Global Existence of Classical Solutions for Some Oldroyd-B Model via the Incompressible Limit*

Abstract

In this paper, we prove local and global existence of classical solutions for a system of equations concerning an incompressible viscoelastic fluid of Oldroyd-B type via the incompressible limit when the initial data are sufficiently small.

Keywords

Incompressible limit / Global existence / Oldroyd model / 11R70 / 11R11 / 11R27

References

[1] Alinhac, S., Blowup for nonlinear hyperbolic equations, Birkhäuser, Boston, 1995.

[2] Chemin SIAM J. Math. Anal., 2001, 33: 84

[3] Climent Ann. Univ. Ferrara, Sez. VII, 2001, 47: 23

[4] Guillopé Nonlinear Anal., 1990, 15: 849

[5] Hron Math. Comput. Simulation, 2003, 61: 297

[6] Joseph Arch. Ration. Mech. Anal., 1980/81, 75: 251

[7] Kawashima Comm. Math. Phys., 1992, 148: 189

[8] Klainerman Comm. Pure Appl. Math., 1985, 38: 321

[9] Klainerman, S., The null condition and global existence to nonlinear wave equations, Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N. M., 1984), 293–326; Lectures in Appl. Math., Vol. 23, A. M. S., Providence, RI, 1986.

[10] Klainerman Comm. Pure Appl. Math., 1981, 34: 481

[11] Lin, F. H., Liu, C. and Zhang, P., On hydrohynamics of viscoelastic fluids, Comm. Pure Appl. Math., 2004, to appear.

[12] Lions Chin. Ann. Math., 2000, 21B: 131

[13] Liu Arch. Rat. Mech Anal., 2001, 159: 229

[14] Málek Arch. Ration. Mech. Anal., 2002, 165: 243

[15] Schowalter, W. R., Mechanics of Non-Newtonian Fluids, Pergamon Press, Oxford, 1978.

[16] Sideris Ann. of Math., 2000, 151: 849

[17] Sideris Comm. Pure Appl. Math., 2004, 57: 1

[18] Slemrod Arch. Ration. Mech. Anal., 1999, 146: 73

[19] Teman, R., Navier-Stokes Equations, North Holland, Amsterdam, 1977.