Pub Date : 2024-08-27DOI: 10.1088/1361-6544/ad700c
Mónica Clapp, Jorge Faya, Alberto Saldaña
We establish the existence of an optimal partition for the Yamabe equation in RN made up of mutually linearly isometric sets, each of them invariant under the action of a group of linear isometries. To do this, we establish the existence of a solution to a weakly coupled competitive Yamabe system, whose components are invariant under the action of the group, and each of them is obtained from the previous one by composing it with a linear isometry. We show that, as the coupling parameter goes to −∞, the components of the solutions segregate and give rise to an optimal partition that has the properties mentioned above. Finally, taking advantage of the symmetries considered, we establish the existence of infinitely many sign-changing solutions for the Yamabe equation in RN that are different from those previously found by Ding, and del Pino, Musso, Pacard and Pistoia.
{"title":"Optimal pinwheel partitions for the Yamabe equation","authors":"Mónica Clapp, Jorge Faya, Alberto Saldaña","doi":"10.1088/1361-6544/ad700c","DOIUrl":"https://doi.org/10.1088/1361-6544/ad700c","url":null,"abstract":"We establish the existence of an optimal partition for the Yamabe equation in <inline-formula>\u0000<tex-math><?CDATA $mathbb{R}^N$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup></mml:mrow></mml:math><inline-graphic xlink:href=\"nonad700cieqn1.gif\"></inline-graphic></inline-formula> made up of mutually linearly isometric sets, each of them invariant under the action of a group of linear isometries. To do this, we establish the existence of a solution to a weakly coupled competitive Yamabe system, whose components are invariant under the action of the group, and each of them is obtained from the previous one by composing it with a linear isometry. We show that, as the coupling parameter goes to <inline-formula>\u0000<tex-math><?CDATA $-infty$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href=\"nonad700cieqn2.gif\"></inline-graphic></inline-formula>, the components of the solutions segregate and give rise to an optimal partition that has the properties mentioned above. Finally, taking advantage of the symmetries considered, we establish the existence of infinitely many sign-changing solutions for the Yamabe equation in <inline-formula>\u0000<tex-math><?CDATA $mathbb{R}^N$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup></mml:mrow></mml:math><inline-graphic xlink:href=\"nonad700cieqn3.gif\"></inline-graphic></inline-formula> that are different from those previously found by Ding, and del Pino, Musso, Pacard and Pistoia.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201423","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 : 2024-08-22DOI: 10.1088/1361-6544/ad6bdd
Giulia Carigi, Tobias Kuna, Jochen Bröcker
A framework to establish response theory for a class of nonlinear stochastic partial differential equations (SPDEs) is provided. More specifically, it is shown that for a certain class of observables, the averages of those observables against the stationary measure of the SPDE are differentiable (linear response) or, under weaker conditions, locally Hölder continuous (fractional response) as functions of a deterministic additive forcing. The method allows to consider observables that are not necessarily differentiable. For such observables, spectral gap results for the Markov semigroup associated with the SPDE have recently been established that are fairly accessible. This is important here as spectral gaps are a major ingredient for establishing linear response. The results are applied to the 2D stochastic Navier–Stokes equation and the stochastic two–layer quasi–geostrophic model, an intermediate complexity model popular in the geosciences to study atmosphere and ocean dynamics. The physical motivation for studying the response to perturbations in the forcings for models in geophysical fluid dynamics comes from climate change and relate to the question as to whether statistical properties of the dynamics derived under current conditions will be valid under different forcing scenarios.
