Pub Date : 2024-08-01DOI: 10.1088/1361-6544/ad61b4
Wenchao Dong
This study focuses on the stability of the viscous contact wave for the one-dimensional full Navier–Stokes–Korteweg equations with density-temperature dependent transport coefficients and large perturbation. Our findings demonstrate a Nishida–Smoller type result, indicating that the solution remains stable under large perturbation as long as is sufficiently small. Notably, the smallness of the capillary coefficient is unnecessary. We then employ the initial layer analysis technique to investigate the asymptotic behaviour in the norm. We show that the capillary term has a smoothing effect, which implies that the strong solution is indeed a smooth one. Our results represent an improvement over those previously reported in Chen and Sheng (2019 Nonlinearity32 395–444). Furthermore, by applying the method in this study to the isothermal case, we can achieve a better outcome than Germain and LeFloch (2016 Commun. Pure Appl. Math.69 3–61).
{"title":"Stability of viscous contact wave for the full compressible Navier–Stokes–Korteweg equations with large perturbation","authors":"Wenchao Dong","doi":"10.1088/1361-6544/ad61b4","DOIUrl":"https://doi.org/10.1088/1361-6544/ad61b4","url":null,"abstract":"This study focuses on the stability of the viscous contact wave for the one-dimensional full Navier–Stokes–Korteweg equations with density-temperature dependent transport coefficients and large perturbation. Our findings demonstrate a Nishida–Smoller type result, indicating that the solution remains stable under large perturbation as long as is sufficiently small. Notably, the smallness of the capillary coefficient is unnecessary. We then employ the initial layer analysis technique to investigate the asymptotic behaviour in the norm. We show that the capillary term has a smoothing effect, which implies that the strong solution is indeed a smooth one. Our results represent an improvement over those previously reported in Chen and Sheng (2019 Nonlinearity32 395–444). Furthermore, by applying the method in this study to the isothermal case, we can achieve a better outcome than Germain and LeFloch (2016 Commun. Pure Appl. Math.69 3–61).","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"21 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882353","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 : 2024-08-01DOI: 10.1088/1361-6544/ad6052
Gaétan Leclerc
Let M be a closed manifold, and let be a Axiom A diffeomorphism. Suppose that f has an attractor Ω with codimension 1 stable lamination. Under a generic nonlinearity condition and a suitable bunching condition, we prove polynomial Fourier decay in the unstable direction for a large class of invariant measures on Ω. Our result applies in particular for the measure of maximal entropy. We construct in the appendix an explicit solenoid that satisfies the nonlinearity and bunching assumption.
设 M 是封闭流形，设 f 是公理 A 差分变形。假设 f 有一个具有标度为 1 的稳定层理的吸引子 Ω。在一般非线性条件和合适的束化条件下，我们证明了Ω上一大类不变度量在不稳定方向上的多项式傅里叶衰减。我们的结果尤其适用于最大熵的度量。我们在附录中构建了一个满足非线性和束状假设的显式螺线管。
{"title":"Fourier decay of equilibrium states for bunched attractors","authors":"Gaétan Leclerc","doi":"10.1088/1361-6544/ad6052","DOIUrl":"https://doi.org/10.1088/1361-6544/ad6052","url":null,"abstract":"Let M be a closed manifold, and let be a Axiom A diffeomorphism. Suppose that f has an attractor Ω with codimension 1 stable lamination. Under a generic nonlinearity condition and a suitable bunching condition, we prove polynomial Fourier decay in the unstable direction for a large class of invariant measures on Ω. Our result applies in particular for the measure of maximal entropy. We construct in the appendix an explicit solenoid that satisfies the nonlinearity and bunching assumption.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"79 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882352","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 : 2024-07-31DOI: 10.1088/1361-6544/ad64a4
Lumin Geng, Jianxun Hu, Chao-Zhong Wu
By using pseudo-differential operators containing two derivations, we extend the Kadomtsev–Petviashvili (KP) hierarchy to a certain KP-mKP hierarchy. For the KP-mKP hierarchy, we derive its Bäcklund transformations, bilinear equations of Baker–Akhiezer functions and Hirota equations of tau functions. Moreover, we show that this hierarchy is equivalent to a subhierarchy of the dispersive Whitham hierarchy associated to the Riemann sphere with its infinity point and one movable point marked.
