scholarly journals Global Existence and Long-Time Behavior of Solutions to the Full Compressible Euler Equations with Damping and Heat Conduction in ℝ 3

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Yunshun Wu ◽  
Yong Wang ◽  
Rong Shen
Keyword(s):  
Heat Conduction ◽  
Euler Equations ◽  
Three Dimensional ◽  
Compressible Euler Equations ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Algebraic Decay ◽  
Compressible Euler ◽  
The Cauchy Problem ◽  
Long Time

We study the Cauchy problem of the three-dimensional full compressible Euler equations with damping and heat conduction. We prove the existence and uniqueness of the global small H N N ≥ 3 solution; in particular, we only require that the H 4 norms of the initial data be small when N ≥ 5 . Moreover, we use a pure energy method to show that the global solution converges to the constant equilibrium state with an optimal algebraic decay rate as time goes to infinity.

Local existence with low regularity for the 2D compressible Euler equations

2021 ◽  
Vol 18 (03) ◽  
pp. 701-728
Author(s):  
Huali Zhang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Euler Equations ◽  
Local Existence ◽  
Compressible Euler Equations ◽  
Two Dimensional ◽  
Spatial Derivative ◽  
Compressible Euler ◽  
The Cauchy Problem ◽  
Local Solutions

We prove the local existence, uniqueness and stability of local solutions for the Cauchy problem of two-dimensional compressible Euler equations, where the initial data of velocity, density, specific vorticity [Formula: see text] and the spatial derivative of specific vorticity [Formula: see text].

Global smooth solutions to 3D irrotational Euler equations for Chaplygin gases

2020 ◽  
Vol 17 (03) ◽  
pp. 613-637
Author(s):  
Changhua Wei ◽  
Yu-Zhu Wang
Keyword(s):  
Euler Equations ◽  
Compressible Euler Equations ◽  
Wave System ◽  
Smooth Solutions ◽  
Small Perturbations ◽  
Global Smooth Solutions ◽  
Compressible Euler ◽  
The Cauchy Problem ◽  
Constant State ◽  
New Formulation

We study here the Cauchy problem associated with the isentropic and compressible Euler equations for Chaplygin gases. Based on the new formulation of the compressible Euler equations in J. Luk and J. Speck [The hidden null structure of the compressible Euler equations and a prelude to applications, J. Hyperbolic Differ. Equ. 17 (2020) 1–60] we show that the wave system satisfied by the modified density and the velocity for Chaplygin gases satisfies the weak null condition. We then prove the global existence of smooth solutions to the irrotational and isentropic Chaplygin gases without introducing a potential function, when the initial data are small perturbations to a constant state.

scholarly journals SHOCK FORMATION IN THE COMPRESSIBLE EULER EQUATIONS AND RELATED SYSTEMS

2013 ◽  
Vol 10 (01) ◽  
pp. 149-172 ◽  
Author(s):  
GENG CHEN ◽  
ROBIN YOUNG ◽  
QINGTIAN ZHANG
Keyword(s):  
Euler Equations ◽  
Space Dimension ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Compressible Euler Equations ◽  
Shock Formation ◽  
Compressible Euler ◽  
Systems Of Conservation Laws ◽  
The One ◽  
Special Case

We prove shock formation results for the compressible Euler equations and related systems of conservation laws in one space dimension, or three dimensions with spherical symmetry. We establish an L∞ bound for C1 solutions of the one-dimensional (1D) Euler equations, and use this to improve recent shock formation results of the authors. We prove analogous shock formation results for 1D magnetohydrodynamics (MHD) with orthogonal magnetic field, and for compressible flow in a variable area duct, which has as a special case spherically symmetric three-dimensional (3D) flow on the exterior of a ball.

scholarly journals EXTREME LONG-TIME DYNAMIC MONTE CARLO SIMULATIONS FOR METASTABLE DECAY IN THE d=3 ISING FERROMAGNET

2003 ◽  
Vol 14 (01) ◽  
pp. 121-131 ◽  
Author(s):  
MIROSLAV KOLESIK ◽  
M. A. NOVOTNY ◽  
PER ARNE RIKVOLD
Keyword(s):  
Metastable Phase ◽  
Three Dimensional ◽  
Glauber Dynamics ◽  
Simulation Method ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Time Dynamic ◽  
Dynamic Monte Carlo ◽  
Long Time ◽  
Good Agreement

We study the extreme long-time behavior of the metastable phase of the three-dimensional Ising model with Glauber dynamics in an applied magnetic field and at a temperature below the critical temperature. For these simulations, we use the advanced simulation method of projective dynamics. The algorithm is described in detail, together with its application to the escape from the metastable state. Our results for the field dependence of the metastable lifetime are in good agreement with theoretical expectations and span more than 50 decades in time.

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