CONVERGENCE AND ERROR ESTIMATES FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS

Thierry Gallouët (1)

1 I2M - Institut de Mathématiques de Marseille

We are interested in the paper by the discretization of the (unsteady and stationary) compressible (isentropic) Navier-Stokes Equations with the Marker-And-Cell scheme. We present recent results for the convergence (as the discretization parameter goes to zero) of the approximate solutions to a weak solution of the continuous equations and error estimates when the solution of the continuous equations is regular enough. I present in this paper some results obtained with R. Eymard, R. Herbin, J. C. Latché, D. Maltese and A. Novotny. Let Ω be a bounded open connected set of R 3 with a Lipschitz continuous boundary, T > 0, γ > 3/2, u 0 ∈ L 2 (Ω), ρ 0 ∈ L γ (Ω) and f ∈ L 2 (]0, T [, L 2 (Ω) 3). The compressible Navier-Stokes equations read ∂ t ρ + div(ρu) = 0 in Ω×]0, T [, (mass equation) (1) ∂ t (ρu) + div(ρu ⊗ u) − ∆u + grad p = f in Ω×]0, T [, (momentum equation) (2) p = ρ γ in Ω×]0, T [. (Equation Of State) (3) To this system, we add a Dirichlet boundary condition, u = 0 on ∂Ω×]0, T [, (4) and an initial condition u(·, 0) = u 0 , ρ(·, 0) = ρ 0 on ∂Ω. (5) The main unknowns of Problem (1)-(5) are u and ρ (then, p is given with (3)). Under the assumption ρ 0 > 0 a.e. on Ω and Ω (1 2 ρ 0 |u 0 | 2 + ρ γ 0 /(γ − 1))dx < +∞, existence of a weak solution (u, ρ) to (1)-(5) is known (but no uniqueness in general) since the works of P.-L. Lions [18] and E. Feireisl and coauthors [5], [6]. This weak solution sastifies ρ ∈ L ∞ (]0, T [, L γ (Ω)), ρ ≥ 0 a.e., u ∈ L 2 ([0, T [, H 1 0 (Ω) 3) and ρ|u| 2 ∈ L ∞ (]0, T [, L 1 (Ω)). Futhermore, Ω ρ(x, t)dx = Ω ρ 0 (x)dx a.e.. In particular, such a weak solution has a finite energy. More precisely, for a.e. t in ]0, T [, if f = 0, Ω (1 2 ρ|u| 2 + ρ γ γ − 1)(t) dx + t 0 Ω | grad u| 2 dxdτ ≤ Ω (1 2 ρ 0 |u 0 | 2 + ρ γ 0 γ − 1)dx. (6)

Domaines: Equations aux dérivées partielles [math.AP]

Thierry Gallouët. CONVERGENCE AND ERROR ESTIMATES FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS. Publicaciones del Seminario Matemático García Galdeano, In press. ⟨hal-02146480⟩