{"ID":23003505,"CreatedAt":"2026-09-17T04:20:54.235930923Z","UpdatedAt":"2026-09-17T04:20:54.235930923Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.18808","arxiv_id":"2609.18808","title":"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 $\\mathbb{R}^3$ are global in time for initial data in the Sobolev space $H^1_σ(\\mathbb{R}^3)$. 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 $L^2_σ(\\mathbb{R}^3) \\cap L^3(\\mathbb{R}^3)$. Even for $u(0, \\cdot) \\in L^2_σ(\\mathbb{R}^3)$, 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 $\\left[ 0, \\infty \\right) \\times \\mathbb{R}^3$.","short_abstract":"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 $H^1_σ(\\mathbb{R}^3)$. 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 /...","url_abs":"https://arxiv.org/abs/2609.18808","url_pdf":"https://arxiv.org/pdf/2609.18808v1","authors":"[\"Thomas Ruf\"]","published":"2026-09-16T15:20:32Z","proceeding":"math.AP","tasks":"[\"math.AP\",\"math-ph\"]","methods":"[]","has_code":false}