{"title":"A KP-mKP hierarchy via pseudo-differential operators with two derivations","authors":"Lumin Geng, Jianxun Hu, Chao-Zhong Wu","doi":"10.1088/1361-6544/ad64a4","DOIUrl":"https://doi.org/10.1088/1361-6544/ad64a4","url":null,"abstract":"By using pseudo-differential operators containing two derivations, we extend the Kadomtsev–Petviashvili (KP) hierarchy to a certain KP-mKP hierarchy. For the KP-mKP hierarchy, we derive its Bäcklund transformations, bilinear equations of Baker–Akhiezer functions and Hirota equations of tau functions. Moreover, we show that this hierarchy is equivalent to a subhierarchy of the dispersive Whitham hierarchy associated to the Riemann sphere with its infinity point and one movable point marked.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"52 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870564","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 : 2024-07-25DOI: 10.1088/1361-6544/ad5bb3
Gianluca Crippa, Marco Inversi, Chiara Saffirio and Giorgio Stefani
We consider the Vlasov–Poisson system both in the repulsive (electrostatic potential) and in the attractive (gravitational potential) cases. Our first main theorem yields the analog for the Vlasov–Poisson system of Yudovich’s celebrated well-posedness theorem for the Euler equations: we prove the uniqueness and the quantitative stability of Lagrangian solutions whose associated spatial density is potentially unbounded but belongs to suitable uniformly-localised Yudovich spaces. This requirement imposes a condition of slow growth on the function uniformly in time. Previous works by Loeper, Miot and Holding–Miot have addressed the cases of bounded spatial density, i.e. , and spatial density such that for . Our approach is Lagrangian and relies on an explicit estimate of the modulus of continuity of the electric field and on a second-order Osgood lemma. It also allows for iterated-logarithmic perturbations of the linear growth condition. In our second main theorem, we complement the aforementioned result by constructing solutions whose spatial density sharply satisfies such iterated-logarithmic growth. Our approach relies on real-variable techniques and extends the strategy developed for the Euler equations by the first and fourth-named authors. It also allows for the treatment of more general equations that share the same structure as the Vlasov–Poisson system. Notably, the uniqueness result and the stability estimates hold for both the classical and the relativistic Vlasov–Poisson systems.
{"title":"Existence and stability of weak solutions of the Vlasov–Poisson system in localised Yudovich spaces","authors":"Gianluca Crippa, Marco Inversi, Chiara Saffirio and Giorgio Stefani","doi":"10.1088/1361-6544/ad5bb3","DOIUrl":"https://doi.org/10.1088/1361-6544/ad5bb3","url":null,"abstract":"We consider the Vlasov–Poisson system both in the repulsive (electrostatic potential) and in the attractive (gravitational potential) cases. Our first main theorem yields the analog for the Vlasov–Poisson system of Yudovich’s celebrated well-posedness theorem for the Euler equations: we prove the uniqueness and the quantitative stability of Lagrangian solutions whose associated spatial density is potentially unbounded but belongs to suitable uniformly-localised Yudovich spaces. This requirement imposes a condition of slow growth on the function uniformly in time. Previous works by Loeper, Miot and Holding–Miot have addressed the cases of bounded spatial density, i.e. , and spatial density such that for . Our approach is Lagrangian and relies on an explicit estimate of the modulus of continuity of the electric field and on a second-order Osgood lemma. It also allows for iterated-logarithmic perturbations of the linear growth condition. In our second main theorem, we complement the aforementioned result by constructing solutions whose spatial density sharply satisfies such iterated-logarithmic growth. Our approach relies on real-variable techniques and extends the strategy developed for the Euler equations by the first and fourth-named authors. It also allows for the treatment of more general equations that share the same structure as the Vlasov–Poisson system. Notably, the uniqueness result and the stability estimates hold for both the classical and the relativistic Vlasov–Poisson systems.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"26 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771718","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 : 2024-07-25DOI: 10.1088/1361-6544/ad6126
Xian Zhang, Zhongjie Zhang, Yantao Wang and Xin Wang
For a class of nonlinear differential systems with heterogeneous time-varying delays, including distributed, leakage and transmission time-varying delays, a novel global exponential stability (GES) analysis method was developed. Based on the GES definition, some sufficient conditions and rigorous convergence analysis of nonlinear delayed differential systems are presented directly, which ensure all states to be globally exponentially convergent. The proposed analysis method not only avoids the construction of the Lyapunov–Krasovskii functional, but also uses some simple integral reduction techniques to determine the global exponential convergence rate. Furthermore, the main advantages and low calculation complexity are demonstrated through a theoretical comparison. Finally, three numerical examples are provided to verify the effectiveness of the theoretical results.
