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The divergence, curl and Stokes operators in exterior domains of $$\mathbb{R}^ 3$$. (English) Zbl 0837.35115
Galdi, G. P. (ed.) et al., Recent developments in theoretical fluid mechanics. Papers presented at the 2nd winter school in fluid dynamics, held in Paseky, Czech Republic, in November 29-December 4, 1992. Harlow: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 291, 34-77 (1993).
The author studies the divergence, curl and Stokes operators between weighted Sobolev spaces of the form $W^m_k(\Omega)= \{v\in {\mathcal D}'(\Omega)\mid \rho(r)^{|\alpha|- m+ k} \partial^\alpha v\in L^2(\Omega),\;0\leq |\alpha|\leq m\},$ where $$\Omega\subset \mathbb{R}^3$$, $$m\in \mathbb{N}$$, $$k\in \mathbb{Z}$$ and $$\rho(r)= \sqrt{1+ r^2}$$. If $$- m\in \mathbb{N}$$ then $$W^{- m}_{- k}(\Omega)$$ is defined as the dual space of $$\mathring W^m_k(\Omega)= \overline{C^\infty_0(\Omega)}^{|\cdot|_{W^m_k}}$$. It is shown that the divergence is an isomorphism from $$W^m_{m+ k}(\mathbb{R}^3)/ V^m_{m+ k}(\mathbb{R}^3)$$ onto $$W^{m- 1}_{m+ k}(\mathbb{R}^3)$$ (or $$W^{m- 1}_{m+ k}(\mathbb{R}^3)/\mathbb{R}$$ if $$k\geq 1$$). Here, $$V^m_k(\mathbb{R}^3)= \{v\in W^m_k(\mathbb{R}^3)^3\mid \text{div } v= 0\}$$.
Combining this result with a corresponding statement for bounded domains, the author is able to prove that the divergence is also an isomorphism from $$\mathring W^m_{m+ k}(\Omega)/V^m_{m+ k}(\Omega)$$ onto $$\mathring W^{m- 1}_{m+ k}(\Omega)$$ (or $$\mathring W^{m- 1}_{m+ k}(\Omega)/\mathbb{R}$$, if $$k\geq 1$$) for an exterior domain $$\Omega$$. $$\Omega$$ is allowed to have a complement $$\Omega'= \mathbb{R}^3\backslash \Omega$$ which is not necessarily connected.
After that the existence and uniqueness of a vector potential for a given divergence-free vector field in $$W^m_{m+ k}(\Omega)$$ is studied.
In the last part of the paper, existence, uniqueness and regularity for the nonhomogeneous Stokes problem $-\nu \Delta u+ \nabla p= f,\quad - \text{div } u= h\quad\text{in }\Omega,\quad u= g\quad\text{on }\partial\Omega$ in the spaces $$W^m_k(\Omega)$$, ($$\Omega$$ an exterior domain) is discussed. The argument is again split up into the case $$\Omega= \mathbb{R}^3$$ and the case of a bounded domain.
For the entire collection see [Zbl 0809.00020].

MSC:
 35Q30 Navier-Stokes equations 46N20 Applications of functional analysis to differential and integral equations