Blowup for the compressible isothermal Euler equations with non-vacuum initial data

2018 ◽  
Vol 99 (4) ◽  
pp. 585-595
Author(s):  
Jianwei Dong
Keyword(s):  
Initial Data ◽  
Euler Equations ◽  
Isothermal Euler Equations

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 On the Impossibility of First-Order Phase Transitions in Systems Modeled by the Full Euler Equations

Entropy ◽  
2019 ◽  
Vol 21 (11) ◽  
pp. 1039
Author(s):  
Maren Hantke ◽  
Ferdinand Thein
Keyword(s):  
Euler Equations ◽  
Single Phase ◽  
Two Phase Flows ◽  
Two Phase ◽  
First Order ◽  
Engineering Sciences ◽  
The One ◽  
First Order Phase ◽  
Isothermal Euler Equations ◽  
Appl Math

Liquid–vapor flows exhibiting phase transition, including phase creation in single-phase flows, are of high interest in mathematics, as well as in the engineering sciences. In two preceding articles the authors showed on the one hand the capability of the isothermal Euler equations to describe such phenomena (Hantke and Thein, arXiv, 2017, arXiv:1703.09431). On the other hand they proved the nonexistence of certain phase creation phenomena in flows governed by the full system of Euler equations, see Hantke and Thein, Quart. Appl. Math. 2015, 73, 575–591. In this note, the authors close the gap for two-phase flows by showing that the two-phase flows considered are not possible when the flow is governed by the full Euler equations, together with the regular Rankine-Hugoniot conditions. The arguments rely on the fact that for (regular) fluids, the differences of the entropy and the enthalpy between the liquid and the vapor phase of a single substance have a strict sign below the critical point.

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 A general existence result for isothermal two-phase flows with phase transition

2019 ◽  
Vol 16 (04) ◽  
pp. 595-637
Author(s):  
Maren Hantke ◽  
Ferdinand Thein
Keyword(s):  
Phase Transition ◽  
Euler Equations ◽  
Equations Of State ◽  
Linear Equations ◽  
Two Phase Flows ◽  
Kinetic Relation ◽  
Two Phase ◽  
Uniqueness Results ◽  
Wide Range ◽  
Isothermal Euler Equations

Liquid–vapor flows with phase transitions have a wide range of applications. Isothermal two-phase flows described by a single set of isothermal Euler equations, where the mass transfer is modeled by a kinetic relation, have been investigated analytically in [M. Hantke, W. Dreyer and G. Warnecke, Exact solutions to the Riemann problem for compressible isothermal Euler equations for two-phase flows with and without phase transition, Quart. Appl. Math. 71(3) (2013) 509–540]. This work was restricted to liquid water and its vapor modeled by linear equations of state. The focus of this work lies on the generalization of the primary results to arbitrary substances, arbitrary equations of state and thus a more general kinetic relation. We prove existence and uniqueness results for Riemann problems. In particular, nucleation and cavitation are discussed.

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