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.

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 A counterexample to well-posedness of entropy solutions to the compressible Euler system

2014 ◽  
Vol 11 (03) ◽  
pp. 493-519 ◽  
Author(s):  
Elisabetta Chiodaroli
Keyword(s):  
Space Dimension ◽  
Initial Density ◽  
Euler System ◽  
Entropy Solutions ◽  
Periodic Case ◽  
Time Interval ◽  
Finite Time Interval ◽  
Compressible Euler ◽  
The Cauchy Problem ◽  
Spatially Periodic

We consider entropy solutions to the Cauchy problem for the isentropic compressible Euler equations in the spatially periodic case. In more than one space dimension, the methods developed by De Lellis–Székelyhidi enable us to show here failure of uniqueness on a finite time-interval for entropy solutions starting from any continuously differentiable initial density and suitably constructed bounded initial linear momenta.

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.

Global smooth solutions for a class of quasilinear hyperbolic systems with dissipative terms

1997 ◽  
Vol 127 (6) ◽  
pp. 1311-1324 ◽  
Author(s):  
Tong Yang ◽  
Changjiang Zhu ◽  
Huijiang Zhao
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Existence Theorem ◽  
Hyperbolic Systems ◽  
A Priori ◽  
Smooth Solutions ◽  
Quasilinear Hyperbolic Systems ◽  
Global Smooth Solutions ◽  
The Cauchy Problem ◽  
Dissipative Terms

In this paper we prove an existence theorem of global smooth solutions for the Cauchy problem of a class of quasilinear hyperbolic systems with nonlinear dissipative terms under the assumption that only the C0-norm of the initial data is sufficiently small, while the C1-norm of the initial data can be large. The analysis is based on a priori estimates, which are obtained by a generalised Lax transformation.

Some stability results on global solutions to the Navier–Stokes equations

Analysis ◽  
2015 ◽  
Vol 35 (3) ◽  
Author(s):  
Isabelle Gallagher
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Global Solutions ◽  
Navier Stokes ◽  
Smooth Solutions ◽  
Navier Stokes Equations ◽  
Global Smooth Solutions ◽  
Whole Space ◽  
The Stability ◽  
Stability Results

AbstractIn these notes we present some results concerning the existence of global smooth solutions to the three-dimensional Navier–Stokes equations set in the whole space. We are particularly interested in the stability of the set of initial data giving rise to a global smooth solution.

BLOW-UP OF SMOOTH SOLUTIONS TO THE NAVIER–STOKES EQUATIONS OF COMPRESSIBLE VISCOUS HEAT-CONDUCTING FLUIDS

2010 ◽  
Vol 88 (2) ◽  
pp. 239-246 ◽  
Author(s):  
ZHONG TAN ◽  
YANJIN WANG
Keyword(s):  
Blow Up ◽  
Stokes Equations ◽  
Initial Density ◽  
Navier Stokes ◽  
Heat Conducting ◽  
Smooth Solutions ◽  
Navier Stokes Equations ◽  
The Cauchy Problem ◽  
Heat Conducting Fluids ◽  
Conducting Fluids

AbstractWe give a simpler and refined proof of some blow-up results of smooth solutions to the Cauchy problem for the Navier–Stokes equations of compressible, viscous and heat-conducting fluids in arbitrary space dimensions. Our main results reveal that smooth solutions with compactly supported initial density will blow up in finite time, and that if the initial density decays at infinity in space, then there is no global solution for which the velocity decays as the reciprocal of the elapsed time.

MODIFIED SCATTERING FOR BIPOLAR NONLINEAR SCHRÖDINGER–POISSON EQUATIONS

2004 ◽  
Vol 14 (10) ◽  
pp. 1481-1494 ◽  
Author(s):  
CHENGCHUN HAO ◽  
LING HSIAO ◽  
HAILIANG LI
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Small Data ◽  
Scattering Operator ◽  
Smooth Solutions ◽  
Nonlinear Schrödinger ◽  
Poisson Equations ◽  
The Cauchy Problem

In this paper, we study the asymptotic behavior in time and the existence of the modified scattering operator of the globally defined smooth solutions to the Cauchy problem for the bipolar nonlinear Schrödinger–Poisson equations with small data in the space ℝ3.

Non-existence of global smooth solutions to symmetrizable nonlinear hyperbolic systems

2003 ◽  
Vol 133 (3) ◽  
pp. 719-728 ◽  
Author(s):  
Tong Yang ◽  
Changjiang Zhu
Keyword(s):  
Blow Up ◽  
Dimensional Space ◽  
Hyperbolic Systems ◽  
Shock Formation ◽  
Sufficient Condition ◽  
Smooth Solutions ◽  
Physical Systems ◽  
Global Smooth Solutions ◽  
The Cauchy Problem ◽  
Nonlinear Hyperbolic Systems

In this paper, we consider the Cauchy problem of general symmetrizable hyperbolic systems in multi-dimensional space. When some components of the initial data have compact support, we give a sufficient condition on the non-existence of global C1 solutions. This non-existence theorem can be applied to some physical systems, such as Euler equations for compressible flow in multi-dimensional space. The blow-up phenomena here can come from the singularity developed at the interface, such as vacuum boundary, rather than the shock formation as studied in the previous works on strictly hyperbolic systems. Therefore, the systems considered here include those which are non-strictly hyperbolic.

scholarly journals Incompressible limit of Euler equations with damping

2021 ◽  
Vol 30 (1) ◽  
pp. 126-139
Author(s):  
Fei Shi ◽  
Keyword(s):  
Cauchy Problem ◽  
Mach Number ◽  
Euler System ◽  
Existence Results ◽  
Incompressible Limit ◽  
Low Mach Number ◽  
Uniform Estimates ◽  
Low Mach Number Limit ◽  
Compressible Euler ◽  
The Cauchy Problem

<abstract><p>The Cauchy problem for the compressible Euler system with damping is considered in this paper. Based on previous global existence results, we further study the low Mach number limit of the system. By constructing the uniform estimates of the solutions in the well-prepared initial data case, we are able to prove the global convergence of the solutions in the framework of small solutions.</p></abstract>

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