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Asymptotic expansions in time for rotating incompressible viscous fluids. (English) Zbl 1464.35182

Summary: We study the three-dimensional Navier-Stokes equations of rotating incompressible viscous fluids with periodic boundary conditions. The asymptotic expansions, as time goes to infinity, are derived in all Gevrey spaces for any Leray-Hopf weak solutions in terms of oscillating, exponentially decaying functions. The results are established for all non-zero rotation speeds, and for both cases with and without the zero spatial average of the solutions. Our method makes use of the Poincaré waves to rewrite the equations, and then implements the Gevrey norm techniques to deal with the resulting time-dependent bi-linear form. Special solutions are also found which form infinite dimensional invariant linear manifolds.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35C20 Asymptotic expansions of solutions to PDEs
76E07 Rotation in hydrodynamic stability
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35D30 Weak solutions to PDEs
76U05 General theory of rotating fluids
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