{"title":"New global exponential stability conditions for nonlinear delayed differential systems with three kinds of time-varying delays","authors":"Xian Zhang, Zhongjie Zhang, Yantao Wang and Xin Wang","doi":"10.1088/1361-6544/ad6126","DOIUrl":"https://doi.org/10.1088/1361-6544/ad6126","url":null,"abstract":"For a class of nonlinear differential systems with heterogeneous time-varying delays, including distributed, leakage and transmission time-varying delays, a novel global exponential stability (GES) analysis method was developed. Based on the GES definition, some sufficient conditions and rigorous convergence analysis of nonlinear delayed differential systems are presented directly, which ensure all states to be globally exponentially convergent. The proposed analysis method not only avoids the construction of the Lyapunov–Krasovskii functional, but also uses some simple integral reduction techniques to determine the global exponential convergence rate. Furthermore, the main advantages and low calculation complexity are demonstrated through a theoretical comparison. Finally, three numerical examples are provided to verify the effectiveness of the theoretical results.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"35 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771720","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 : 2024-07-24DOI: 10.1088/1361-6544/ad6053
Mark Pollicott and Julia Slipantschuk
We present a practical and effective method for rigorously estimating quantities associated to top eigenvalues of transfer operators to very high precision. The method combines explicit error bounds of the Lagrange-Chebyshev approximation with an established min-max method. We illustrate its applicability by significantly improving rigorous estimates on various ergodic quantities associated to the Bolyai–Rényi map.
{"title":"Effective estimates of ergodic quantities illustrated on the Bolyai-Rényi map","authors":"Mark Pollicott and Julia Slipantschuk","doi":"10.1088/1361-6544/ad6053","DOIUrl":"https://doi.org/10.1088/1361-6544/ad6053","url":null,"abstract":"We present a practical and effective method for rigorously estimating quantities associated to top eigenvalues of transfer operators to very high precision. The method combines explicit error bounds of the Lagrange-Chebyshev approximation with an established min-max method. We illustrate its applicability by significantly improving rigorous estimates on various ergodic quantities associated to the Bolyai–Rényi map.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"198 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771721","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 : 2024-07-23DOI: 10.1088/1361-6544/ad6127
Arein Duaibes and Yulia Karpeshina
The goal is construction of stationary solutions close to non-trivial combinations of two plane waves at high energies for a periodic non-linear Schrödinger Equation in dimension two. The corresponding isoenergetic surface is described for any sufficiently large energy k2. It is shown that the isoenergetic surface corresponding to k2 is essentially different from that for the zero potential even for small potentials. We use a combination of the perturbative results obtained earlier for the linear case and a method of successive approximation.
{"title":"Resonant solutions of the non-linear Schrödinger equation with periodic potential *","authors":"Arein Duaibes and Yulia Karpeshina","doi":"10.1088/1361-6544/ad6127","DOIUrl":"https://doi.org/10.1088/1361-6544/ad6127","url":null,"abstract":"The goal is construction of stationary solutions close to non-trivial combinations of two plane waves at high energies for a periodic non-linear Schrödinger Equation in dimension two. The corresponding isoenergetic surface is described for any sufficiently large energy k2. It is shown that the isoenergetic surface corresponding to k2 is essentially different from that for the zero potential even for small potentials. We use a combination of the perturbative results obtained earlier for the linear case and a method of successive approximation.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"21 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771722","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 : 2024-07-22DOI: 10.1088/1361-6544/ad5fd8
Gloria Marí Beffa
In this paper we define a Poisson pencil associated to a lattice Wm-algebras defined in a recent paper by Izosimov and Marí Beffa (2023 Int. Math. Res. Not.2023 17021–59). We then prove that this Poisson pencil is equal to the one defined in 2013 by Marí Beffa and Wang (2013 Nonlinearity26 2515) and the author using a type of discrete Drinfel’d–Sokolov reduction. We then show that, much as in the continuous case, a family of Hamiltonians defined by fractional powers of difference operators commute with respect to both structures, defining the kernel of one of them and creating an integrable hierarchy in the Liouville sense.
