Non linear problemswith

Fractional Diffusions

Luis A. Caffarelli

The University of Texas at Austin

Non linear problems involving fractional diffusionsappear 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

•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 inan extension theorem:

It 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 Rn+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 meansthat v is harmonic across y=0, reducing regularity properties of u to those of the harmonic function v

This can be interpreted as an extension into a space of “fractional dimension” and suggest the correct formof homogeneous solutions, monotonicity formulas,truncated test functions, etc

In fact, any other fractional power of the Laplacian of agiven function u(x) can be realized as the normal derivative of an appropriate extension v(x,y).

This “harmonic” extension has the virtue of reducing many global issues and arguments to local, more familiar methods of the calculus of variations. The global properties of the solutions are somehow encoded in the restriction of the extension v(x,y) to unit ball in one more dimension. (L.C and L.Silvestre, arXiv.org, 07)

i) The quasi-geostrophic equation

See also:Kiselev, Nasarov, Volberg, arXiv.org’06

ii) Problems with constrains or Free Boundary problems

iv) Random Homogenization

See the work of D. Cioranescu and F. Murat (1982) where the Homogenized equation was derived for periodic media.

Fully non-linear equations with fractional diffusion

. . .

Formally, the solution u0 to a fully non-linear equation, its first derivatives and its second derivatives all satisfy equations or inequalities like (1) above.

This implies that u0 is classical (Evans – Krylov)

.

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