Fractional Order Sobolev Spaces Research Articles

Local existence of classical solutions for the Einstein–Euler system using weighted Sobolev spaces of fractional order

We prove the existence of a class of local in time solutions, including static solutions, of the Einstein–Euler system. This result is the relativistic generalisation of a similar result for the Euler–Poisson system obtained by Gamblin (1993). As in his case the initial data of the density do not have compact support but fall off at infinity in an appropriate manner. An essential tool in our approach is the construction and use of weighted Sobolev spaces of fractional order. Moreover, these new spaces allow us to improve the regularity conditions for the solutions of evolution equations. The details of this construction, the properties of these spaces and results on elliptic and hyperbolic equations will be presented in a forthcoming article. To cite this article: U. Brauer, L. Karp, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

A Note on the Relaxation-Time Limit of the Isothermal Euler Equations

This work is concerned with the relaxation-time limit of the multidimensional isothermal Euler equations with relaxation. We show that Coulombel-Goudon's results (2007) can hold in the weaker and more general Sobolev space of fractional order. The method of proof used is the Littlewood-Paley decomposition.

Construction of Sobolev spaces of fractional order with sub-riemannian vector fields

Étant donné une famille de champs de vecteurs 𝒵 vérifiant la condition de Chow-Hörmander de rang 2 et une hypothèse de régularité, on montre que les espaces de Sobolev d’ordre fractionnaire construits par les procédés standards de l’analyse fonctionnelle admettent une caractérisation géométrique à l’aide de la distance sous-riemannienne induite par la famille 𝒵. L’approche proposée est basée sur une analyse microlocale des opérateurs de translation dans un contexte anisotrope. Elle utilise aussi des estimations classiques du noyau de la chaleur associé à un Laplacien sous-elliptique.

Regularity in Sobolev spaces of steady flows of fluids with shear‐dependent viscosity

AbstractThe system is considered on a bounded three‐dimensional domain under no‐stick boundary value conditions, where S has p‐structure for some p<2 and D(u) is the symmetrized gradient of u. Various regularity results for the velocity u and the pressure π in fractional order Sobolev and Nikolskii spaces are obtained. Copyright © 2006 John Wiley & Sons, Ltd.

Supraconvergence of a Finite Difference Scheme for Elliptic Boundary Value Problems of the Third Kind in Fractional Order Sobolev Spaces

AbstractIn this paper, we study the convergence of the finite difference discretization of a second order elliptic equation with variable coefficients subject to general boundary conditions. We prove that the scheme exhibits the phenomenon of supraconvergence on nonuniform grids, i.e., although the truncation error is in general of the first order alone, one has second order convergence. All error estimates are strictly local. Another result of the paper is a close relationship between finite difference scheme and linear finite element methods combined with a special kind of quadrature. As a consequence, the results of the paper can be viewed as the introduction of a fully discrete finite element method for which the gradient is superclose. A numerical example is given.

Global Regularity in Fractional Order Sobolev Spaces for the p-Laplace Equation on Polyhedral Domains

The p -Laplace equation is considered for p > 2 on a n -dimensional convex polyhedral domain under a Dirichlet boundary value condition. Global regularity of weak solutions in weighted Sobolev spaces and in fractional order Nikolskij and Sobolev spaces are proven.

Global regularity in Nikolskij spaces for elliptic equations with p-structure on polyhedral domains

Nonlinear elliptic equations with p-structure on non-convex polyhedral domains under homogeneous Dirichlet boundary values are considered. Global regularity in fractional order Nikolskij and Sobolev spaces is proved.

Wavelet-based functional reconstruction and extrapolation of fractional random fields

Least-squares linear functional reconstruction and extrapolation of a random field defining the random input of a linear system represented by an integral equation is considered. This problem is solved for a class of random fields with reproducing kernel Hilbert space norm equivalent to the norm of a Sobolev space of an appropriate fractional order. More specifically, functional reconstruction and extrapolation formulae are derived from generalized wavelet-based orthogonal expansions of the input and output random fields in the class considered (see Angulo and Ruiz-Medina, 1999, for the ordinary case). In the Gaussian and ordinary case, the results derived also provide sample-path functional reconstruction and extrapolation formulae. Simulation studies are carried out for systems defined in terms of fractional integration of fractional Brownian motion.

Regularity in Sobolev spaces for doubly nonlinear parabolic equations

The doubly nonlinear parabolic equation u t= div[| ∇(|u| m−1u)| p−2 ∇(|u| m−1u)] (m>1,m(p−1)>1) is considered in several dimensions and regularity results in fractional order Sobolev spaces are obtained. The main tools in the proof are a difference quotient technique and the imbedding theorem of Nikolskii spaces into Sobolev spaces.

Mixed Boundary Value Problems for Nonlinear Elliptic Systems withp-Structure in Polyhedral Domains

The nonlinear elliptic system is investigated on a non-smooth domain. Mixed boundary value conditions are given. The left-hand side of the system has p-structure (e.g., it is the p-Laplacian and 1 < p < ∞). Global regularity results of u and |∇u|p/2 in fractional order Sobolev spaces are proven.

Sobolev Spaces on Non Smooth Domains and Dirichlet Forms Related to Subordinate Reflecting Diffusions

Let Ω be a bounded domain with fractal boundary, for instance von Koch's snowflake domain. First we determine the range and the kernel of the trace on ∂Ω of Sobolev spaces of fractional order defined on Ω. This extends some earlier results of H. Wallin and J. Marschall Secondly we apply these results in studying Dirichlet forms related to subordinate reflecting diffusions in non–smooth domains.

