$L_2$-theory for two incompressible fluids separated by a free interface
Keywords
Two-phase problem, viscous incompressible fluids, interface problem with suface tension, Navier-Stokes system, Sobolev-Slobodetskiĭ spacesAbstract
The paper is devoted to the problem of non-stationary motion of two viscous incompressible fluids separated by a free surface and contained in a bounded vessel. It is assumed that the fluids are subject to mass forces and capillary forces at the interface. We prove the stability of a rest state under the assumption that initial velocities are small, a free interface is close to a sphere at an initial instant of time, and mass forces decay as $t\to\infty$.References
H. Abels, On general solutions of two-phase flows for viscous incompressible fluids, Interfaces Free Bound. 9 (2007), 31–65.
I.V. Denisova, The motion of a drop in a flow of a liquid, Dinamika Sploshn. Sredy 93/94 (1989), 32–37. (in Russian)
I.V. Denisova, A priori estimates of the solution of a linear time dependent problem connected with the motion of a drop in a fluid medium, Trudy Mat. Inst. Steklov. 188 (1990), 3–21; English transl.: Proc. Steklov Inst. Math. 3 (1991), 1–24.
I.V. Denisova, Problem of the motion of two viscous incompressible fluids separated by a closed free interface, Acta Appl. Math. 37 (1994), 31–40.
I.V. Denisova, Global solvability of a problem on two fluid motion without surface tension, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 348 (2007), 19–39; English transl.: J. Math. Sci. 152 (2008), no. 5, 625–637.
I.V. Denisova, Global L2 -solvability of a problem governing two-phase fluid motion without surface tension, Port. Math. 71 (2014), 1–24.
I.V. Denisova, Global classical solvability of an interface problem on the motion of two fluids, RIMS Kôkyûroku Series, Kyoto University 1875 (2014), 84–108.
I.V. Denisova and V.A. Solonnikov, Solvability of the linearized problem on the motion of a drop in a liquid flow, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 171 (1989), 53–65; English transl.: J. Soviet Math. 56 (1991), no. 2, 2309–2316.
I.V. Denisova and V.A. Solonnikov, Global solvability of a problem governing the motion of two incompressible capillary fluids, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 397 (2011), 20–52; English transl.: J. Math. Sci. 185 (2012), no. 5, 668–686.
I.V. Denisova and V.A. Solonnikov, Classical well-posedness of free boundary problems in viscous incompressible fluid mechanics, Handbook of Mathematical Analysis in
J. Giga and Sh. Takahashi, On global weak solutions of the nonstationary two-phase Stokes flow, SIAM J. Math. Anal. 25 (1994), 876–893.
M. Köhne, J. Prüss and M. Wilke, Qualitative behaviour of solutions for the two-phase Navier–Stokes equations with surface tension, Math. Ann. 356 (2013), no. 2, 737–792.
M. Padula, On the exponential stability of the rest state of a viscous compressible fluid, J. Math. Fluid Mech. 1 (1999), 62–77.
M. Padula and V.A. Solonnikov, On the local solvability of free boundary problem for the Navier–Stokes equations, Problemy Mat. Analiza 50 (2010), 87–112; J. Math. Sci. 170 (4), 522–553.
J. Prüss and G. Simonett, On the two-phase Navier–Stokes equations with surface tension, Interfaces Free Bound., 12 (2010), no. 3, 311–345.
Yo. Shibata and S. Shimizu, Maximal Lp –Lq regularity for the two-phase Stokes equations. Model problems, J. Differential Equations 251 (2011), 373–419.
S. Shimizu, Local solvability of free boundary problems for the two-phase Navier–Stokes equations with surface tension in the whole space, Progr. Nonlinear Differential Equations Appl. 80 (2011), 647–686.
V.A. Solonnikov, On non-stationary motion of a finite liquid mass bounded by a free surface, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 152 (1986), 137–157; English transl.: J. Soviet Math. 40 (1988), no. 5, 672–686.
V.A. Solonnikov, Generalized energy estimates in a free boundary problem for a viscous incompressible fluid, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 282 (2001), 216–243; English transl.: J. Math. Sci. 120 (2004), no. 5, 1766–1783.
V.A. Solonnikov, Lectures on evolution free boundary problems: classical solutions, Lecture Notes in Math. 1812 (2003), 123–175.
V.A. Solonnikov, On problem of stability of equilibrium figures of uniformly rotating viscous incompressible liquid, Instability in models connected with fluid flows II, (C. Bardos, A. Fursikov, eds.), Int. Math. Ser., Vol 7, Springer, New York, 2008, 189–254.
V.A. Solonnikov, On the linear problem arising in the study of a free boundary problem for the Navier–Stokes equations, Algebra i Analiz 22 (2010), no. 6, 235–269; English transl.: St. Petersburg Math. J. 22 (2011), no. 6, 1023–1049.
V.A. Solonnikov, Lp -theory of the problem of motion of two incompressible capillary fluids in a container, Probl. Mat. Anal. 75 (2014), 93–152; English. transl.: J. Math. Sci. 198 (2014), no. 6, 761–827.
Sh. Takahashi, On global weak solutions of the nonstationary two-phase Navier–Stokes flow, Adv. Math. Sci. Appl. 5 (1995), 321–342.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0