By bifurcation and topological methods, we study the existence/nonexistence and multiplicity of one-sign or nodal solutions of the following k-th mean curvature problem in Minkowski spacetime r N − k v ′ 1 − v ′ 2 k ′ = λ N C N k r N − 1 H k ( r , v ) in ( 0 , R ) , | v ′ | < 1 in ( 0 , R ) , v ′ ( 0 ) = v ( R ) = 0 . As a previous step, we investigate the spectral structure of its linearized problem at zero. Moreover, we also obtain a priori bounds and the asymptotic behaviors of solutions with respect to λ.
利用分岔方法和拓扑方法，研究了Minkowski时空r N−k v ' 1−v ' 2 k ' = λ N C N k r N−1 H k (r, v) in (0, r)， | v ' <中k-平均曲率问题的一符号或节点解的存在性/不存在性和多重性;1 in (0, R) v ' (0) = v (R) = 0。作为前一步，我们研究了它在零处线性化问题的谱结构。此外，我们还得到了关于λ的先验界和解的渐近性质。
{"title":"Spectrum, bifurcation and hypersurfaces of prescribed k-th mean curvature in Minkowski space","authors":"Guowei Dai, Zhitao Zhang","doi":"10.3233/asy-231877","DOIUrl":"https://doi.org/10.3233/asy-231877","url":null,"abstract":"By bifurcation and topological methods, we study the existence/nonexistence and multiplicity of one-sign or nodal solutions of the following k-th mean curvature problem in Minkowski spacetime r N − k v ′ 1 − v ′ 2 k ′ = λ N C N k r N − 1 H k ( r , v ) in ( 0 , R ) , | v ′ | < 1 in ( 0 , R ) , v ′ ( 0 ) = v ( R ) = 0 . As a previous step, we investigate the spectral structure of its linearized problem at zero. Moreover, we also obtain a priori bounds and the asymptotic behaviors of solutions with respect to λ.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135137212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, the Landau–Lifshitz–Baryakhtar (LLBar) equation for magnetization dynamics in ferrimagnets is considered. We prove global existence of a periodic solutions as well as local existence and uniqueness of regular solutions. We also study the relationships between the Landau–Lifshitz–Baryakhtar equation and both Landau–Lifshitz–Bloch and harmonic map equations.
{"title":"Existence results for the Landau–Lifshitz–Baryakhtar equation","authors":"C. Ayouch, D. Meskine, M. Tilioua","doi":"10.3233/asy-231874","DOIUrl":"https://doi.org/10.3233/asy-231874","url":null,"abstract":"In this paper, the Landau–Lifshitz–Baryakhtar (LLBar) equation for magnetization dynamics in ferrimagnets is considered. We prove global existence of a periodic solutions as well as local existence and uniqueness of regular solutions. We also study the relationships between the Landau–Lifshitz–Baryakhtar equation and both Landau–Lifshitz–Bloch and harmonic map equations.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135137217","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive the dynamic boundary condition for the heat equation as a limit of boundary layer problems. We study convergence of their weak and strong solutions as the width of the layer tends to zero. We also discuss Γ-convergence of the functionals generating these flows. Our analysis of strong solutions depends on a new version of the Reilly identity.
