153 research outputs found
On regular solutions of the 3-D compressible isentropic Euler-Boltzmann equations with vacuum
In this paper, we discuss the Cauchy Problem for the compressible isentropic
Euler-Boltzmann equations with vacuum in radiation hydrodynamics. Firstly, we
establish the local existence of regular solutions by the fundamental methods
in the theory of quasi-linear symmetric hyperbolic systems under some physical
assumptions. Then we give the non-global existence of regular solutions caused
by the effect of vacuum for . Finally, we extend our result to
the initial-boundary value problem under some suitable boundary conditions.
These blow-up results tell us that the radiation cannot prevent the formation
of singularities caused by the appearance of vacuum.Comment: 28 page
Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum
We study the Cauchy problem for multi-dimensional compressible radiation
hydrodynamics equations with vacuum. First, we present some sufficient
conditions on the blow-up of smooth solutions in multi-dimensional space. Then,
we obtain the invariance of the support of density for the smooth solutions
with compactly supported initial mass density by the property of the system
under the vacuum state. Based on the above-mentioned results, we prove that we
cannot get a global classical solution, no matter how small the initial data
are, as long as the initial mass density is of compact support. Finally, we
will see that some of the results that we obtained are still valid for the
isentropic flows with degenerate viscosity coefficients as well as 1-D case.Comment: 31page
On classical solutions to 2D Shallow water equations with degenerate viscosities
In this paper, the -D isentropic Navier-Stokes systems for compressible
fluids with density-dependent viscosity coefficients are considered. In
particular, we assume that the viscosity coefficients are proportional to
density. These equations, including several models in -D shallow water
theory, are degenerate when vacuum appears. We introduce the notion of regular
solutions and prove the local existence of solutions in this class allowing the
initial vacuum in the far field. This solution is further shown to be stable
with respect to initial data in sense. A Beal-Kato-Majda type blow-up
criterion is also established.Comment: 43pages. arXiv admin note: substantial text overlap with
arXiv:1407.782
On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum
In this paper, we consider the three-dimensional isentropic Navier-Stokes
equations for compressible fluids with viscosities depending on density in a
power law and allowing initial vacuum. We introduce the notion of regular
solutions and prove the local-in-time well-posedness of solutions with
arbitrarily large initial data and vacuum in this class, which is a
long-standing open problem due to the very high degeneracy caused by vacuum.
Moreover, for certain classes of initial data with local vacuum, we show that
the regular solution that we obtained will break down in finite time, no matter
how small and smooth the initial data are.Comment: 46page
New fast divide-and-conquer algorithms for the symmetric tridiagonal eigenvalue problem
In this paper, two accelerated divide-and-conquer algorithms are proposed for
the symmetric tridiagonal eigenvalue problem, which cost {flops} in
the worst case, where is the dimension of the matrix and is a modest
number depending on the distribution of eigenvalues. Both of these algorithms
use hierarchically semiseparable (HSS) matrices to approximate some
intermediate eigenvector matrices which are Cauchy-like matrices and are
off-diagonally low-rank. The difference of these two versions lies in using
different HSS construction algorithms, one (denoted by {ADC1}) uses a
structured low-rank approximation method and the other ({ADC2}) uses a
randomized HSS construction algorithm. For the ADC2 algorithm, a method is
proposed to estimate the off-diagonal rank. Numerous experiments have been done
to show their stability and efficiency. These algorithms are implemented in
parallel in a shared memory environment, and some parallel implementation
details are included. Comparing the ADCs with highly optimized multithreaded
libraries such as Intel MKL, we find that ADCs could be more than 6x times
faster for some large matrices with few deflations
Existence results for compressible radiation hydrodynamics equations with vacuum
In this paper, we consider the 3-D compressible isentropic radiation
hydrodynamics (RHD) equations. The local existence of strong solutions with
vacuum is firstly established when the initial data is arbitrarily large,
contains vacuum and satisfy some initial layer compatibility condition. The
