Dissipative solutions for the Navier-Stokes-Korteweg system and their numerical treatment

Many problems in computational fluid dynamics are described via the compressible Euler or Navier-Stokes (NS) equations. Recently, dissipative weak (DW) solutions have been introduced as a generalization to classical solution concepts. In a series of works, DW have been established as a meaningful concept from the analytical and numerical point of views. DW solutions do not have to fulfill the equations weakly but up to some defect and oscillations measures, They can be identified as limits of consistent and stable approximations and convergence towards DW solutions have been demonstrated for several structure-preserving numerical methods. Further, they are a natural extension of classical solutions since DW solutions coincide with them if either the classical solution exists, referred to as weak-strong uniqueness principle, or if they enjoy certain smoothness. A further extension to Navier-Stokes is the Navier-Stokes-Korteweg (NSK) which includes capillarity terms in the equations. Our motivation of considering NSK in our project is driven by the recent observations made by Slemrod that the rigorous passage of solutions from the mesoscopic equations (Boltzmann) to macroscopic systems, known as Hilbert sixth’s problem, will fail for the classical systems (NS and Euler) because of the appearance of van der Waals–Korteweg capillarity terms in a macroscopic description. Therefore, Korteweg systems are more suitable, for describing real fluid motions. In our project, we will extend the framework of DW solutions to NSK equations. We will define DW solutions in such a way that they form a natural extension of classical solutions, i.e. such that they satisfy a weak strong uniqueness principle, for the local and non-local NSK model. In addition, we plan to prove their global existence by demonstrating convergence of structure-preserving numerical schemes. First, structure-preserving low order schemes will be constructed and analyzed, whereas later we focus as well on the development and convergence properties of higher-order finite element (FE) schemes. Our investigations extend classical concepts and give new insights in the description of fluid- flows both theoretically and numerically, and lies in the scope of the SPP 2410 program.