{"title":"The heat equation with the dynamic boundary condition as a singular limit of problems degenerating at the boundary","authors":"Yoshikazu Giga, Michał Łasica, Piotr Rybka","doi":"10.3233/asy-231862","DOIUrl":"https://doi.org/10.3233/asy-231862","url":null,"abstract":"We derive the dynamic boundary condition for the heat equation as a limit of boundary layer problems. We study convergence of their weak and strong solutions as the width of the layer tends to zero. We also discuss Γ-convergence of the functionals generating these flows. Our analysis of strong solutions depends on a new version of the Reilly identity.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135087348","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we introduce the (unipolar) pressureless Euler–Poisswell equation for electrons as the O ( 1 / c ) semi-relativistic approximation and the (unipolar) pressureless Euler–Darwin equation as the O ( 1 / c 2 ) semi-relativistic approximation of the (unipolar) pressureless Euler–Maxwell equation. In the “infinity-ion-mass” limit, the unipolar Euler–Maxwell equation arises from the bipolar Euler–Maxwell equation, describing a coupled system for a plasma of electrons and ions. The non-relativistic limit c → ∞ of the Euler–Maxwell equation is the repulsive Euler–Poisson equation with electric force. We propose that the Euler–Poisswell equation, where the Euler equation with electric force is coupled to the magnetostatic O ( 1 / c ) approximation of Maxwell’s equations, is the correct semi-relativistic O ( 1 / c ) approximation of the Euler–Maxwell equation. In the Euler–Poisswell equation the fluid dynamics are decoupled from the magnetic field since the Lorentz force reduces to the electric force. The first non-trivial interaction with the magnetic field is at the O ( 1 / c 2 ) level in the Euler–Darwin equation. This is a consistent and self-consistent model of order O ( 1 / c 2 ) and includes the full Lorentz force, which is relativistic at O ( 1 / c 2 ). The Euler–Poisswell equation also arises as the semiclassical limit of the quantum Pauli–Poisswell equation, which is the O ( 1 / c ) approximation of the relativistic Dirac–Maxwell equation. We also present a local wellposedness theory for the Euler–Poisswell equation. The Euler–Maxwell system couples the non-relativistic Euler equation and the relativistic Maxwell equations and thus it is not fully consistent in 1 / c. The consistent fully relativistic model is the relativistic Euler–Maxwell equation where Maxwell’s equations are coupled to the relativistic Euler equation – a model that is neglected in the mathematics community. We also present the relativistic Euler–Darwin equation resulting as a O ( 1 / c 2 ) approximation of this model.
本文将电子的(单极)无压欧拉-泊斯威尔方程作为O (1 / c)半相对论近似，将(单极)无压欧拉-达尔文方程作为(单极)无压欧拉-麦克斯韦方程的O (1 / c)半相对论近似引入。在“无限离子质量”极限中，单极欧拉-麦克斯韦方程由双极欧拉-麦克斯韦方程衍生而来，描述了电子和离子等离子体的耦合系统。欧拉-麦克斯韦方程的非相对论极限c→∞是带电磁力的排斥欧拉-泊松方程。我们提出欧拉-泊斯韦尔方程是欧拉-麦克斯韦方程的正确的半相对论性O (1 / c)近似，其中欧拉方程与电磁力耦合到麦克斯韦方程的静磁O (1 / c)近似。在欧拉-泊斯韦尔方程中，流体动力学与磁场解耦，因为洛伦兹力减小为电磁力。在欧拉-达尔文方程中，第一个与磁场的非平凡相互作用是在0 (1 / c 2)水平上。这是一个O (1 / c 2)阶的一致和自一致的模型，并包括完整的洛伦兹力，它在O (1 / c 2)是相对论性的。欧拉-泊斯韦尔方程也是量子泡利-泊斯韦尔方程的半经典极限，它是相对论性狄拉克-麦克斯韦方程的0 (1 / c)近似。我们也给出了Euler-Poisswell方程的局部适定性理论。欧拉-麦克斯韦系统耦合了非相对论性欧拉方程和相对论性麦克斯韦方程，因此它在1 / c中不是完全一致的。一致的完全相对论性模型是相对论性欧拉-麦克斯韦方程，其中麦克斯韦方程与相对论性欧拉方程耦合-一个在数学界被忽视的模型。我们还提出了相对论欧拉-达尔文方程，作为该模型的O (1 / c 2)近似值。
{"title":"The Euler–Poisswell/Darwin equation and the asymptotic hierarchy of the Euler–Maxwell equation","authors":"Jakob Möller, Norbert J. Mauser","doi":"10.3233/asy-231864","DOIUrl":"https://doi.org/10.3233/asy-231864","url":null,"abstract":"In this paper we introduce the (unipolar) pressureless Euler–Poisswell equation for electrons as the O ( 1 / c ) semi-relativistic approximation and the (unipolar) pressureless Euler–Darwin equation as the O ( 1 / c 2 ) semi-relativistic approximation of the (unipolar) pressureless Euler–Maxwell equation. In the “infinity-ion-mass” limit, the unipolar Euler–Maxwell equation arises from the bipolar Euler–Maxwell equation, describing a coupled system for a plasma of electrons and ions. The non-relativistic limit c → ∞ of the Euler–Maxwell equation is the repulsive Euler–Poisson equation with electric force. We propose that the Euler–Poisswell equation, where the Euler equation with electric force is coupled to the magnetostatic O ( 1 / c ) approximation of Maxwell’s equations, is the correct semi-relativistic O ( 1 / c ) approximation of the Euler–Maxwell equation. In the Euler–Poisswell equation the fluid dynamics are decoupled from the magnetic field since the Lorentz force reduces to the electric force. The first non-trivial interaction with the magnetic field is at the O ( 1 / c 2 ) level in the Euler–Darwin equation. This is a consistent and self-consistent model of order O ( 1 / c 2 ) and includes the full Lorentz force, which is