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This work discusses nonlinear problems involving fractional diffusions across various applied mathematics fields, including boundary diffusion and fluid dynamics. It particularly emphasizes the relationship between fractional Laplacians and the normal derivatives of function extensions in higher dimensions. The study highlights applications in areas such as quasi-geostrophic equations and stochastic processes with discontinuities. By applying extension theorems, it simplifies complex global issues to more familiar local calculus of variations methods.
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Non linear problems with Fractional Diffusions Luis A. Caffarelli The University of Texas at Austin
Non linear problems involving fractional diffusions appear I in several areas of applied mathematics: Boundary diffusion (see for instance Duvaut and Lions) (Or more generally calculus of variations when the energy integrals involved correspond to fractional derivatives). Fluid dynamics like in the quasi-geostrophic equation modeling ocean atmospheric interaction, or in the case of turbulent transport
In stochastic processes of discontinuous nature (Levy processes) in applications for which random walks have jumps at many different scales. (Stocks, insurance)
Remark: the work just described is strongly based in an extension theorem that identifies the fractional Laplacian of a given function u(x) in Rn with the normal derivative of an extension v(x,y) of u(x) into the upper half space, (y>0), of R(n+1). The classical example is the ½ Laplacian: If v is the harmonic extension of u ( convolution with the Poison kernel), then the normal derivative of v at y=0 is exactly the half Laplacian of u. In particular, u being “half harmonic” simply means that v is harmonic across y=0, reducing regularity properties of u to those of the harmonic function v
In fact, any other fractional power of the Laplacian of a given function u(x) can be realized as the normal derivative of an apropriate extension v(x,y). The equation satisfied now by the extension v has a power of y as diffusion coefficient. This can be interpreted as an extension into a space of “fractional dimension” and suggest the correct form of homogeneous solutions, monotonicity formulas, truncated test functions,etc
This has the virtue of reducing many global issues and arguments to local, more familiar methods of the calculus of variations. (C-Silvestre, arXiv.org)
