A very short proof of a new energy identity for strong solutions to the Navier-Stokes system in 3D

RESUMO

We prove that strong solutions to the incompressible, homogeneous Navier-Stokes system on $\mathbb{R}^3$ are global in time for initial data in the Sobolev space $H^1_\sigma(\mathbb{R}^3)$. The central ingredient of the proof is a new energy estimate that rules out finite-time blow-up in $L^r(\mathbb{R}^3)$ for $r = 2 + 2 / \sqrt{3}$. This estimate is based crucially on properties of the material derivative in combination with the divergence-free constraint, together with an intricate estimate that harnesses the delicate interplay between the viscous coercivity of the new energy functional and the control of a perturbing pressure term. Notably, the estimate does not carry over to systems that merely share the standard energy structure, such as T. Tao's averaged Navier-Stokes system. As a consequence of the new energy estimate, every Leray-Hopf weak solution $u$ to the homogeneous Navier-Stokes system is globally unique whenever the initial value $u(0, \cdot)$ belongs to $L^2_\sigma(\mathbb{R}^3) \cap L^3(\mathbb{R}^3)$. Even for $u(0, \cdot) \in L^2_\sigma(\mathbb{R}^3)$, the only possible non-uniqueness in the class of Leray-Hopf weak solutions is initial branching, and every such solution is $C^\infty$-smooth on $\left( 0, \infty \right) \times \mathbb{R}^3$. Moreover, if $u(0, \cdot)$ is smooth with derivatives of all orders decaying rapidly at infinity, we show that $u$ is in fact $C^\infty$-smooth on all of $\left[ 0, \infty \right) \times \mathbb{R}^3$.