| Abstract: |
| In this talk, large-time behavior of the incompressible Navier-Stokes flow in n-dimensional whole space is discussed. For this theme, Carpio (1996) and Fujigaki and Miyakawa (2001) derived the asymptotic expansion up to n-th order by employing the theory via Escobedo and Zuazua (1991). In those expansion, integrability of the moments of solution is required. However, on the next profile, the moment is growing logarithmically in time. Not only that, but there is also a problem with spatial integrability since the nonlinear effect contains the nonlocal operator. More precisely it contains the Riesz transform which is coming from the solenoidal conditions. To omit those difficulties, we employ the renormalization together with Biot-Savart law. By employing this method, asymptotic expansion up to 2n-th order is derived. Any terms on the expansion have their own parabolic-scalings. The parabolic scaling guarantees uniqueness of the expansion. Furthermore the above logarithmic evolution is specified on this expansion. |
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