaled solutions of the damped Kapila system converge to the solutions of the new two-phase porous media system as the time-relaxation parameter tends to zero. In addition, we also prove the convergence of the Baer–Nunziato system to the same two-phase porous media system as both relaxation parameters tend to zero. For each relaxation limit, we exhibit sharp rates of convergence in a critical regularity setting. Our proof is based on an elaborate low-frequency and high-frequency analysis via the Littlewood–Paley decomposition and includes three main ingredients: a refined spectral analysis of the linearized problem to determine the frequency threshold explicitly in terms of the time-relaxation parameter, the introduction of an effective flux in the low-frequency region to overcome the loss of parameters due to the overdamping phenomenon, and renormalized energy estimates in the high-frequency region to cancel higher-order nonlinear terms. To justify the convergence rates, we discover several auxiliary unknowns allowing us to recover crucial bounds.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"4 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148311","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 : 2024-05-19DOI: 10.1088/1361-6544/ad4843
Guopeng Li
In this paper, we study the low regularity convergence problem for the intermediate long wave equation (ILW), with respect to the depth parameter δ > 0, on the real line and the circle. As a natural bridge between the Korteweg–de Vries (KdV) and the Benjamin–Ono (BO) equations, the ILW equation is of physical interest. We prove that the solutions of ILW converge in the Hs-Sobolev space for , to those of BO in the deep-water limit (as ), and to those of KdV in the shallow-water limit (as δ → 0). This improves previous convergence results by Abdelouhab et al (1989 Physica D 40 360–92), which required in the deep-water limit and in the shallow-water limit. Moreover, the convergence results also apply to the generalised ILW equation, i.e. with nonlinearity for . Furthermore, this work gives the first convergence results of generalised ILW solutions on the circle with regularity . Overall, this study provides mathematical insights for the behaviour of the ILW equation and its solutions in different water depths, and has implications for predicting and modelling wave behaviour in various environments.
{"title":"Deep-water and shallow-water limits of the intermediate long wave equation","authors":"Guopeng Li","doi":"10.1088/1361-6544/ad4843","DOIUrl":"https://doi.org/10.1088/1361-6544/ad4843","url":null,"abstract":"In this paper, we study the low regularity convergence problem for the intermediate long wave equation (ILW), with respect to the depth parameter δ > 0, on the real line and the circle. As a natural bridge between the Korteweg–de Vries (KdV) and the Benjamin–Ono (BO) equations, the ILW equation is of physical interest. We prove that the solutions of ILW converge in the Hs-Sobolev space for , to those of BO in the deep-water limit (as ), and to those of KdV in the shallow-water limit (as δ → 0). This improves previous convergence results by Abdelouhab et al (1989 Physica D 40 360–92), which required in the deep-water limit and in the shallow-water limit. Moreover, the convergence results also apply to the generalised ILW equation, i.e. with nonlinearity for . Furthermore, this work gives the first convergence results of generalised ILW solutions on the circle with regularity . Overall, this study provides mathematical insights for the behaviour of the ILW equation and its solutions in different water depths, and has implications for predicting and modelling wave behaviour in various environments.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"4 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148387","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 : 2024-05-14DOI: 10.1088/1361-6544/ad45a1
Jiayin Du, Lu Xu and Yong Li
In this paper, we consider a classical Hamiltonian normal form with degeneracy in the normal direction. In previous results, one needs to assume that the perturbation satisfies certain non-degenerate conditions in order to remove the degeneracy in the normal form. Instead of that, we introduce a topological degree condition and a weak convexity condition, which are easy to verify, and we prove the persistence of lower dimensional tori without any restriction on perturbation but only smallness and analyticity.
{"title":"An infinite dimensional KAM theorem with normal degeneracy","authors":"Jiayin Du, Lu Xu and Yong Li","doi":"10.1088/1361-6544/ad45a1","DOIUrl":"https://doi.org/10.1088/1361-6544/ad45a1","url":null,"abstract":"In this paper, we consider a classical Hamiltonian normal form with degeneracy in the normal direction. In previous results, one needs to assume that the perturbation satisfies certain non-degenerate conditions in order to remove the degeneracy in the normal form. Instead of that, we introduce a topological degree condition and a weak convexity condition, which are easy to verify, and we prove the persistence of lower dimensional tori without any restriction on perturbation but only smallness and analyticity.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"7 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939509","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 : 2024-05-14DOI: 10.1088/1361-6544/ad46bf
Qiangchang Ju and Jianjun Xu
We investigate the long time existence of smooth solutions to the initial boundary value problem for the non-isentropic slightly compressible Navier–Stokes equations with slip or non-slip boundary conditions on the velocity. We verify that the compressible Navier–Stokes equations with boundary conditions admit a unique smooth solution on the time interval where the smooth solution of the incompressible Navier–Stokes equations exists, when the Mach number is sufficiently small. Moreover, we obtain the uniform convergence of smooth solutions for the compressible system toward those for the corresponding incompressible system on that time interval.
