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Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. (English) Zbl 0823.35140
The existence, uniqueness and regularity of the solution of the Stokes problem $-\Delta u+ \nabla p=f, \quad \text{div } u=\varphi \quad \text{in }\Omega, \qquad u=g \quad \text{on } \partial \Omega$ in arbitrary dimension is considered. The authors propose to solve this problem by relating it to a Helmholtz decomposition.
First, they apply a simplified version of De Rham’s theorem, which gives a characterization of some distributions by means of their gradient and is an extension of Necas’ theorem. They show that the divergence operator is an isomorphism between adequate spaces, which is a generalization of the “inf-sup” condition of Babuska and Brezzi. This and Heron’s theorems on the trace of the divergence of functions in $$W^{m,r} (\Omega)$$ allow them to construct functions in $$W^{m,r} (\Omega)$$ with prescribed divergence and trace.
Applying a result of Agmon, Douglis and Nirenberg valid for elliptic systems, the authors establish a Helmholtz decomposition of the space $$W^{m,r} (\Omega)\cap W_ 0^{1,r} (\Omega)$$ for $$m\geq 2$$, and derive the existence and uniqueness of the solution of Stokes problem in $$W^{m,r} (\Omega) \times W^{m-1,r} (\Omega)$$. This result is carried over to $$m=0$$ by duality argument introduced by Lions and Magenes and for $$m=1$$ by interpolation between $$m=0$$ and $$m=2$$. Finally, using the penalty method introduced by Temam the pressure is decoupled from the velocity. This method replaces the Stokes system by $-\Delta u_ \varepsilon- {\textstyle {1\over \varepsilon}} \nabla \text{ div } u_ \varepsilon=f \quad \text{in }\Omega, \qquad u_ \varepsilon=0 \quad \text{on }\partial \Omega,$ where $$\varepsilon>0$$ tend to zero. The pressure is approximated by $$p_ \varepsilon=- {1\over \varepsilon} \text{ div } u_ \varepsilon$$ and $$u_ \varepsilon$$ is the approximation of $${u}$$.
The last theorem says that for $${f}$$ given in $$W^{m-2,r} (\Omega)$$, if $$({u}, {p})$$ is the solution of the homogeneous Stokes problem, then $$u_ \varepsilon$$ tends to $$u$$ in $$W^{m,r} (\Omega)$$, $$p_ \varepsilon=- {1\over \varepsilon} \text{ div } u_ \varepsilon$$ tends to $$p$$ in $$W^{m-1, r} (\Omega)$$ as $$\varepsilon$$ tends to 0.

MSC:
 35Q30 Navier-Stokes equations 76D07 Stokes and related (Oseen, etc.) flows
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References:
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