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The energy equality and regularity results for the stationary Navier-Stokes equations on the exterior of a rotating obstacle. (English) Zbl 1421.35247

The author investigates the following stationary Navier-Stokes system \[ \begin{cases} -\nu \Delta u + (u\cdot \nabla)u - (\omega \wedge x)\cdot \nabla u + \omega \wedge u + \nabla p = f & \ \ \text{ in } \Omega, \\ \operatorname{div} u=0 & \ \ \text{ in } \Omega, \\ u=\omega \wedge x & \ \ \text{ on } \partial \Omega , \\ u(x)\rightarrow ke_1 & \ \ \text{ as } |x|\rightarrow \infty , \end{cases} \] where \(\Omega \subset \mathbb{R}^3\) is the exterior of a rotating rigid body, which ia also moving in the direction of the axis of rotation with constant velocity \(-ke_1\). Here \(f= div \, F\), where \(F\in L^2(\Omega)\). The existence of at least one weak solution satisfying \(\nabla u \in L^2(\Omega)\) follows by the classical Galerkin approximation. Using the additional regularity property \(u-ke_1\in L^4(\Omega)\), the author shows that the generalized energy equality is valid, so the uniqueness of the weak solutions follows for small data.

MSC:

35Q30 Navier-Stokes equations
35D30 Weak solutions to PDEs
35D35 Strong solutions to PDEs
76D07 Stokes and related (Oseen, etc.) flows
35B65 Smoothness and regularity of solutions to PDEs
76U05 General theory of rotating fluids
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