Abstract
<p style='text-indent:20px;'>This work concerns the initial value problem for the three dimensional compressible Navier-Stokes equations (both isentropic and polytropic). By exploiting the famous Fujita-Kato theorem to the Classical incompressible Navier-Stokes equations, we prove the existence of global-in-time unique solutions under as weak as possible smallness conditions in the scaling invariant spaces. In particular, our results improve the classical theorems obtained by Danchin [Invent. Math., <b>141</b>, 579–614, 2000] and Danchin [Arch. Ration. Mech. Anal., <b>160</b>, 1–39, 2001].</p>
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