An important problem in gas and fluid dynamics is to understand the behavior of vacuum states, namely the behavior of the system in the presence of vacuum. In particular, physical vacuum, in which the boundary moves with a nontrivial finite normal acceleration, naturally arises in the study of the motion of gaseous stars or shallow water. Despite its importance, there are only few mathematical results available near vacuum. The main difficulty lies in the fact that the physical systems become degenerate along the vacuum boundary. In this paper, we establish the local-in-time well-posedness of threedimensional compressible Euler equations for polytropic gases with physical vacuum by considering the problem as a free boundary problem.

An important problem in the theory of compressible gas flows is to understand the singular behavior of vacuum states. The main difficulty lies in the fact that the system becomes degenerate at the vacuum boundary, where the characteristic speeds u˙c coincide and have unbounded spatial derivative since c behaves like x 1=2 close to the boundary. In this paper, we overcome this difficulty by presenting a new formulation and new energy spaces. We establish the local-in-time well-posedness of one-dimensional compressible Euler equations for isentropic flows with the physical vacuum singularity in some spaces adapted to the singularity.

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...

This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and the upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces. We establish a sharp nonlinear global-in-time stability criterion and give the explicit decay rates to the equilibrium. When the upper fluid is heavier than the lower fluid along the equilibrium interface, we characterize the set of surface tension values in which the equilibrium is nonlinearly stable. Remarkably, this set is non-empty, i.e. sufficiently large surface tension can prevent the onset of the Rayleigh-Taylor instability. When the lower fluid is heavier than the upper fluid, we show that the equilibrium is stable for all non-negative surface tensions and we establish the zero surface tension limit.