Existence of global smooth solutions for Euler equations with symmetry (II)

2000 ◽  
Vol 41 (1-2) ◽  
pp. 187-203 ◽  
Author(s):  
Tong Yang ◽  
Changjiang Zhu ◽  
Yongshu Zheng
Keyword(s):  
Euler Equations ◽  
Smooth Solutions ◽  
Global Smooth Solutions

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 Circulation and Energy Theorem Preserving Stochastic Fluids

2019 ◽  
Vol 150 (6) ◽  
pp. 2776-2814 ◽  
Author(s):  
Theodore D. Drivas ◽  
Darryl D. Holm
Keyword(s):  
Variational Principle ◽  
Energy Conservation ◽  
Euler Equations ◽  
Navier Stokes ◽  
Fluid Models ◽  
Smooth Solutions ◽  
Stochastic Fluid Models ◽  
Natural Extensions ◽  
Stochastic Fluid ◽  
Incompressible Euler

AbstractSmooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier–Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin–Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler–Poincaré and stochastic Navier–Stokes–Poincaré equations respectively. The stochastic Euler–Poincaré equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems.

Singularities of solutions for first order quasilinear hyperbolic systems

1983 ◽  
Vol 94 (1-2) ◽  
pp. 137-147 ◽  
Author(s):  
Lee Da-tsin(Li Ta-tsien) ◽  
Shi Jia-hong
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Hyperbolic Systems ◽  
Smooth Solutions ◽  
Small Initial Data ◽  
Quasilinear Hyperbolic Systems ◽  
First Order ◽  
Global Smooth Solutions ◽  
The Cauchy Problem

SynopsisIn this paper, the existence of global smooth solutions and the formation of singularities of solutions for strictly hyperbolic systems with general eigenvalues are discussed for the Cauchy problem with essentially periodic small initial data or nonperiodic initial data. A result of Klainerman and Majda is thus extended to the general case.

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