Inspired by a recent work of Sideris on affine motions of compactly supported moving ellipsoids, we construct global-in-time solutions to the vacuum free boundary three-dimensional isentropic compressible Euler equations when γ ∈ (1, 5 3 ] for initial configurations that are sufficiently close to the affine motions, and satisfy the physical vacuum boundary condition. The support of these solutions expands at a linear rate in time, they remain smooth in the interior of their support, and no shocks are formed in the evolution. We impose no symmetry assumptions on our initial data. We prove the existence of such solutions by reformulating the problem as a nonlinear stability question in suitably rescaled variables, wherein the stabilizing effect of the fluid expansion becomes visible in the range γ ∈ (1, 5 3 ].where we have set the entropy constant to be 1. We additionally demand that the initial density satisfies the physical vacuum boundary condition [16,25]:is the speed of the sound. We shall refer to the system of equations (1.1)-(1.3) as the Euler system and denote it by E γ .Due to the inherent lack of smoothness of the enthalpy c 2 s at the vacuum boundary (implied by the assumption (1.3)), a rigorous understanding of the existence of physical vacuum states in compressible fluid dynamics has been a challenging problem. Only recently, a successful local-in-time well-posedness theory for the E γ system was developed in [5,18] using the Lagrangian formulation of the Euler system in the vacuum free boundary framework. The fundamental unknown is the flow map ζ defined as a solution of the ordinary differential equations (ODE) Theorem 1.1 (Global existence in the vicinity of affine motions). Assume that γ ∈ (1, 5 3 ]. Then small perturbations of the expanding affine motions given by (1.5)-(1.7) give rise to unique globally-in-time defined solutions of the Euler system E γ . Moreover, their support expands at a linear rate and they remain close to the underlying "moduli" space S of affine motions.Remark 1.2. A precise statement of Theorem 1.1 specifying the function spaces and the notions of "small" and "close" in the statement above is provided in Theorem 2.3. Remark 1.3. Theorem 1.1 covers a range of adiabatic exponents of physical importance. The exponent γ = 5 3 is commonly used in the description of a monatomic gas, γ = 7 5 corresponds to a diatomic gas, and γ = 4 3 is often referred to as the radiative case. Remark 1.4. As shown in [41], the affine motions exist even when γ > 5 3 . It would be interesting to understand whether one can go beyond the γ = 5 3 threshold in our theorem. The assumption γ ≤ 5 3 is certainly optimal for our method, as the case γ > 5 3 yields an anti-damping effect that can in principle lead to an instability.Remark 1.5. The theorem shows that in the forward time direction, in the vicinity of S the solution naturally splits into an affine component (i.e. an element of S ) and a remainder, which as we shall show, remains small due to the underlying expansion associated with the elements...
We prove nonlinear stability of compactly supported expanding star solutions of the mass‐critical gravitational Euler‐Poisson system. These special solutions were discovered by Goldreich and Weber in 1980. The expansion rate of such solutions can be either self‐similar or non‐self‐similar (linear), and we treat both types. An important outcome of our stability results is the existence of a new class of global‐in‐time radially symmetric solutions, which are not homologous and therefore not encompassed by the existing works. Using Lagrangian coordinates we reformulate the associated free‐boundary problem as a degenerate quasilinear wave equation on a compact spatial domain. The problem is mass‐critical with respect to an invariant rescaling and the analysis is carried out in similarity variables. © 2017 Wiley Periodicals, Inc.
Using recent developments in the theory of globally defined expanding compressible gases, we construct a class of global-in-time solutions to the compressible 3-D Euler-Poisson system without any symmetry assumptions in both the gravitational and the plasma case. Our allowed range of adiabatic indices includes, but is not limited to all γ of the form γ = 1 + 1 n , n ∈ N \ {1}. The constructed solutions have initially small densities and a compact support. As t → ∞ the density scatters to zero and the support grows at a linear rate in t.The three dimensional compressible Euler-Poisson system couples the equation for a compressible gas to a self-consistent force field created by the gas particles: if the interaction is gravitational, we refer to the model as the gravitational Euler-Poisson system and if the interaction is electrostatic we talk about the electrostatic Euler-Poisson system. In the gravitational case we obtain a model of a Newtonian star [45,1,2], while in the case of repelling forces between the particles, we arrive at a model for plasmas [12,14]. We shall work with the free-boundary formulation of the problem, wherein a moving *
ABSTRACT. The classical one-phase Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition, such as ice melting to water. This is accomplished by solving the heat equation on a time-dependent domain whose boundary is transported by the normal derivative of the temperature along the evolving and a priori unknown free-boundary. We establish a global-in-time stability result for nearly spherical geometries and small temperatures, using a novel hybrid methodology, which combines energy estimates, decay estimates, and Hopf-type inequalities.
