32 research outputs found
Stochastic Lagrangian path for Leray solutions of 3D Navier-Stokes equations
In this paper we show the existence of stochastic Lagrangian particle
trajectory for Leray's solution of 3D Navier-Stokes equations. More precisely,
for any Leray's solution of 3D-NSE and each
, we show the existence of weak
solutions to the following SDE, which has a density belonging
to provided with
:
where is a three dimensional standard Brownian motion, is the
viscosity constant. Moreover, we also show that for Lebesgue almost all
, the solution of the above SDE associated with the
mollifying velocity field weakly converges to
so that is a Markov process in almost sure sense.Comment: 25page
On Distribution depend SDEs with singular drifts
We investigate the well-posedness of distribution dependent SDEs with
singular coefficients. Existence is proved when the diffusion coefficient
satisfies some non-degeneracy and mild regularity assumptions, and the drift
coefficient satisfies an integrability condition and a continuity condition
with respect to the (generalized) total variation distance. Uniqueness is also
obtained under some additional Lipschitz type continuity assumptions.Comment: 26 pages. All comments are welcom
L\'evy-type operators with low singularity kernels: regularity estimates and martingale problem
We consider the linear non-local operator denoted by Here is bounded and is the jumping kernel of a L\'evy
process, which only has a low-order singularity near the origin and does not
allow for standard scaling. The aim of this work is twofold. Firstly, we
introduce generalized Orlicz-Besov spaces tailored to accommodate the analysis
of elliptic equations associated with , and establish regularity
results for the solutions of such equations in these spaces. Secondly, we
investigate the martingale problem associated with . By utilizing
analytic results, we prove the well-posedness of the martingale problem under
mild conditions. Additionally, we obtain a new Krylov-type estimate for the
martingale solution through the use of a Morrey-type inequality for generalized
Orlicz-Besov spaces.Comment: 43 page