initial mass density needs not be bounded away from zero, it may vanish in some
open set or decay at infinity. We also prove that if the initial vacuum is not
so irregular, then the compatibility condition of the initial data is necessary
and sufficient to guarantee the existence of a unique strong solution. Finally,
we prove a blow-up criterion for the local strong solution. The similar result
also holds for the general barotropic flow with pressure law .Comment: 31page
On the global-in-time inviscid limit of the 3D isentropic compressible Navier-Stokes equations with degenerate viscosities and vacuum
In the recent paper, the global-in-time inviscid limit of the
three-dimensional (3D) isentropic compressible Navier-Stokes equations is
considered. First, when viscosity coefficients are given as a constant multiple
of density's power ( with ), for regular
solutions to the corresponding Cauchy problem, via introducing one
"quasi-symmetric hyperbolic"--"degenerate elliptic" coupled structure to
control the behavior of the velocity near the vacuum, we establish the uniform
energy estimates for the local sound speed in and
in with respect to the viscosity
coefficients for arbitrarily large time under some smallness assumption on the
initial density. Second, by making full use of this structure's quasi-symmetric
property and the weak smooth effect on solutions, we prove the strong
convergence of the regular solutions of the degenerate viscous flow to that of
the inviscid flow with vacuum in for arbitrarily large time. The result
here applies to a class of degenerate density-dependent viscosity coefficients,
is independent of the B-D relation for viscosities, and seems to be the first
on the global-in-time inviscid limit of smooth solutions which have large
velocities and contain vacuum state for compressible flow in three space
dimensions without any symmetric assumption.Comment: arXiv admin note: text overlap with arXiv:1806.0238
Vanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid Flow with Vacuum
We establish the vanishing viscosity limit of the Navier-Stokes equations to
the Euler equations for three-dimensional compressible isentropic flow in the
whole space. It is shown that there exists a unique regular solution of
compressible Navier-Stokes equations with density-dependent viscosities,
arbitrarily large initial data and vacuum, whose life span is uniformly
positive in the vanishing viscosity limit. It is worth paying special attention
that, via introducing a "quasi-symmetric hyperbolic"--"degenerate elliptic"
coupled structure, we can also give some uniformly bounded estimates of
in space and
in space (adiabatic exponent and
), which lead the strong convergence of the
regular solution of the viscous flow to that of the inviscid flow in
(for any ) space with the rate of
. Further more, we point out that our framework in this
paper is applicable to other physical dimensions, say 1 and 2, with some minor
modifications. This paper is based on our early preprint in 2015.Comment: arXiv admin note: text overlap with arXiv:1503.05644; and text
overlap with arXiv:0910.2360, arXiv:1005.2713 by other author
A New High Performance and Scalable SVD algorithm on Distributed Memory Systems
This paper introduces a high performance implementation of \texttt{Zolo-SVD}
algorithm on distributed memory systems, which is based on the polar
decomposition (PD) algorithm via the Zolotarev's function (\texttt{Zolo-PD}),
originally proposed by Nakatsukasa and Freund [SIAM Review, 2016]. Our
implementation highly relies on the routines of ScaLAPACK and therefore it is
portable. Compared with the other PD algorithms such as the QR-based
dynamically weighted Halley method (\texttt{QDWH-PD}), \texttt{Zolo-PD} is
naturally parallelizable and has better scalability though performs more
floating-point operations. When using many processes, \texttt{Zolo-PD} is
usually 1.20 times faster than \texttt{QDWH-PD} algorithm, and
\texttt{Zolo-SVD} can be about two times faster than the ScaLAPACK routine
\texttt{\texttt{PDGESVD}}. These numerical experiments are performed on
Tianhe-2 supercomputer, one of the fastest supercomputers in the world, and the
tested matrices include some sparse matrices from particular applications and
some randomly generated dense matrices with different dimensions. Our
\texttt{QDWH-SVD} and \texttt{Zolo-SVD} implementations are freely available at
https://github.com/shengguolsg/Zolo-SVD.Comment: 25 pages, 3 figure
Updating quadratic palindromic models with no spillover effect on unmeasured spectral data
This paper concerns the model updating problems with no spillover of the
quadratic palindromic system Comment: I find some errors in the proof of some results in this manuscrip
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