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 $1<\gamma\leq 3$. 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 $2$-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 $2$-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 $H^2$ 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 $O(N^2r)$ {flops} in the worst case, where $N$ is the dimension of the matrix and $r$ 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 $p_m=p_m(\rho)\in C^1(\mathbb{\overline{R}}^+)$.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 ($(\rho^\epsilon)^\delta$ with $\delta>1$), 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 $H^3$ and $(\rho^\epsilon)^{\frac{\delta-1}{2}}$ in $H^2$ 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 $H^2$ 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 $\displaystyle\Big(\rho^{\frac{\gamma-1}{2}}, u\Big)$ in $H^3$ space and $\rho^{\frac{\delta-1}{2}}$ in $H^2$ space (adiabatic exponent $\gamma>1$ and $1<\delta \leq \min\{3, \gamma\}$), which lead the strong convergence of the regular solution of the viscous flow to that of the inviscid flow in $L^{\infty}([0, T]; H^{s'})$ (for any $s'\in [2, 3)$) space with the rate of $\epsilon^{2(1-s'/3)}$. 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 $P(\lambda)=\lambda^2 A^{\star}+\lambda Q+A$Comment: I find some errors in the proof of some results in this manuscrip