Finite energy solutions to the isentropic Euler equations with geometric effects

Abstract

Considering the isentropic Euler equations of compressible fluid dynamics with geometric effects included, we establish the existence of entropy solutions for a large class of initial data. We cover fluid flows in a nozzle or in spherical symmetry when the origin r = 0 is included. These partial differential equations are hyperbolic, but fail to be strictly hyperbolic when the fluid mass density vanishes and vacuum is reached. Furthermore, when geometric effects are taken into account, the sup-norm of solutions can not be controlled since there exist no invariant regions. To overcome these difficulties and to establish an existence theory for solutions with arbitrarily large amplitude, we search for solutions with finite mass and total energy. Our strategy of proof takes advantage of the particular structure of the Euler equations, and leads to a versatile framework covering general compressible fluid problems. We establish first higherintegrability estimates for the mass density and the total energy. Next, we use arguments from the theory of compensated compactness and Young measures, extended here to sequences of solutions with finite mass and total energy. The third ingredient of the proof is a characterization of the unbounded support of entropy admissible Young measures. This requires the study of singular products involving measures and principal values.The unknowns of this system are the density ρ 0 and the velocity u, which are functions of the independent variables (t, x) ∈ [0, ∞) × R. The pressure P (ρ) is related to the internal energy U (ρ) by the relationfor all ρ 0. We restrict ourselves to polytropic perfect gases, for which U (ρ) = κ γ−1 ρ γ and P (ρ) = κρ γ .Here γ > 1 is the adiabatic coefficient, and κ := θ 2 /γ with θ := (γ − 1)/2 are constants. The case of general pressure laws will be addressed in future work. R ψ(s) χ(s|ρ, u), σ(s|ρ, u) ds, (1.10) and we impose the entropy inequalitiesin the distribution sense. We use the notation g ,ρ := ∂ ρ g for all functions g. Definition 1.1. Let (ρ, u) be given initial data with finite mass and total energy. A pair of measurable functions (ρ, u) : [0, ∞) × Ω −→ [0, ∞) × R is called an entropy solution with finite mass and energy (or a finite energy solution, for short) to the Cauchy problem (1.1) & (1.7) if the following is true: 1. The total mass is conserved in time: for almost every (a.e.) t M [ ρ ](t) = M . 2. The total energy is bounded in time: for a.e. t E[ ρ, u ](t) E.

Finite Energy Method for Compressible Fluids: The Navier‐Stokes‐Korteweg Model

“…The proposed method allows us to validate the singular limit problem associated with these models -in presence of cavitation and shock waves-and, specifically, to establish the convergence with finite energy solutions to the augmented models toward finite energy solutions to the Euler system. It originates from pioneering works by DiPerna [18,19] on bounded solutions and by LeFloch and Westdickenberg [47] on finite energy solutions.…”

“…Several years ago, LeFloch and Westdickenberg [47] opened the way to constructing solutions within the broader class of solutions with finite energy and realized that working within such a large class of solutions was necessary in order to overcome certain limitations in DiPerna's theory. In this class, they established the first existence result of radially symmetric fluid flows -including the singularity at the center of radial coordinates.…”

“…In the case κ ǫ ≡ 0 and restricting attention to polytropic fluids p(ρ) ≃ ρ γ (with 1 < γ < 3), Chen and Perepelitsa [13] first established a convergence result of the form above; they restricted attention to the viscosity coefficient µ ǫ (ρ) = ǫ and, by following LeFloch and Westdickenberg's method [47], observed that the diffusion term can be controled by a priori estimate derived earlier by Kanel [36] for a different purpose. Next, Huang et al [32] treated the viscosity functions µ ǫ (ρ) = ǫρ α and polytropic fluids with 2 3 < α < γ.…”

Shallow water equations: viscous solutions and inviscid limit

“…for any continuous function ψ(s), where the weak entropy kernel χ(ρ, s − u) is determined by 12) with the Dirac mass δ u=s concentrated at u = s. This implies that the weak entropy kernel as the unique solution of (1.12) is…”

“…Also see LeFlochWestdickenberg [12] for the reduction theorem for the case 1 < γ ≤ 5/3 and ChenPerepelitsa [4] for the general case γ > 1 with a simpler proof for the Young measures with unbounded support.…”