Abstract
We prove global well-posedness for instationary Navier–Stokes equations with initial data in Besov space \({B^{0}_{n,\infty}(\Omega)}\) in whole and half space, and bounded domains of \({{\mathbb R}^{n}}\), \({n \geq 3}\). To this end, we prove maximal \({L^{\infty}_{\gamma}}\) -regularity of the sectorial operators in some Banach spaces and, in particular, maximal \({L^{\infty}_{\gamma}}\) -regularity of the Stokes operator in little Nikolskii spaces \({b^{s}_{q,\infty}(\Omega)}\), \({s \in (-1, 2)}\), which are of independent significance. Then, based on the maximal regularity results and \({b^{s_{1}}_{q_{1},\infty}-B^{s_{2}}_{q_{2,1}}}\) estimates of the Stokes semigroups, we prove global well-posedness for Navier–Stokes equations under smallness condition on \({\|u_{0}\|_{B^{0}_{n,\infty}(\Omega)}}\) via a fixed point argument using Banach fixed point theorem.
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Communicated by Y. Giga
M.-H. Ri was supported by 2012 CAS-TWAS Postdoctoral Fellowship with Grant No. 3240267229.
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Ri, MH., Zhang, P. & Zhang, Z. Global Well-Posedness for Navier–Stokes Equations with Small Initial Value in \({B^{0}_{n,\infty}(\Omega)}\) . J. Math. Fluid Mech. 18, 103–131 (2016). https://doi.org/10.1007/s00021-015-0243-4
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DOI: https://doi.org/10.1007/s00021-015-0243-4