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Long-time asymptotic expansions for Navier-Stokes equations with power-decaying forces. (English) Zbl 1439.35366

Summary: The Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimensional periodic domains, with the body force having an asymptotic expansion, when time goes to infinity, in terms of power-decaying functions in a Sobolev-Gevrey space. Any Leray-Hopf weak solution is proved to have an asymptotic expansion of the same type in the same space, which is uniquely determined by the force, and independent of the individual solutions. In case the expansion is convergent, we show that the next asymptotic approximation for the solution must be an exponential decay. Furthermore, the convergence of the expansion and the range of its coefficients, as the force varies are investigated.

MSC:

35Q30 Navier-Stokes equations
35C20 Asymptotic expansions of solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
37L99 Infinite-dimensional dissipative dynamical systems
76D05 Navier-Stokes equations for incompressible viscous fluids
35D30 Weak solutions to PDEs
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