What question did this study set out to answer?

The aim is to provide a complete proof of the global existence and smoothness of the Navier–Stokes equations in three-dimensional space.

February 8, 2026Open Access

NAVIER–STOKES EXISTENCE AN SMOOTHNESS: Complete Proof

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Key Points

The aim is to provide a complete proof of the global existence and smoothness of the Navier–Stokes equations in three-dimensional space.
Constructed a structured proof framework organized into numbered modules or nodes.
Each node has explicit dependencies to facilitate verification without external context.
Included a consolidated bibliography for reference.
Successfully demonstrated the conditions under which the Navier–Stokes equations remain valid.
Established rigorous pathways to confirm the existence and smoothness of solutions in three dimensions.

Abstract

This document is a strict proof architecture for global existence and smoothness of the three-dimensional incompressible Navier–Stokes equations on R3. The argument is organized into numbered modules (“nodes”) with explicit dependencies and a claim register, so a reader can verify the chain node-by-node without relying on external narrative context. The snapshot is designed for immutable release fixing and includes a consolidated bibliography.

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Authors

Vojtěch Svěrák

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Institutions

Charles University

Czech Academy of Sciences

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NAVIER–STOKES EXISTENCE AN SMOOTHNESS: Complete Proof

Key Points

Abstract

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Discussion

Authors

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References and Citations

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