Pub Date : 2024-06-11DOI: 10.1088/1361-6544/ad5131
Jie Zhang and Wenjun Liu
In this paper, we give a rigorous justification for the derivation of the primitive equations with only horizontal viscosity and magnetic diffusivity (PEHM) as the small aspect ratio limit of the incompressible three-dimensional scaled horizontal viscous magnetohydrodynamics (SHMHD) equations. Choosing an aspect ratio parameter , we consider the case that if the orders of the horizontal and vertical viscous coefficients µ and ν are and , and the orders of magnetic diffusion coefficients κ and σ are and , with α > 2, then the limiting system is the PEHM as ɛ goes to zero. For -initial data, we prove that the global weak solutions of the SHMHD equations converge strongly to the local-in-time strong solutions of the PEHM, as ɛ tends to zero. For -initial data with additional regularity , we slightly improve the well-posed result in Cao et al (2017 J. Funct. Anal.272 4606–41) to extend the local-in-time strong convergences to the global-in-time one. For -initial data, we show that the global-in-time strong solutions of the SHMHD equations converge strongly to the global-in-time strong solutions of the PEHM, as ɛ goes to zero. Moreover, the rate of convergence is of the order , where with . It should be noted that in contrast to the case α > 2, the case α = 2 has been investigated by Du and Li in (2022 arXiv:2208.01985), in which they consider the primitive equations with magnetic field (PEM) and the rate of global-in-time convergences is of the order .
{"title":"The horizontal magnetic primitive equations approximation of the anisotropic MHD equations in a thin 3D domain","authors":"Jie Zhang and Wenjun Liu","doi":"10.1088/1361-6544/ad5131","DOIUrl":"https://doi.org/10.1088/1361-6544/ad5131","url":null,"abstract":"In this paper, we give a rigorous justification for the derivation of the primitive equations with only horizontal viscosity and magnetic diffusivity (PEHM) as the small aspect ratio limit of the incompressible three-dimensional scaled horizontal viscous magnetohydrodynamics (SHMHD) equations. Choosing an aspect ratio parameter , we consider the case that if the orders of the horizontal and vertical viscous coefficients µ and ν are and , and the orders of magnetic diffusion coefficients κ and σ are and , with α > 2, then the limiting system is the PEHM as ɛ goes to zero. For -initial data, we prove that the global weak solutions of the SHMHD equations converge strongly to the local-in-time strong solutions of the PEHM, as ɛ tends to zero. For -initial data with additional regularity , we slightly improve the well-posed result in Cao et al (2017 J. Funct. Anal.272 4606–41) to extend the local-in-time strong convergences to the global-in-time one. For -initial data, we show that the global-in-time strong solutions of the SHMHD equations converge strongly to the global-in-time strong solutions of the PEHM, as ɛ goes to zero. Moreover, the rate of convergence is of the order , where with . It should be noted that in contrast to the case α > 2, the case α = 2 has been investigated by Du and Li in (2022 arXiv:2208.01985), in which they consider the primitive equations with magnetic field (PEM) and the rate of global-in-time convergences is of the order .","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"9 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528948","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}
We establish existence and multiplicity of one-peaked and multi-peaked positive bound states for nonlinear Schrödinger equations on general compact and noncompact metric graphs. Precisely, we construct solutions concentrating at every vertex of odd degree greater than or equal to 3. We show that these solutions are not minimizers of the associated action and energy functionals. To the best of our knowledge, this is the first work exhibiting solutions concentrating at vertices with degree different than 1. The proof is based on a suitable Ljapunov–Schmidt reduction.
