Convergence of a low order non-local Navier–Stokes–Korteweg system: The order-parameter model

Article type: Research Article

Authors: Charve, Frédéric

Affiliations: Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), Université Paris-Est Créteil, 61 Avenue du Général de Gaulle, 94 010 Créteil Cedex, France. E-mail: frederic.charve@u-pec.fr

Abstract: In the present article we consider a capillary compressible system introduced by C. Rohde after works of Bandon, Lin and Rogers, called the order-parameter model, and whose aim is to reduce the numerical difficulties generated by the classical local Korteweg system (involving derivatives of order three) or the non-local system (also introduced by Rohde after works of Van der Waals, and which involves a convolution operator). We prove that this system has a unique global solution for initial data close to an equilibrium and we obtain the convergence of this solution towards the local Korteweg model as well as a convergence rate with respect to the order parameter, in accordance to what conjectured C. Rohde. As a by-product, the a priori estimates we obtain allow to provide global existence results in the Lp setting (p≠2).

Keywords: Compressible Navier–Stokes system, local and non-local capillarity, Besov spaces, Lagrangian change of variables

DOI: 10.3233/ASY-161390

Journal: Asymptotic Analysis, vol. 100, no. 3-4, pp. 153-191, 2016