Expand all Collapse all | Results 1 - 5 of 5 |
1. CJM 2012 (vol 64 pp. 1415)
Global Well-Posedness and Convergence Results for 3D-Regularized Boussinesq System Analytical study to the regularization of the Boussinesq system is
performed in frequency space using Fourier theory. Existence and
uniqueness of weak solution with minimum regularity requirement are
proved. Convergence results of the unique weak solution of the
regularized Boussinesq system to a weak Leray-Hopf solution of the
Boussinesq system are established as the regularizing parameter
$\alpha$ vanishes. The proofs are done in the frequency space and use
energy methods, ArselÃ -Ascoli compactness theorem and a Friedrichs
like approximation scheme.
Keywords:regularizing Boussinesq system, existence and uniqueness of weak solution, convergence results, compactness method in frequency space Categories:35A05, 76D03, 35B40, 35B10, 86A05, 86A10 |
2. CJM 2010 (vol 63 pp. 153)
Asymptotics for Functions Associated with Heat Flow on the Sierpinski Carpet
We establish the asymptotic behaviour of the partition function, the
heat content, the integrated eigenvalue counting function, and, for
certain points, the on-diagonal heat kernel of generalized
Sierpinski carpets. For all these functions the leading term is of
the form $x^{\gamma}\phi(\log x)$ for a suitable exponent $\gamma$
and $\phi$ a periodic function. We also discuss similar results for
the heat content of affine nested fractals.
Categories:35K05, 28A80, 35B40, 60J65 |
3. CJM 2005 (vol 57 pp. 1193)
Some Conditions for Decay of Convolution Powers and Heat Kernels on Groups Let $K$ be a function on a unimodular locally compact group
$G$, and denote by $K_n = K*K* \cdots * K$ the $n$-th convolution
power of $K$.
Assuming that $K$ satisfies certain operator estimates in $L^2(G)$,
we give estimates of
the norms $\|K_n\|_2$ and $\|K_n\|_\infty$
for large $n$.
In contrast to previous methods for estimating $\|K_n\|_\infty$,
we do not need to assume that
the function $K$ is a probability density or non-negative.
Our results also adapt for continuous time semigroups on $G$.
Various applications are given, for example, to estimates of
the behaviour of heat kernels on Lie groups.
Categories:22E30, 35B40, 43A99 |
4. CJM 2004 (vol 56 pp. 794)
Semi-Classical Behavior of the Scattering Amplitude for Trapping Perturbations at Fixed Energy We study the semi-classical behavior as $h\rightarrow 0$ of the scattering
amplitude $f(\theta,\omega,\lambda,h)$ associated to a Schr\"odinger operator
$P(h)=-\frac 1 2 h^2\Delta +V(x)$ with short-range trapping
perturbations. First we realize a spatial localization in the general case
and we deduce a bound of the scattering amplitude on the real
line. Under an additional assumption on the resonances, we show that
if we modify the potential $V(x)$ in a domain lying behind the
barrier $\{x:V(x)>\lambda\}$, the scattering amplitude
$f(\theta,\omega,\lambda,h)$ changes by a term of order
$\O(h^{\infty})$. Under an escape assumption on the classical
trajectories incoming with fixed direction $\omega$, we obtain
an asymptotic development of $f(\theta,\omega,\lambda,h)$
similar to the one established in thenon-trapping case.
Categories:35P25, 35B34, 35B40 |
5. CJM 2000 (vol 52 pp. 522)
On Multiple Mixed Interior and Boundary Peak Solutions for Some Singularly Perturbed Neumann Problems |
On Multiple Mixed Interior and Boundary Peak Solutions for Some Singularly Perturbed Neumann Problems We consider the problem
\begin{equation*}
\begin{cases}
\varepsilon^2 \Delta u - u + f(u) = 0, u > 0 & \mbox{in } \Omega\\
\frac{\partial u}{\partial \nu} = 0 & \mbox{on } \partial\Omega,
\end{cases}
\end{equation*}
where $\Omega$ is a bounded smooth domain in $R^N$, $\ve>0$ is a small
parameter and $f$ is a superlinear, subcritical nonlinearity. It is
known that this equation possesses multiple boundary spike solutions
that concentrate, as $\epsilon$ approaches zero, at multiple critical
points of the mean curvature function $H(P)$, $P \in \partial \Omega$.
It is also proved that this equation has multiple interior spike solutions
which concentrate, as $\ep\to 0$, at {\it sphere packing\/} points in $\Om$.
In this paper, we prove the existence of solutions with multiple spikes
{\it both\/} on the boundary and in the interior. The main difficulty
lies in the fact that the boundary spikes and the interior spikes usually
have different scales of error estimation. We have to choose a special set
of boundary spikes to match the scale of the interior spikes in a
variational approach.
Keywords:mixed multiple spikes, nonlinear elliptic equations Categories:35B40, 35B45, 35J40 |