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.
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Alinhac, S., Blowup for nonlinear hyperbolic equations, Birkhäuser, Boston, 1995.
Chemin, J. Y. and Masmoudi, N., About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33(1), 2001, 84–112.
Climent Ezquerra, B. and Guillén González, F., Global in time solutions for the Poiseuille flow of Oldroyd type in 3D domains, Ann. Univ. Ferrara, Sez. VII (N. S.), 47, 2001, 23–40.
Guillopé, C. and Saut, J. C., Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal., 15, 1990, 849–869.
Hron, J., Málek, J., Neěas, J. and Rajagopal, K. R., Numerical simulations and global existence of solutions of two-dimensional flows of fluids with pressure- and shear- dependent viscosities, Math. Comput. Simulation, 61(3-6), 2003, 297–315; MODELLING 2001, Pilsen.
Joseph, D. D., Instability of the rest state of fluids of arbitrary grade greater than one, Arch. Ration. Mech. Anal., 75(3), 1980/81, 251–256.
Kawashima, S. and Shibata, Y., Global existence and exponetial stability of small solutions to nonlinear viscoelasticy, Comm. Math. Phys., 148, 1992, 189–208.
Klainerman, S., Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math., 38, 1985, 321–332.
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.
Klainerman, S. and Majda, A., Singular limits of quasilinear hyperbolic system with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34, 1981, 481–524.
Lin, F. H., Liu, C. and Zhang, P., On hydrohynamics of viscoelastic fluids, Comm. Pure Appl. Math., 2004, to appear.
Lions, P. L. and Masmoudi, N., Global solutions for some Oldroyd models of non-Newtonian flows, Chin. Ann. Math., 21B(2), 2000, 131–146.
Liu, C and Walkington, N. J., An Eulerian description of fluids containing visco-hyperelastic particles, Arch. Rat. Mech Anal., 159, 2001, 229–252.
Málek, J., Nečas, J. and Rajagopal, K. R., Global analysis of solutions of the flows of fluids with pressuredependent viscosities, Arch. Ration. Mech. Anal., 165(3), 2002, 243–269.
Schowalter, W. R., Mechanics of Non-Newtonian Fluids, Pergamon Press, Oxford, 1978.
Sideris, T. C., Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math., 151, 2000, 849–874.
Sideris, T. C. and Thomases, B., Global existence for 3D incompressible isotropic elastodynamics via the incompressible limit, Comm. Pure Appl. Math., 57, 2004, 1–39.
Slemrod, M., Constitutive relations for Rivlin-Erichsen fluids bases on generalized rational approximation, Arch. Ration. Mech. Anal., 146(1), 1999, 73–93.
Teman, R., Navier-Stokes Equations, North Holland, Amsterdam, 1977.
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*Project supported by the National Natural Science Foundation of China (No.10225102), the 973 Project of the Ministry of Science and Technology of China and the Doctoral Program Foundation of the Ministry of Education of China.
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Lei, Z. Global Existence of Classical Solutions for Some Oldroyd-B Model via the Incompressible Limit*. Chin. Ann. Math. Ser. B 27, 565–580 (2006). https://doi.org/10.1007/s11401-005-0041-z
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DOI: https://doi.org/10.1007/s11401-005-0041-z