# On convergence of approximate solutions to the compressible Euler system

@article{Feireisl2020OnCO,
title={On convergence of approximate solutions to the compressible Euler system},
author={Eduard Feireisl and Martina Hofmanov'a},
journal={Annals of PDE},
year={2020}
}
• Published 7 May 2019
• Mathematics
• Annals of PDE
We consider a sequence of approximate solutions to the compressible Euler system admitting uniform energy bounds and/or satisfying the relevant field equations modulo an error vanishing in the asymptotic limit. We show that such a sequence either (i) converges strongly in the energy norm, or (ii) the limit is not a weak solution of the associated Euler system. This is in sharp contrast to the incompressible case, where (oscillatory) approximate solutions may converge weakly to solutions of the…
Approximating viscosity solutions of the Euler system
• Physics, Mathematics
Math. Comput.
• 2022
It is shown how to efficiently compute a viscosity solution of the Euler system as the S-limit of numerical solutions obtained by the Viscosity Finite Volume method.
Convergence of a stochastic collocation finite volume method for the compressible Navier-Stokes system
• Mathematics
ArXiv
• 2021
We propose a stochastic collocation method based on the piecewise constant interpolation on the probability space combined with a finite volume method to solve the compressible Navier–Stokes system
(S)-convergence and approximation of oscillatory solutions in fluid dynamics
We propose a new concept of (S)-convergence applicable to numerical methods as well as other consistent approximations of the Euler system in gas dynamics. (S)-convergence, based on averaging in the
Asymptotic properties of solutions to equations of fluid dynamics
The goal of this lecture series is to give an overview of some recent results and new directions in continuum fluid dynamics. After a brief introduction of the basic system of balance laws, we focus
Minimal acceleration for the multi-dimensional isentropic Euler equations
Among all dissipative solutions of the multi-dimensional isentropic Euler equations there exists at least one that minimizes the acceleration, which implies that the solution is as close to being a
Analysis of the generalised Aw-Rascle model
• Mathematics, Computer Science
• 2022
It is proved the existence of globalin-time measure-valued solutions for arbitrary large class of initial data and the periodic domain and using the relative energy technique, it is shown that the measure- valued solutions coincide with the classical solutions as long as the latter exist.
Limit of a Consistent Approximation to the Complete Compressible Euler System
The goal of the present paper is to prove that if a weak limit of a consistent approximation scheme of the compressible complete Euler system in full space $$\mathbb {R}^d,\; d=2,3$$ R d
Randomness in Compressible Fluid Flows Past an Obstacle
• Mathematics
Journal of Statistical Physics
• 2022
We consider a statistical limit of solutions to the compressible Navier–Stokes system in the high Reynolds number regime in a domain exterior to a rigid body. We investigate to what extent this
Generalized solutions to models of inviscid fluids
• Mathematics
Discrete & Continuous Dynamical Systems - B
• 2020
We discuss several approaches to generalized solutions of problems describing the motion of inviscid fluids. We propose a new concept of dissipative solution to the compressible Euler system based on

## References

SHOWING 1-10 OF 46 REFERENCES
On Admissibility Criteria for Weak Solutions of the Euler Equations
• Mathematics
• 2007
We consider solutions to the Cauchy problem for the incompressible Euler equations satisfying several additional requirements, like the global and local energy inequalities. Using some techniques
Non–uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data
• Mathematics
• 2018
We consider the isentropic Euler equations of gas dynamics in the whole two-dimensional space and we prove the existence of a $C^\infty$ initial datum which admits infinitely many bounded admissible
Global Ill‐Posedness of the Isentropic System of Gas Dynamics
• Mathematics
• 2015
We consider the isentropic compressible Euler system in 2 space dimensions with pressure law p (ρ) = ρ2 and we show the existence of classical Riemann data, i.e. pure jump discontinuities across a
Oscillations and concentrations in weak solutions of the incompressible fluid equations
• Mathematics
• 1987
The authors introduce a new concept of measure-valued solution for the 3-D incompressible Euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of
Computing oscillatory solutions of the Euler system via K-convergence
• Mathematics
ArXiv
• 2019
The approach is based on the concept of K−convergence adapted to sequences of parametrized measures, which is strong in space and time whereas the measures converge narrowly or in the Wasserstein distance to the corresponding limit.
Measure-valued solutions to the complete Euler system
• Mathematics
Journal of the Mathematical Society of Japan
• 2018
We introduce the concept of dissipative measure-valued solution to the complete Euler system describing the motion of an inviscid compressible fluid. These solutions are characterized by a
Vanishing dissipation limit for the Navier-Stokes-Fourier system
We consider the motion of a compressible, viscous, and heat conducting fluid in the regime of small viscosity and heat conductivity. It is shown that weak solutions of the associated Navier-
𝒦-convergence as a new tool in numerical analysis
• Mathematics
IMA Journal of Numerical Analysis
• 2019
We adapt the concept of $\mathscr{K}$-convergence of Young measures to the sequences of approximate solutions resulting from numerical schemes. We obtain new results on pointwise convergence of
Vanishing viscosity limit for the compressible Navier–Stokes system via measure-valued solutions
We identify a class of measure-valued solutions of the barotropic Euler system on a general (un-bounded) spatial domain as a vanishing viscosity limit for the compressible Navier-Stokes system. Then
Vanishing Viscosity Solutions of the Compressible Euler Equations with Spherical Symmetry and Large Initial Data
• Mathematics
• 2014
We are concerned with spherically symmetric solutions of the Euler equations for multidimensional compressible fluids, which are motivated by many important physical situations. Various evidences