On the hyperbolicity and causality of the relativistic Euler system under the kinetic equation of state

The classical non-ideal gas shows that the two original concepts of the pressure based of the motion and the forces have eventually developed into drift and dissipation contributions. Collisions of realistic particles are nonlocal and non-instant. A collision delay characterizes the effective duration of collisions, and three displacements, describe its effective non-locality. Consequently, the scattering integral of kinetic equation is nonlocal and non-instant. The non-instant and nonlocal corrections to the scattering integral directly result in the virial corrections to the equation of state. The interaction of particles via long-range potential tails is approximated by a mean field which acts as an external field. The effect of the mean field on free particles is covered by the momentum drift. The effect of the mean field on the colliding pairs causes the momentum and the energy gains which enter the scattering integral and lead to an internal mechanism of energy conversion. The entropy production is shown and the nonequilibrium hydrodynamic equations are derived. Two concepts of quasiparticle, the spectral and the variational one, are explored with the help of the virial of forces.

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Concentration and cavitation phenomena of Riemann solutions for the isentropic Euler system with the logarithmic equation of state

Nonlinear Analysis Real World Applications ◽

10.1016/j.nonrwa.2019.103068 ◽

2020 ◽

Vol 53 ◽

pp. 103068 ◽

Cited By ~ 9

Author(s):

Meina Sun

Keyword(s):

Equation Of State ◽

Euler System ◽

Riemann Solutions ◽

Logarithmic Equation

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Asymptotic analysis of Vlasov-type equations under strong local alignment regime

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202515500542 ◽

2015 ◽

Vol 25 (11) ◽

pp. 2153-2173 ◽

Cited By ~ 13

Author(s):

Moon-Jin Kang ◽

Alexis F. Vasseur

Keyword(s):

Kinetic Equation ◽

Hyperbolic Conservation Laws ◽

Hydrodynamic Limit ◽

Local Equilibrium ◽

Euler System ◽

Entropy Method ◽

Local Alignment ◽

Hyperbolic Conservation ◽

Self Organized ◽

Relative Entropy Method

We consider the hydrodynamic limit of a collisionless and non-diffusive kinetic equation under strong local alignment regime. The local alignment is first considered by Karper, Mellet and Trivisa in [On strong local alignment in the kinetic Cucker–Smale model, in Hyperbolic Conservation Laws and Related Analysis with Applications, Springer Proceedings in Mathematics & Statistics, Vol. 49 (Springer, 2014), pp. 227–242], as a singular limit of an alignment force proposed by Motsch and Tadmor in [A new model for self-organized dynamics and its flocking behavior, J. Statist. Phys. 141 (2011) 923–947]. As the local alignment strongly dominates, a weak solution to the kinetic equation under consideration converges to the local equilibrium, which has the form of mono-kinetic distribution. We use the relative entropy method and weak compactness to rigorously justify the weak convergence of our kinetic equation to the pressureless Euler system.

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The limiting behavior of the Riemann solution to the isentropic Euler system for logarithmic equation of state with a source term

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.7254 ◽

2021 ◽

Author(s):

Anupam Sen ◽

T. Raja Sekhar

Keyword(s):

Equation Of State ◽

Source Term ◽

Euler System ◽

Limiting Behavior ◽

Riemann Solution ◽

Logarithmic Equation

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Turning Point Principle for Relativistic Stars

Communications in Mathematical Physics ◽

10.1007/s00220-021-04197-6 ◽

2021 ◽

Vol 387 (2) ◽

pp. 729-759

Author(s):

Mahir Hadžić ◽

Zhiwu Lin

Keyword(s):

Equation Of State ◽

Gravitational Collapse ◽

Turning Point ◽

Euler System ◽

Gravitation Theory ◽

Spherically Symmetric ◽

University Of Chicago ◽

Relativistic Stars ◽

The University ◽

University Of Chicago Press

AbstractUpon specifying an equation of state, spherically symmetric steady states of the Einstein-Euler system are embedded in 1-parameter families of solutions, characterized by the value of their central redshift. In the 1960’s Zel’dovich (Voprosy Kosmogonii 9:157–170, 1963) and Harrison et al. (Gravitation Theory and Gravitational Collapse. The University of Chicago press, Chicago, 1965) formulated a turning point principle which states that the spectral stability can be exchanged to instability and vice versa only at the extrema of mass along the mass-radius curve. Moreover the bending orientation at the extrema determines whether a growing mode is gained or lost. We prove the turning point principle and provide a detailed description of the linearized dynamics. One of the corollaries of our result is that the number of growing modes grows to infinity as the central redshift increases to infinity.

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