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

TL;DR

This paper establishes a sharp non-uniqueness result for weak solutions of the Navier-Stokes equations in three-dimensional space, extending previous results from torus to the whole space and highlighting the limits of classical criteria.

Contribution

It introduces a new iterative scheme that demonstrates non-uniqueness of weak solutions in L^p spaces, extending sharp non-uniqueness results to the entire space.

Findings

Non-uniqueness in L^p([0,T];L^ Infty(\ ^3)) for 1 ≤ p < 2

Extension of sharp non-uniqueness from torus to R^3

Development of a novel iterative scheme balancing Reynolds stress and solution support

Abstract

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.