Chiun-Chuan Chen

ESTIMATES OF THE CONFORMAL SCALAR CURVATURE EQUATION VIA THE METHOD OF MOVING PLANES

In this paper we derive a local estimate of a positive singular solution u near its singular set Z of the conformal equation where K(x) is a positive continuous function, Z is a compact subset of , and g satisfies that is nonincreasing for t > 0. Assuming that the order of flatness at critical points of K on Z is no less than , we prove that, through the application of the method of moving planes, the inequality holds for any solution of (0.1) with Cap(Z) = 0. By the same method, we also derive a Harnack-type inequality for smooth positive solutions. Let u satisfy Assume that the order of flatness at critical points of K is no less than n - 2; then the inequality holds for R ≤ 1. We also show by examples that the assumption about the flatness at critical points is optimal for validity of the inequality (0.4).

Profile of Blow-up Solutions to Mean Field Equations with Singular Data

Abstract Motivated by the study of selfdual vortices in gauge field theory, we consider a class of Mean Field equations of Liouville-type on compact surfaces involving singular data assigned by Dirac measures supported at finitely many points (the so called vortex points). According to the applications, we need to describe the blow-up behavior of solution-sequences which concentrate exactly at the given vortex points. We provide accurate pointwise estimates for the profile of the bubbling sequences as well as “sup + inf” estimates for solutions. Those results extend previous work of Li [Li, Y. Y. (1999). Harnack type inequality: The method of moving planes. Comm. Math. Phys. 200:421–444] and Brezis et al. [Brezis, H., Li, Y. Shafrir, I. (1993). A sup + inf inequality for some nonlinear elliptic equations involving the exponential nonlinearities. J. Funct. Anal. 115: 344–358] relative to the “regular” case, namely in absence of singular sources.

Lower Bounds on the Blow-Up Rate of the Axisymmetric Navier–Stokes Equations II

Consider axisymmetric strong solutions of the incompressible Navier–Stokes equations in ℝ3 with non-trivial swirl. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies, for some 0 ≤ ϵ ≤ 1, |v (x, t)| ≤ C * r −1+ϵ |t|−ϵ/2 for − T 0 ≤ t < 0 and 0 < C * < ∞ allowed to be large. We prove that v is regular at time zero.

Lower Bound on the Blow-up Rate of the Axisymmetric Navier–Stokes Equations

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in

Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations

\R^3

EXISTENCE OF BOUNDARY LAYER SOLUTIONS TO THE BOLTZMANN EQUATION

with non-trivial swirl. Such solutions are not known to be globally defined, but it is shown in \cite{MR673830} that they could only blow up on the axis of symmetry. Let

Blowing up with infinite energy of conformal metrics on Sn

z

A maximum principle for diffusive Lotka–Volterra systems of two competing species

denote the axis of symmetry and

Simple pde model of spot replication in any dimension

r

Distribution of first arrival position in molecular communication

measure the distance to the z-axis. Suppose the solution satisfies the pointwise scale invariant bound