relativistic at O ( 1 / c 2 ). The Euler–Poisswell equation also arises as the semiclassical limit of the quantum Pauli–Poisswell equation, which is the O ( 1 / c ) approximation of the relativistic Dirac–Maxwell equation. We also present a local wellposedness theory for the Euler–Poisswell equation. The Euler–Maxwell system couples the non-relativistic Euler equation and the relativistic Maxwell equations and thus it is not fully consistent in 1 / c. The consistent fully relativistic model is the relativistic Euler–Maxwell equation where Maxwell’s equations are coupled to the relativistic Euler equation – a model that is neglected in the mathematics community. We also present the relativistic Euler–Darwin equation resulting as a O ( 1 / c 2 ) approximation of this model.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135088512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Barbara Brandolini, Ida de Bonis, Vincenzo Ferone, Bruno Volzone
We provide symmetrization results in the form of mass concentration comparisons for fractional singular elliptic equations in bounded domains, coupled with homogeneous external Dirichlet conditions. Two types of comparison results are presented, depending on the summability of the right-hand side of the equation. The maximum principle arguments employed in the core of the proofs of the main results offer a nonstandard, flexible alternative to the ones described in (Arch. Ration. Mech. Anal. 239 (2021) 1733–1770, Theorem 31). Some interesting consequences are L p regularity results and nonlocal energy estimates for solutions.
{"title":"Comparison results for a nonlocal singular elliptic problem","authors":"Barbara Brandolini, Ida de Bonis, Vincenzo Ferone, Bruno Volzone","doi":"10.3233/asy-231860","DOIUrl":"https://doi.org/10.3233/asy-231860","url":null,"abstract":"We provide symmetrization results in the form of mass concentration comparisons for fractional singular elliptic equations in bounded domains, coupled with homogeneous external Dirichlet conditions. Two types of comparison results are presented, depending on the summability of the right-hand side of the equation. The maximum principle arguments employed in the core of the proofs of the main results offer a nonstandard, flexible alternative to the ones described in (Arch. Ration. Mech. Anal. 239 (2021) 1733–1770, Theorem 31). Some interesting consequences are L p regularity results and nonlocal energy estimates for solutions.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135087589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper is devoted to the derivation, by linearization, of simplified homogenized models of an immiscible incompressible two-phase flow in double porosity media in the case of thin fissures. In a simplified double porosity model derived previously by the authors the matrix-fracture source term is approximated by a convolution type source term. This approach enables to exclude the cell problem, in form of the imbibition equation, from the global double porosity model. In this paper we propose a new linear version of the imbibition equation which leads to a new simplified double porosity model. We also present numerical simulations which show that the matrix-fracture exchange term based on this new linearization procedure gives a better approximation of the exact one than the corresponding exchange term obtained earlier by the authors.
{"title":"Some remarks on simplified double porosity model of immiscible incompressible two-phase flow","authors":"M. Jurak, L. Pankratov, A. Vrbaški","doi":"10.3233/asy-231866","DOIUrl":"https://doi.org/10.3233/asy-231866","url":null,"abstract":"The paper is devoted to the derivation, by linearization, of simplified homogenized models of an immiscible incompressible two-phase flow in double porosity media in the case of thin fissures. In a simplified double porosity model derived previously by the authors the matrix-fracture source term is approximated by a convolution type source term. This approach enables to exclude the cell problem, in form of the imbibition equation, from the global double porosity model. In this paper we propose a new linear version of the imbibition equation which leads to a new simplified double porosity model. We also present numerical simulations which show that the matrix-fracture exchange term based on this new linearization procedure gives a better approximation of the exact one than the corresponding exchange term obtained earlier by the authors.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135088347","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we consider a viscoelastic plate equation with a logarithmic nonlinearity in the presence of nonlinear frictional damping term. Here we prove the existence of the solution to the problem using the Faedo–Galerkin method. Also, we prove few general decay rate results.