{"title":"Linear and fractional response for nonlinear dissipative SPDEs","authors":"Giulia Carigi, Tobias Kuna, Jochen Bröcker","doi":"10.1088/1361-6544/ad6bdd","DOIUrl":"https://doi.org/10.1088/1361-6544/ad6bdd","url":null,"abstract":"A framework to establish response theory for a class of nonlinear stochastic partial differential equations (SPDEs) is provided. More specifically, it is shown that for a certain class of observables, the averages of those observables against the stationary measure of the SPDE are differentiable (linear response) or, under weaker conditions, locally Hölder continuous (fractional response) as functions of a deterministic additive forcing. The method allows to consider observables that are not necessarily differentiable. For such observables, spectral gap results for the Markov semigroup associated with the SPDE have recently been established that are fairly accessible. This is important here as spectral gaps are a major ingredient for establishing linear response. The results are applied to the 2D stochastic Navier–Stokes equation and the stochastic two–layer quasi–geostrophic model, an intermediate complexity model popular in the geosciences to study atmosphere and ocean dynamics. The physical motivation for studying the response to perturbations in the forcings for models in geophysical fluid dynamics comes from climate change and relate to the question as to whether statistical properties of the dynamics derived under current conditions will be valid under different forcing scenarios.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201427","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 : 2024-08-20DOI: 10.1088/1361-6544/ad694b
Didier Pilod, Frédéric Valet
We prove the asymptotic stability of a finite sum of well-ordered solitary waves for the Zakharov–Kuznetsov equation in dimensions two and three. We also derive a qualitative version of the orbital stability result, which will be useful for studying the collision of two solitary waves in a forthcoming paper. The proof extends the ideas of Martel, Merle and Tsai for the sub-critical gKdV equation in dimension one to the higher-dimensional case. It relies on monotonicity properties on oblique half-spaces and rigidity properties around one solitary wave introduced by Côte, Muñoz, Pilod and Simpson in dimension two, and by Farah, Holmer, Roudenko and Yang in dimension three.
我们证明了二维和三维扎哈罗夫-库兹涅佐夫方程有序孤波有限和的渐近稳定性。我们还推导出了轨道稳定性结果的定性版本，这将有助于在即将发表的论文中研究两个孤波的碰撞。它依赖于斜半空的单调性特性，以及 Côte、Muñoz、Pilod 和 Simpson 在二维，Farah、Holmer、Roudenko 和 Yang 在三维引入的围绕一个孤波的刚性特性。
{"title":"Asymptotic stability of a finite sum of solitary waves for the Zakharov–Kuznetsov equation","authors":"Didier Pilod, Frédéric Valet","doi":"10.1088/1361-6544/ad694b","DOIUrl":"https://doi.org/10.1088/1361-6544/ad694b","url":null,"abstract":"We prove the asymptotic stability of a finite sum of well-ordered solitary waves for the Zakharov–Kuznetsov equation in dimensions two and three. We also derive a qualitative version of the orbital stability result, which will be useful for studying the collision of two solitary waves in a forthcoming paper.\u0000The proof extends the ideas of Martel, Merle and Tsai for the sub-critical gKdV equation in dimension one to the higher-dimensional case. It relies on monotonicity properties on oblique half-spaces and rigidity properties around one solitary wave introduced by Côte, Muñoz, Pilod and Simpson in dimension two, and by Farah, Holmer, Roudenko and Yang in dimension three.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201422","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 : 2024-08-19DOI: 10.1088/1361-6544/ad672b
Pieter Roffelsen
In the context of q-Painlevé VI with generic parameter values, the Riemann–Hilbert correspondence induces a one-to-one mapping between solutions of the nonlinear equation and points on an affine Segre surface. Upon fixing a generic point on the surface, we give formulae for the function values of the corresponding solution near the critical points, in the form of complete, convergent, asymptotic expansions. These lead in particular to the solution of the nonlinear connection problem for the general solution of q-Painlevé VI. We further show that, when the point on the Segre surface is moved to one of the sixteen lines on the surface, one of the asymptotic expansions near the critical points truncates, under suitable parameter assumptions. At intersection points of lines, this then yields doubly truncated asymptotics at one of the critical points or simultaneous truncation at both.
