scholarly journals Optimal L p – L q -Type Decay Rates of Solutions to the Three-Dimensional Nonisentropic Compressible Euler Equations with Relaxation

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Rong Shen ◽  
Yong Wang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Euler Equations ◽  
Existence And Uniqueness ◽  
Three Dimensional ◽  
Decay Rates ◽  
Compressible Euler Equations ◽  
Compressible Euler ◽  
Formula Type ◽  
Math Phys

In this paper, we consider the three-dimensional Cauchy problem of the nonisentropic compressible Euler equations with relaxation. Following the method of Wu et al. (2021, Adv. Math. Phys. Art. ID 5512285, pp. 1–13), we show the existence and uniqueness of the global small H k k ⩾ 3 solution only under the condition of smallness of the H 3 norm of the initial data. Moreover, we use a pure energy method with a time-weighted argument to prove the optimal L p – L q 1 ⩽ p ⩽ 2 , 2 ⩽ q ⩽ ∞ -type decay rates of the solution and its higher-order derivatives.

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 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 The blowup of solutions for 3-D axisymmetric compressible Euler equations

1999 ◽  
Vol 154 ◽  
pp. 157-169 ◽  
Author(s):  
Huicheng Yin ◽  
Qingjiu Qiu
Keyword(s):  
Asymptotic Expansion ◽  
Initial Data ◽  
Euler Equations ◽  
Blow Up ◽  
Three Dimensional ◽  
Compressible Euler Equations ◽  
Classical Solutions ◽  
Complete Asymptotic Expansion ◽  
Blowup Of Solutions ◽  
Compressible Euler

AbstractIn this paper, for three dimensional compressible Euler equations with small perturbed initial data which are axisymmetric, we prove that the classical solutions have to blow up in finite time and give a complete asymptotic expansion of lifespan.

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 SHOCK FORMATION IN THE COMPRESSIBLE EULER EQUATIONS AND RELATED SYSTEMS

2013 ◽  
Vol 10 (01) ◽  
pp. 149-172 ◽  
Author(s):  
GENG CHEN ◽  
ROBIN YOUNG ◽  
QINGTIAN ZHANG
Keyword(s):  
Euler Equations ◽  
Space Dimension ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Compressible Euler Equations ◽  
Shock Formation ◽  
Compressible Euler ◽  
Systems Of Conservation Laws ◽  
The One ◽  
Special Case

We prove shock formation results for the compressible Euler equations and related systems of conservation laws in one space dimension, or three dimensions with spherical symmetry. We establish an L∞ bound for C1 solutions of the one-dimensional (1D) Euler equations, and use this to improve recent shock formation results of the authors. We prove analogous shock formation results for 1D magnetohydrodynamics (MHD) with orthogonal magnetic field, and for compressible flow in a variable area duct, which has as a special case spherically symmetric three-dimensional (3D) flow on the exterior of a ball.

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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