Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces

Abstract The Keller-Segel-Navier-Stokes system in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:math> {{\mathbb{R}}}^{N} is considered, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>N</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:math> N\ge 3 . We show the existence and uniqueness of local mild solutions for arbitrary initial data and gravitational potential in scaling invariant Lorentz spaces. Although such a result has already been shown by Kozono, Miura, and Sugiyama ( Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid , J. Funct. Anal. 270 (2016), no. 5, 1663–1683), we reveal the precise regularities of mild solutions by showing the smoothing estimates of the heat semigroup on Lorentz spaces. The method is based on the real interpolation. In addition, we prove that the mild solutions exist globally in time, provided that the initial data are sufficiently small. Compared with the usual result, a part of the smallness conditions is reduced. We also obtain the asymptotic behavior of the global mild solutions. In the proof of the asymptotic behavior, to overcome a lack of density for the space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {L}^{\infty }\left({{\mathbb{R}}}^{N}) to which one <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:math> {c}_{0} of the initial data belongs, we show the decay of the global solutions without any approximation for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:math> {c}_{0} .