The fractional Laplacian in R d, which we write as (− Δ) α/2 with α∈(0, 2), has multiple
equivalent characterizations. Moreover, in bounded domains, boundary conditions must be …

Let L=− div x (A (x)∇ x) be a uniformly elliptic operator in divergence form in a bounded
domain Ω. We consider the fractional nonlocal equations {L su= f, in Ω, u= 0, on∂ Ω, and {L …

The method of semigroups is a unifying, widely applicable, general technique to formulate
and analyze fundamental aspects of fractional powers of operators L and their regularity …

The analysis of nonlocal discrete equations driven by fractional powers of the discrete
Laplacian on a mesh of size h> 0 (− Δ h) su= f, for u, f: Z h→ R, 0< s< 1, is performed. The …

We consider conservative and gradient flows for N-particle Riesz energies with meanfield
scaling on the torus Td, for d 1, and with thermal noise of McKean–Vlasov type. We prove …

We prove the existence of a compact global attractor for the dynamics of the forced critical
surface quasi-geostrophic equation (SQG) and prove that it has finite fractal (box-counting) …

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Fractional differential operators provide an attractive mathematical tool to model effects with
limited regularity properties. Particular examples are image processing and phase field …

We prove the ill-posedness of Leray solutions to the Cauchy problem for the hypodissipative
Navier–Stokes equations, when the dissipative term is a fractional Laplacian (-Δ)^ α (-Δ) α …

We develop the regularity theory for solutions to space-time nonlocal equations driven by
fractional powers of the heat operator (\partial_t-Δ)^su(t,x)=f(t,x)~for~0<s<1. This nonlocal …