We study uniqueness of mild solutions to the two--dimensional incompressible Navier-Stokes equations on the torus in borderline spatial classes. While Lorentz-space methods yield uniqueness in $C([0,T);L^{ 2,1}(\mathbb{ T}^2))$ via real interpolation and weak $L^2$ control, extending such arguments to larger Lorentz spaces $L^{ 2,q}$, $1<q<2$, encounters endpoint obstructions. In this paper we prove that uniqueness in $C([0,T);L^{ 2,q}(\mathbb{ T}^2))$ holds provided one assumes a short-time $L^\infty$ smoothing property at every restart time, namely \[ \lim_{ \delta\downarrow 0}\sup_{ t\in(T_0,T_0+\delta]}\sqrt{ t-T_0}\,\|v(t)\|_{ L^\infty(\mathbb{ T}^2)}=0, \quad \text{ for all } T_0\in[0,T). \] The proof combines the restart mild formulation, the $L^1$ bound for the periodic Oseen kernel of $e^{ t\Delta}\mathbb{ P}\nabla\cdot$, and an explicit Beta-function computation yielding a strict $L^2$ contraction on short intervals. The smoothing assumption is natural in Kato and Koch-Tataru type critical well-posedness frameworks and clarifies how parabolic regularization can replace Lorentz endpoint structure in uniqueness arguments.
Categories: math.AP