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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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