Equivalence of Sobolev Spaces

We repair the proof of equivalence of certain L2-Sobolev spaces on manifolds with bounded curvature of all orders from [4]. The results are extended to generalized compatible Dirac operators, fractional order Sobolev spaces and weighted Sobolev spaces. A certain way of doing coordinate free computations is presented.

Nonlinear elliptic problems with $p$-structure under mixed boundary value conditions in polyhedral domains

Nonlinear elliptic systems with $p$-structure for $p \! \geq \! 2$, such as \[ -\mbox{div}\left((1+|\nabla u|^{p-2})\nabla u\right) =f(x)+\sum_{i=1}^n\partial_if_i(x) ,\] are considered under mixed boundary value conditions on nonsmooth domains. Regularity results in fractional-order Sobolev spaces are proven, e.g., $u\in W^{r,p}(\Omega)$ for all $r <1+\frac{1}{p}$ and $|\nabla u|^p\in W^{s,1}(\Omega)$ for some $s>1$.

Scales of ε-uniform a priori estimates for nonstationary convection-dominated diffusion problems

Convection-diffusion problems of practical interest are often convection-dominated in the sense that the diffusion parameter e satisfies 0<e≪1. Standard error estimates for numerical approximations of such problems mostly contain constants which depend either reciprocally on e or at least on higher order norms of the solution. These norms possibly increase to infinity in the hyperbolic limit e→0. For the Lagrange-Galerkin scheme such error estimates have been proven by Douglas and Russell, Pironneau or Suli. In the particular case of the Navier-Stokes equations Johnson, Rannacher and Boman have observed that some of those constants which arise in existing analyses depend exponentially on the Reynolds number. Consequently, standard error analyses have no real meaning in the case of convection-dominance. Recently, Bause and Knabner have developed a rigorous e-uniform convergence theory for finite element and Lagrange-Galerkin approximations of convection-dominated diffusion problems. The error analysis is heavily based on e-uniform a priori estimates for the solution of the continuous problems. In their work such e-uniform estimates are established in the Sobolev spacesL2(Ω) andH2(Ω) in a Lagrangian framework by transforming the convection-diffusion problem into subcharacteristic coordinates. To obtain the e-uniformH2-bound, strong conditions about the right-hand side which cannot realistically be assumed in practice are supposed. In this paper we demonstrate how the method of complex interpolation of Banach spaces can be employed to derive elegantly e-uniform a priori estimates in intermediate Sobolev spaces of fractional order. We will observe that in particularHα-regularity of the data has to be supposed to get an e-uniformHα-bound for the solution. To derive the desired estimates we use several complex interpolation results which we provide first.

Stokes waves in Hardy spaces and as distributions

This paper deals with some functional-analytic questions which arise when the Stokes-wave problem, for the free boundary of a steady irrotational water wave, is formulated as a quadratic equation for a 2 π -periodic, real-valued function w on R which need not be weakly differentiable. It is shown how any solution w of bounded variation which lies in the fractional order Sobolev space H 1/2 must be real-analytic and describes the profile of a steady water wave. The investigation involves only elementary real and complex Hardy space theory.

Convergence of Galerkin method solutions of the integral equation for thin wire antennas

In this paper we consider the Pocklington integro–differential equation for the current induced on a straight, thin wire by an incident harmonic electromagnetic field. We show that this problem is well posed in suitable fractional order Sobolev spaces and obtain a coercive or Garding type inequality for the associated operator. Combining this coercive inequality with a standard abstract formulation of the Galerkin method we obtain rigorous convergence results for Galerkin type numerical solutions of Pocklington's equation, and we demonstrate that certain convergence rates hold for these methods.

On the existence of solutions of nonlinear boundary value problems at resonance in Sobolev spaces of fractional order

On the existence of solutions of nonlinear boundary value problems at resonance in Sobolev spaces of fractional order

Low Energy Scattering for Nonlinear Schrödinger Equations in Fractional Order Sobolev Spaces

We consider the scattering problem for the nonlinear Schrödinger equations with interactions behaving as a power p at zero. In the critical and subcritical cases (s≥n/2-2/(p-1)≥0), we prove the existence and asymptotic completeness of wave operators in the sense of Sobolev norm of order s on a set of asymptotic states with small homogeneous norm of order n/2-2/(p-1) in space dimension n≥1.

Necessary conditions on composition operators acting on Sobolev spaces of fractional order. The critical case 1&lt;s&lt;n/p

Let G : R→ R be a sufficiently smooth function. Denote by TG the corresponding composition operator which sends f to G(f). Then we prove necessary conditions on s, p, r, and t such that the inclusion

Semilinear parabolic equations with distributions as initial data

We study the local Cauchy problem for the semilinear parabolic equations <p align="center"> $\partial _t U-\Delta U=P(D)F(U), \quad (t,x) \in [0,T[ \times \mathbb{R}^n $ <p align="left" class="times"> with initial data in Sobolev spaces of fractional order $H^s_p(\mathbb{R}^n)$. The techniques that we use allow us to consider measures but also distributions as initial data ($s<0$). We also prove some smoothing effects and $L^q([0,T[,L^p)$ estimates for the solutions of such equations.