Expanding large global solutions of the equations of compressible fluid mechanics

Abstract

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...

A Class of Global Solutions to the Euler–Poisson System

“…One may further decompose the solution in the form A δ (t) = tb δ + a δ (t), where a δ , b δ are 3 × 3 matrices such that b δ ∈ GL + (3) and moreover lim t→∞ a δ (t) 1+t = lim t→∞ȧ δ (t) = 0. For a concise proof of these statements see [44] or Lemma A.1 of [17]. However, the solution and therefore all the constants in the aforementioned bounds depend on the small parameter δ.…”

“…By contrast to the Lane-Emden stars, one may wonder whether there exist dynamic regimes, wherein the dynamics is effectively driven by the pressure term. In the absence of gravity, Sideris [44] constructed a family of special globally defined affine motions, which can be realised as steady states of quasiconformally rescaled Euler system [17]. It is thus natural to investigate the behaviour of the EP γ -system under this rescaling.…”

“…Our aim is to prove various statements about the asymptotic behaviour of the solution with constants that can be chosen uniformly-in-δ. We use the notation [44,17,42]) that initial value problem (2.31)-(2.32) possesses a global 3 . One may further decompose the solution in the form A δ (t) = tb δ + a δ (t), where a δ , b δ are 3 × 3 matrices such that b δ ∈ GL + (3) and moreover lim t→∞ a δ (t) 1+t = lim t→∞ȧ δ (t) = 0.…”

“…It is realised for a famous class of steady states of the gravitational EP γ system, known as the Lane-Emden stars. For the history of the physical vacuum condition, its physical significance, and its remarkable role played in a rigorous development of the well-posedness theory for vacuum free boundary fluids, we refer the reader to [31,30,32,33,5,3,4,25,26,27,28,35,11,41,43,16,17,18].…”

“…Our basic new insight is that scaling symmetries and a suitable notion of criticality developed in our recent works [16,17,18] allow one to identify an open set of initial data in the phase space which lead to global-in-time solutions. If interpreted correctly, for such a choice of data the gravitational/electrostatic interaction can be viewed as subcritical with respect to the gas pressure.…”