scholarly journals Vanishing Viscosity Solutions of the Compressible Euler Equations with Spherical Symmetry and Large Initial Data

2015 ◽  
Vol 338 (2) ◽  
pp. 771-800 ◽  
Author(s):  
Gui-Qiang G. Chen ◽  
Mikhail Perepelitsa
Keyword(s):  
Initial Data ◽  
Euler Equations ◽  
Viscosity Solutions ◽  
Spherical Symmetry ◽  
Compressible Euler Equations ◽  
Large Initial Data ◽  
Vanishing Viscosity ◽  
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 FORMATION OF SINGULARITY AND SMOOTH WAVE PROPAGATION FOR THE NON-ISENTROPIC COMPRESSIBLE EULER EQUATIONS

2011 ◽  
Vol 08 (04) ◽  
pp. 671-690 ◽  
Author(s):  
GENG CHEN
Keyword(s):  
Shock Wave ◽  
Wave Propagation ◽  
Initial Data ◽  
Differential Equations ◽  
Euler Equations ◽  
Hyperbolic Systems ◽  
Ideal Gas ◽  
Compressible Euler Equations ◽  
Compressible Euler ◽  
Shock Wave Formation

We define the notion of compressive and rarefactive waves and derive the differential equations describing smooth wave steepening for the compressible Euler equations with a varying entropy profile and general pressure laws. Using these differential equations, we directly generalize Lax's singularity (shock wave) formation results (established in 1964 for hyperbolic systems with two variables) to the 3 × 3 compressible Euler equations for a polytropic ideal gas. Our results are valid globally without restriction on the size of the variation of initial data.

scholarly journals Blowup Phenomenon of Solutions for the IBVP of the Compressible Euler Equations in Spherical Symmetry

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Ka Luen Cheung ◽  
Sen Wong
Keyword(s):  
Positive Integer ◽  
Boundary Value Problem ◽  
Euler Equations ◽  
Spherical Symmetry ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Compressible Euler Equations ◽  
Compressible Euler ◽  
Isothermal Case ◽  
Initial Boundary Value

The blowup phenomenon of solutions is investigated for the initial-boundary value problem (IBVP) of theN-dimensional Euler equations with spherical symmetry. We first show that there are only trivial solutions when the velocity is of the formc(t)xα-1x+b(t)(x/x)for any value ofα≠1or any positive integerN≠1. Then, we show that blowup phenomenon occurs whenα=N=1andc2(0)+c˙(0)<0. As a corollary, the blowup properties of solutions with velocity of the form(a˙t/at)x+b(t)(x/x)are obtained. Our analysis includes both the isentropic case(γ>1)and the isothermal case(γ=1).

scholarly journals Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data

2004 ◽  
Vol 175 ◽  
pp. 125-164 ◽  
Author(s):  
Huicheng Yin
Keyword(s):  
Shock Wave ◽  
Initial Data ◽  
Euler Equations ◽  
Blow Up ◽  
Three Dimensional ◽  
Compressible Euler Equations ◽  
Lipschitz Continuous ◽  
Compressible Euler ◽  
Nondegeneracy Conditions ◽  
Weak Entropy Solution

AbstractIn this paper, the problem on formation and construction of a shock wave for three dimensional compressible Euler equations with the small perturbed spherical initial data is studied. If the given smooth initial data satisfy certain nondegeneracy conditions, then from the results in [22], we know that there exists a unique blowup point at the blowup time such that the first order derivatives of a smooth solution blow up, while the solution itself is still continuous at the blowup point. From the blowup point, we construct a weak entropy solution which is not uniformly Lipschitz continuous on two sides of a shock curve. Moreover the strength of the constructed shock is zero at the blowup point and then gradually increases. Additionally, some detailed and precise estimates on the solution are obtained in a neighbourhood of the blowup point.

scholarly journals Non-uniqueness of energy-conservative solutions to the isentropic compressible two-dimensional Euler equations

2018 ◽  
Vol 15 (04) ◽  
pp. 721-730 ◽  
Author(s):  
Christian Klingenberg ◽  
Simon Markfelder
Keyword(s):  
Initial Data ◽  
Euler Equations ◽  
Weak Solutions ◽  
Gas Dynamics ◽  
Energy Inequality ◽  
Compressible Euler Equations ◽  
Two Dimensional ◽  
Energy Equality ◽  
Compressible Euler ◽  
Appl Math

We consider the 2-d isentropic compressible Euler equations. It was shown in [E. Chiodaroli, C. De Lellis and O. Kreml, Global ill-posedness of the isentropic system of gas dynamics, Comm. Pure Appl. Math. 68(7) (2015) 1157–1190] that there exist Riemann initial data as well as Lipschitz initial data for which there exist infinitely many weak solutions that fulfill an energy inequality. In this paper, we will prove that there is Riemann initial data for which there exist infinitely many weak solutions that conserve energy, i.e. they fulfill an energy equality. As in the aforementioned paper, we will also show that there even exist Lipschitz initial data with the same property.

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