ABSTRACT. We study small perturbations of the well-known family of Friedman-Lemaître-RobertsonWalker (FLRW) solutions to the dust-Einstein system with a positive cosmological constant in the case that the spacelike Cauchy hypersurfaces are diffeomorphic to T 3 . These solutions model a quiet pressureless fluid in a dynamic spacetime undergoing accelerated expansion. We show that the FLRW solutions are nonlinearly globally future-stable under small perturbations of their initial data. Our analysis takes place relative to a harmonic-type coordinate system, in which the cosmological constant results in the presence of dissipative terms in the evolution equations. Our result extends the results of [40,46,32], where analogous results were proved for the Euler-Einstein system under the equations of state p = c 2 s ρ, 0 < c 2 s ≤ 1/3. The dust-Einstein system is the Euler-Einstein system with c s = 0. The main difficulty that we overcome is that the energy density of the dust loses one degree of differentiability compared to the cases 0 < c 2 s ≤ 1/3. Because the dustEinstein equations are coupled, this loss of differentiability introduces new obstacles for deriving estimates for the top-order derivatives of all solution variables. To resolve this difficulty, we commute the equations with a well-chosen differential operator and derive a collection of elliptic estimates that complement the energy estimates of [40,46]. An important feature of our analysis is that we are able to close our estimates even though the top-order derivatives of all solution variables can grow much more rapidly than in the cases 0 < c 2 s ≤ 1/3. Our results apply in particular to small compact perturbations of the vanishing dust state.
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 prove that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in the 1960s by Zel'dovich & Podurets (for the Einstein-Vlasov system) and by Harrison, Thorne, Wakano, & Wheeler (for the Einstein-Euler system). Our results are in sharp contrast to the corresponding nonrelativistic, Newtonian setting. We carry out a careful analysis of the linearized dynamics around the above steady states and prove an exponential trichotomy result and the corresponding index theorems for the stable/unstable invariant spaces. Finally, in the case of the Einstein-Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve.
Using numerical integration, in 1969 Penston (Mon Not R Astr Soc 144:425–448, 1969) and Larson (Mon Not R Astr Soc 145:271–295, 1969) independently discovered a self-similar solution describing the collapse of a self-gravitating asymptotically flat fluid with the isothermal equation of state $$p=k\varrho $$ p = k ϱ , $$k>0$$ k > 0 , and subject to Newtonian gravity. We rigorously prove the existence of such a Larson–Penston solution.
The stability of static solutions of the spherically symmetric, asymptotically flat Einstein–Vlasov system is studied using a Hamiltonian approach based on energy-Casimir functionals. The main results are a coercivity estimate for the quadratic part of the expansion of the natural energy-Casimir functional about an isotropic steady state, and the linear stability of such steady states. The coercivity bound shows in a quantified way that this quadratic part is positive definite on a class of linearly dynamically accessible perturbations, provided the particle distribution of the steady state is a strictly decreasing function of the particle energy and provided the steady state is not too relativistic. In contrast to the stability theory for isotropic steady states of the gravitational Vlasov-Poisson system the monotonicity of the particle distribution alone does not determine the stability character of the state, a fact which was observed by Ze'ldovitch et al. in the 1960's. The result in an essential way exploits the non-linear structure of the Einstein equations satisfied by the steady state and is not just a perturbation result of the analogous coercivity bounds for the Newtonian case. It should be an essential step in a fully non-linear stability analysis for the Einstein–Vlasov system.
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