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July 15, 2026Open Access

Global Regularity for the Three-Dimensional Incompressible Navier–Stokes Equations

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This randomized trial investigates local solvability in the Navier–Stokes equations, indicating new insights for mathematical fluid dynamics.

The study aims to analyze the global regularity of the three-dimensional incompressible Navier–Stokes equations using a dyadic Littlewood–Paley framework.
Developed local strong solvability for Hs when s > 5/2.
Established energy inequality and higher-order Sobolev differential bounds.
Introduced frequency-localized commutator and paraproduct estimates for controlling nonlinear interactions.
Confirmed local well-posedness in the Sobolev space, establishing a foundation for global continuation.
Utilized a shell defect envelope for organizing dyadic interactions effectively.
Demonstrated a limiting passage back to the classical Navier–Stokes system, supporting proposed regularity results.

Daphne Garrido (2026) studied this question.

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This randomized trial investigates local solvability in the Navier–Stokes equations, indicating new insights for mathematical fluid dynamics.