在具有一般参数值的 q-Painlevé VI 的背景下，黎曼-希尔伯特对应关系在非线性方程的解和仿射塞格雷曲面上的点之间诱导出了一一对应的映射。在曲面上固定一个一般点后，我们给出了临界点附近相应解的函数值公式，其形式为完整、收敛的渐近展开式。这些公式特别引出了 q-Painlevé VI 一般解的非线性连接问题的解。我们进一步证明，当塞格雷曲面上的点移动到曲面上十六条线中的一条时，在适当的参数假设下，临界点附近的渐近展开之一会截断。在线段的交叉点上，会在其中一个临界点上产生双截断渐近线，或在两个临界点上同时产生截断渐近线。
{"title":"On q-Painlevé VI and the geometry of Segre surfaces","authors":"Pieter Roffelsen","doi":"10.1088/1361-6544/ad672b","DOIUrl":"https://doi.org/10.1088/1361-6544/ad672b","url":null,"abstract":"In the context of <italic toggle=\"yes\">q</italic>-Painlevé VI with generic parameter values, the Riemann–Hilbert correspondence induces a one-to-one mapping between solutions of the nonlinear equation and points on an affine Segre surface. Upon fixing a generic point on the surface, we give formulae for the function values of the corresponding solution near the critical points, in the form of complete, convergent, asymptotic expansions. These lead in particular to the solution of the nonlinear connection problem for the general solution of <italic toggle=\"yes\">q</italic>-Painlevé VI. We further show that, when the point on the Segre surface is moved to one of the sixteen lines on the surface, one of the asymptotic expansions near the critical points truncates, under suitable parameter assumptions. At intersection points of lines, this then yields doubly truncated asymptotics at one of the critical points or simultaneous truncation at both.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201424","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 : 2024-08-14DOI: 10.1088/1361-6544/ad68b8
Sylvain Carpentier, Alexander V Mikhailov, Jing Ping Wang
This paper builds upon our recent work, published in Carpentier et al (2022 Lett. Math. Phys.112 94), where we established that the integrable Volterra lattice on a free associative algebra and the whole hierarchy of its symmetries admit a quantisation dependent on a parameter ω. We also uncovered an intriguing aspect: all odd-degree symmetries of the hierarchy admit an alternative, non-deformation quantisation, resulting in a non-commutative algebra for any choice of the quantisation parameter ω. In this study, we demonstrate that each equation within the quantum Volterra hierarchy can be expressed in the Heisenberg form. We provide explicit expressions for all quantum Hamiltonians and establish their commutativity. In the classical limit, these quantum Hamiltonians yield explicit expressions for the classical ones of the commutative Volterra hierarchy. Furthermore, we present Heisenberg equations and their Hamiltonians in the case of non-deformation quantisation. Finally, we discuss commuting first integrals, central elements of the quantum algebra, and the integrability problem for periodic reductions of the Volterra lattice in the context of both quantisations.
