In this paper the author studies the singular limit c to infinity of the family of Euler-Nordström systems indexed by the parameters kappa and c (EN_kappa,c), where kappa^2 > 0 is the cosmological constant and c is the speed of light. Using Christodoulou's techniques to generate energy currents, the author develops Sobolev estimates that show that for initial data belonging to an appropriate Sobolev space, as c tends to infinity, the solutions to the EN_kappa,c system converge uniformly on a spacetime slab [0,T]xR^3 to the solution of the Euler-Poisson system with the cosmological constant kappa^2.
The Nordström-Vlasov system provides an interesting relativistic generalization of the Vlasov-Poisson system in the gravitational case, even though there is no direct physical application. The study of this model will probably lead to a better mathematical understanding of the class of non-linear systems consisting of hyperbolic and transport equations. In this paper it is shown that solutions of the Nordström-Vlasov system converge to solutions of the Vlasov-Poisson system in a pointwise sense as the speed of light tends to infinity, providing a further and rigorous justification of this mode…
In this paper the author considers the motion of a relativistic perfect fluid with self-interaction mediated by Nordstrom's scalar theory of gravity. The evolution of the fluid is determined by a quasilinear hyperbolic system of PDEs, and a cosmological constant is introduced in order to ensure the existence of non-zero constant solutions. Accordingly, the initial value problem for a compact perturbation of an infinitely extended quiet fluid is studied. Although the system is neither symmetric hyperbolic nor strictly hyperbolic, Christodoulou's constructive results on the existence of energy c…
The Vlasov-Nordström-Fokker-Planck system describes the evolution of self-gravitating matter experiencing collisions with a fixed background of particles in the framework of a relativistic scalar theory of gravitation. We study the spatially-homogeneous system and prove global existence and uniqueness of solutions for the corresponding initial value problem in three momentum dimensions. Additionally, we study the long time asymptotic behavior of the system and prove that even in the absence of friction, solutions possess a non-trivial asymptotic profile. An exact formula for the long time limi…
In this paper, we study the blowup phenomena for the regular solutions of the isentropic relativistic Euler-Poisson equations with a vacuum state in spherical symmetry. Using a general family of testing functions, we obtain new blowup conditions for the relativistic Euler-Poisson equations. We also show that the proposed blowup conditions are valid regardless of the speed requirement, which was one of the key constraints stated in "Y. Geng, Singularity Formation for Relativistic Euler and Euler-Poisson Equations with Repulsive Force, Commun. Pure Appl. Anal., 14 (2015), 549--564.".
We study the relativistic Euler equations on the Minkowski spacetime background. We make assumptions on the equation of state and the initial data that are relativistic analogs of the well-known physical vacuum boundary condition, which has played an important role in prior work on the non-relativistic compressible Euler equations. Our main result is the derivation, relative to Lagrangian (also known as co-moving) coordinates, of local-in-time a priori estimates for the solution. The solution features a fluid-vacuum boundary, transported by the fluid four-velocity, along which the hyperbolicit…
We analyze the relativistic Euler equations of conservation laws of baryon number and momentum with a general pressure law. The existence of global-in-time bounded entropy solutions for the system is established by developing a compensated compactness framework. The proof relies on a careful analysis of the entropy and entropy-flux functions, which are represented by the fundamental solutions of the entropy and entropy-flux equations for the relativistic Euler equations. Based on the careful entropy analysis, we establish a compactness framework for sequences of both exact solutions and approx…
In this paper, from an investigation of a symmetric hyperbolic system, a prior derivation of the logarithm equation of state is provided. Through a diffeomorphism transforming the classical Euler equations to the symmetric hyperbolic system, we show that the logarithm pressure co-exists with existing barotropic equations of state without applying any physical laws. In this connection, we also establish a local existence of classical solutions for the relativistic Euler equations with the logarithm equation of state.
In this paper, we are concerned with the non-relativistic limit of a class of computable approximation models for radiation hydrodynamics. The models consist of the compressible Euler equations coupled with moment closure approximations to the radiative transfer equation. They are first-order partial differential equations with source terms. As hyperbolic relaxation systems, they are showed to satisfy the structural stability condition proposed by W.-A. Yong (1999). Base on this, we verify the non-relativistic limit by combining an energy method with a formal asymptotic analysis.
