35,577 research outputs found
Structural Analysis of the Support System for a Large Compressor Driven by a Synchronous Electric Motor
For economic reasons, the steam drive for a large compressor was replaced by a large synchronous electric motor. Due to the resulting large increase in mass and because the unit was mounted on a steel frame approximately 18 feet above ground level, it was deemed necessary to determine if a steady state or transient vibration problem existed. There was a definite possibility that a resonant or near resonant condition could be encountered. The ensuing analysis, which led to some structural changes as the analysis proceeded, did not reveal any major steady state vibration problems. However, the analysis did indicate that the system would go through several natural frequencies of the support structure during start-up and shutdown. This led to the development of special start-up and shutdown procedures to minimize the possibility of exciting any of the major structural modes. A coast-down could result in significant support structure and/or equipment damage, especially under certain circumstances. In any event, dynamic field tests verified the major analytical results. The unit has now been operating for over three years without any major vibration problems
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Symmetric and Asymmetric Multiple Clusters In a Reaction-Diffusion System
We consider the Gierer-Meinhardt system in
the interval (-1,1) with Neumann boundary
conditions for small diffusion constant
of the activator and finite diffusion
constant of the inhibitor.
A cluster is a combination of several spikes
concentrating at the same point.
In this paper, we rigorously show the existence
of symmetric and asymmetric multiple clusters.
This result is new for systems and seems not
to occur for single equations.
We reduce the problem to the computation of two
matrices which depend on the coefficient of
the inhibitor as well as the number of different clusters and the number of spikes within each
cluster
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On the stationary Cahn-Hilliard equation: Interior spike solutions
We study solutions of the stationary Cahn-Hilliard equation in a bounded smooth domain which have a spike in the interior. We show that a large class of interior points (the "nondegenerate peak" points) have the following property: there exist such solutions whose spike lies close to a given nondegenerate peak point. Our construction uses among others the methods of viscosity solution, weak convergence of measures and Liapunov-Schmidt reduction
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Existence, classification and stability analysis of multiple-peaked solutions for the gierer-meinhardt system in R^1
We consider the Gierer-Meinhardt system in R^1.
where the exponents (p, q, r, s) satisfy
1< \frac{ qr}{(s+1)( p-1)} < \infty, 1 <p < +\infty,
and where \ep<<1, 0<D<\infty, \tau\geq 0,
D and \tau are constants which are independent of \ep.
We give a rigorous and unified approach to show that the existence and stability of N-peaked steady-states can be reduced to computing two
matrices in terms of the coefficients D, N, p, q, r, s. Moreover, it is shown that N-peaked steady-states are generated by exactly two types of peaks, provided their mutual distance is bounded away from zero
Symmetry of Nodal Solutions for Singularly Perturbed Elliptic Problems on a Ball
In [40], it was shown that the following singularly perturbed Dirichlet problem
\ep^2 \Delta u - u+ |u|^{p-1} u=0, \ \mbox{in} \ \Om,\]
\[ u=0 \ \mbox{on} \ \partial \Om
has a nodal solution u_\ep which has the least energy among all nodal solutions. Moreover, it is shown that u_\ep has exactly one local maximum point P_1^\ep with a positive value
and one local minimum point P_2^\ep with a negative value and, as \ep \to 0,
\varphi (P_1^\ep, P_2^\ep) \to \max_{ (P_1, P_2) \in \Om \times \Om } \varphi (P_1, P_2),
where \varphi (P_1, P_2)= \min (\frac{|P_1-P_2}{2}, d(P_1, \partial \Om), d(P_2, \partial \Om)). The following question naturally arises: where is the {\bf nodal surface} \{ u_\ep (x)=0 \}? In this paper, we give an answer in the case of the unit ball \Om=B_1 (0).
In particular, we show that for \epsilon sufficiently small, P_1^\ep, P_2^\ep and the origin must lie on a line. Without loss of generality, we may assume that this line is the x_1-axis.
Then u_\ep must be even in x_j, j=2, ..., N, and odd in x_1.
