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- The Lefschetz
property,
formality and
blowing up in
symplectic
geometry: (6 Apr 2005)In
this paper we
study the
behaviour of
the Lefschetz
property under
the blow-up
construction.
We show that
it is possible
to reduce the
dimension of
the kernel of
the Lefschetz
map if we blow
up along a
suitable
submanifold
satisfying the
Lefschetz
property. We
use that,
together with
results about
Massey
products, to
construct
nonformal
(simply
connected)
symplectic
manifolds
satisfying the
Lefschetz
property.
Source: (6 Apr 2005) - Blowup of
smooth
solutions for
relativistic
Euler
equations: (18 March
2005)We study
the
singularity
formation of
smooth
solutions of
the
relativistic
Euler
equations in
$(3+1)$-dimens
ional
spacetime for
both finite
initial energy
and infinite
initial
energy. For
the finite
initial energy
case, we prove
that any
smooth
solution, with
compactly
supported
non-trivial
initial data,
blows up in
finite time.
For the case
of infinite
initial
energy, we
first prove
the existence,
uniqueness and
stability of a
smooth
solution if
the initial
data is in the
subluminal
region away
from the
vacuum. By
further
assuming the
initial data
is a smooth
compactly
supported
perturbation
around a
non-vacuum
constant
background, we
prove the
property of
finite
propagation
speed of such
a
perturbation.
The smooth
solution is
shown to blow
up in finite
time provided
that the
radial
component of
the initial
"generalized"
momentum is
sufficiently
large.
Source: (18 March 2005) - Blow-up in
finite time
for the dyadic
model of the
Navier-Stokes
equations: (4 Jan 2006)We
study the
dyadic model
of the
Navier-Stokes
equations
introduced by
Katz and
Pavlovic. They
showed a
finite time
blow-up in the
case where the
dissipation
degree $?$ is
less than 1/4.
In this paper
we prove the
existence of
weak solutions
for all $?$,
energy
inequality for
every weak
solution with
nonnegative
initial datum
starting from
any time,
local
regularity for
$? > 1/3$, and
global
regularity for
$? ? 1/2$. In
addition, we
prove a finite
time blow-up
in the case
where $?
Source: (4 Jan 2006) - Nonexistence
of
self-similar
singularities
for the 3D
incompressible
Euler
equations: (4 Jan 2006)We
prove that
there exists
no
self-similar
finite time
blowing up
solution to
the 3D
incompressible
Euler
equations. The
proof uses the
vorticity
transport
formula
represented in
terms of the
back to label
map. By
similar method
we also show
nonexistence
of
self-similar
blowing up
solutions to
the
divergence-fre
e transport
equation in
$\Bbb R^n$.
This result
has direct
applications
to the density
dependent
Euler
equations, the
Boussinesq
system, and
the
quasi-geostrop
hic equations,
for which we
also show
nonexistence
of
self-similar
blowing up
solutions.
Source: (4 Jan 2006) - Nonexistence
of Local
Self-Similar
Blow-up for
the 3D
Incompressible
Navier-Stokes
Equations: (6 Mar 2006)We
prove the
nonexistence
of local
self-similar
solutions of
the three
dimensional
incompressible
Navier-Stokes
equations. The
local
self-similar
solutions we
consider here
are different
from the
global
self-similar
solutions. The
self-similar
scaling is
only valid in
an inner core
region which
shrinks to a
point
dynamically as
the time, $t$,
approaches the
singularity
time, $T$. The
solution
outside the
inner core
region is
assumed to be
regular. Under
the assumption
that the local
self-similar
velocity
profile
converges to a
limiting
profile as $t
\to T$ in
$L^p$ for some
$p ? (3,?)$,
we prove that
such local
self-similar
blow-up is not
possible for
any finite
time.
Source: (6 Mar 2006) - On the blow-up
problem for
the
axisymmetric
3D Euler
equations: (12 Mar
2008)In this
paper we study
the finite
time blow-up
problem for
the
axisymmetric
3D
incompressible
Euler
equations with
swirl. The
evolution
equations for
the
deformation
tensor and the
vorticity are
reduced
considerably
in this case.
Under the
assumption of
local minima
for the
pressure on
the axis of
symmetry with
respect to the
radial
variations we
show that the
solution
blows-up in
finite time.
If we further
assume that
the second
radial
derivative
vanishes on
the axis, then
system reduces
to the form of
Constantin-Lax
-Majda
equations, and
can be
integrated
explicitly.
Source: (12 Mar 2008) - The finite
time blow-up
for the
Euler-Poisson
equations in
$\Bbb R^n$: (12 Mar
2008)We prove
the finite
time blow-up
for $C^1$
solutions to
the
Euler-Poisson
equations in
$\Bbb R^n$,
$n? 1$,
with/without
background
density for
initial data
satisfying
suitable
conditions. We
also find a
sufficient
condition for
the initial
data such that
$C^3$ solution
breaks down in
finite time
for the
compressible
Euler
equations for
polytropic gas
flows.
Source: (12 Mar 2008)
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