Sharp non-uniqueness for the Navier-Stokes equations in R^3

In this paper, we prove a sharp and strong non-uniqueness for a class of weak solutions to the incompressible Navier-Stokes equations in $\R^3$. To be more precise, we exhibit the non-uniqueness result in a strong sense, that is, any weak solution is non-unique in L^p([0,T];L^\infty(\R^3)) with 1\le p<2. Moreover, this non-uniqueness result is sharp with regard to the classical Ladyzhenskaya-Prodi-Serrin criteria at endpoint (2, \infty), which extends the sharp nonuniqueness for the Navier-Stokes equations on torus $\TTT^3$ in the recent groundbreaking work (Cheskidov and Luo, Invent. Math., 229 (2022), pp. 987-1054) to the setting of the whole space. The key ingredient is developing a new iterative scheme that balances the compact support of the Reynolds stress error with the non-compact support of the solution via introducing incompressible perturbation fluid.