{"title":"Global existence and asymptotic behaviour for a viscoelastic plate equation with nonlinear damping and logarithmic nonlinearity","authors":"Bhargav Kumar Kakumani, Suman Prabha Yadav","doi":"10.3233/asy-231859","DOIUrl":"https://doi.org/10.3233/asy-231859","url":null,"abstract":"In this article, we consider a viscoelastic plate equation with a logarithmic nonlinearity in the presence of nonlinear frictional damping term. Here we prove the existence of the solution to the problem using the Faedo–Galerkin method. Also, we prove few general decay rate results.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135087591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We discuss the existence and non-existence of periodic in one variable and compactly supported in the other variables least energy solutions for equations with non-Lipschitz nonlinearity of the form: − Δ u = λ u p − u q in R N + 1 , where 0 < q < p < 1 and λ ∈ R. The approach is based on the Nehari manifold method supplemented by a one-sided constraint given through the functional of the suitable Pohozaev identity. The limit value of the parameter λ, where the approach is applicable, corresponds to the existence of periodic in one variable and compactly supported in the other variables least energy solutions. This value is found through the extrem values of nonlinear generalized Rayleigh quotients and the so-called curve of the critical exponents of p, q. Important properties of the solutions are derived for suitable ranges of the parameters, such as that they are not trivial with respect to the periodic variable and do not coincide with compactly supported solutions on the entire space R N + 1 .
我们讨论了一类非lipschitz非线性方程的最小能量解的存在性和不存在性:−Δ u = λ up−u q in R N + 1，其中0 <问& lt;p & lt;该方法基于Nehari流形方法，辅以通过适当的Pohozaev恒等式的泛函给出的单侧约束。当该方法适用时，参数λ的极限值对应于一个变量周期解的存在性和其他变量最小能量解的紧支持性。这个值是通过非线性广义瑞利商的极值和所谓的p, q的临界指数曲线得到的。在适当的参数范围内，得到了解的重要性质，例如它们对周期变量不平凡，并且不与整个空间rn + 1上的紧支持解相一致。
{"title":"On periodic and compactly supported least energy solutions to semilinear elliptic equations with non-Lipschitz nonlinearity","authors":"Jacques Giacomoni, Yavdat Il’yasov, Deepak Kumar","doi":"10.3233/asy-231878","DOIUrl":"https://doi.org/10.3233/asy-231878","url":null,"abstract":"We discuss the existence and non-existence of periodic in one variable and compactly supported in the other variables least energy solutions for equations with non-Lipschitz nonlinearity of the form: − Δ u = λ u p − u q in R N + 1 , where 0 < q < p < 1 and λ ∈ R. The approach is based on the Nehari manifold method supplemented by a one-sided constraint given through the functional of the suitable Pohozaev identity. The limit value of the parameter λ, where the approach is applicable, corresponds to the existence of periodic in one variable and compactly supported in the other variables least energy solutions. This value is found through the extrem values of nonlinear generalized Rayleigh quotients and the so-called curve of the critical exponents of p, q. Important properties of the solutions are derived for suitable ranges of the parameters, such as that they are not trivial with respect to the periodic variable and do not coincide with compactly supported solutions on the entire space R N + 1 .","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135137214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article completes the study of the influence of the intensity parameter α in the boundary condition ε ∂ ν ε u ε − u ε V ε → · ν ε = ε α φ ε given on the boundary of a thin three-dimensional graph-like network consisting of thin cylinders that are interconnected by small domains (nodes) with diameters of order O ( ε ). Inside of the thin network a time-dependent convection-diffusion equation with high Péclet number of order O ( ε − 1 ) is considered. The novelty of this article is the case of α < 1, which indicates a strong intensity of physical processes on the boundary, described by the inhomogeneity φ ε (the cases α = 1 and α > 1 were previously studied by the same authors). A complete Puiseux asymptotic expansion is constructed for the solution u ε as ε → 0, i.e., when the diffusion coefficients are eliminated and the thin network shrinks into a graph. Furthermore, the corresponding uniform pointwise and energy estimates are proved, which provide an approximation of the solution with a given accuracy in terms of the parameter ε.
