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 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 Limits of Riemann Solutions to the Relativistic Euler Systems for Chaplygin Gas as Pressure Vanishes

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Gan Yin ◽  
Kyungwoo Song
Keyword(s):  
Shock Wave ◽  
Rarefaction Wave ◽  
Wave Solution ◽  
Chaplygin Gas ◽  
Euler System ◽  
Delta Shock Wave ◽  
Riemann Solutions ◽  
Euler Systems ◽  
Delta Shock ◽  
Shock Wave Solution

Vanishing pressure limits of Riemann solutions to relativistic Euler system for Chaplygin gas are identified and analyzed in detail. Unlike the polytropic or barotropic gas case, as the parameter decreases to a critical value, the two-shock solution converges firstly to a delta shock wave solution to the same system. It is shown that, as the parameter decreases, the strength of the delta shock increases. Then as the pressure vanishes ultimately, the solution is nothing but the delta shock wave solution to the zero pressure relativistic Euler system. Meanwhile, the two-rarefaction wave solution and the solution containing one-rarefaction wave and one-shock wave tend to the vacuum solution and the contact discontinuity solution to the zero pressure relativistic Euler system, respectively.

Conical shock wave for non-isentropic compressible Euler system of equations

2016 ◽  
Vol 13 (02) ◽  
pp. 215-231 ◽  
Author(s):  
Dening Li ◽  
Zheng Zhang
Keyword(s):  
Shock Wave ◽  
Supersonic Flow ◽  
Shock Waves ◽  
Gas Dynamics ◽  
Euler System ◽  
System Of Equations ◽  
Conical Shock ◽  
Compressible Euler ◽  
Isentropic Gas Dynamics ◽  
Isentropic Gas

Conical shock wave is generated when a sharp conical projectile flies supersonically in the air. We study the linear stability and existence of steady conical shock waves in supersonic flow for the equations of complete Euler system in 3D non-isentropic gas-dynamics.

STABILITY OF RAREFACTION WAVES AND VACUUM STATES FOR THE MULTIDIMENSIONAL EULER EQUATIONS

2007 ◽  
Vol 04 (01) ◽  
pp. 105-122 ◽  
Author(s):  
GUI-QIANG CHEN ◽  
JUN CHEN
Keyword(s):  
Initial Data ◽  
Large Class ◽  
Euler Equations ◽  
Riemann Problem ◽  
Global Attractors ◽  
Compressible Fluids ◽  
Entropy Solutions ◽  
Rarefaction Waves ◽  
One Dimensional ◽  
Vacuum States

We are interested in properties of the multidimensional Euler equations for compressible fluids. Rarefaction waves are the unique solutions that may contain vacuum states in later time, in the context of one-dimensional Riemann problem, even when the Riemann initial data are away from the vacuum. For the multidimensional Euler equations describing isentropic or adiabatic fluids, we prove that plane rarefaction waves and vacuum states are stable within a large class of entropy solutions that may contain vacuum states. Rarefaction waves and vacuum states are also shown to be global attractors of entropy solutions in L∞, provided initial data are L∞ ∩ L1 perturbations of Riemann initial data. Our analysis applies to entropy solutions with arbitrarily large oscillation, and no bounded variation regularity is required.

scholarly journals Non-equilibrium steady state formation in 3+1 dimensions

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Christian Ecker ◽  
Johanna Erdmenger ◽  
Wilke van der Schee
Keyword(s):  
Steady State ◽  
Energy Density ◽  
Rarefaction Wave ◽  
Riemann Problem ◽  
State Formation ◽  
Numerical Solutions ◽  
Entanglement Entropy ◽  
Rarefaction Waves ◽  
Combined System ◽  
Non Equilibrium

We present the first holographic simulations of non-equilibrium steady state formation in strongly coupled \mathcal{N}=4𝒩=4 SYM theory in 3+1 dimensions. We initially join together two thermal baths at different temperatures and chemical potentials and compare the subsequent evolution of the combined system to analytical solutions of the corresponding Riemann problem and to numerical solutions of ideal and viscous hydrodynamics. The time evolution of the energy density that we obtain holographically is consistent with the combination of a shock and a rarefaction wave: A shock wave moves towards the cold bath, and a smooth broadening wave towards the hot bath. Between the two waves emerges a steady state with constant temperature and flow velocity, both of which are accurately described by a shock+rarefaction wave solution of the Riemann problem. In the steady state region, a smooth crossover develops between two regions of different charge density. This is reminiscent of a contact discontinuity in the Riemann problem. We also obtain results for the entanglement entropy of regions crossed by shock and rarefaction waves and find both of them to closely follow the evolution of the energy density.

scholarly journals On the density of “wild” initial data for the compressible Euler system

2020 ◽  
Vol 59 (5) ◽  
Author(s):  
Eduard Feireisl ◽  
Christian Klingenberg ◽  
Simon Markfelder
Keyword(s):  
Initial Data ◽  
Weak Solutions ◽  
Euler System ◽  
Convex Integration ◽  
Compressible Euler ◽  
Dense Set

Abstract We consider a class of “wild” initial data to the compressible Euler system that give rise to infinitely many admissible weak solutions via the method of convex integration. We identify the closure of this class in the natural $$L^1$$ L 1 -topology and show that its complement is rather large, specifically it is an open dense set.

On the passage of a shock wave through a dusty-gas layer

1983 ◽  
Vol 385 (1788) ◽  
pp. 85-105 ◽  
Keyword(s):  
Shock Wave ◽  
Rarefaction Wave ◽  
Contact Surface ◽  
Equations Of Motion ◽  
Operator Splitting ◽  
Dusty Gas ◽  
Rarefaction Waves ◽  
Splitting Technique ◽  
Gas Layer ◽  
First Contact

The flow resulting from the passage of a shock wave through a dusty-gas layer is studied theoretically. On the basis of an idealized equilibrium-gas approximation, the criteria for the wave reflexion at the contact surface separating the pure gas from the dusty-gas layer are obtained in terms of the properties of the gas and the dusty gas. For the cases treated here, a shock wave is reflected at the first contact surface and a shock wave stronger than the incident one is transmitted into the dusty-air layer. Subsequently, a rarefaction wave is reflected at the second contact surface and the shock wave transmitted into the free air is weakened by this nonlinear interaction. The induced rarefaction wave reflects later at the first contact surface as a compression wave, which runs through the layer to overtake the transmitted shock wave in air. The final emergent shock wave from the dusty air has almost the same strength as the original shock wave entering the layer. The time-dependent transition properties through the shock waves, contact surfaces and rarefaction waves are found by solving the equations of motion numerically by a modified random-choice method with an operator-splitting technique.

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