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.

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

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 role of multidimensional instabilities in direct initiation of gaseous detonations in free space

2017 ◽  
Vol 813 ◽  
Author(s):  
Hua Shen ◽  
Matteo Parsani
Keyword(s):  
Euler Equations ◽  
Chemical Reaction ◽  
Critical Energy ◽  
Combustible Mixture ◽  
Compressible Euler Equations ◽  
Two Dimensional ◽  
One Dimensional ◽  
Direct Initiation ◽  
Compressible Euler ◽  
The One

We numerically investigate the direct initiation of detonations driven by the propagation of a blast wave into a unconfined gaseous combustible mixture to study the role played by multidimensional instabilities in direct initiation of stable and unstable detonations. To this end, we first model the dynamics of unsteady propagation of detonation using the one-dimensional compressible Euler equations with a one-step chemical reaction model and cylindrical geometrical source terms. Subsequently, we use two-dimensional compressible Euler equations with just the chemical reaction source term to directly model cylindrical detonations. The one-dimensional results suggest that there are three regimes in the direct initiation for stable detonations, that the critical energy for mildly unstable detonations is not unique, and that highly unstable detonations are not self-sustainable. These phenomena agree well with one-dimensional theories and computations available in the literature. However, our two-dimensional results indicate that one-dimensional approaches are valid only for stable detonations. In mildly and highly unstable detonations, one-dimensional approaches break down because they cannot take the effects and interactions of multidimensional instabilities into account. In fact, instabilities generated in multidimensional settings yield the formation of strong transverse waves that, on one hand, increase the risk of failure of the detonation and, on the other hand, lead to the initiation of local over-driven detonations that enhance the overall self-sustainability of the global process. The competition between these two possible outcomes plays an important role in the direct initiation of detonations.

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.

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.

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