Onsager critical solutions of the forced Navier-Stokes equations

We answer positively to [BDL22, Question 2.4] by building new examples of solutions to the forced 3d-Navier-Stokes equations with vanishing viscosity, which exhibit anomalous dissipation and which enjoy uniform bounds in the space $L_t^3{C}_x^{1\mathrm{/}3-\epsilon }$, for any fixed $\epsilon >0$. Our construction combines ideas of [BDL22] and [CCS22].