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.

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.

scholarly journals Uniqueness of rarefaction waves in multidimensional compressible Euler system

2015 ◽  
Vol 12 (03) ◽  
pp. 489-499 ◽  
Author(s):  
Eduard Feireisl ◽  
Ondřej Kreml
Keyword(s):  
Shock Wave ◽  
Rarefaction Wave ◽  
Riemann Problem ◽  
Euler System ◽  
Entropy Solutions ◽  
Infinitely Many Solutions ◽  
Rarefaction Waves ◽  
Convex Integration ◽  
Wave Solutions ◽  
Compressible Euler

We show that 1D rarefaction wave solutions are unique in the class of bounded entropy solutions to the multidimensional compressible Euler system. Such a result may be viewed as a counterpart of the recent examples of non-uniqueness of the shock wave solutions to the Riemann problem, where infinitely many solutions are constructed by the method of convex integration.

Global entropy solutions to an inhomogeneous isentropic compressible euler system

2016 ◽  
Vol 36 (4) ◽  
pp. 1215-1224 ◽  
Author(s):  
Wentao CAO ◽  
Feimin HUANG ◽  
Tianhong LI ◽  
Huimin YU
Keyword(s):  
Euler System ◽  
Entropy Solutions ◽  
Compressible Euler

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>

Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data

1986 ◽  
Vol 103 (3-4) ◽  
pp. 301-315 ◽  
Author(s):  
David Hoff
Keyword(s):  
Space Dimension ◽  
Stokes Equations ◽  
Initial Density ◽  
Navier Stokes ◽  
Jump Conditions ◽  
Simple Analysis ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
Isentropic Flow ◽  
The Cauchy Problem

We prove the global existence of weak solutions for the Cauchy problem for the Navier-Stokes equations for one-dimensional, isentropic flow when the initial velocity is in L2 and the initial density is in L2 ∩ BV. Solutions are obtained as limits of approximations obtained by building heuristic jump conditions into a semi-discrete difference scheme. This allows for a rather simple analysis in which pointwise control is achieved through piecewise H1 and total variation estimates.

scholarly journals Global existence of a radiative Euler system coupled to an electromagnetic field

2018 ◽  
Vol 8 (1) ◽  
pp. 1158-1170
Author(s):  
Xavier Blanc ◽  
Bernard Ducomet ◽  
Šárka Nečasová
Keyword(s):  
Cauchy Problem ◽  
Electromagnetic Field ◽  
Global Existence ◽  
Smooth Solution ◽  
Euler System ◽  
Singular Limit ◽  
System Of Equations ◽  
Global Smooth Solution ◽  
Compressible Euler ◽  
The Cauchy Problem

Abstract We study the Cauchy problem for a system of equations corresponding to a singular limit of radiative hydrodynamics, namely, the 3D radiative compressible Euler system coupled to an electromagnetic field. Assuming smallness hypotheses for the data, we prove that the problem admits a unique global smooth solution and study its asymptotics.

scholarly journals Uniform regularity for the free surface compressible Navier–Stokes equations with or without surface tension

2017 ◽  
Vol 28 (02) ◽  
pp. 259-336
Author(s):  
Yu Mei ◽  
Yong Wang ◽  
Zhouping Xin
Keyword(s):  
Surface Tension ◽  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Stokes System ◽  
Euler System ◽  
Navier Stokes ◽  
Time Interval ◽  
Uniform Estimates ◽  
Compressible Euler

In this paper, we investigate the uniform regularity of solutions to the three-dimensional isentropic compressible Navier–Stokes system with free surfaces and study the corresponding asymptotic limits of such solutions to that of the compressible Euler system for vanishing viscosity and surface tension. It is shown that there exists a unique strong solution to the free boundary problem for the compressible Navier–Stokes system in a finite time interval which is independent of the viscosity and the surface tension. The solution is uniformly bounded both in [Formula: see text] and a conormal Sobolev space. It is also shown that the boundary layer for the density is weaker than the one for the velocity field. Based on such uniform estimates, the asymptotic limits, to the free boundary problem for the ideal compressible Euler system with or without surface tension as both the viscosity and the surface tension tend to zero, are established by a strong convergence argument.

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