Analytic solutions for the two-phase navier-stokes equations with surface tension and gravity

Progress in Nonlinear Differential Equations and Their Application

Volumn 80, Issue , 2011, Pages 507-540

  Prüss, Jan ^a   Simonett, Gieri ^b  

^a MARTIN LUTHER UNIVERSITY HALLE WITTENBERG   (Germany)

^b VANDERBILT UNIVERSITY   (United States)

Author keywords

Analyticity; Free boundary problem; Gravity; Navier Stokes equations; Rayleigh Taylor instability; Surface tension; Well posedness

Indexed keywords

EID: 85016600894     PISSN: 14211750     EISSN: 23740280     Source Type: Book Series    
DOI: 10.1007/978-3-0348-0075-4_26     Document Type: Chapter

References (37)

1. G. Allain, Small-time existence for the Navier-Stokes equations with a free surface. Appl. Math. Optim. 16 (1987), 37–50.
2. H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory. Monographs in Mathematics 89, Birkhäuser, Boston, 1995.
3. ∞-calculus for elliptic operators. Differential Integral Equations 7 (1994), 613–653.
4. S. Angenent, Nonlinear analytic semiflows. Proc.Roy.Soc.Edinburgh115A (1990), 91–107.
5. S. Angenent, Parabolic equations for curves on surfaces, Part I. Curves with p-integrable curvature. Annals of Math. 132 (1990), 451–483.
6. J. Bergh, J. Löfström, Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.
7. J.T. Beale, Large-time regularity of viscous surface waves. Arch. Rational Mech. Anal. 84, (1983/84), 304–352.
8. J.T. Beale, T. Nishida, Large-time behavior of viscous surface waves. Recent topics in nonlinear PDE, II (Sendai, 1984), 1–14, North-Holland Math. Stud., 128, North-Holland, Amsterdam, 1985.
9. D. Bothe, J. Prüss, G. Simonett, Well-posedness of a two-phase flow with soluble surfactant. Nonlinear elliptic and parabolic problems, Progress Nonlinear Differential Equations Appl., 64, Birkhäuser, Basel, 2005, 37–61.
10. p-Theory for a class of non-Newtonian fluids. SIAM J. Math. Anal. 39 (2007), 379–421.
11. I.V. Denisova, A priori estimates for the solution of the linear nonstationary problem connected with the motion of a drop in a liquid medium. (Russian) Trudy Mat. Inst. Steklov 188 (1990), 3–21. Translated in Proc. Steklov Inst. Math. 1991, no. 3, 1–24.
12. I.V. Denisova, Problem of the motion of two viscous incompressible fluids separated by a closed free interface. Mathematical problems for Navier-Stokes equations (Centro, 1993). Acta Appl. Math. 37 (1994), 31–40.
13. I.V. Denisova, V.A. Solonnikov, Classical solvability of the problem of the motion of two viscous incompressible fluids. (Russian) Algebra i Analiz 7 (1995), no. 5, 101– 142. Translation in St. Petersburg Math. J. 7 (1996), no. 5, 755–786.
14. R. Denk, M. Hieber, J. Prüss, R-boundedness, Fourier multipliers, and problems of elliptic and parabolic type. AMS Memoirs 788, Providence, R.I. (2003).
15. R. Denk, G. Dore, M. Hieber, J. Prüss, A. Venni, Some new thoughts on old results of R.T. Seeley. Math. Annalen 328 (2004) 545–583.
16. G. Dore, A. Venni, On the closedness of the sum of two closed operators. Math. Z. 196 (1987), no. 2, 189–201.
17. J. Escher, G. Simonett, Analyticity of the interface in a free boundary problem. Math. Ann. 305 (1996), no. 3, 439–459.
18. J. Escher, G. Simonett, Analyticity of solutions to fully nonlinear parabolic evolution equations on symmetric spaces. Dedicated to Philippe Bénilan. J. Evol. Equ. 3 (2003), no. 4, 549–576.
19. J. Escher, J. Prüss, G. Simonett, Analytic solutions for a Stefan problem with Gibbs-Thomson correction. J. Reine Angew. Math. 563 (2003), 1–52.
20. J. Escher, J. Prüss, G. Simonett, A new approach to the regularity of solutions for parabolic equations. Evolution equations, 167–190, Lecture Notes in Pure and Appl. Math., 234, Dekker, New York, 2003.
21. G.P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I. Linearized steady problems. Springer Tracts in Natural Philosophy, 38. Springer-Verlag, New York, 1994.
22. ∞-calculus and sums of closed operators. Math. Ann. 321 (2001), 319–345.
23. ∞-functional calculus. Functional analytic methods for evolution equations, 65–311, Lecture Notes in Math., 1855, Springer, Berlin, 2004.
24. K. Masuda, On the regularity of solutions of the nonstationary Navier-Stokes equations, in: Approximation Methods for Navier-Stokes Problems, 360–370, Lecture Notes in Mathematics 771, Springer-Verlag, Berlin, 1980.
25. J. Prüss, J. Saal, G. Simonett, Existence of analytic solutions for the classical Stefan problem. Math. Ann. 338 (2007), 703–755.
26. ∞-calculus for the sum of noncommuting operators. Trans. Amer. Math. Soc. 359 (2007), no. 8, 3549–3565.
27. J. Prüss, G. Simonett, Analysis of the boundary symbol for the two-phase Navier-Stokes equations with surface tension. Banach Center Publ. 86 (2009), 265–285.
28. J. Prüss, G. Simonett, On the two-phase Navier-Stokes equations wit surface tension. arXiv:0908.3327. Interfaces Free Bound. 12 (2010), 311–345.
29. J. Prüss, G. Simonett, On the Rayleigh-Taylor instability for the two-phase Navier-Stokes equations. arXiv:0908.3334. Indiana Univ. Math. J., to appear.
30. J. Prüss, H. Sohr, On operators with bounded imaginary powers in Banach spaces. Math. Z. 203 (1990), 429–452.
31. Y. Shibata, S. Shimizu, On a free boundary problem for the Navier-Stokes equations. Differential Integral Equations 20 (2007), no. 3, 241–276.
32. Y. Shibata, S. Shimizu, Local solvability of free boundary problems for the Navier-Stokes equations with surface tension. Preprint.
33. V.A. Solonnikov, Lectures on evolution free boundary problems: classical solutions. Mathematical aspects of evolving interfaces (Funchal, 2000), 123–175, Lecture Notes in Math., 1812, Springer, Berlin, 2003.
34. N. Tanaka, Two-phase free boundary problem for viscous incompressible thermo-capillary convection. Japan J. Mech. 21 (1995), 1–41.
35. A. Tani, Small-time existence for the three-dimensional Navier-Stokes equations for an incompressible fluid with a free surface. Arch. Rational Mech. Anal. 133 (1996), 299–331.
36. A. Tani, N. Tanaka, Large-time existence of surface waves in incompressible viscous fluids with or without surface tension. Arch. Rat. Mech. Anal. 130 (1995), 303–304.
37. H. Triebel, Theory of function spaces. Monographs in Mathematics, 78. Birkhäuser Verlag, Basel, 1983.