We prove a global existence theorem for the $3\times 3$ system of relativistic Euler equations in one spacial dimension. It is shown that in the ultra-relativistic limit, there is a family of equations of state that satisfy the second law of thermodynamics for which solutions exist globally. With this limit and equation of state, which includes equations of state for both an ideal gas and one dominated by radiation, the relativistic Euler equations can be analyzed by a Nishida-type method leading to a large data existence theorem, including the entropy and particle number evolution, using a Glimm scheme. Our analysis uses the fact that the equations of state are of the form $p=p(n,S)$, but whose form simplifies to $p=a^{2}ρ$ when viewed as a function of $ρ$ alone.
Our results provide a first step to make rigorous the formal analysis in terms of $\frac{1}{c^2}$ proposed by Chandrasekhar \cite{Chandra65b}, \cite{Chandra65a}, motivated by the methods of Einstein, Infeld and Hoffmann, see Thorne \cite{Thorne1}. We consider the non-relativistic limit for the local smooth solutions to the free boundary value problem of the cylindrically symmetric relativistic Euler equations, when the mass energy density includes the vacuum states at the free boundary. For large enough (rescaled) speed of light $c$ and suitably small time $T,$ we obtain uniform, with respect …
We prove a local existence theorem for the free boundary problem for a relativistic fluid in a fixed spacetime. Our proof involves an a priori estimate which only requires control of derivatives tangential to the boundary, which holds also in the Newtonian compressible case.
The relativistic full Euler equations for a Chaplygin gas are studied. The Riemann problem is solved constructively. There are two kinds of Riemann solutions, in which one consists of three contact discontinuities and the other involves a delta shock wave on which both state variables the rest mass density and the proper energy density simultaneously contain the Dirac delta functions. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. The formation mechanism, generalized Rankine-Hugoniot relation and entropy condition are clarified …
This paper contributes to the study of large data problems for $C^1$ solutions of the relativistic Euler equations. In the $(1+1)$-dimensional spacetime setting, if the initial data are away from vacuum, a key difficulty in proving the global well-posedness or finite time blow-up is coming up with a way to obtain sharp enough control on the lower bound of the mass-energy density function $ρ$. First, for $C^1$ solutions of the 1-dimensional classical isentropic compressible Euler equations in the Eulerian setting, we show a novel idea of obtaining a mass density time-dependent lower bound by st…
Hydrodynamic limit and Newtonian limit from the relativistic Vlasov-Maxwell-Boltzmann system to the classical Euler-Poisson system
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）
We derive a new formulation of the relativistic Euler equations that exhibits remarkable properties. This new formulation consists of a coupled system of geometric wave, transport, and elliptic equations, sourced by nonlinearities that are null forms relative to the acoustical metric. Our new formulation is well-suited for various applications, in particular for the study of stable shock formation, as it is surveyed in the paper. Moreover, using the new formulation presented here, we establish a local well-posedness result showing that the vorticity and the entropy of the fluid are one degree …
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.".
In this paper we provide a complete local well-posedness theory for the free boundary relativistic Euler equations with a physical vacuum boundary on a Minkowski background. Specifically, we establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and continuous dependence on the data; (ii) low regularity solutions: our uniqueness result holds at the level of Lipschitz velocity and density, while our rough solutions, obtained as unique limits of smooth solutions, have regularity only a half derivative above scaling; (iii) stability: ou…
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.
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）
The purpose of this paper is to study the phenomenon of singularity formation in large data problems for classical solutions to the Cauchy problem of the relativistic Euler equations. The classical theory established by P. D. Lax in 1964 (J. Math. Phys. 5: 611-614) shows that, for 2x2 hyperbolic systems, the break-down of classical solutions occurs in finite time if initial data contain any compression in some truly nonlinear characteristic field under some additional conditions, which include genuine nonlinearity and the strict positivity of the difference between two corresponding eigenvalue…
In this paper, we consider the compressible Euler-Maxwell equations arising in semiconductor physics, which take the form of Euler equations for the conservation laws of mass density and current density for electrons, coupled to Maxwell's equations for self-consistent electromagnetic field. We study the global well-posedness in critical spaces and the limit to zero of some physical parameters in the scaled Euler-Maxwell equations. More precisely, using high- and low-frequency decomposition methods, we first construct uniform (global) classical solutions (around constant equilibrium) to the Cau…
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…
We study the non relativistic and ultra relativistic limits in the two-dimensional nonlinear damped Klein-Gordon equation driven by a space-time white noise on the torus. In order to take the limits, it is crucial to clarify the parameter dependence in the estimates of solution. In this paper we present two methods to confirm this parameter dependence. One is the classical, simple energy method. Another is the method via Strichartz estimates.
We introduce a new wave formulation for the relativistic Euler equations with vacuum boundary conditions that consists of a system of non-linear wave equations in divergence form with a combination of acoustic and Dirichlet boundary conditions. We show that solutions of our new wave formulation determine solutions of the relativistic Euler equations that satisfy the vacuum boundary conditions provided the initial data is chosen to make a specific set of constraints vanish on the initial hypersurface. Moreover, we prove that these constraints propagate. This article is the first step of a two s…
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 …
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 kap…
The fundamental "two-fluid" model for describing plasma dynamics is given by the Euler-Maxwell system, in which compressible ion and electron fluids interact with their own self-consistent electromagnetic field. We prove global stability of a constant neutral background, in the sense that irrotational, smooth and localized perturbations of a constant background with small amplitude lead to global smooth solutions in three space dimensions for the Euler-Maxwell system. Our construction applies equally well to other plasma models such as the Euler-Poisson system for two-fluids and a relativistic…
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 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 establish the future stability of nonlinear perturbations of a class of homogeneous solutions to the relativistic Euler equations with a linear equation of state $p=Kρ$ on exponentially expanding FLRW spacetimes for the equation of state parameter values $1/3 < K < 1/2$.
We prove that for large enough data, the life span of smooth solutions to the Cauchy problem for the following two quasilinear hyperbolic systems is finite: (1) equations of relativistic compressible fluid dynamics, (2) equations of plasma dynamics (Euler-Maxwell equations with background charge) in the spherically symmetric case.
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…
