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1. CMB 2011 (vol 55 pp. 176)
Linear Dispersive Decay Estimates for the 3+1 Dimensional Water Wave Equation with Surface Tension
We consider the linearization of the three-dimensional water waves
equation with surface tension about a flat interface. Using
oscillatory integral methods, we prove that solutions of this equation
demonstrate dispersive decay at the somewhat surprising rate of
$t^{-5/6}$. This rate is due to competition between surface tension
and gravitation at $O(1)$ wave numbers and is connected to the fact
that, in the presence of surface tension, there is a so-called "slowest
wave". Additionally, we combine our dispersive estimates with $L^2$
type energy bounds to prove a family of Strichartz estimates.
Keywords:oscillatory integrals, water waves, surface tension, Strichartz estimates Categories:76B07, 76B15, 76B45 |
2. CMB 2010 (vol 54 pp. 147)
Generalized Quandle Polynomials
We define a family of generalizations of the two-variable quandle polynomial.
These polynomial invariants generalize in a natural way to eight-variable
polynomial invariants of finite biquandles. We use these polynomials to define
a family of link invariants that further generalize the quandle counting
invariant.
Keywords:finite quandles, finite biquandles, link invariants Categories:57M27, 76D99 |
3. CMB 2008 (vol 51 pp. 386)
Positive Solutions of the Falkner--Skan Equation Arising in the Boundary Layer Theory The well-known Falkner--Skan equation is one of the most important
equations in laminar boundary layer theory and is used to describe
the steady two-dimensional flow of a slightly viscous
incompressible fluid past wedge shaped bodies of angles related to
$\lambda\pi/2$, where $\lambda\in \mathbb R$ is a parameter
involved in the equation. It is known that there exists
$\lambda^{*}<0$ such that the equation with suitable boundary
conditions has at least one positive solution for each $\lambda\ge
\lambda^{*}$ and has no positive solutions for
$\lambda<\lambda^{*}$. The known numerical result shows
$\lambda^{*}=-0.1988$. In this paper, $\lambda^{*}\in
[-0.4,-0.12]$ is proved analytically by establishing a singular
integral equation which is equivalent to the Falkner--Skan
equation. The equivalence result
provides new techniques to study properties and existence of solutions of
the Falkner--Skan equation.
Keywords:Falkner-Skan equation, boundary layer problems, singular integral equation, positive solutions Categories:34B16, 34B18, 34B40, 76D10 |
4. CMB 2007 (vol 50 pp. 547)
Inverse Laplace Transforms Encountered in Hyperbolic Problems of Non-Stationary Fluid-Structure Interaction |
Inverse Laplace Transforms Encountered in Hyperbolic Problems of Non-Stationary Fluid-Structure Interaction The paper offers a study of the inverse Laplace
transforms of the functions $I_n(rs)\{sI_n^{'}(s)\}^{-1}$ where
$I_n$ is the modified Bessel function of the first kind and $r$ is
a parameter. The present study is a continuation of the author's
previous work %[\textit{Canadian Mathematical Bulletin} 45]
on the
singular behavior of the special case of the functions in
question, $r$=1. The general case of $r \in [0,1]$ is addressed,
and it is shown that the inverse Laplace transforms for such $r$
exhibit significantly more complex behavior than their
predecessors, even though they still only have two different types
of points of discontinuity: singularities and finite
discontinuities. The functions studied originate from
non-stationary fluid-structure interaction, and as such are of
interest to researchers working in the area.
Categories:44A10, 44A20, 33C10, 40A30, 74F10, 76Q05 |
5. CMB 2004 (vol 47 pp. 30)
Existence of Leray's Self-Similar Solutions of the Navier-Stokes Equations In $\mathcal{D}\subset\mathbb{R}^3$ |
Existence of Leray's Self-Similar Solutions of the Navier-Stokes Equations In $\mathcal{D}\subset\mathbb{R}^3$ Leray's self-similar solution of the Navier-Stokes equations is
defined by
$$
u(x,t) = U(y)/\sqrt{2\sigma (t^*-t)},
$$
where $y = x/\sqrt{2\sigma (t^*-t)}$, $\sigma>0$. Consider the
equation for $U(y)$ in a smooth bounded domain $\mathcal{D}$ of
$\mathbb{R}^3$ with non-zero boundary condition:
\begin{gather*}
-\nu \bigtriangleup U + \sigma U +\sigma y \cdot \nabla U + U\cdot
\nabla U + \nabla P = 0,\quad y \in \mathcal{D}, \\
\nabla \cdot U = 0, \quad y \in \mathcal{D}, \\
U = \mathcal{G}(y), \quad y \in \partial \mathcal{D}.
\end{gather*}
We prove an existence theorem for the Dirichlet problem in Sobolev
space $W^{1,2} (\mathcal{D})$. This implies the local existence of
a self-similar solution of the Navier-Stokes equations which blows
up at $t=t^*$ with $t^* < +\infty$, provided the function
$\mathcal{G}(y)$ is permissible.
Categories:76D05, 76B03 |