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viscoelastic fluid; Oseen problem; steady transport equation; weighted estimates
We consider the steady plane flow of certain classes of viscoelastic fluids in exterior domains with a non-zero velocity prescribed at infinity. We study existence as well as asymptotic behaviour of solutions near infinity and show that for sufficiently small data the solution decays near infinity as fast as the fundamental solution to the Oseen problem.
[1] J. Baranger, C. Guillopé and J. C.  Saut: Mathematical analysis of differential models for viscoelastic fluids. In: Rheology for Polymer Melts Processing, Chapt. II, Elsevier Science, Amsterdam, 1996.
[2] P.  Dutto: Solutions physiquement raisonnables des équations de Navier-Stokes compressibles stationnaires dans un domaine extérieur du plan. Ph.D. thesis, University of Toulon, 1998.
[3] R. Farwig, A. Novotný and M. Pokorný: The fundamental solution of a modified Oseen problem. Z.  Anal. Anwendungen 19 (2000), 713–728. DOI 10.4171/ZAA/976 | MR 1784127
[4] G. P. Galdi: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I: Linearized Steady Problems. Springer Tracts in Natural Philosophy, Vol. 38. Springer-Verlag, New York, 1994. MR 1284205
[5] S. Kračmar, M. Novotný and M. Pokorný: Estimates of Oseen kernels in weighted $L^p$ spaces. Journal of Mathematical Society of Japan 53 (2001), 59–111. DOI 10.2969/jmsj/05310059 | MR 1800524
[6] A. Novotný: About the steady transport equation. In: Proceedings of Fifth Winter School at Paseky, Pitman Research Notes in Mathematics, 1998. MR 1692347
[7] A.  Novotný: Some Topics in the Mathematical Theory of Compressible Navier-Stokes Equations. Lecture Notes, Lipschitz Vorlesung. Univ. Bonn, to appear.
[8] A.  Novotný, M. Padula: Physically reasonable solutions to steady Navier-Stokes equations in 3-D exterior domains II ($v_\infty \ne 0$). Math. Ann. 308 (1997), 439–489. DOI 10.1007/s002080050084 | MR 1457741
[9] A.  Novotný, M.  Pokorný: Three-dimensional steady flow of viscoelastic fluid past an obstacle. J.  Math. Fluid Mech. 2 (2000), 294–314. DOI 10.1007/PL00000956 | MR 1781917
[10] M.  Pokorný: Asymptotic behaviour of solutions to certain PDE’s describing the flow of fluids in unbounded domains. Ph.D. thesis, Charles University, Prague & University of Toulon and Var, Toulon-La Garde, 1999.
[11] M.  Renardy: Existence of slow steady flows of viscoelastic fluid with differential constitutive equations. Z. Angew. Math. Mech. 65 (1985), 449–451. DOI 10.1002/zamm.19850650919 | MR 0814684
[12] D. R.  Smith: Estimates at infinity for stationary solutions of the Navier-Stokes equations in two dimensions. Arch. Rational Mech. Anal. 20 (1965), 341–372. DOI 10.1007/BF00282357 | MR 0185926 | Zbl 0149.44701
[13] B. O.  Turesson: Nonlinear Potential Theory and Weighted Sobolev Spaces. Lecture Notes in Mathematics Vol. 1736. Springer-Verlag, Berlin-Heidelberg, 2000. DOI 10.1007/BFb0103912 | MR 1774162
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