3,857 research outputs found
Small data global regularity for simplified 3-D Ericksen-Leslie's compressible hyperbolic liquid crystal model
In this article, we consider the Ericksen-Leslie's hyperbolic system for
compressible liquid crystal model in three spatial dimensions. Global
regularity for small and smooth initial data near equilibrium is proved for the
case that the system is a nonlinear coupling of compressible Navier-Stokes
equations with wave map to . Our argument is a combination of
vector field method and Fourier analysis. The main strategy to prove global
regularity relies on an interplay between the control of high order energies
and decay estimates, which is based on the idea inspired by the method of
space-time resonances. In particular the different behaviors of the decay
properties of the density and velocity field for compressible fluids at
different frequencies play a key role.Comment: 61 pages; all comments wellcom
Stability in Bondy's theorem on paths and cycles
In this paper, we study the stability result of a well-known theorem of
Bondy. We prove that for any 2-connected non-hamiltonian graph, if every vertex
except for at most one vertex has degree at least , then it contains a cycle
of length at least except for some special families of graphs. Our
results imply several previous classical theorems including a deep and old
result by Voss. We point out our result on stability in Bondy's theorem can
directly imply a positive solution (in a slight stronger form) to the following
problem: Is there a polynomial time algorithm to decide whether a 2-connected
graph on vertices has a cycle of length at least
. This problem originally motivates the recent study on
algorithmic aspects of Dirac's theorem by Fomin, Golovach, Sagunov and Simonov,
although a stronger problem was solved by them by completely different methods.
Our theorem can also help us to determine all extremal graphs for wheels on odd
number of vertices. We also discuss the relationship between our results and
some previous problems and theorems in spectral graph theory and generalized
Tur\'{a}n problem.Comment: 17 page
- …