Strong well-posedness for a Korteweg-type model for the dynamics of a compressible non-isothermal fluid Matthias Kotschote Mathematical Institute, University of Leipzig, Johannisgasse 26, 04109 Leipzig, Germany Email: kotschote@math.uni-leipzig.de Abstract The aim of this work is to prove an existence and uniqueness result for a non- isothermal model of capillary compressible fluids derived by J.E. Dunn and J. Serrin (1985). The proof is essentially based on the maximal regularity result of the associ- ated linear problem, where we can fall back upon useful results proved before. Using the maximal regularity the nonlinear problem can be approached by the contraction mapping principle. Mathematics Subject Classification 2000: 76N10, 35K50, 35K55, 35L65, 35M10 Keywords: Korteweg model, compressible fluids, capillarity, parabolic systems, maximal regularity, inhomogeneous boundary conditions 1. Introduction The purpose of this work is to prove existence and uniqueness of local strong solutions for a system of partial differential equations governing motions of compressible viscous capillary fluids that allow for thermal effects. The model we consider here originates in the classical papers of van der Waals and of Korteweg and was rigorously derived by Dunn and Serrin in [11]. The proof draws upon results in [15] where strong solutions were established in the isothermal case. To the author’s knowledge, all previously published results on existence and uniqueness for dynamic Korteweg-type concern the isothermal case, see e.g. [9], [13], [14], [6], [7], and in the Euler-Korteweg case [4], [5]. The fluid is characterised by its density ρ, velocity field u ∈ R n , temperature θ, energy density e, and pressure π. Introducing the Helmholtz free energy density ψ, the energy density and entropy density are given by e = ψ − θψ θ and s = −ψ θ , respectively. Since the Helmholtz free energy is objective under a frame change, this function can also depend on ∇ρ but only through its squared magnitude φ := |∇ρ| 2 , i.e. ψ = ψ(ρ,θ,φ) and thus e, s as well. The unknown functions u, ρ are governed by equations of momentum and mass conservation in [0,T 0 ] × Ω, 0 <T 0 , reading ∂ t (ρu)+ ∇· (ρu ⊗ u)= ∇· (S + K)+ ρf, ∂ t ρ + ∇· (ρu)=0, (1.1)