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Finite-Time Blowup Demonstrated for Averaged 3D Navier-Stokes Equation
Key points
- Paper submitted to J. Amer. Math. Soc. on Navier-Stokes blowup.
- Constructs a modified 3D Navier-Stokes equation with finite-time blowup.
- Modified equation maintains function space estimates and energy identity.
- First 3D Navier-Stokes type equation with energy identity to show blowup.
New Research on Navier-Stokes Equations
A new paper titled "Finite time blowup for an averaged three-dimensional Navier-Stokes equation" has been uploaded to arXiv and submitted to J. Amer. Math. Soc. This work addresses the global regularity problem for the Navier-Stokes equation, a significant challenge in mathematics and physics.
Formalizing the Supercriticality Barrier
The paper's main purpose is to formalize the "supercriticality barrier." This concept suggests that global regularity cannot be established through abstract approaches relying solely on upper bound function space estimates for the nonlinear part of the equation combined with the energy identity. The research constructs a modification of the Navier-Stokes equations to demonstrate this barrier.
Demonstrating Finite-Time Blowup
The modified Navier-Stokes equation features a nonlinearity that adheres to essential function space estimates and the energy identity, similar to the true Navier-Stokes nonlinearity. However, for this modified equation, solutions can be constructed that exhibit finite-time blowup. This is a novel result for a three-dimensional Navier-Stokes type equation that also preserves the energy identity.
Implications for True Navier-Stokes Equations
Previous blowup results for variants of the Navier-Stokes equation either lacked the energy identity or were for higher-dimensional dyadic analogues. The method of proof used in this paper hints at a possible route for establishing blowup for the true Navier-Stokes equations, potentially for a very small set of initial data.
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Primary sources
GitHub WordPress/WordPress arXiv 1402.0290 arXiv 1502.01206 arXiv 1509.08766 arXiv 1409.3093 arXiv 1204.5444Reporting from
A new paper demonstrates finite-time blowup for a modified three-dimensional Navier-Stokes equation that retains key properties of the original, including the energy identity. This research formalizes the "supercriticality barrier" for the global regularity problem and suggests a potential path for proving blowup in the true Navier-Stokes equations.