As a consequence, we show that \{ u_\ep (x)=0 \} = \{ x \in B_1 (0) | x_1=0 \}. Our proof
is divided into two steps:
first, by using the method of moving planes, we show that P_1^\ep, P_2^\ep and the origin must lie on the x_1-axis and u_\ep must be even in x_j, j=2, ..., N. Then,
using the Liapunov-Schmidt reduction method, we prove
the uniqueness of u_\ep (which implies the odd symmetry of u_\ep in x_1). Similar results are also proved for the problem with Neumann boundary conditions
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Existence and stability of multiple spot solutions for the gray-scott model in R^2
We study the Gray-Scott model in a bounded two dimensional domain and establish the existence and stability of {\bf symmetric} and {\bf asymmetric} multiple spotty patterns. The Green's function and its derivatives
together with two nonlocal eigenvalue problems
both play a major role in the analysis.
For symmetric spots, we establish a threshold behavior for stability:
If a certain inequality for the parameters holds
then we get stability, otherwise we get instability of multiple spot solutions.
For asymmetric spots, we show that they can be stable within a narrow parameter range
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On a Cubic-Quintic Ginzburg-Landau Equation with Global Coupling
We study standing wave solutions in a
Ginzburg-Landau equation which consists of
a cubic-quintic equation stabilized by global coupling
A_t= \Delta A +\mu A + c A^3 -A^5 -k A
(\int_{R^n} A^2\,dx).
We classify the existence and stability of
all possible standing wave solutions
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Stability of monotone solutions for the shadow Gierer-Meinhardt system with finite diffusivity
We consider the following shadow system of the Gierer-Meinhardt model:
\left\{\begin{array}{l}
A_t= \epsilon^2 A_{xx} -A +\frac{A^p}{\xi^q},\, 00,\\
\tau \xi_t= -\xi + \xi^{-s} \int_0^1 A^2 \,dx,\\
A>0,\, A_x (0,t)= A_x(1, t)=0,
\end{array}
\right.
where 1<p<+\infty,\,
\frac{2q}{p-1} >s+1,\, s\geq 0, and \tau >0.
It is known that a nontrivial monotone steady-state solution exists if and only if
\ep < \frac{\sqrt{p-1}}{\pi}.
In this paper, we show that for any \ep < \frac{\sqrt{p-1}}{\pi}, and
p=2 or p=3, there exists a unique \tau_c>0 such that for
\tau\tau_c it is linearly unstable. (This result is optimal.)
The transversality of this Hopf bifurcation is proven.
Other cases for the exponents as well as extensions to higher
dimensions are also considered. Our proof makes use of functional analysis and the properties of Weierstrass functions and elliptic integrals
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On the Gierer-Meinhardt System with Saturation
We consider the following shadow Gierer-Meinhardt system with saturation:
\left\{\begin{array}{l}
A_t=\epsilon^2 \Delta A -A + \frac{A^2}{ \xi (1+k A^2)} \ \ \mbox{in} \ \Omega \times (0, \infty),\\
\tau \xi_t= -\xi +\frac{1}{|\Omega|} \int_\Om A^2\,dx
\ \ \mbox{in} \ (0, +\infty),
\frac{\partial A}{\partial \nu} =0 \ \mbox{on} \ \partial \Omega\times(0,\infty),
\end{array}
\right.
where \ep>0 is a small parameter, k$ for the existence and stability of stable spiky solutions.
In the one-dimensional case we have a complete answer to the stability behavior.
Central to our study are a parameterized ground-state equation
and the associated nonlocal eigenvalue problem (NLEP)
which is solved by functional analysis arguments and the continuation method
Multi-interior-spike solutions for the Cahn-Hilliard equation with arbitrarily many peaks
We study the Cahn-Hilliard equation in a bounded smooth
domain without any symmetry
assumptions. We prove that for any fixed positive integer K there
exist interior --spike solutions
whose peaks have maximal possible distance from the boundary and
from one another. This implies that for any bounded and smooth
domain there
exist interior K-peak solutions.
The central ingredient of our analysis is the novel derivation
and exploitation of a reduction of the energy to finite dimensions
(Lemma 5.5) with variables which are closely related to the location of
the peaks.
We do not assume nondegeneracy of the points of
maximal distance to the boundary but can do with a global condition instead
which in many cases is weaker
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