{"title":"Wm -algebras and fractional powers of difference operators","authors":"Gloria Marí Beffa","doi":"10.1088/1361-6544/ad5fd8","DOIUrl":"https://doi.org/10.1088/1361-6544/ad5fd8","url":null,"abstract":"In this paper we define a Poisson pencil associated to a lattice Wm-algebras defined in a recent paper by Izosimov and Marí Beffa (2023 Int. Math. Res. Not.2023 17021–59). We then prove that this Poisson pencil is equal to the one defined in 2013 by Marí Beffa and Wang (2013 Nonlinearity26 2515) and the author using a type of discrete Drinfel’d–Sokolov reduction. We then show that, much as in the continuous case, a family of Hamiltonians defined by fractional powers of difference operators commute with respect to both structures, defining the kernel of one of them and creating an integrable hierarchy in the Liouville sense.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"28 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141754117","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 : 2024-07-22DOI: 10.1088/1361-6544/ad5924
Ethan Dudley
Necessary and sufficient conditions for the third order Kolmogorov universal scaling flux laws are derived for the stochastically forced incompressible Navier Stokes equations on the torus in 2D and 3D. This paper rigorously generalises the result of (Bedrossian 2019 Commun. Math. Phys.367 1045–75) to functions which are heavy-tailed in Fourier space or have local finite time singularities in the inviscid limit. In other words, we have rigorously derived the existence of the well known physical relationships, the direct and inverse cascades. Furthermore we show that the rate of the direct cascade is proportional to the amount of energy ‘escaping to infinity’ in spectral space as well as a measure of the total singularities within the solution. Similarly, an inverse cascade is proportional to the amount of energy that moves towards the k = 0 Fourier mode in the invisicid limit.
{"title":"Necessary and sufficient conditions for Kolmogorov’s flux laws on T2 and T3","authors":"Ethan Dudley","doi":"10.1088/1361-6544/ad5924","DOIUrl":"https://doi.org/10.1088/1361-6544/ad5924","url":null,"abstract":"Necessary and sufficient conditions for the third order Kolmogorov universal scaling flux laws are derived for the stochastically forced incompressible Navier Stokes equations on the torus in 2D and 3D. This paper rigorously generalises the result of (Bedrossian 2019 Commun. Math. Phys.367 1045–75) to functions which are heavy-tailed in Fourier space or have local finite time singularities in the inviscid limit. In other words, we have rigorously derived the existence of the well known physical relationships, the direct and inverse cascades. Furthermore we show that the rate of the direct cascade is proportional to the amount of energy ‘escaping to infinity’ in spectral space as well as a measure of the total singularities within the solution. Similarly, an inverse cascade is proportional to the amount of energy that moves towards the k = 0 Fourier mode in the invisicid limit.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"75 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141754115","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 : 2024-07-21DOI: 10.1088/1361-6544/ad6054
Keefer Rowan
We show that by ‘accelerating’ relaxation enhancing flows, one can construct a flow that is smooth on but highly singular at t = 1 so that for any positive diffusivity, the advection–diffusion equation associated to the accelerated flow totally dissipates solutions, taking arbitrary initial data to the constant function at t = 1.
我们的研究表明，通过 "加速 "松弛增强流，我们可以构造出一种在 t = 1 时平滑但高度奇异的流，这样，对于任何正扩散率，加速流相关的平流-扩散方程都会完全耗散解，在 t = 1 时取任意初始数据为常数函数。
{"title":"Accelerated relaxation enhancing flows cause total dissipation","authors":"Keefer Rowan","doi":"10.1088/1361-6544/ad6054","DOIUrl":"https://doi.org/10.1088/1361-6544/ad6054","url":null,"abstract":"We show that by ‘accelerating’ relaxation enhancing flows, one can construct a flow that is smooth on but highly singular at t = 1 so that for any positive diffusivity, the advection–diffusion equation associated to the accelerated flow totally dissipates solutions, taking arbitrary initial data to the constant function at t = 1.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"105 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743821","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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