A steady-state exterior Navier-Stokes problem that is not well-posed

Galdi, Giovanni

doi:10.1090/S0002-9939-08-09658-5

A steady-state exterior Navier-Stokes problem that is not well-posed
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by Giovanni P. Galdi

Proc. Amer. Math. Soc. 137 (2009), 679-684

DOI: https://doi.org/10.1090/S0002-9939-08-09658-5

Published electronically: August 15, 2008

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Abstract:

We prove that the exterior Navier-Stokes problem with zero velocity at infinity is not well-posed in homogeneous Sobolev spaces. This result complements and clarifies well-known previous results obtained by various authors.

References

N. Aronszajn and E. Gagliardo, Interpolation spaces and interpolation methods, Ann. Mat. Pura Appl. (4) 68 (1965), 51–117. MR 226361, DOI 10.1007/BF02411022
Wolfgang Borchers and Tetsuro Miyakawa, On stability of exterior stationary Navier-Stokes flows, Acta Math. 174 (1995), no. 2, 311–382. MR 1351321, DOI 10.1007/BF02392469
Giovanni P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Springer Tracts in Natural Philosophy, vol. 38, Springer-Verlag, New York, 1994. Linearized steady problems. MR 1284205, DOI 10.1007/978-1-4612-5364-8
Giovanni P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Springer Tracts in Natural Philosophy, vol. 38, Springer-Verlag, New York, 1994. Linearized steady problems. MR 1284205, DOI 10.1007/978-1-4612-5364-8
G. P. Galdi and M. Padula, Existence of Steady Incompressible Flows Past an Obstacle, Mathematical Analysis of Phenomena in Fluid and Plasma Dynamics, RIMS Kokyuroku, Kyoto, vol. 745, 1991, 87–101.
Giovanni P. Galdi and Christian G. Simader, New estimates for the steady-state Stokes problem in exterior domains with applications to the Navier-Stokes problem, Differential Integral Equations 7 (1994), no. 3-4, 847–861. MR 1270107
Hideo Kozono and Masao Yamazaki, Exterior problem for the stationary Navier-Stokes equations in the Lorentz space, Math. Ann. 310 (1998), no. 2, 279–305. MR 1602012, DOI 10.1007/s002080050149
Hideo Kozono and Hermann Sohr, On stationary Navier-Stokes equations in unbounded domains, Ricerche Mat. 42 (1993), no. 1, 69–86. MR 1283806
Hideo Kozono, Hermann Sohr, and Masao Yamazaki, Representation formula, net force and energy relation to the stationary Navier-Stokes equations in $3$-dimensional exterior domains, Kyushu J. Math. 51 (1997), no. 1, 239–260. MR 1437320, DOI 10.2206/kyushujm.51.239
J. Leray, Étude de Diverses Équations Intégrales non Linéaires et de Quelques Problèmes que Pose l’Hydrodynamique, J. Math. Pures Appl., 12, 1933, 1–82.
Tetsuro Miyakawa, On uniqueness of steady Navier-Stokes flows in an exterior domain, Adv. Math. Sci. Appl. 5 (1995), no. 2, 411–420. MR 1360998
S. Smale, An infinite dimensional version of Sard’s theorem, Amer. J. Math. 87 (1965), 861–866. MR 185604, DOI 10.2307/2373250
Hermann Sohr, The Navier-Stokes equations, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2001. An elementary functional analytic approach. MR 1928881, DOI 10.1007/978-3-0348-8255-2
Eberhard Zeidler, Applied functional analysis, Applied Mathematical Sciences, vol. 109, Springer-Verlag, New York, 1995. Main principles and their applications. MR 1347692