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On the asymptotic structure of \(D\)-solutions to steady Navier-Stokes equations in exterior domains. (English) Zbl 0794.35111
Galdi, Giovanni Paolo (ed.), Mathematical problems relating to the Navier-Stokes equation. River Edge, NJ: World Scientific Publishing Co. Ser. Adv. Math. Appl. Sci. 11, 81-104 (1992).
The author proves that every solution with bounded Dirichlet integral \((D\)-solution) \({\mathbf v}\) to the steady Navier-Stokes problem in a three dimensional exterior domain \(\Omega\) corresponding to a nonzero velocity \({\mathbf v}_ \infty\) at infinity and to a sufficiently smooth data with the body force of bounded support, presents the same asymptotic structure of the fundamental solution of the associated linearized problem, i.e. as \(|{\mathbf x} | \to \infty\), \[ {\mathbf v} (x)={\mathbf v}_ \infty+\bigl\{ {\mathbf m} \cdot E(x) \bigr\}+O \left( | {\mathbf x} |^{-3/2}+\delta \right) \] where \({\mathbf m}\) is a constant vector, \(E\) is the Oseen fundamental tensor and \(\delta\) is an arbitrary positive number.
This result was already obtained by K. I. Babenko [Mat. Sb. 91(133), 3-26 (1973), English transl. in Math. USSR, Sb. 20, 1-25 (1974; Zbl 0285.76009)] by a completely different and more complicated technique. However the present proof appears to be more elegant and understandable (to the reviewer) than Babenko’s one and allows to see more clearly the deep relationships between the linearized problem and the fully nonlinear one.
For the entire collection see [Zbl 0780.00006].
Reviewer: F.Rosso (Firenze)

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35A08 Fundamental solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs