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| Important equations in fluid dynamics involve nonlocal diffusions typically modeled by the fractional Laplacian
\cite{Constantin-Majda-Tabak}.
These kind of nonlocal operators received a great deal of attention in the past years, especially since the appearance of
the works by Luis Caffarelli and collaborators \cite{Caffarelli-Salsa-Silvestre, Caffarelli-Silvestre,
Caffarelli-Vasseur}.
In this talk we shall discuss a novel point of view to understand fractional Laplacians: the \textit{semigroup language}
approach. This method was introduced in \cite{Stinga}. With this method at hand we can generalize the Caffarelli--Silvestre
extension problem to positive operators other than the Laplacian in the whole space \cite{Stinga-Torrea} and then prove
Harnack's inequalities for a large class of fractional operators \cite{Stinga-Zhang}. Moreover, we are able to define
the fractional powers of the discrete Laplacian on a mesh of size $h$
and to show that it converges to the fractional Laplacian on the whole space in the discrete supremum norm as $h\to0$ \cite{Discreto}.
Bibliography
\bibitem{Caffarelli-Salsa-Silvestre} L. Caffarelli, S. Salsa and L. Silvestre,
{Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian},
\textit{Invent. math.}
\textbf{171} (2008), 425--461.
\bibitem{Caffarelli-Silvestre} L. Caffarelli and L. Silvestre,
{An extension problem related to the fractional Laplacian},
\textit{Comm. Partial Differential Equations}
\textbf{32} (2007), 1245--1260.
\bibitem{Caffarelli-Vasseur} L. Caffarelli and A. Vasseur,
{Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation},
\textit{Ann. of Math. (2)}
\textbf{171} (2010), 1903--1930.
\bibitem{Discreto} \'O. Ciaurri, L. Roncal, P. R. Stinga, J. L. Torrea and J. L. Varona,
{The discrete fractional Laplacian},
\textit{preprint} (2014).
\bibitem{Constantin-Majda-Tabak} P. Constantin, A. J. Majda and E. Tabak,
{Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar},
\textit{Nonlinearity}
\textbf{7} (1994), 1495--1533.
\bibitem{Stinga} P. R. Stinga,
\textit{Fractional Powers of Second Order Partial Differential Operators: Extension Problem and Regularity Theory},
PhD Thesis, Universidad Aut\'onoma de Madrid, 2010.
\bibitem{Stinga-Torrea} P. R. Stinga and J. L. Torrea,
{Extension problem and Harnack's inequality for some fractional operators},
\textit{Comm. Partial Differential Equations}
\textbf{35} (2010), 2092--2122.
\bibitem{Stinga-Zhang} P. R. Stinga and C. Zhang,
{Harnack's inequality for fractional nonlocal equations},
\textit{Discrete Contin. Dyn. Syst.}
\textbf{33} (2013), 3153--3170. |
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