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Local energy decay of solutions to the Oseen equation in the exterior domains. (English) Zbl 1088.35048
The authors prove a local energy decay of the Oseen semigroup in the \(n\)-dimensional exterior domain \((n\geq 3)\). To formulate the Oseen equation in the framework of semigroup theory, the authors introduce the Helmholtz decomposition. The local energy decay is a crucial step to obtain \(L^p\)-\(L^q\) estimates of the Oseen semigroup, which in turn enables to prove the unique existence of global in time solutions to the Navier-Stokes equation in an exterior domain with small initial data in the \(L_m\) framework, and their properties of time decay.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35M20 PDE of composite type (MSC2000)
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