Large forward discretely self-similar solutions to the two-dimensional Navier-Stokes equations with rough data

Abstract: We construct forward discretely self-similar (DSS) solutions of two-dimensional Navier-Stokes equations with arbitrarily large data in $L<sup>2_{loc}(\mathbb{R}<sup>2\setminus{0})$. For such rough initial data, we first establish a sharp linear estimate under the discrete scaling, which identifies the (L^2)-norm of the initial datum on a fundamental annulus with the Dirichlet energy of its caloric lift over one period in logarithmic time. This, together with a structural cancellation between the linear profile and the nonlinear remainder over one period in logarithmic time, yields a uniform (\dot{H}^1)-bound for the remainder, where the natural $L<sup>2_{loc}$-regularity of the initial data seems to be sharp. Since the two-dimensional Leray operator has no (L^2)-coercivity, we recover uniform energy control for the remainder by exploiting (L^p)-estimates ($1<p<2)$, together with a key bootstrap argument and periodicity. We then first construct smooth DSS solutions using further weighted estimates and a fixed-point argument as long as the initial data belong to $C<sup>2_{loc}(\mathbb{R}<sup>2\setminus{0})$. These stronger weighted bounds are not uniform under approximation of rough data. Instead, the passage to (L^2_{\rm loc}) data relies only on the unweighted energy estimates. Finally, we use strong time continuity of the limiting periodic profile to recover the uniform spatial tightness needed to verify the initial data.