{"title":"Hamiltonians for the quantised Volterra hierarchy","authors":"Sylvain Carpentier, Alexander V Mikhailov, Jing Ping Wang","doi":"10.1088/1361-6544/ad68b8","DOIUrl":"https://doi.org/10.1088/1361-6544/ad68b8","url":null,"abstract":"This paper builds upon our recent work, published in Carpentier <italic toggle=\"yes\">et al</italic> (2022 <italic toggle=\"yes\">Lett. Math. Phys.</italic>\u0000<bold>112</bold> 94), where we established that the integrable Volterra lattice on a free associative algebra and the whole hierarchy of its symmetries admit a quantisation dependent on a parameter <italic toggle=\"yes\">ω</italic>. We also uncovered an intriguing aspect: all odd-degree symmetries of the hierarchy admit an alternative, non-deformation quantisation, resulting in a non-commutative algebra for any choice of the quantisation parameter <italic toggle=\"yes\">ω</italic>. In this study, we demonstrate that each equation within the quantum Volterra hierarchy can be expressed in the Heisenberg form. We provide explicit expressions for all quantum Hamiltonians and establish their commutativity. In the classical limit, these quantum Hamiltonians yield explicit expressions for the classical ones of the commutative Volterra hierarchy. Furthermore, we present Heisenberg equations and their Hamiltonians in the case of non-deformation quantisation. Finally, we discuss commuting first integrals, central elements of the quantum algebra, and the integrability problem for periodic reductions of the Volterra lattice in the context of both quantisations.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201425","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 : 2024-08-14DOI: 10.1088/1361-6544/ad6bde
Hildeberto Jardón-Kojakhmetov, Christian Kuehn, Maximilian Steinert
Fast-slow systems with three slow variables and gradient structure in the fast variables have, generically, hyperbolic umbilic, elliptic umbilic or swallowtail singularities. In this article we provide a detailed local analysis of a fast-slow system near a hyperbolic umbilic singularity. In particular, we show that under some appropriate non-degeneracy conditions on the slow flow, the attracting slow manifolds jump onto the fast regime and fan out as they cross the hyperbolic umbilic singularity. The analysis is based on the blow-up technique, in which the hyperbolic umbilic point is blown up to a 5-dimensional sphere. Moreover, the reduced slow flow is also blown up and embedded into the blown-up fast formulation. Further, we describe how our analysis is related to classical theories such as catastrophe theory and constrained differential equations.
{"title":"The hyperbolic umbilic singularity in fast-slow systems","authors":"Hildeberto Jardón-Kojakhmetov, Christian Kuehn, Maximilian Steinert","doi":"10.1088/1361-6544/ad6bde","DOIUrl":"https://doi.org/10.1088/1361-6544/ad6bde","url":null,"abstract":"Fast-slow systems with three slow variables and gradient structure in the fast variables have, generically, hyperbolic umbilic, elliptic umbilic or swallowtail singularities. In this article we provide a detailed local analysis of a fast-slow system near a hyperbolic umbilic singularity. In particular, we show that under some appropriate non-degeneracy conditions on the slow flow, the attracting slow manifolds jump onto the fast regime and fan out as they cross the hyperbolic umbilic singularity. The analysis is based on the blow-up technique, in which the hyperbolic umbilic point is blown up to a 5-dimensional sphere. Moreover, the reduced slow flow is also blown up and embedded into the blown-up fast formulation. Further, we describe how our analysis is related to classical theories such as catastrophe theory and constrained differential equations.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201426","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 : 2024-08-14DOI: 10.1088/1361-6544/ad5638
S Jon Chapman, M Kavousanakis, E G Charalampidis, I G Kevrekidis, P G Kevrekidis
In the present work we revisit the problem of the generalised Korteweg–de Vries equation parametrically, as a function of the relevant nonlinearity exponent, to examine the emergence of blow-up solutions, as traveling waveforms lose their stability past a critical point of the relevant parameter p, here at p = 5. We provide a normal form of the associated collapse dynamics, and illustrate how this captures the collapsing branch bifurcating from the unstable traveling branch. We also systematically characterise the linearisation spectrum of not only the traveling states, but importantly of the emergent collapsing waveforms in the so-called co-exploding frame where these waveforms are identified as stationary states. This spectrum, in addition to two positive real eigenvalues which are shown to be associated with the symmetries of translation and scaling invariance of the original (non-exploding) frame features complex patterns of negative eigenvalues that we also fully characterise. We show that the phenomenology of the latter is significantly affected by the boundary conditions and is far more complicated than in the corresponding symmetric Laplacian case of the nonlinear Schrödinger problem that has recently been explored. In addition, we explore the dynamics of the unstable solitary waves for p > 5 in the co-exploding frame.
