Integral representation for fractional Laplace–Beltrami operators

Abstract

In this paper we provide an integral representation of the fractional Laplace-Beltrami operator for general riemannian manifolds which has several interesting applications. We give two different proofs, in two different scenarios, of essentially the same result. One of them deals with compact manifolds with or without boundary, while the other approach treats the case of riemannian manifolds without boundary whose Ricci curvature is uniformly bounded below.

Continuity of weak solutions of the critical surface quasigeostrophic equation on S2

“…For the sake of completeness, in the appendix 8 we provide a proof of the existence of global weak solutions to the Cauchy problem for the critical quasigeostrophic equation. We need to use a fractional Sobolev embedding on compact manifolds which is not easy to find in the literature, but a detailed proof of that fact was included in a recent paper of the authors [1] (cf. [2]).…”

“…might hold true, which is almost the kind of non linear inequality we would like to use. 1 The estimate of the remaining term, η k−2 θ * k−2 2(n+1)/n L 2(n+1)/n (M ) , can be reduced to the above. However, since this is not inmediate, let us show first that for any t θ * k+1 (x, t, z) ≤ (η k θ k ) * (x, t, z) for any (x, z) ∈ B * g (h(1 + 2 −k−1 ), hδ k ) holds provided the claim is true.…”

Existence and maximalLp-regularity of solutions for the porous medium equation on manifolds with conical singularities

“…This is called Restricted Fractional Laplacian (RFL). Approaches based on RFL are similar to those in (2).…”

Quasigeostrophic Equations for Fractional Powers of Infinitesimal Generators

“…For example, in [3,4], A. Córdoba and D. Córdoba studied regularity and -decay for solutions. In [5] the well-posedness of quasigeostrophic equation was treated on the sphere, on general riemannian manifolds in [6] or the 2D stochastic quasigeostrophic equation on the torus T 2 in [7]. This equation is also denominated as advection-fractional diffusion; see for example [8], or it may be classified as a fractional Fokker-Planck equation [9].…”