{"title":"Existence and multiplicity of peaked bound states for nonlinear Schrödinger equations on metric graphs","authors":"Haixia Chen, Simone Dovetta, Angela Pistoia, Enrico Serra","doi":"10.1088/1361-6544/ad5133","DOIUrl":"https://doi.org/10.1088/1361-6544/ad5133","url":null,"abstract":"We establish existence and multiplicity of one-peaked and multi-peaked positive bound states for nonlinear Schrödinger equations on general compact and noncompact metric graphs. Precisely, we construct solutions concentrating at every vertex of odd degree greater than or equal to 3. We show that these solutions are not minimizers of the associated action and energy functionals. To the best of our knowledge, this is the first work exhibiting solutions concentrating at vertices with degree different than 1. The proof is based on a suitable Ljapunov–Schmidt reduction.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"32 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141516159","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 : 2024-06-07DOI: 10.1088/1361-6544/ad5132
M Bendahmane, K H Karlsen, F Mroué
We analyse a system of nonlinear stochastic partial differential equations (SPDEs) of mixed elliptic-parabolic type that models the propagation of electric signals and their effect on the deformation of cardiac tissue. The system governs the dynamics of ionic quantities, intra and extra-cellular potentials, and linearised elasticity equations. We introduce a framework called the active strain decomposition, which factors the material gradient of deformation into an active (electrophysiology-dependent) part and an elastic (passive) part, to capture the coupling between muscle contraction, biochemical reactions, and electric activity. Under the assumption of linearised elastic behaviour and a truncation of the nonlinear diffusivities, we propose a stochastic electromechanical bidomain model, and establish the existence of weak solutions for this model. To prove existence through the convergence of approximate solutions, we employ a stochastic compactness method in tandem with an auxiliary non-degenerate system and the Faedo–Galerkin method. We utilise a stochastic adaptation of de Rham’s theorem to deduce the weak convergence of the pressure approximations.
{"title":"Stochastic electromechanical bidomain model *","authors":"M Bendahmane, K H Karlsen, F Mroué","doi":"10.1088/1361-6544/ad5132","DOIUrl":"https://doi.org/10.1088/1361-6544/ad5132","url":null,"abstract":"We analyse a system of nonlinear stochastic partial differential equations (SPDEs) of mixed elliptic-parabolic type that models the propagation of electric signals and their effect on the deformation of cardiac tissue. The system governs the dynamics of ionic quantities, intra and extra-cellular potentials, and linearised elasticity equations. We introduce a framework called the active strain decomposition, which factors the material gradient of deformation into an active (electrophysiology-dependent) part and an elastic (passive) part, to capture the coupling between muscle contraction, biochemical reactions, and electric activity. Under the assumption of linearised elastic behaviour and a truncation of the nonlinear diffusivities, we propose a stochastic electromechanical bidomain model, and establish the existence of weak solutions for this model. To prove existence through the convergence of approximate solutions, we employ a stochastic compactness method in tandem with an auxiliary non-degenerate system and the Faedo–Galerkin method. We utilise a stochastic adaptation of de Rham’s theorem to deduce the weak convergence of the pressure approximations.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"30 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550080","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 : 2024-06-03DOI: 10.1088/1361-6544/ad4ade
Dimitrios Giannakis and Claire Valva
Koopman operators and transfer operators represent dynamical systems through their induced linear action on vector spaces of observables, enabling the use of operator-theoretic techniques to analyze nonlinear dynamics in state space. The extraction of approximate Koopman or transfer operator eigenfunctions (and the associated eigenvalues) from an unknown system is nontrivial, particularly if the system has mixed or continuous spectrum. In this paper, we describe a spectrally accurate approach to approximate the Koopman operator on L2 for measure-preserving, continuous-time systems via a ‘compactification’ of the resolvent of the generator. This approach employs kernel integral operators to approximate the skew-adjoint Koopman generator by a family of skew-adjoint operators with compact resolvent, whose spectral measures converge in a suitable asymptotic limit, and whose eigenfunctions are approximately periodic. Moreover, we develop a data-driven formulation of our approach, utilizing data sampled on dynamical trajectories and associated dictionaries of kernel eigenfunctions for operator approximation. The data-driven scheme is shown to converge in the limit of large training data under natural assumptions on the dynamical system and observation modality. We explore applications of this technique to dynamical systems on tori with pure point spectra and the Lorenz 63 system as an example with mixing dynamics.