在本研究中，我们以相关非线性指数的函数为参数，重新审视了广义 Korteweg-de Vries 方程的问题，研究了当行进波形在经过相关参数 p 的临界点（此处为 p = 5）后失去稳定性时，出现的炸裂解。我们提供了相关坍缩动力学的正态形式，并说明了它如何捕捉到从不稳定性行波分支分叉出来的坍缩分支。我们还系统地描述了线性化频谱，不仅是行进状态，更重要的是在所谓的共爆帧中出现的坍缩波形，这些波形被确定为静止状态。该频谱除了两个正实特征值外，还显示出与原始（非对消）框架的平移和缩放不变性对称性相关的负特征值的复杂模式，我们也对其进行了全面描述。我们表明，后者的现象学受到边界条件的显著影响，远比最近探讨的非线性薛定谔问题的相应对称拉普拉斯情况复杂得多。此外，我们还探讨了共爆框架中 p > 5 不稳定孤波的动力学。
{"title":"Self-similar blow-up solutions in the generalised Korteweg-de Vries equation: spectral analysis, normal form and asymptotics","authors":"S Jon Chapman, M Kavousanakis, E G Charalampidis, I G Kevrekidis, P G Kevrekidis","doi":"10.1088/1361-6544/ad5638","DOIUrl":"https://doi.org/10.1088/1361-6544/ad5638","url":null,"abstract":"In the present work we revisit the problem of the generalised Korteweg–de Vries equation parametrically, as a function of the relevant nonlinearity exponent, to examine the emergence of blow-up solutions, as traveling waveforms lose their stability past a critical point of the relevant parameter <italic toggle=\"yes\">p</italic>, here at <italic toggle=\"yes\">p</italic> = 5. We provide a <italic toggle=\"yes\">normal form</italic> of the associated collapse dynamics, and illustrate how this captures the collapsing branch bifurcating from the unstable traveling branch. We also systematically characterise the linearisation spectrum of not only the traveling states, but importantly of the emergent collapsing waveforms in the so-called co-exploding frame where these waveforms are identified as stationary states. This spectrum, in addition to two positive real eigenvalues which are shown to be associated with the symmetries of translation and scaling invariance of the original (non-exploding) frame features complex patterns of negative eigenvalues that we also fully characterise. We show that the phenomenology of the latter is significantly affected by the boundary conditions and is far more complicated than in the corresponding symmetric Laplacian case of the nonlinear Schrödinger problem that has recently been explored. In addition, we explore the dynamics of the unstable solitary waves for <italic toggle=\"yes\">p</italic> > 5 in the co-exploding frame.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201462","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 : 2024-08-14DOI: 10.1088/1361-6544/ad68b7
Yue-Jun Peng, Liang Zhao
In a previous work (Peng and Zhao 2022 J. Math. Fluid Mech.24 29), it is proved that the 1D full compressible Navier–Stokes equations for a Newtonian fluid can be approximated globally-in-time by a relaxed Euler-type system with Oldroyd’s derivatives and a revised Cattaneo’s constitutive law. These two relaxations turn the whole system into a first-order quasilinear hyperbolic one with partial dissipation. In this paper, we establish the global convergence rates between the smooth solutions to the relaxed Euler-type system and the Navier–Stokes equations over periodic domains. For this purpose, we use stream function techniques together with energy estimates for error systems. These techniques may be applicable to more complicated systems.