The Nordström-Vlasov system is a relativistic Lorentz invariant generalization of the Vlasov-Poisson system in the gravitational case. The asymptotic behavior of solutions and the non-linear stability of steady states are investigated. It is shown that solutions of the Nordström-Vlasov system with energy grater or equal to the mass satisfy a dispersion estimate in terms of the conformal energy. When the energy is smaller than the mass, we prove existence and non-linear (orbital) stability of a class of static solutions (isotropic polytropes) against general perturbations. The proof of orbital …
We study the singularity formation of smooth solutions of the relativistic Euler equations in $(3+1)$-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any smooth solution, with compactly supported non-trivial initial data, blows up in finite time. For the case of infinite initial energy, we first prove the existence, uniqueness and stability of a smooth solution if the initial data is in the subluminal region away from the vacuum. By further assuming the initial data is a smooth compactly supported perturbation …
We study the free boundary problem for the equations of compressible Euler equations with a vacuum boundary condition. Our main goal is to recover in Eulerian coordinates the earlier well-posedness result obtained by Lindblad [Lindblad H., Commun. Math. Phys. 260 (2005), 319-392] for the isentropic Euler equations and extend it to the case of full gas dynamics. For technical simplicity we consider the case of an unbounded domain whose boundary has the form of a graph and make short comments about the case of a bounded domain. We prove the local-in-time existence in Sobolev spaces by the techni…
Semi-classical limit of the massive Klein-Gordon-Maxwell system toward the relativistic Euler-Maxwell system via an adapted modulated energy method
Centre de mathématiques Laurent-Schwartz, École polytechnique（洛朗·施瓦茨数学中心，巴黎综合理工学院）；Département de mathématiques et applications, École normale supérieure（数学与应用系，巴黎高等师范学院）
In this paper, we study the Riemann problem of relativistic Euler system for rarefied monatomic and diatomic gases when the constitutive equation for the energy is the Synge equation that is the only one compatible with the relativistic kinetic theory. The Synge equation is involved with modified Bessel functions of the second kind and this makes the relativistic Euler system quite complex. Based on delicate estimates of the modified Bessel functions of the second kind, we provide a detailed investigation of basic hyperbolic properties and the structure of elementary waves, especially for the …
We show that a pair of conjectures raised in [11] concerning the construction of normal solutions to the relativistic Boltzmann equation are valid. This ensures that the results in [11] hold for any range of positive temperatures and that the relativistic Euler system under the kinetic equation of state is hyperbolic and the speed of sound cannot overcome $c/\sqrt{3}$.
We investigate the initial-value problem for the relativistic Euler equations governing isothermal perfect fluid flows, and generalize an approach introduced by LeFloch and Shelukhin in the non-relativistic setting. We establish the existence of globally defined, bounded measurable, entropy solutions with arbitrary large amplitude. An earlier result by Smoller and Temple for the same system covered solutions with bounded variation that avoid the vacuum state. The new framework proposed here provides entropy solutions in a larger function space and allows for the mass density to vanish and the …
We study the local-in-time hydrodynamic limit of the relativistic Boltzmann equation using a Hilbert expansion. More specifically, we prove the existence of local solutions to the relativistic Boltzmann equation that are nearby the local relativistic Maxwellian constructed from a class of solutions to the relativistic Euler equations that includes a large subclass of near-constant, non-vacuum fluid states. In particular, for small Knudsen number, these solutions to the relativistic Boltzmann equation have dynamics that are effectively captured by corresponding solutions to the relativistic Eul…
We reformulate the relativistic perfect fluid system on curved space-time. Using standard variables, the velocity field $u$,energy density $ρ$ and pressure $p$, the covariant Euler-Lagrange equation is obtained from variational principle. This leads to the Euler equation and the equation of continuity in reparametrization invariant form.
We consider steady state solutions of the massive, asymptotically flat, spherically symmetric Einstein-Vlasov system, i.e., relativistic models of galaxies or globular clusters, and steady state solutions of the Einstein-Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift $κ>0$. We prove their linear instability when $κ$ is sufficiently large, i.e., when they are strongly relativistic, and that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability propo…
We consider a self-gravitating collisionless gas as described by the Vlasov-Poisson or Einstein-Vlasov system or a self-gravitating fluid ball as described by the Euler-Poisson or Einstein-Euler system. We give a simple proof for the finite extension of spherically symmetric equilibria, which covers all these models simultaneously. In the Vlasov case the equilibria are characterized by a local growth condition on the microscopic equation of state, i.e., on the dependence of the particle distribution on the particle energy, at the cut-off energy E_0, and in the Euler case by the corresponding g…
We consider the relativistic Vlasov-Maxwell and Vlasov-Nordström systems which describe large particle ensembles interacting by either electromagnetic fields or a relativistic scalar gravity model. For both systems we derive a radiation formula analogous to the Einstein quadrupole formula in general relativity.
A relativistic kinetic Fokker-Planck equation that has been recently proposed in the physical literature is studied. It is shown that, in contrast to other existing relativistic models, the one considered in this paper is invariant under Lorentz transformations in the absence of friction. A similar property (invariance by Galilean transformations in the absence of friction) is verified in the non-relativistic case. In the first part of the paper some fundamental mathematical properties of the relativistic Fokker-Planck equation are established. In particular, it is proved that the model is com…
This note presents Godunov variables and 4-potentials for the relativistic Euler equations of barotropic fluids. The associated additional conservation/ production law has different interpretations for different fluids. In particular it refers to ENTROPY in the case of thermobarotropic fluids, and to MATTER in the case of isentropic fluids. The paper also presents an explicit formula for the generating function of the Euler equations in the case of ideal gases. It pursues ideas on symmetric hyperbolicity going back to Godunov (cf. also Lax and Friedrichs as well as Boillat) that were elaborate…
We develop first-principles theory of relativistic fluid turbulence at high Reynolds and Péclet numbers. We follow an exact approach pioneered by Onsager, which we explain as a non-perturbative application of the principle of renormalization-group invariance. We obtain results very similar to those for non-relativistic turbulence, with hydrodynamic fields in the inertial-range described as distributional or "coarse-grained" solutions of the relativistic Euler equations. These solutions do not, however, satisfy the naive conservation-laws of smooth Euler solutions but are afflicted with dissipa…