{"title":"Long time existence of the non-isentropic slightly compressible Navier-Stokes equations with boundary conditions","authors":"Qiangchang Ju and Jianjun Xu","doi":"10.1088/1361-6544/ad46bf","DOIUrl":"https://doi.org/10.1088/1361-6544/ad46bf","url":null,"abstract":"We investigate the long time existence of smooth solutions to the initial boundary value problem for the non-isentropic slightly compressible Navier–Stokes equations with slip or non-slip boundary conditions on the velocity. We verify that the compressible Navier–Stokes equations with boundary conditions admit a unique smooth solution on the time interval where the smooth solution of the incompressible Navier–Stokes equations exists, when the Mach number is sufficiently small. Moreover, we obtain the uniform convergence of smooth solutions for the compressible system toward those for the corresponding incompressible system on that time interval.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"120 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939518","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 : 2024-05-13DOI: 10.1088/1361-6544/ad45a2
Jiajie Chen
We show that the constructions of asymptotically self-similar singularities for the three-dimensional (3D) Euler equations by Elgindi, and for the 3D Euler equations with large swirl and 2D Boussinesq equations with boundary by Chen-Hou can be extended to construct singularity with velocity that is not smooth at only one point. The proof is based on a carefully designed small initial perturbation to the blowup profile, and a BKM-type continuation criterion for the one-point nonsmoothness. We establish the criterion using weighted Hölder estimates with weights vanishing near the singular point. Our results are inspired by the recent work of Cordoba, Martinez-Zoroa and Zheng that it is possible to construct a singularity for the 3D axisymmetric Euler equations without swirl and with velocity .
{"title":"Remarks on the smoothness of the C1,α asymptotically self-similar singularity in the 3D Euler and 2D Boussinesq equations","authors":"Jiajie Chen","doi":"10.1088/1361-6544/ad45a2","DOIUrl":"https://doi.org/10.1088/1361-6544/ad45a2","url":null,"abstract":"We show that the constructions of asymptotically self-similar singularities for the three-dimensional (3D) Euler equations by Elgindi, and for the 3D Euler equations with large swirl and 2D Boussinesq equations with boundary by Chen-Hou can be extended to construct singularity with velocity that is not smooth at only one point. The proof is based on a carefully designed small initial perturbation to the blowup profile, and a BKM-type continuation criterion for the one-point nonsmoothness. We establish the criterion using weighted Hölder estimates with weights vanishing near the singular point. Our results are inspired by the recent work of Cordoba, Martinez-Zoroa and Zheng that it is possible to construct a singularity for the 3D axisymmetric Euler equations without swirl and with velocity .","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"7 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939536","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 : 2024-05-13DOI: 10.1088/1361-6544/ad4504
Diego Chamorro and Claudiu Mîndrilă
We address here the problem of regularity for weak solutions of the 3D Boussinesq equation. By introducing the new notion of partial suitable solutions, which imposes some conditions over the velocity field only, we show a local gain of regularity for the two variables and θ.
{"title":"A new approach for the regularity of weak solutions of the 3D Boussinesq system","authors":"Diego Chamorro and Claudiu Mîndrilă","doi":"10.1088/1361-6544/ad4504","DOIUrl":"https://doi.org/10.1088/1361-6544/ad4504","url":null,"abstract":"We address here the problem of regularity for weak solutions of the 3D Boussinesq equation. By introducing the new notion of partial suitable solutions, which imposes some conditions over the velocity field only, we show a local gain of regularity for the two variables and θ.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"139 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939517","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 : 2024-05-12DOI: 10.1088/1361-6544/ad4505
Rémi Carles
We consider the Schrödinger equation with an external potential and a cubic nonlinearity, in the semiclassical limit. The initial data are sums of WKB states, with smooth phases and smooth, compactly supported initial amplitudes, with disjoint supports. We show that like in the linear case, a superposition principle holds on some time interval independent of the semiclassical parameter, in several régimes in term of the size of initial data with respect to the semiclassical parameter. When nonlinear effects are strong in terms of the semiclassical parameter, we invoke properties of compressible Euler equations. For weaker nonlinear effects, we show that there may be no nonlinear interferences on some time interval independent of the semiclassical parameter, and interferences for later time, thanks to explicit computations available for particular phases.
{"title":"On nonlinear effects in multiphase WKB analysis for the nonlinear Schrödinger equation *","authors":"Rémi Carles","doi":"10.1088/1361-6544/ad4505","DOIUrl":"https://doi.org/10.1088/1361-6544/ad4505","url":null,"abstract":"We consider the Schrödinger equation with an external potential and a cubic nonlinearity, in the semiclassical limit. The initial data are sums of WKB states, with smooth phases and smooth, compactly supported initial amplitudes, with disjoint supports. We show that like in the linear case, a superposition principle holds on some time interval independent of the semiclassical parameter, in several régimes in term of the size of initial data with respect to the semiclassical parameter. When nonlinear effects are strong in terms of the semiclassical parameter, we invoke properties of compressible Euler equations. For weaker nonlinear effects, we show that there may be no nonlinear interferences on some time interval independent of the semiclassical parameter, and interferences for later time, thanks to explicit computations available for particular phases.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"42 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939538","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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