Uniqueness in Lorentz Spaces of the 2d Navier-Stokes equation

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 $L^{\mathrm{\infty }}$ smoothing property at every restart time, namely $$\underset{\delta \downarrow 0}{\lim }\underset{t\in (T_0,T_0+\delta ]}{\sup }\sqrt{t-T_0}\mathrm{\parallel }v(t){\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathbb{T}}^2)}=0,\text{for all }{T}_0\in [0,T)\mathrm{.}$$ The proof combines the restart mild formulation, the $L^1$ bound for the periodic Oseen kernel of $e^{t\mathrm{\Delta }}\mathbb{P}\mathrm{\nabla }\cdot$, 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.