2012/01/01 by Matthias Kotschote
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations
paper · doi:10.1137/110821202
crossref issued 2012/01/01 · crossref published 2012/01/01 · crossref published-print 2012/01/01 · openalex publication_date 2012/01/01 · crossref created 2012/01/14 · crossref deposited 2021/12/26 · openalex created_date 2025/10/10 · openalex updated_date 2025/11/06 · crossref indexed 2026/08/03
The equations of motion for compressible fluids of Korteweg type as derived by Dunn and Serrin in 1985 are studied in their full generality: the Korteweg tensor is assumed to be an arbitrary function of the form K := ( - ρ2 ∂ρ ψ + ρ ∇ ⋅ ( κ ∇ ρ) ) I - κ ∇ ρ ⊗ ∇ ρ, κ := 2 ρ ∂φ ψ(ρ,θ,φ), φ:=|∇ ρ|2, where ψ denotes Helmholtz free energy density and the capillarity κ is subject only to the natural positivity conditions κ(ρ,θ,φ) >0, κ(ρ,θ,φ) + 2 φ ∂φ κ(ρ,θ,φ) > 0, ρ, θ, φ ≥ 0. The viscous stress is supposed to be of generalized Newtonian type. The main result of the paper establishes well-posedness on domains with compact boundaries; the proof is based on refined methods of maximal regularity.