Maximal Lp-Lq regularity for the two-phase Stokes equations; Model problems

Journal of Differential Equations

Volumn 251, Issue 2, 2011, Pages 373-419

  Shibata, Yoshihiro ^a   Shimizu, Senjo ^b  

^a WASEDA UNIVERSITY   (Japan)

^b SHIZUOKA UNIVERSITY   (Japan)

Author keywords

Gravity; Maximal regularity; Resolvent estimate; Stokes equation; Surface tension; Two phase problem

Indexed keywords

EID: 79955740035     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2011.04.005     Document Type: Article

References (33)

1. Abels H. On general solutions of two-phase flows for viscous incompressible fluids. Interfaces Free Bound. 2007, 9:31-65.
2. Amann H. Linear and Quasilinear Parabolic Problems, vol. I 1995, Birkhäuser, Basel.
3. Amann H. Anisotropic Function Spaces and Maximal Regularity for Parabolic Problem. Part 1. Function Spaces. Jindrich Nacas Center Math. Modelling Lecture Notes, Prague 2009, vol. 6.
4. Arendt W., Batty C.J.K., Hieber M., Neubrander F. Vector-Valued Laplace Transforms and Cauchy Problems. Monogr. Math. 2001, vol. 96. Birkhäuser Verlag, Basel, Boston, Berlin.
5. Bourgain J. Vector-valued singular integrals and the H1-BMO duality. Probability Theory and Harmonic Analysis 1968, 1-19. Marcel Dekker, New York. D. Borkholdeer (Ed.).
6. Calderón A.P. Lebesgue spaces of differentiable functions and distributions. Proc. Sympos. Pure Math. 1961, vol. 4:33-49. Amer. Math. Soc.
7. Denisova I.V. A priori estimates for the solution of a linear time-dependent problem connected with the motion of a drop in a fluid medium. Tr. Mat. Inst. Steklova. Proc. Steklov Inst. Math. 1991, 188:1-24. (in Russian); English transl.:.
8. Denisova I.V. Problem of the motion of two viscous incompressible fluids separated by a closed free interface. Acta Appl. Math. 1994, 37:31-40.
9. Denisova I.V., Solonnikov V.A. Solvability in Hölder spaces of a model initial-boundary value problem generated by a problem on the motion of two fluids. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI). J. Math. Sci. 1994, 70:1717-1746. (in Russian); English transl.:.
10. Denisova I.V., Solonnikov V.A. Classical solvability of the problem on the motion of two viscous incompressible fluids. Algebra i Analiz. St. Petersburg Math. J. 1996, 7:755-786. (in Russian); English transl.:.
11. Denk R., Hieber M., Prüß J. R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc. 2003, 166(788).
12. Desch W., Hieber M., Prüß J. Lp-Theory of the Stokes equation in a half space. J. Evol. Equ. 2001, 1:115-142.
13. Farwig R., Sohr H. Generalized resolvent estimates for the Stokes system in bounded and unbounded domains. J. Math. Soc. Japan 1994, 46:607-642.
14. Giga Y., Takahashi Sh. On global weak solutions of the nonstationary two-phase Stokes flow. SIAM J. Math. Anal. 1994, 25:876-893.
15. Mihlin S.G. On the multipliers of Fourier integrals. Dokl. Akad. Nauk 1956, 109:701-703. (in Russian).
16. Mihlin S.G. Fourier integrals and multiple singular integrals. Vestnik Leningrad Univ. Ser. Mat. 1957, 12:143-145. (in Russian).
17. Kalton N.J., Lancien G. Lp-maximal regularity on Banach spaces with a Schauder basis. Arch. Math. 2002, 78:397-408.
18. Nouri A., Poupaud F. An existence theorem for the multifluid Navier-Stokes problem. J. Differential Equations 1995, 123:71-88.
19. Prüß J., Simonett G. Analysis of the boundary symbol for the two-phase Navier-Stokes equations with surface tension. Proc. Banach Center Publ. 2009, 86:265-285.
20. Prüß J., Simonett G. On the two-phase Navier-Stokes equations with surface tension. Interface Free Bound. 2010, 12:311-345.
21. J. Prüß, G. Simonett, Analytic solutions for the two-phase Navier-Stokes equations with surface tension and gravity, Preprint.
22. ∞-calculus for the Stokes operator, PhD thesis, TU Darmstadt, Logos Verlag, Berlin, 2003.
23. Shibata Y., Shimada R. On a generalized resolvent estimate for the stokes system with Robin boundary condition. J. Math. Soc. Japan 2007, 59:469-519.
24. Shibata Y., Shimizu S. On resolvent estimate of the interface problem for the Stokes system in a bounded domain. J. Differential Equations 2003, 191:408-444.
25. Shibata Y., Shimizu S. On the Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain. J. Reine Angew. Math. 2008, 615:157-209.
26. Shibata Y., Shimizu S. On a resolvent estimate of the Stokes system in a half space arising from a free boundary problem for the Stokes equations. Math. Nachr. 2009, 282:482-499.
27. Y. Shibata, S. Shimizu, On the Lp-Lq maximal regularity of the Stokes problem with first order boundary condition; Model problem, J. Math. Soc. Japan, in press.
28. Shimizu S. Lp-Lq maximal regularity of the interface problem for the Stokes system in unbounded domain. RIMS Kokyuroku 2007, 1536:58-72.
29. Shimizu S. Maximal regularity and viscous incompressible flows with free interface. Banach Center Publ. 2008, 81:471-480.
30. S. Shimizu, Local solvability of free boundary problems for the two-phase Navier-Stokes equations with surface tension in the whole space, Preprint.
31. Takahashi Sh. On global weak solutions of the nonstationary two-phase Navier-Stokes flow. Adv. Math. Sci. Appl. 1995, 5:321-342.
32. Tanaka N. Global existence of two phase non-homogeneous viscous incompressible fluid flow. Comm. Partial Differential Equations 1993, 18:41-81.
33. Weis L. Operator-valued Fourier multiplier theorems and maximal Lp-regularity. Math. Ann. 2001, 319:735-758.