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

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 $L^{p}$ setting ( $p \neq 2$ ).

Keywords

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

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