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For initial data $f$ in a subcritical Lorentz space $L^{p,q}(\mathbb{R}^{n})\hookrightarrow \dot B^{-\frac np}_{\infty,\infty}(\mathbb{R}^n)$($n<p<\infty$, $1\leq q \leq \infty$), we prove results which imply inparticular that a local in time mild Navier-Stokes solution cannot becomeunbounded in the $L^{p,q}(\mathbb{R}^{n})$-norm before it becomes unbounded inthe norm of the larger subcritical Besov space $\dot B^{-\fracnp}_{\infty,\infty}(\mathbb{R}^n)$. In view of the known local theory in suchlarge Besov spaces, this can be thought of as a `propagation of regularity'type of result; here, we provide a self-contained local theory (includingscaling-appropriate `blow-up estimates', similar to those established by J.Leray in the Lebesgue setting) in the Lorentz setting, along with uniquenessresults which imply such propagation of regularity. Our existence results arebased on the method of T. Kato (1984) with Lorentz spaces replacing Lebesguespaces throughout, and are given without any reference to the Besov framework.The uniqueness results are similarly self-contained, extending certain Lebesguespace methods of Fabes-Jones-Riviere (1972) to the Lorentz setting. We alsoestablish certain continuity properties of the solutions which are constructed.(A more detailed abstract is provided in the article itself.)

Properties of Navier-Stokes mild solutions with initial data in subcritical Lorentz spaces