scholarly journals Nonuniqueness of Admissible Weak Solution to the Riemann Problem for the Full Euler System in Two Dimensions

2020 ◽  
Vol 52 (2) ◽  
pp. 1729-1760 ◽  
Author(s):  
Hind Al Baba ◽  
Christian Klingenberg ◽  
Ondřej Kreml ◽  
Václav Mácha ◽  
Simon Markfelder
Keyword(s):  
Weak Solution ◽  
Riemann Problem ◽  
Euler System ◽  
Two Dimensions

Generalised Riemann problem for Euler system

2017 ◽  
Vol 60 (4) ◽  
pp. 581-592
Author(s):  
ShuXing Chen ◽  
DeNing Li
Keyword(s):  
Riemann Problem ◽  
Euler System ◽  
Generalised Riemann Problem

THE RIEMANN PROBLEM FOR THE MULTI-PRESSURE EULER SYSTEM

2005 ◽  
Vol 02 (03) ◽  
pp. 745-782 ◽  
Author(s):  
C. CHALONS ◽  
F. COQUEL
Keyword(s):  
Riemann Problem ◽  
Stokes Equations ◽  
Traveling Wave Solutions ◽  
Viscosity Coefficient ◽  
Continuous Functions ◽  
Euler System ◽  
Small Scale ◽  
Riemann Solutions ◽  
Lipschitz Continuous ◽  
Kinetic Relations

We prove the existence and uniqueness of the Riemann solutions to the Euler equations closed by N independent constitutive pressure laws. This model stands as a natural asymptotic system for the multi-pressure Navier–Stokes equations in the regime of infinite Reynolds number. Due to the inherent lack of conservation form in the viscous regularization, the limit system exhibits measure-valued source terms concentrated on shock discontinuities. These non-positive bounded measures, called kinetic relations, are known to provide a suitable tool to encode the small-scale sensitivity in the singular limit. Considering N independent polytropic pressure laws, we show that these kinetic relations can be derived by solving a simple algebraic problem which governs the endpoints of the underlying viscous shock profiles, for any given but prescribed ratio of viscosity coefficient in the viscous perturbation. The analysis based on traveling wave solutions allows us to introduce the asymptotic Euler system in the setting of piecewise Lipschitz continuous functions and to study the Riemann problem.

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.

scholarly journals On two-dimensional Riemann problem for pressure-gradient equations of the Euler system

1998 ◽  
Vol 4 (4) ◽  
pp. 609-634 ◽  
Author(s):  
Peng Zhang ◽  
◽  
Jiequan Li ◽  
Tong Zhang ◽  
◽  
...  
Keyword(s):  
Pressure Gradient ◽  
Riemann Problem ◽  
Euler System ◽  
Two Dimensional ◽  
Pressure Gradient Equations

scholarly journals Limit of a Consistent Approximation to the Complete Compressible Euler System

2021 ◽  
Vol 23 (4) ◽  
Author(s):  
Nilasis Chaudhuri
Keyword(s):  
Weak Solution ◽  
Initial Data ◽  
Heat Conductivity ◽  
Approximation Scheme ◽  
Approximate Solutions ◽  
Weak Limit ◽  
Euler System ◽  
Vanishing Viscosity ◽  
Consistent Approximation ◽  
Compressible Euler

AbstractThe 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 , d = 2 , 3 is a weak solution of the system, then the approximate solutions eventually converge strongly in suitable norms locally under a minimal assumption on the initial data of the approximate solutions. The class of consistent approximate solutions is quite general and includes the vanishing viscosity and heat conductivity limit. In particular, they may not satisfy the minimal principle for entropy.

A NUMERICAL STUDY OF NONLINEAR PROPAGATION OF DISTURBANCES IN TWO-DIMENSIONS

1996 ◽  
Vol 04 (03) ◽  
pp. 291-319 ◽  
Author(s):  
TONY W.H. SHEU ◽  
C.C. FANG
Keyword(s):  
Acoustic Field ◽  
Numerical Study ◽  
Element Model ◽  
Analytic Solutions ◽  
Finite Amplitude ◽  
Euler System ◽  
Two Dimensions ◽  
Nonlinear Propagation ◽  
Characteristic Model ◽  
Transport Problems

In the spirit of the method of characteristics, we present in this paper a generalized Taylor-Galerkin finite element model to simulate the nonlinear propagation of finite-amplitude disturbances. In a nonlinear Euler system, the multi-dimensional formulation is constructed through the conservation variables. Noticeable is that the scheme is found to exhibit high-phase-accuracy, together with minimal numerical damping. This scheme, therefore, is best-suited to simulation of disturbances in an acoustic field. To begin with, we validate the characteristic model by simulating two transport problems amenable to analytic solutions. Motivated by the apparent success, we apply the proposed third-order accurate upwind model to investigate a truly nonlinear acoustic field. The present analysis is intended to elucidate to what extent the nondissipative, nondispersive and isotropic characteristics pertaining to three wave modes of the acoustic system are still valid.

Analysis of a mean field model of superconducting vortices

1999 ◽  
Vol 10 (4) ◽  
pp. 319-352 ◽  
Author(s):  
R. SCHÄTZLE ◽  
V. STYLES
Keyword(s):  
Weak Solution ◽  
Steady State ◽  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Field Model ◽  
Mean Field ◽  
Two Dimensions ◽  
Mean Field Model ◽  
Superconducting Vortices

We study a mean-field model of superconducting vortices in one and two dimensions. The existence of a weak solution and a steady-state solution of the model are proved. A special case of the steady-state problem is shown to be of the form of a free boundary problem. The solutions of this free boundary problem are investigated. It is also shown that the weak solution of the one-dimensional model is unique and satisfies an entropy inequality.

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