{"title":"Consistent spectral approximation of Koopman operators using resolvent compactification","authors":"Dimitrios Giannakis and Claire Valva","doi":"10.1088/1361-6544/ad4ade","DOIUrl":"https://doi.org/10.1088/1361-6544/ad4ade","url":null,"abstract":"Koopman operators and transfer operators represent dynamical systems through their induced linear action on vector spaces of observables, enabling the use of operator-theoretic techniques to analyze nonlinear dynamics in state space. The extraction of approximate Koopman or transfer operator eigenfunctions (and the associated eigenvalues) from an unknown system is nontrivial, particularly if the system has mixed or continuous spectrum. In this paper, we describe a spectrally accurate approach to approximate the Koopman operator on L2 for measure-preserving, continuous-time systems via a ‘compactification’ of the resolvent of the generator. This approach employs kernel integral operators to approximate the skew-adjoint Koopman generator by a family of skew-adjoint operators with compact resolvent, whose spectral measures converge in a suitable asymptotic limit, and whose eigenfunctions are approximately periodic. Moreover, we develop a data-driven formulation of our approach, utilizing data sampled on dynamical trajectories and associated dictionaries of kernel eigenfunctions for operator approximation. The data-driven scheme is shown to converge in the limit of large training data under natural assumptions on the dynamical system and observation modality. We explore applications of this technique to dynamical systems on tori with pure point spectra and the Lorenz 63 system as an example with mixing dynamics.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"19 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141253924","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 : 2024-05-30DOI: 10.1088/1361-6544/ad4cd2
Jianjun Liu and Duohui Xiang
Rational normal form is a powerful tool to deal with Hamiltonian partial differential equations without external parameters. In this paper, we build rational normal form with exact global control of small divisors. As an application to nonlinear Schrödinger equations in Gevrey spaces, we prove sub-exponentially long time stability results for generic small initial data.
{"title":"Exact global control of small divisors in rational normal form *","authors":"Jianjun Liu and Duohui Xiang","doi":"10.1088/1361-6544/ad4cd2","DOIUrl":"https://doi.org/10.1088/1361-6544/ad4cd2","url":null,"abstract":"Rational normal form is a powerful tool to deal with Hamiltonian partial differential equations without external parameters. In this paper, we build rational normal form with exact global control of small divisors. As an application to nonlinear Schrödinger equations in Gevrey spaces, we prove sub-exponentially long time stability results for generic small initial data.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"62 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141195945","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 : 2024-05-28DOI: 10.1088/1361-6544/ad4c4a
Ryan Hynd and Adrian Tudorascu
We study the long-time asymptotic behavior of the Sticky Particles dynamics on the real line. The time average of the Sticky Particles Lagrangian map has a limit which arises as a general property of projections onto closed convex cones in Hilbert spaces. More notably, we prove that the map itself has an asymptotic limit in the case where the sticky particles dynamics is confined to a compact set.
{"title":"Asymptotics of the sticky particles evolution","authors":"Ryan Hynd and Adrian Tudorascu","doi":"10.1088/1361-6544/ad4c4a","DOIUrl":"https://doi.org/10.1088/1361-6544/ad4c4a","url":null,"abstract":"We study the long-time asymptotic behavior of the Sticky Particles dynamics on the real line. The time average of the Sticky Particles Lagrangian map has a limit which arises as a general property of projections onto closed convex cones in Hilbert spaces. More notably, we prove that the map itself has an asymptotic limit in the case where the sticky particles dynamics is confined to a compact set.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"62 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141168575","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 : 2024-05-28DOI: 10.1088/1361-6544/ad4cd0
Begoña Barrios, Leandro Del Pezzo, Alexander Quaas and Julio D Rossi
In this paper we study the evolution problem associated with the first fractional eigenvalue. We prove that the Dirichlet problem with homogeneous boundary condition is well posed for this operator in the framework of viscosity solutions (the problem has existence and uniqueness of a solution and a comparison principle holds). In addition, we show that solutions decay to zero exponentially fast as with a bound that is given by the first eigenvalue for this problem that we also study.
