arXiv subjects
Publications and source records attributed to Bryan Carrillo.
A version of the classical Buffon problem in the plane naturally extends to the setting of any Riemannian surface with constant Gaussian curvature. The Buffon probability determines a Buffon deficit. The relationship between Gaussian curvature and the Buffon deficit is similar to the relationship that the Bertrand-Diguet-Puiseux Theorem establishes between Gaussian curvature and both circumference and area deficits.
An old problem since Leray \cite{Le:1} asks whether homogeneous D solutions of the 3 dimensional Navier-Stokes equation in $\mathbb{R}^3$ or some noncompact domains are 0. In this paper, we give a positive solution to the problem in two special cases: (1) when the solution is axially symmetric and periodic in the vertical variable; (2) full 3 dimensional slab case $\mathbb{R}^2 \times [0, 1]$ with Dirichlet boundary condition. Other partial results are also presented. The paper is self contained comparing with the first part \cite{CPZ:1} although the general idea is related.
We study axially symmetric D-solutions of the 3 dimensional Navier-Stokes equations. The first result is an a priori decay estimate of the velocity for general domains. The second is an a priori decay estimate of the vorticity in $\bR^3$, which improves the corresponding results in the literature. In addition, we prove a similar decay of full 3d solutions except for a small set of angles. Next we turn to D-solutions which are periodic in the third variable and prove vanishing result under a reasonable condition. As a corollary we prove that axially symmetric D-solutions in the slab $\bR^2 \times I$ with suitable boundary condition is $0$. Here $I$ is any finite interval. To the best of our knowledge, this seems to be the first vanishing result on a 3 dimensional D-solution without extra integral or decay or smallness assumption on the solution. The tools used include Brezis-Gallouet inequality, dimension reduction, scaling, Green's function bound and Liouville theorems for Navier-Stokes equations.