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The fundamental solution of the linearized Navier-Stokes equations for spinning bodies in three spatial dimensions-time dependent case. (English) Zbl 1125.35076
Summary: Explicit formulae for the fundamental solution of the linearized time dependent Navier-Stokes equations in three spatial dimensions are obtained. The linear equations considered in this paper include those used to model rigid bodies that are translating and rotating at a constant velocity. Estimates extending those obtained by V. A. Solonnikov [Tr. Mat. Inst. Steklova 70, 213–317 (1964; Zbl 0163.33803)] for the fundamental solution of the time dependent Stokes equations, corresponding to zero translational and angular velocity, are established. Existence and uniqueness of solutions of these linearized problems is obtained for a class of functions that includes the classical Lebesgue spaces \(L^{p}(\mathbb R^3)\), \(1 < p < \infty\). Finally, the asymptotic behavior and semigroup properties of the fundamental solution are established.

MSC:
35Q35 PDEs in connection with fluid mechanics
35A08 Fundamental solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
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