For the two-phase incompressible Navier–Stokes equations with surface tension, we derive an appropriate weak formulation incorporating a variational formulation using divergence-free test functions. We prove a consistency result to justify our definition and, under reasonable regularity assumptions, we reconstruct the pressure function from the weak formulation.
H.Abels, On generalized solutions of two-phase flows for viscous incompressible fluids, Interfaces Free Bound.9(1) (2007), 31–65. doi:10.4171/IFB/155.
2.
H.Abels and H.Garcke, Weak solutions and diffuse interface models for incompressible two-phase flows, in: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, 2018, pp. 1268–1327.
3.
A.Alphonse, C.M.Elliott and B.Stinner, An abstract framework for parabolic PDEs on evolving spaces, Port. Math. (NS)72(1) (2015), 1–46. doi:10.4171/PM/1955.
4.
A.Alphonse, C.M.Elliott and B.Stinner, On some linear parabolic PDEs on moving hypersurfaces, Interfaces Free Bound.17(2) (2015), 157–187. doi:10.4171/IFB/338.
5.
H.Amann, Linear and Quasilinear Parabolic Problems. Vol. 1: Abstract Linear Theory, Birkhäuser, Basel, 1995.
6.
L.Ambrosio, N.Fusco and D.Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, 2000.
7.
F.Boyer and P.Fabrie, Mathematical Tools for the Study of the Incompressible Navier–Stokes Equations and Related Models, Springer, New York, NY, 2013.
8.
J.Daube, Sharp-interface limit for the Navier–Stokes–Korteweg equations, PhD thesis, University of Freiburg, 2016. doi:10.6094/UNIFR/11679.
9.
K.Deckelnick, G.Dziuk and C.M.Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numerica14 (2005), 139–232. doi:10.1017/S0962492904000224.
10.
I.V.Denisova and V.A.Solonnikov, Solvability in Hölder spaces of a model initial-boundary value problem generated by a problem on the motion of two fluids, Journal of Mathematical Sciences70(3) (1991), 1717–1746. doi:10.1007/BF02149145.
11.
I.V.Denisova and V.A.Solonnikov, Global solvability of a problem governing the motion of two incompressible capillary fluids in a container, Journal of Mathematical Sciences185(5) (2012), 668–686. doi:10.1007/s10958-012-0951-8.
12.
L.C.Evans and R.F.Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992.
13.
G.P.Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems, 2nd edn, Springer, New York, NY, 2011.
14.
D.Gilbarg and N.S.Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edn, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin, 2001, XIII, 513 p.
15.
M.Kimura, Geometry of hypersurfaces and moving hypersurfaces in for the study of moving boundary problems, in: Topics in Mathematical Modeling, Jindr˘ich Nec˘as Cent. Math. Model. Lect. Notes., Vol. 4, Matfyzpress, Prague, 2008, pp. 39–93.
16.
M.Köhne, J.Prüss and M.Wilke, Qualitative behaviour of solutions for the two-phase Navier–Stokes equations with surface tension, Math. Ann.356(2) (2013), 737–792. doi:10.1007/s00208-012-0860-7.
17.
J.L.Lions and E.Magenes, Problèmes aux Limites Non Homogenes et Applications, Vol. 1, Dunod 1, Paris, 1968, XIX.
18.
P.Nägele, Monotone operator theory for unsteady problems on non-cylindrical domains, PhD thesis, University of Freiburg, 2015. doi:10.6094/UNIFR/10187.
19.
P.I.Plotnikov, Generalized solutions of a problem on the motion of a non-Newtonian fluid with a free boundary, Sibirsk. Mat. Zh.34(4) (1993), 127–141, iii, ix.
20.
P.I.Plotnikov, Compressible Stokes flow driven by capillarity on a free surface, in: Navier–Stokes Equations and Related Nonlinear ProblemsPalanga, 1997, VSP, Utrecht, 1998, pp. 217–238.
21.
J.Prüss and G.Simonett, On the two-phase Navier–Stokes equations with surface tension, Interfaces Free Bound.12(3) (2010), 311–345. doi:10.4171/IFB/237.
J.Saal, Maximal regularity for the Stokes system on noncylindrical space-time domains, J. Math. Soc. Japan58(3) (2006), 617–641. doi:10.2969/jmsj/1156342030.
24.
H.Sohr, The Navier–Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser, Basel, 2001.
