Existence of Global Smooth Solutions to Euler Equations for an Isentropic Perfect Gas

1999 ◽  
pp. 395-400
Author(s):  
M. Grassin
Keyword(s):  
Euler Equations ◽  
Smooth Solutions ◽  
Perfect Gas ◽  
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.

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

scholarly journals The global existence issue for the compressible Euler system with Poisson or Helmholtz couplings

2021 ◽  
Vol 18 (01) ◽  
pp. 169-193
Author(s):  
Xavier Blanc ◽  
Raphaël Danchin ◽  
Bernard Ducomet ◽  
Šárka Nečasová
Keyword(s):  
Initial Density ◽  
Euler System ◽  
Decay Estimates ◽  
Smooth Solutions ◽  
Perfect Gas ◽  
Poisson Equations ◽  
Global Smooth Solutions ◽  
Compressible Euler ◽  
The Cauchy Problem ◽  
Whole Space

We consider the Cauchy problem for the barotropic Euler system coupled to Helmholtz or Poisson equations, in the whole space. We assume that the initial density is small enough, and that the initial velocity is close to some reference vector field [Formula: see text] such that the spectrum of [Formula: see text] is positive and bounded away from zero. We prove the existence of a global unique solution with (fractional) Sobolev regularity, and algebraic time decay estimates. Our work extends Grassin and Serre’s papers [Existence de solutions globales et régulières aux équations d’Euler pour un gaz parfait isentropique, C. R. Acad. Sci. Paris Sér. I 325 (1997) 721–726, 1997; Global smooth solutions to Euler equations for a perfect gas, Indiana Univ. Math. J. 47 (1998) 1397–1432; Solutions classiques globales des équations d’Euler pour un fluide parfait compressible, Ann. Inst. Fourier Grenoble 47 (1997) 139–159] dedicated to the compressible Euler system without coupling and with integer regularity exponents.

LOCAL SMOOTH SOLUTIONS OF A THIN SPRAY MODEL WITH COLLISIONS

2010 ◽  
Vol 20 (02) ◽  
pp. 191-221 ◽  
Author(s):  
JULIEN MATHIAUD
Keyword(s):  
Boltzmann Equation ◽  
Initial Data ◽  
Euler Equations ◽  
Existence And Uniqueness ◽  
Well Posedness ◽  
Liquid Droplets ◽  
Smooth Solutions ◽  
Perfect Gas ◽  
Complex Flows

Sprays are complex flows made of liquid droplets surrounded by a gas. The aim of this paper is to study the local in time well-posedness of a collisional thin spray model, that is a coupling between Euler equations for a perfect gas and a Vlasov–Boltzmann equation for the droplets. We prove the existence and uniqueness of (local in time) solutions for this problem as soon as initial data are smooth enough.

GLOBAL CENTERED WAVES AND CONTACT DISCONTINUITIES OF THE AXISYMMETRIC EULER EQUATIONS

2006 ◽  
Vol 03 (01) ◽  
pp. 143-193 ◽  
Author(s):  
PAUL GODIN
Keyword(s):  
Initial Data ◽  
Euler Equations ◽  
Contact Discontinuity ◽  
Existence Results ◽  
Smooth Solutions ◽  
Perfect Gas ◽  
Rotation Invariant ◽  
Contact Discontinuities ◽  
Positive Time ◽  
Two Space Dimensions

Global existence results have been obtained by Serre and Grassin–Serre for smooth solutions to the Euler equations of a perfect gas, provided the initial data belong to suitable spaces, the initial sound speed is small, and the initial velocity forces particles to spread out. We work in two space dimensions and start with initial data which are rotation invariant around 0 and of the type considered by Serre and Grassin–Serre. We then consider slightly perturbed initial data which are also rotation invariant around 0 and jump across a given circle centered at 0, in such a way that there is a solution with these perturbed initial data which presents two centered waves (in radial coordinates) and one contact discontinuity for small positive time. We show that this solution is global in positive time and keeps the same structure.

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