{"title":"The evolution problem associated with the fractional first eigenvalue","authors":"Begoña Barrios, Leandro Del Pezzo, Alexander Quaas and Julio D Rossi","doi":"10.1088/1361-6544/ad4cd0","DOIUrl":"https://doi.org/10.1088/1361-6544/ad4cd0","url":null,"abstract":"In this paper we study the evolution problem associated with the first fractional eigenvalue. We prove that the Dirichlet problem with homogeneous boundary condition is well posed for this operator in the framework of viscosity solutions (the problem has existence and uniqueness of a solution and a comparison principle holds). In addition, we show that solutions decay to zero exponentially fast as with a bound that is given by the first eigenvalue for this problem that we also study.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"59 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141168593","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 : 2024-05-28DOI: 10.1088/1361-6544/ad4947
Georg A Gottwald and Ian Melbourne
We give a complete description and clarification of the structure of the Lévy area correction to Itô/Stratonovich stochastic integrals arising as limits of time-reversible deterministic dynamical systems. In particular, we show that time-reversibility forces the Lévy area to vanish only in very specific situations that are easily classified. In the absence of such obstructions, we prove that there are no further restrictions on the Lévy area and that it is typically nonvanishing and far from negligible.
{"title":"Time-reversibility and nonvanishing Lévy area","authors":"Georg A Gottwald and Ian Melbourne","doi":"10.1088/1361-6544/ad4947","DOIUrl":"https://doi.org/10.1088/1361-6544/ad4947","url":null,"abstract":"We give a complete description and clarification of the structure of the Lévy area correction to Itô/Stratonovich stochastic integrals arising as limits of time-reversible deterministic dynamical systems. In particular, we show that time-reversibility forces the Lévy area to vanish only in very specific situations that are easily classified. In the absence of such obstructions, we prove that there are no further restrictions on the Lévy area and that it is typically nonvanishing and far from negligible.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"94 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141168679","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 : 2024-05-27DOI: 10.1088/1361-6544/ad4c49
Stefanos Georgiadis and Ansgar Jüngel
The dynamics of multicomponent gas mixtures with vanishing barycentric velocity is described by Maxwell–Stefan equations with mass diffusion and heat conduction. The equations consist of the mass and energy balances, coupled to an algebraic system that relates the partial velocities and driving forces. The global existence of weak solutions to this system in a bounded domain with no-flux boundary conditions is proved by using the boundedness-by-entropy method. A priori estimates are obtained from the entropy inequality which originates from the consistent thermodynamic modelling. Furthermore, a conditional weak–strong uniqueness property is shown by using the relative entropy method.
{"title":"Global existence of weak solutions and weak–strong uniqueness for nonisothermal Maxwell–Stefan systems *","authors":"Stefanos Georgiadis and Ansgar Jüngel","doi":"10.1088/1361-6544/ad4c49","DOIUrl":"https://doi.org/10.1088/1361-6544/ad4c49","url":null,"abstract":"The dynamics of multicomponent gas mixtures with vanishing barycentric velocity is described by Maxwell–Stefan equations with mass diffusion and heat conduction. The equations consist of the mass and energy balances, coupled to an algebraic system that relates the partial velocities and driving forces. The global existence of weak solutions to this system in a bounded domain with no-flux boundary conditions is proved by using the boundedness-by-entropy method. A priori estimates are obtained from the entropy inequality which originates from the consistent thermodynamic modelling. Furthermore, a conditional weak–strong uniqueness property is shown by using the relative entropy method.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"59 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141168856","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 : 2024-05-27DOI: 10.1088/1361-6544/ad4aaf
Livio Flaminio and Davide Ravotti
We consider Abelian covers of compact hyperbolic surfaces. We establish an asymptotic expansion of the correlations for the horocycle flow on -covers, thus proving a strong form of Krickeberg mixing. We also prove that the spectral measures around 0 of the Casimir operators on any increasing sequence of finite Abelian covers converge weakly to an absolutely continuous measure.
{"title":"Abelian covers of hyperbolic surfaces: equidistribution of spectra and infinite volume mixing asymptotics for horocycle flows","authors":"Livio Flaminio and Davide Ravotti","doi":"10.1088/1361-6544/ad4aaf","DOIUrl":"https://doi.org/10.1088/1361-6544/ad4aaf","url":null,"abstract":"We consider Abelian covers of compact hyperbolic surfaces. We establish an asymptotic expansion of the correlations for the horocycle flow on -covers, thus proving a strong form of Krickeberg mixing. We also prove that the spectral measures around 0 of the Casimir operators on any increasing sequence of finite Abelian covers converge weakly to an absolutely continuous measure.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"59 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141168590","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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