1. Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations Dietmar Kröner, Freiburg Paris, Nov.2, 2009 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAA

2. Co-workers: • D. Diehl • A. Dressel • K. Hermsdörfer • C. Kraus

3. Outline • Introduction, numerical experiments for the NSK system • Jump conditions across the interface for NSK, static case • The pressure jump for the incompressible Navier-Stokes equations • Low Mach number limit for the compressible Navier-Stokes system • Low Mach number limit for the NSK system • NSK system dynamical case • Phase field like scaling

6. ρψ(ρ) Double well Pressure p β1 β2 ρ β1 ρ β2

7. ρψ(ρ) Double well Pressure p β1 β2 ρ β1 ρ β2

9. Initial data

10. Theoretical results

11. Danchin, Desjardin: Existence of solutions for compressible fluid models of Korteweg type. Annales de l'IHP, Analyse non lineaire, 18,(2001), 97-133. Global existence result for initial data close to stable equilibrium, d=2,3; local in time existence for Bresch, Desjardin, Lin: Existence of global weak solutions in energy spaces if Preprint 2002 (?) H. Hattori, D. Li: The existence of global solutions to a fluid dynamic model for materials for Korteweg type. J. Partial Differential Equations 9 (4) (1996) 323-342. H. Hattori, D. Li: Global solutions of a high-dimensional system for Korteweg materials. J. Math. Anal. Appl. 198, No. 1, (1996), 84-97. R. Danchin, B. Desjardin, : Existence of solutions for compressible fluid models of Korteweg type. Annales de l'IHP, Analyse nonlineaire, 18,(2001), 97-133.

12. D. Bresch, B. Desjardins, C.-K. Lin. On some compressible fluid models: Korteweg, lubrication and shallow water systems. Commun Partial Differ. Equations, 28(3):843-868, 2003. S. Benzoni-Gavage, R. Danchin, S. Descombes: Well-posedness of one-dimensional Korteweg models, preprint 2004 S. Benzoni-Gavage, R. Danchin, S. Descombes: On the well-posedness for the Euler-Korteweg model in several space dimensions, preprint 2005 M. Kotschote: Strong well-posedness for a Korteweg-type model for the dynamics of a compressible non-isothermal fluid. Preprint Leipzig 2006. (Initial boundary value)

13. Numerical results PhD Thesis Dennis Diehl

20. Jump conditions across the interface (static case)

21. Stationary case: (Luckhaus, Modica, Dreyer, Kraus)

40. liquid vapor Jump conditions: ??????

42. Multiply by a smooth testfunction ψ Integration by parts

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48. =:R