2 research outputs found
Stochastic Lagrangian Particle Approach to Fractal Navier-Stokes Equations
In this article we study the fractal Navier-Stokes equations by using
stochastic Lagrangian particle path approach in Constantin and Iyer
\cite{Co-Iy}. More precisely, a stochastic representation for the fractal
Navier-Stokes equations is given in terms of stochastic differential equations
driven by L\'evy processes. Basing on this representation, a self-contained
proof for the existence of local unique solution for the fractal Navier-Stokes
equation with initial data in \mW^{1,p} is provided, and in the case of two
dimensions or large viscosity, the existence of global solution is also
obtained. In order to obtain the global existence in any dimensions for large
viscosity, the gradient estimates for L\'evy processes with time dependent and
discontinuous drifts is proved.Comment: 19 page
On degenerate stochastic equations of Itô type with jumps
The one-dimensional stochastic equation , where are Borel measurable functions, W is a Brownian motion, and Z is a symmetric stable process of index 0