A very short proof of a new energy identity for strong solutions to the Navier-Stokes system in 3D
Abstract
We prove that strong solutions to the incompressible, homogeneous Navier-Stokes system on ℝ3 are global in time for initial data in the Sobolev space H1σ(ℝ3). The central ingredient of the proof is a new energy estimate that rules out finite-time blow-up in Lr(ℝ3) for r = 2 + 2 / √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, ·) belongs to L2σ(ℝ3) ∩ L3(ℝ3). Even for u(0, ·) ∈ L2σ(ℝ3), the only possible non-uniqueness in the class of Leray-Hopf weak solutions is initial branching, and every such solution is C∞-smooth on ( 0, ∞ ) × ℝ3. Moreover, if u(0, ·) is smooth with derivatives of all orders decaying rapidly at infinity, we show that u is in fact C∞-smooth on all of [ 0, ∞ ) × ℝ3.
Related
- Local and Global Existence of Strong Solutions to the One-Dimensional Compressible Navier--Stokes Equations with General Pressure Law
- Global strong solutions of the three-dimensional Cahn--Hilliard--Navier--Stokes system via finite-dimensional scalar forcing
- Global Existence of Strong and Weak Solutions to the 2D Compressible Navier-Stokes-Poisson System in Bounded Domains with Large Data and Vacuum
- Global existence of smooth solutions to the Navier-Stokes equations for initial data with infinite energy
- Global strong solutions to the 1D full compressible Navier--Stokes equations with temperature-dependent coefficients and vacuum
BibTeX
@misc{ruf2026very,
title = {{A very short proof of a new energy identity for strong solutions to the Navier-Stokes system in 3D}},
author = {Thomas Ruf},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18808},
eprint = {2609.18808},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18808},
}
RIS
TY - UNPB
TI - A very short proof of a new energy identity for strong solutions to the Navier-Stokes system in 3D
AU - Thomas Ruf
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.18808
UR - https://arxiv.org/abs/2609.18808
AB - We prove that strong solutions to the incompressible, homogeneous Navier-Stokes system on ℝ^3 are global in time for initial data in the Sobolev space H^1_σ(ℝ^3). The central ingredient of the proof is a new energy estimate that rules out finite-time blow-up in L^r(ℝ^3) for r = 2 + 2 / √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, ·) belongs to L^2_σ(ℝ^3) ∩ L^3(ℝ^3). Even for u(0, ·) ∈ L^2_σ(ℝ^3), the only possible non-uniqueness in the class of Leray-Hopf weak solutions is initial branching, and every such solution is C^∞-smooth on ( 0, ∞ ) × ℝ^3. Moreover, if u(0, ·) is smooth with derivatives of all orders decaying rapidly at infinity, we show that u is in fact C^∞-smooth on all of [ 0, ∞ ) × ℝ^3.
ER -