Abstract
In this paper, we prove that the incompressible inhomogeneous Navier–Stokes equations have a unique global solution with initial data \({(a_0,u_0)}\) in critical Besov spaces \({\dot{B}_{q,1}^{\frac{n}{q}}(\mathbb{R}^{n})\times\dot{B}_{p,1}^{\frac{n}{p}-1}(\mathbb{R}^{n})}\) satisfying a nonlinear smallness condition for all \({(p,q)\in[1,2n)\times[1,\infty)}\), \({-\frac1n\leq\frac1p-\frac1q\leq\frac1n}\) and \({\frac1p+\frac1q > \frac1n}\). We also construct an initial data satisfying that nonlinear smallness condition, but the norm of each component of the initial velocity field can be arbitrarily large in \({\dot{B}_{p,1}^{\frac{n}{p}-1}(\mathbb{R}^{n})}\) with \({n < p < 2n}\).
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Communicated by D. Chae
This work is supported by NSFC under Grant numbers 11171116 and 11571118, and by the Fundamental Research Funds for the Central Universities of China under the Grant number 2014ZG0031.
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Xu, H., Li, Y. & Chen, F. Global Solution to the Incompressible Inhomogeneous Navier–Stokes Equations with Some Large Initial Data. J. Math. Fluid Mech. 19, 315–328 (2017). https://doi.org/10.1007/s00021-016-0282-5
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DOI: https://doi.org/10.1007/s00021-016-0282-5