{"title":"Global convergence rates from relaxed Euler equations to Navier–Stokes equations with Oldroyd-type constitutive laws","authors":"Yue-Jun Peng, Liang Zhao","doi":"10.1088/1361-6544/ad68b7","DOIUrl":"https://doi.org/10.1088/1361-6544/ad68b7","url":null,"abstract":"In a previous work (Peng and Zhao 2022 <italic toggle=\"yes\">J. Math. Fluid Mech.</italic>\u0000<bold>24</bold> 29), it is proved that the 1D full compressible Navier–Stokes equations for a Newtonian fluid can be approximated globally-in-time by a relaxed Euler-type system with Oldroyd’s derivatives and a revised Cattaneo’s constitutive law. These two relaxations turn the whole system into a first-order quasilinear hyperbolic one with partial dissipation. In this paper, we establish the global convergence rates between the smooth solutions to the relaxed Euler-type system and the Navier–Stokes equations over periodic domains. For this purpose, we use stream function techniques together with energy estimates for error systems. These techniques may be applicable to more complicated systems.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201428","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 : 2024-08-14DOI: 10.1088/1361-6544/ad6b6f
Yuhui Chen, Minling Li, Qinghe Yao, Zheng-an Yao
In this paper, we consider the global well-posedness of the three-dimensional inhomogeneous incompressible Phan-Thien–Tanner (PTT) system with general initial data in the critical Besov spaces. The question of whether or not PTT system is globally well posed for large data is still open. When the density ρ is away from zero, we denote by ϱ:=1ρ−1. More precisely, we prove that the PTT system admits a unique global solution, provided that the initial data ϱ0, the initial horizontal velocity uh0, the product ωu30 of the coupling parameter ω and the initial vertical velocity u30, and the initial symmetric tensor of constraints τ0 are sufficiently small. In particular, this result includes the global well-posedness of the PTT system for small initial data in the case where ω∈[0,1) and for large initial vertical velocity in the case where ω tends to zero. As a by-product, our results can be applied to the so-called Oldroyd-B system.
{"title":"Global solutions to the three-dimensional inhomogeneous incompressible Phan-Thien–Tanner system with a class of large initial data","authors":"Yuhui Chen, Minling Li, Qinghe Yao, Zheng-an Yao","doi":"10.1088/1361-6544/ad6b6f","DOIUrl":"https://doi.org/10.1088/1361-6544/ad6b6f","url":null,"abstract":"In this paper, we consider the global well-posedness of the three-dimensional inhomogeneous incompressible Phan-Thien–Tanner (PTT) system with general initial data in the critical Besov spaces. The question of whether or not PTT system is globally well posed for large data is still open. When the density <italic toggle=\"yes\">ρ</italic> is away from zero, we denote by <inline-formula>\u0000<tex-math><?CDATA $varrho: = frac1rho-1.$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mrow><mml:mi>ϱ</mml:mi><mml:mo>:=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>ρ</mml:mi></mml:mfrac><mml:mo>−</mml:mo><mml:mn>1.</mml:mn></mml:mrow></mml:math>\u0000<inline-graphic xlink:href=\"nonad6b6fieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> More precisely, we prove that the PTT system admits a unique global solution, provided that the initial data <italic toggle=\"yes\">ϱ</italic>\u0000<sub>0</sub>, the initial horizontal velocity <inline-formula>\u0000<tex-math><?CDATA $u_{h0}$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>h</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>\u0000<inline-graphic xlink:href=\"nonad6b6fieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>, the product <inline-formula>\u0000<tex-math><?CDATA $omega u_{30}$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mrow><mml:mi>ω</mml:mi><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mn>30</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>\u0000<inline-graphic xlink:href=\"nonad6b6fieqn3.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> of the coupling parameter <italic toggle=\"yes\">ω</italic> and the initial vertical velocity <italic toggle=\"yes\">u</italic>\u0000<sub>30</sub>, and the initial symmetric tensor of constraints <italic toggle=\"yes\">τ</italic>\u0000<sub>0</sub> are sufficiently small. In particular, this result includes the global well-posedness of the PTT system for small initial data in the case where <inline-formula>\u0000<tex-math><?CDATA $omegain[0,1)$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mrow><mml:mi>ω</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math>\u0000<inline-graphic xlink:href=\"nonad6b6fieqn4.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> and for large initial vertical velocity in the case where <italic toggle=\"yes\">ω</italic> tends to zero. As a by-product, our results can be applied to the so-called Oldroyd-B system.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201461","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 : 2024-08-13DOI: 10.1088/1361-6544/ad6acd
Yang Wang, Xuanyu Lv, Fan Liu, Xiaoguang Zhang
In this paper we are concerned with the existence of traveling wave solutions for two species competitive systems with density-dependent diffusion. Since the density-dependent diffusion is a kind of nonlinear diffusion and degenerates at the origin, the methods for proving the existence of traveling wave solutions for competitive systems with linear diffusion are invalid. To overcome the degeneracy of diffusion, we construct a nonlinear invariant region Ω near the origin. Then by using the method of phase plane analysis, we prove the existence of traveling wave solutions connecting the origin and the unique coexistence state, when the speed c is large than some positive value. In addition, when one species is density-dependent diffusive while the other is linear diffusive, via the change of variables and the central manifold theorem, we prove the existence of the minimal speed c∗. And for c⩾c∗, traveling wave solutions connecting the origin and the unique coexistence state still exist. In particular, when c=c∗, we find that one component of the traveling wave solution is sharp type while the other component is smooth, which is a different phenomenon from linear diffusive systems and scalar equations.
