Nonlinear Diffusion with Fractional Laplacian Operators

Vázquez, Juan Luís

doi:10.1007/978-3-642-25361-4_15

Cited by 147 publications

(107 citation statements)

References 62 publications

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“…We obtain then finally for the constant of (7.12), , 0 < α < 2, (7.17) which is in accordance with the (inverse) normalization constant given in the literature, e.g. in [40] (and see the references therein) which occurs by defining the (negative) fractional Laplacian (−Δ) (even powers) contributes. By using (7.22)-(7.29) we obtain for (7.31) the series…”

Section: Determination Of a Nαsupporting

confidence: 83%

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The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion

Michelitsch

Maugin

Nowakowski

et al. 2013

fcaa

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We analyze the "fractional continuum limit" and its generalization to n dimensions of a self-similar discrete spring model which we introduced recently [21]. Application of Hamilton's (variational) principle determines in rigorous manner a self-similar and as consequence non-local Laplacian operator. In the fractional continuum limit the discrete self-similar Laplacian takes the form of the fractional Laplacian −(−Δ) α 2 with 0 < α < 2. We analyze the fundamental link of fractal vibrational features of the discrete self-similar spring model and the smooth regular ones of the corresponding fractional continuum limit model in n dimensions: We find a characteristic scaling law for the density of normal modes ∼ ω 828of Lévi flights in n-dimensions. In the limit of "large scaled times" ∼ t/r α >> 1 we show that all distributions exhibit the same asymptotically algebraic decay ∼ t −n/α → 0 independent from the initial distribution and spatial position. This universal scaling depends only on the ratio n/α of the dimension n of the physical space and the Lévi parameter α.MSC 2010: 28A80, 35Q84, 35R11, 60E07, 26A33, 82C31 Keywords and phrases: fractional Laplacian, self-similarity, power laws, scale invariance, fractals, Weierstrass-Mandelbrot function, continuum limit, fractional calculus, Fokker-Planck equation, anomalous diffusion, Lévi flights, Lévi (stable) distributions, non-locality IntroductionSelf-similarity (scaling-invariance) can be found in many problems of physics. There is a need for understanding the dynamics that leads to fractal and self-similarity properties in domains of the physics as varied as flow turbulence and complex materials. In "traditional modelling" the continuum is assumed to have a characteristic length scale which determines the wavelengths where wave fields interact with the microstructure. However, there are numerous materials in nature which are constituted by a scale hierarchy of recurrent microstructure which can be conceived in a good approximation as self-similar. This is true for solids and porous media but also in fluid mechanics, or multicomponent and multiphase flows. As a consequence, there are many areas of modelling that could benefit from a better understanding of the mechanisms underlying the dynamics of objects exhibiting self-similarity properties.In fluid mechanics, synthetic turbulence models are one of many examples of tempering with the input spectrum in order to understand better the physics underlying Lagrangian diffusion [26] and such spectra can be traced to the fractal distributions of velocity accumulation points in the synthetic flow. Fractal approaches are developing rapidly in fluid mechanics. Such approaches consist in either experimentally or numerically interfering with the flow, forcing it through self-similar objects (or through numerical forcing), [36,11,27].On the other hand, there is a large area of research devoted to the so called Fractional Laplacian and a huge number of references exists, employing this operator in various contexts of ...

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Section: Determination Of a Nαsupporting

confidence: 83%

“…[3, 4,5,10,12,13,17,20,29,31,33,34,35,38,40]. In all these analytical models the fractional Laplacian has been introduced in heuristic manner.…”

Section: Introductionmentioning

confidence: 99%

The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion

Michelitsch

Maugin

Nowakowski

et al. 2013

fcaa

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“…See [1] and the references therein for a more detailed introduction. Some interesting models involving the fractional Laplacian have received much attention recently, such as the fractional Schrödinger equation (see [2,8,17,22,23]), the fractional Kirchhoff equation (see [18,25]) and the fractional porous medium equation (see [33]). Another driving force for the study of problem (1.1) arises in the study of the following timedependent local Schrödinger equation…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Fractional NLS equations with magnetic field, critical frequency and critical growth

2017

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Abstract. The paper is devoted to the study of a singularly perturbed fractional Schrödinger equations involving critical frequency and critical growth in the presence of a magnetic field. By using variational methods, we obtain the existence of mountain pass solutions uε which tend to the trivial solution as ε → 0. Moreover, we get infinitely many solutions and sign-changing solutions for the problem in absence of magnetic effects under some extra assumptions.

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“…In all these cases, the nonlocal effect is modeled by the singularity at infinity. For more details and applications, see [6,10,28] and the references therein. In this paper we consider the weighted fractional eigenvalue problem…”

Section: Introductionmentioning

confidence: 99%

Steiner symmetry in the minimization of the first eigenvalue of a fractional eigenvalue problem with indefinite weight

Anedda

Cuccu

Frassu

2020

Can. J. Math.-J. Can. Math.

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Let Ω ⊂ R N , N ≥ 2, be an open bounded connected set. We consider the fractional weighted eigenvalue problem (−∆) s u = λρu in Ω with homogeneous Dirichlet boundary condition, where (−∆) s , s ∈ (0, 1), is the fractional Laplacian operator, λ ∈ R and ρ ∈ L ∞ (Ω). We study weak* continuity, convexity and Gâteaux differentiability of the map ρ → 1/λ 1 (ρ), where λ 1 (ρ) is the first positive eigenvalue. Moreover, denoting by G(ρ 0 ) the class of rearrangements of ρ 0 , we prove the existence of a minimizer of λ 1 (ρ) when ρ varies on G(ρ 0 ). Finally, we show that, if Ω is Steiner symmetric, then every minimizer shares the same symmetry.Recently, great attention has been focused on the study of fractional and nonlocal operators of elliptic type, both for pure mathematical research and in view of concrete real-world applications. This type of operators arises in a quite natural way in many different contexts, such as, among others, the thin obstacle problem,

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Nonlinear Diffusion with Fractional Laplacian Operators

Cited by 147 publications

References 62 publications

The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion

The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion

Fractional NLS equations with magnetic field, critical frequency and critical growth

Steiner symmetry in the minimization of the first eigenvalue of a fractional eigenvalue problem with indefinite weight

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