{"title":"Puiseux asymptotic expansions for convection-dominated transport problems in thin graph-like networks: Strong boundary interactions","authors":"Taras Mel’nyk, Christian Rohde","doi":"10.3233/asy-231876","DOIUrl":"https://doi.org/10.3233/asy-231876","url":null,"abstract":"This article completes the study of the influence of the intensity parameter α in the boundary condition ε ∂ ν ε u ε − u ε V ε → · ν ε = ε α φ ε given on the boundary of a thin three-dimensional graph-like network consisting of thin cylinders that are interconnected by small domains (nodes) with diameters of order O ( ε ). Inside of the thin network a time-dependent convection-diffusion equation with high Péclet number of order O ( ε − 1 ) is considered. The novelty of this article is the case of α < 1, which indicates a strong intensity of physical processes on the boundary, described by the inhomogeneity φ ε (the cases α = 1 and α > 1 were previously studied by the same authors). A complete Puiseux asymptotic expansion is constructed for the solution u ε as ε → 0, i.e., when the diffusion coefficients are eliminated and the thin network shrinks into a graph. Furthermore, the corresponding uniform pointwise and energy estimates are proved, which provide an approximation of the solution with a given accuracy in terms of the parameter ε.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135137224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we extend the spectral method developed (Dechicha and Puel (2023)) to any dimension d ⩾ 1, in order to construct an eigen-solution for the Fokker–Planck operator with heavy tail equilibria, of the form ( 1 + | v | 2 ) − β 2 , in the range β ∈ ] d , d + 4 [. The method developed in dimension 1 was inspired by the work of H. Koch on nonlinear KdV equation (Nonlinearity 28 (2015) 545). The strategy in this paper is the same as in dimension 1 but the tools are different, since dimension 1 was based on ODE methods. As a direct consequence of our construction, we obtain the fractional diffusion limit for the kinetic Fokker–Planck equation, for the correct density ρ : = ∫ R d f d v, with a fractional Laplacian κ ( − Δ ) β − d + 2 6 and a positive diffusion coefficient κ.
在本文中，我们将开发的频谱方法(Dechicha和Puel(2023))扩展到任何维度d大于或等于1，以便为具有重尾平衡的Fokker-Planck算子构建一个特征解，形式为(1 + | v | 2)−β 2，在β∈]d, d + 4[。在维1中开发的方法受到H. Koch关于非线性KdV方程(非线性28(2015)545)的工作的启发。本文中的策略与维度1中的相同，但工具不同，因为维度1是基于ODE方法的。作为我们构造的直接结果，我们得到了动力学Fokker-Planck方程的分数扩散极限，对于正确的密度ρ: =∫R d f d v，具有分数阶拉普拉斯算子κ(−Δ) β - d + 26和正扩散系数κ。
{"title":"Fractional diffusion for Fokker–Planck equation with heavy tail equilibrium: An à la Koch spectral method in any dimension","authors":"Dahmane Dechicha, Marjolaine Puel","doi":"10.3233/asy-231870","DOIUrl":"https://doi.org/10.3233/asy-231870","url":null,"abstract":"In this paper, we extend the spectral method developed (Dechicha and Puel (2023)) to any dimension d ⩾ 1, in order to construct an eigen-solution for the Fokker–Planck operator with heavy tail equilibria, of the form ( 1 + | v | 2 ) − β 2 , in the range β ∈ ] d , d + 4 [. The method developed in dimension 1 was inspired by the work of H. Koch on nonlinear KdV equation (Nonlinearity 28 (2015) 545). The strategy in this paper is the same as in dimension 1 but the tools are different, since dimension 1 was based on ODE methods. As a direct consequence of our construction, we obtain the fractional diffusion limit for the kinetic Fokker–Planck equation, for the correct density ρ : = ∫ R d f d v, with a fractional Laplacian κ ( − Δ ) β − d + 2 6 and a positive diffusion coefficient κ.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135316388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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