scholarly journals RIEMANN PROBLEM WITH DELTA INITIAL DATA FOR THE RELATIVISTIC CHAPLYGIN EULER EQUATIONS

2016 ◽  
Vol 6 (2) ◽  
pp. 376-395 ◽  
Author(s):  
Meixiang Huang ◽  
◽  
Zhiqiang Shao
Keyword(s):  
Initial Data ◽  
Euler Equations ◽  
Riemann Problem

scholarly journals Riemann problem with delta initial data for the two-dimensional steady pressureless isentropic relativistic Euler equations

2021 ◽  
Author(s):  
Yu Zhang ◽  
Yanyan Zhang
Keyword(s):  
Initial Data ◽  
Euler Equations ◽  
Riemann Problem ◽  
Contact Discontinuity ◽  
Global Solutions ◽  
Two Dimensional ◽  
Relativistic Euler Equations ◽  
Riemann Solutions ◽  
Isentropic Relativistic Euler Equations ◽  
The Stability

The Riemann problem for the two-dimensional steady pressureless isentropic relativistic Euler equations with delta initial data is studied. First, the perturbed Riemann problem with three pieces constant initial data is solved. Then, via discussing the limits of solutions to the perturbed Riemann problem, the global solutions of Riemann problem with delta initial data are completely constructed under the stability theory of weak solutions. Interestingly, the delta contact discontinuity is found in the Riemann solutions of the two-dimensional steady pressureless isentropic relativistic Euler equations with delta initial data.

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.

Transonic Shock Formation in a Rarefaction Riemann Problem for the 2D Compressible Euler Equations

2008 ◽  
Vol 69 (3) ◽  
pp. 720-742 ◽  
Author(s):  
James Glimm ◽  
Xiaomei Ji ◽  
Jiequan Li ◽  
Xiaolin Li ◽  
Peng Zhang ◽  
...  
Keyword(s):  
Euler Equations ◽  
Riemann Problem ◽  
Compressible Euler Equations ◽  
Shock Formation ◽  
Transonic Shock ◽  
Compressible Euler

Local existence with low regularity for the 2D compressible Euler equations

2021 ◽  
Vol 18 (03) ◽  
pp. 701-728
Author(s):  
Huali Zhang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Euler Equations ◽  
Local Existence ◽  
Compressible Euler Equations ◽  
Two Dimensional ◽  
Spatial Derivative ◽  
Compressible Euler ◽  
The Cauchy Problem ◽  
Local Solutions

We prove the local existence, uniqueness and stability of local solutions for the Cauchy problem of two-dimensional compressible Euler equations, where the initial data of velocity, density, specific vorticity [Formula: see text] and the spatial derivative of specific vorticity [Formula: see text].

scholarly journals The Global Well-Posedness for Large Amplitude Smooth Solutions for 3D Incompressible Navier–Stokes and Euler Equations Based on a Class of Variant Spherical Coordinates

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1195
Author(s):  
Shu Wang ◽  
Yongxin Wang
Keyword(s):  
Initial Data ◽  
Euler Equations ◽  
Large Amplitude ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Well Posedness ◽  
Spherical Coordinates ◽  
Smooth Solutions ◽  
Euler Systems

This paper investigates the globally dynamical stabilizing effects of the geometry of the domain at which the flow locates and of the geometry structure of the solutions with the finite energy to the three-dimensional (3D) incompressible Navier–Stokes (NS) and Euler systems. The global well-posedness for large amplitude smooth solutions to the Cauchy problem for 3D incompressible NS and Euler equations based on a class of variant spherical coordinates is obtained, where smooth initial data is not axi-symmetric with respect to any coordinate axis in Cartesian coordinate system. Furthermore, we establish the existence, uniqueness and exponentially decay rate in time of the global strong solution to the initial boundary value problem for 3D incompressible NS equations for a class of the smooth large initial data and a class of the special bounded domain described by variant spherical coordinates.

scholarly journals The Limit $\alpha \to 0$ of the $\alpha$-Euler Equations in the Half-Plane with No-Slip Boundary Conditions and Vortex Sheet Initial Data

2020 ◽  
Vol 52 (5) ◽  
pp. 5257-5286
Author(s):  
Adriana V. Busuioc ◽  
Dragos Iftimie ◽  
Milton D. Lopes Filho ◽  
Helena J. Nussenzveig Lopes
Keyword(s):  
Initial Data ◽  
Boundary Conditions ◽  
Euler Equations ◽  
Half Plane ◽  
Vortex Sheet ◽  
Slip Boundary Conditions ◽  
Slip Boundary

scholarly journals Construction of the Global Solutions to the Perturbed Riemann Problem for the Leroux System

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
Pengpeng Ji ◽  
Chun Shen
Keyword(s):  
Initial Data ◽  
Riemann Problem ◽  
Phase Plane ◽  
Wave Interaction ◽  
Global Solutions ◽  
Geometrical Structures ◽  
Small Perturbations ◽  
Riemann Solutions ◽  
Piecewise Constant ◽  
Rarefaction Curves

The global solutions of the perturbed Riemann problem for the Leroux system are constructed explicitly under the suitable assumptions when the initial data are taken to be three piecewise constant states. The wave interaction problems are widely investigated during the process of constructing global solutions with the help of the geometrical structures of the shock and rarefaction curves in the phase plane. In addition, it is shown that the Riemann solutions are stable with respect to the specific small perturbations of the Riemann initial data.

scholarly journals Regularization of the Shock Wave Solution to the Riemann Problem for the Relativistic Burgers Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Ting Zhang ◽  
Chun Shen
Keyword(s):  
Shock Wave ◽  
Initial Data ◽  
Riemann Problem ◽  
Wave Solution ◽  
Burgers Equation ◽  
Explicit Computation ◽  
Convective Term ◽  
Regularization Technique ◽  
Shock Wave Solution

The regularization of the shock wave solution to the Riemann problem for the relativistic Burgers equation is considered. For Riemann initial data consisting of a single decreasing jump, we find that the regularization of nonlinear convective term cannot capture the correct shock wave solution. In order to overcome it, we consider a new regularization technique called the observable divergence method introduced by Mohseni and discover that it can capture the correct shock wave solution. In addition, we take the Helmholtz filter for the fully explicit computation.

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