On Stokes operators with variable viscosity in bounded and unbounded domains. (English) Zbl 1172.35050
The authors consider the Stokes problem with non-constant viscosity \(\nu(x)\) in bounded or unbounded domains in \(\mathbb R^d\). They assume \(\nu(x) = \nu_\infty + \nu'(x)\), \(\nu' \in W^{1,r_1}(\Omega)\) and \(\partial \Omega \in W^{2-1/r_2,r_2}\) with \(d<r_1, r_2\). Further, \(1<q<\infty\) with \(q, q/(q-1) \leq r_1, r_2\). The boundary conditions are on \(\Gamma_1\) the homogeneous Dirichlet condition and on \(\Gamma_2\) generally inhomogeneous Neumann condition (i.e. \((2\nu(x) Dv-pI)\cdot n = a\)). The assumptions on \(\Omega\) imply that the Helmholtz decomposition in \(L^q\) is valid with the given boundary conditions.
7mm
- a)
- In the case of a generally unbounded domain \(\Omega\), the resolvent estimate for the Stokes problem is proved. Further, for the shifted Stokes operator \(A_q\) it is shown that it admits the bounded \(H_\infty\) calculus.
- b)
- If \(\Omega\) is bounded and \(\Gamma_1\) is nonempty, it is in addition shown that the Stokes operator admits the bounded \(H_\infty\) calculus.
- c)
- If \(c\in\mathbb R\) is such that \(A_q+c\) admits the bounded \(H_\infty\) calculus, then the modified evolutionary Stokes problem (with the Stokes operator replaced by \(A_q+c\)) has for \(1<p<\infty\) the maximal \(L^p(L^q)\) regularity.
Reviewer: Milan Pokorný (Praha)
MSC:
| 35Q30 | Navier-Stokes equations |
| 76D07 | Stokes and related (Oseen, etc.) flows |
| 47A60 | Functional calculus for linear operators |
| 47F05 | General theory of partial differential operators |
| 47A10 | Spectrum, resolvent |
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