Non-negative global weak solutions for a degenerated parabolic system approximating the two-phase Stokes problem

Journal of Differential Equations

Volumn 256, Issue 8, 2014, Pages 2659-2676

  Escher, Joachim ^a   Matioc, Bogdan Vasile ^b  

^a LEIBNIZ UNIVERSITY HANNOVER   (Germany)

^b UNIVERSITY OF VIENNA   (Austria)

Author keywords

Degenerated parabolic system; Non negative global weak solutions; Thin Film

Indexed keywords

EID: 84893816222     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2014.01.005     Document Type: Article

References (23)

1. Beretta E., Bertsch M., Dal Passo R. Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation. Arch. Ration. Mech. Anal. 1995, 129(2):175-200.
2. Bernis F. Viscous flows, fourth order nonlinear degenerate parabolic equations and singular elliptic problems. Pitman Res. Notes Math. Ser. 1995, vol. 323:40-56. Longman Sci. Tech., Harlow.
3. Bernis F., Friedman A. Higher order nonlinear degenerate parabolic equations. J. Differential Equations 1990, 83(1):179-206.
4. Bertozzi A.L., Pugh M. The lubrication approximation for thin viscous films: regularity and long-time behavior of weak solutions. Comm. Pure Appl. Math. 1996, 49(2):85-123.
5. Bertsch M., Dal Passo R., Garcke H., Grün G. The thin viscous flow equation in higher space dimensions. Adv. Differential Equations 1998, 3(3):417-440.
6. Blanchet A., Laurençot Ph. The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in Rd, d≥3. Comm. Partial Differential Equations 2013, 38(4):658-686.
7. Escher J., Hillairet M., Laurençot Ph., Walker C. Weak solutions to a thin film model with capillary effects and insoluble surfactant. Nonlinearity 2012, 25:2423-2441.
8. Escher J., Laurençot Ph., Matioc B.-V. Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media. Ann. Inst. H. Poincaré Anal. Non Linéaire 2011, 28(4):583-598.
9. Escher J., Matioc A.-V., Matioc B.-V. Modelling and analysis of the Muskat problem for thin fluid layers. J. Math. Fluid Mech. 2012, 14:267-277.
10. Escher J., Matioc A.-V., Matioc B.-V. Thin-film approximations of the two-phase Stokes problem. Nonlinear Anal. 2013, 73:1-13.
11. Escher J., Matioc B.-V. Existence and stability of solutions for a strongly coupled system modelling thin fluid films. NoDEA Nonlinear Differential Equations Appl. 2013, 20:539-555.
12. Garcke H., Wieland S. Surfactant spreading on thin viscous films: nonnegative solutions of a coupled degenerate system. SIAM J. Math. Anal. 2006, 37(6):2025-2048. (electronic).
13. Giacomelli L., Knüpfer H., Otto F. Smooth zero-contact-angle solutions to a thin-film equation around the steady state. J. Differential Equations 2008, 245(6):1454-1506.
14. Giacomelli L., Otto F. Rigorous lubrication approximation. Interfaces Free Bound. 2003, 5(4):483-529.
15. Gilbarg D., Trudinger N.S. Elliptic Partial Differential Equations of Second Order 2001, Springer Verlag.
16. Grün G. Droplet spreading under weak slippage-existence for the Cauchy problem. Comm. Partial Differential Equations 2004, 29(11-12):1697-1744.
17. Günther M., Prokert G. A justification for the thin film approximation of Stokes flow with surface tension. J. Differential Equations 2008, 245(10):2802-2845.
18. Knüpfer H. Well-posedness for the Navier slip thin-film equation in the case of partial wetting. Comm. Pure Appl. Math. 2011, 64(9):1263-1296.
19. Laurençot Ph., Matioc B.-V. A gradient flow approach to a thin film approximation of the Muskat problem. Calc. Var. Partial Differential Equations 2013, 47:319-341.
20. Laurençot Ph., Matioc B.-V. A thin film approximation of the Muskat problem with gravity and capillary forces. J. Math. Soc. Japan 2013, in press.
21. Matioc B.-V. Non-negative global weak solutions for a degenerate parabolic system modeling thin films driven by capillarity. Proc. Roy. Soc. Edinburgh Sect. A 2012, 142(5):1071-1085.
22. Matioc B.-V., Prokert G. Hele-Shaw flow in thin threads: a rigorous limit result. Interfaces Free Bound. 2012, 14(2):205-230.
23. p(0, T;B). Ann. Mat. Pura Appl. (4) 1987, 146:65-96.