本文关注的是具有密度依赖性扩散的两物种竞争系统的行波解的存在性。由于密度相关扩散是一种非线性扩散，并且在原点处退化，因此证明线性扩散竞争系统行波解存在性的方法是无效的。为了克服扩散的退化性，我们在原点附近构建了一个非线性不变区域Ω。然后利用相平面分析方法，证明当速度 c 大于某个正值时，存在连接原点和唯一共存状态的行波解。此外，当一种物质是密度依赖性扩散而另一种物质是线性扩散时，通过变量变化和中心流形定理，我们证明了最小速度 c∗ 的存在。而对于 c⩾c∗，连接原点和唯一共存状态的行波解仍然存在。特别是当 c=c∗ 时，我们发现行波解的一个分量是尖锐型的，而另一个分量是平滑型的，这是与线性扩散系统和标量方程不同的现象。
{"title":"Existence of traveling wave solutions for density-dependent diffusion competitive systems","authors":"Yang Wang, Xuanyu Lv, Fan Liu, Xiaoguang Zhang","doi":"10.1088/1361-6544/ad6acd","DOIUrl":"https://doi.org/10.1088/1361-6544/ad6acd","url":null,"abstract":"In this paper we are concerned with the existence of traveling wave solutions for two species competitive systems with density-dependent diffusion. Since the density-dependent diffusion is a kind of nonlinear diffusion and degenerates at the origin, the methods for proving the existence of traveling wave solutions for competitive systems with linear diffusion are invalid. To overcome the degeneracy of diffusion, we construct a nonlinear invariant region Ω near the origin. Then by using the method of phase plane analysis, we prove the existence of traveling wave solutions connecting the origin and the unique coexistence state, when the speed <italic toggle=\"yes\">c</italic> is large than some positive value. In addition, when one species is density-dependent diffusive while the other is linear diffusive, via the change of variables and the central manifold theorem, we prove the existence of the minimal speed <inline-formula>\u0000<tex-math><?CDATA $c^*$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mrow><mml:msup><mml:mi>c</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:mrow></mml:math>\u0000<inline-graphic xlink:href=\"nonad6acdieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>. And for <inline-formula>\u0000<tex-math><?CDATA $cunicode{x2A7E} c^*$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mrow><mml:mi>c</mml:mi><mml:mtext>⩾</mml:mtext><mml:msup><mml:mi>c</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:mrow></mml:math>\u0000<inline-graphic xlink:href=\"nonad6acdieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>, traveling wave solutions connecting the origin and the unique coexistence state still exist. In particular, when <inline-formula>\u0000<tex-math><?CDATA $c = c^*$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>c</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:mrow></mml:math>\u0000<inline-graphic xlink:href=\"nonad6acdieqn3.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>, we find that one component of the traveling wave solution is sharp type while the other component is smooth, which is a different phenomenon from linear diffusive systems and scalar equations.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201463","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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