Local existence and singularities formation for isothermal relativistic radiation hydrodynamics equations

2016 ◽  
Vol 13 (04) ◽  
pp. 661-683
Author(s):  
Yongcai Geng
Keyword(s):  
Blow Up ◽  
Hyperbolic Equations ◽  
Local Existence ◽  
Three Dimensional ◽  
Propagation Speed ◽  
Smooth Solutions ◽  
Nonlinear Hyperbolic Equations ◽  
The Cauchy Problem ◽  
Finite Propagation Speed ◽  
Hydrodynamics Equations

We study the Cauchy problem for multi-dimensional compressible relativistic hydrodynamics in the presence of a radiation field. First, based on the theory of quasilinear symmetric hyperbolic, we establish the local existence of smooth solutions for both non-vacuum and vacuum cases. Next, in the spirit of Sideris’ work [T. Sideris, Formation of singularities of solutions to nonlinear hyperbolic equations, Arch. Ration. Mech. Anal. 86 (1984) 369–381; T. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys. 101 (1985) 475–485], we show that smooth solutions blow-up in finite time if the initial data is compactly supported and large enough. Compared with the previous work, the main difficulties of the first problem lie in two aspects, we must first deal with the source terms relying on radiative quantities, and we also need to solve out the new coefficients matrices under the Lorentz transformation for vacuum case. The second difficulty arises on how to verifying that the smooth solution has finite propagation speed..

Local existence and Serrin-type blow-up criterion for strong solutions to the radiation hydrodynamic equations

2020 ◽  
Vol 17 (03) ◽  
pp. 501-557
Author(s):  
Hao Li ◽  
Yachun Li
Keyword(s):  
Blow Up ◽  
Initial Density ◽  
Local Existence ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Hydrodynamic Equations ◽  
Open Set ◽  
The Cauchy Problem ◽  
Local Strong Solutions ◽  
Blow Up Criterion

We consider the Cauchy problem for the three-dimensional, compressible radiation hydrodynamic equations. We establish the existence and uniqueness of local strong solutions for large initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set. Moreover, we establish a Serrin-type blow-up criterion, which is stated in terms of the velocity and density variables [Formula: see text] and is independent of the temperature and the radiation intensity.

A LOGARITHMALLY IMPROVED BLOW-UP CRITERION OF SMOOTH SOLUTIONS FOR THE THREE-DIMENSIONAL MHD EQUATIONS

2012 ◽  
Vol 23 (02) ◽  
pp. 1250027 ◽  
Author(s):  
YU-ZHU WANG ◽  
HENGJUN ZHAO ◽  
YIN-XIA WANG
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Three Dimensional ◽  
Mhd Equations ◽  
Magnetohydrodynamic Equations ◽  
Smooth Solutions ◽  
The Cauchy Problem ◽  
Incompressible Magnetohydrodynamic Equations ◽  
Blow Up Criterion

In this paper we investigate the Cauchy problem for the three-dimensional incompressible magnetohydrodynamic equations. A logarithmal improved blow-up criterion of smooth solutions is obtained.

Three-Dimensional Sloshing of Water on Decks

1988 ◽  
Vol 25 (04) ◽  
pp. 253-261
Author(s):  
Michael S. Pantazopoulos
Keyword(s):  
Hyperbolic Equations ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Nonlinear Hyperbolic Equations ◽  
Complex Flow ◽  
Random Choice ◽  
Three Dimensional Flow ◽  
Random Choice Method ◽  
Wide Range ◽  
Choice Method

A methodology is proposed to solve the problem of the three-dimensional flow of water sloshing on the deck of a vessel, and to calculate the resulting forces and moments at the center of gravity. The Eulerian equations of motion of the water particle for incompressible inviscid shallow water flow are formulated with respect to a system attached to the oscillating vessel. The system of the nonlinear hyperbolic equations of motion is solved numerically using Glimm's method (random-choice method). Complex flow patterns consisting of oblique bores and "swirling" motions of the water on deck were revealed, for a vessel oscillating in roll and pitch motions, for a wide range of excitation frequencies. Large accumulation of water occurs at the corners while parts of the deck become dry. Significant rolling moments due to sloshing are exerted on the vessel. These must be taken into account when the dynamic response of the vessel is studied.

scholarly journals Blow-Up Criterion of Weak Solutions for the 3D Boussinesq Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Zhaohui Dai ◽  
Xiaosong Wang ◽  
Lingrui Zhang ◽  
Wei Hou
Keyword(s):  
Weak Solutions ◽  
Blow Up ◽  
Three Dimensional ◽  
Boussinesq Equations ◽  
Interpolation Inequality ◽  
Embedding Theorem ◽  
The Earth ◽  
The Cauchy Problem ◽  
Homogeneous Besov Space ◽  
Blow Up Criterion

The Boussinesq equations describe the three-dimensional incompressible fluid moving under the gravity and the earth rotation which come from atmospheric or oceanographic turbulence where rotation and stratification play an important role. In this paper, we investigate the Cauchy problem of the three-dimensional incompressible Boussinesq equations. By commutator estimate, some interpolation inequality, and embedding theorem, we establish a blow-up criterion of weak solutions in terms of the pressurepin the homogeneous Besov spaceḂ∞,∞0.

Existence of strong solutions to the rotating shallow water equations with degenerate viscosities

2019 ◽  
Vol 18 (02) ◽  
pp. 333-358
Author(s):  
Ben Duan ◽  
Zhen Luo ◽  
Yan Zhou
Keyword(s):  
Cauchy Problem ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Blow Up ◽  
Local Existence ◽  
Strong Solutions ◽  
Force Term ◽  
The Cauchy Problem ◽  
Rotating Shallow Water ◽  
Rotating Shallow Water Equations

In this paper, we consider the Cauchy problem of a viscous compressible shallow water equations with the Coriolis force term and non-constant viscosities. More precisely, the viscous coefficients are constants multiple of height, the equations are degenerate when vacuum appears. For initial data allowing vacuum, the local existence of strong solution is obtained and a blow-up criterion is established.

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