Finite time blowup for an averaged three-dimensional Navier-Stokes equation (1402.0290v3)

Abstract: The Navier-Stokes equation on the Euclidean space $\mathbf{R}^3$ can be expressed in the form $\partial_t u = \Delta u + B(u,u)$, where $B$ is a certain bilinear operator on divergence-free vector fields $u$ obeying the cancellation property $\langle B(u,u), u\rangle=0$ (which is equivalent to the energy identity for the Navier-Stokes equation). In this paper, we consider a modification $\partial_t u = \Delta u + \tilde B(u,u)$ of this equation, where $\tilde B$ is an averaged version of the bilinear operator $B$ (where the average involves rotations and Fourier multipliers of order zero), and which also obeys the cancellation condition $\langle \tilde B(u,u), u \rangle = 0$ (so that it obeys the usual energy identity). By analysing a system of ODE related to (but more complicated than) a dyadic Navier-Stokes model of Katz and Pavlovic, we construct an example of a smooth solution to such a averaged Navier-Stokes equation which blows up in finite time. This demonstrates that any attempt to positively resolve the Navier-Stokes global regularity problem in three dimensions has to use finer structure on the nonlinear portion $B(u,u)$ of the equation than is provided by harmonic analysis estimates and the energy identity. We also propose a program for adapting these blowup results to the true Navier-Stokes equations.

• The paper demonstrates a finite time blowup for an averaged version of the three-dimensional Navier-Stokes equations, proposed by Terence Tao, to explore barriers in solving the global regularity problem.
• It uses a modified averaged bilinear operator that maintains the energy identity but reveals potential singularities not apparent with standard harmonic analysis methods.
• The results suggest that methods based solely on harmonic analysis and the energy identity are insufficient for solving the Navier-Stokes global regularity problem, indicating a need for new approaches leveraging specific equation structures.

Finite Time Blowup for an Averaged Three-Dimensional Navier-Stokes Equation

The paper explores the longstanding Navier-Stokes global regularity problem by studying an averaged version of the Navier-Stokes equations. Proposed by Terence Tao, the approach utilizes a novel modified equation to demonstrate finite time blowup, representing a significant theoretical understanding of the barriers encountered in resolving the regularity problem. This work highlights that techniques relying solely on harmonic analysis and the energy identity are insufficient for solving the global regularity problem, thus providing critical insights into the requirements of a successful resolution.

Mathematical Framework and Problem Setup

The classical Navier-Stokes equations are defined on Euclidean space $\mathbb{R}^3$ as $\partial_t u = \Delta u + B(u,u)$, where $u$ is a divergence-free vector field, $\Delta$ is the Laplacian, and $B$ represents a bilinear operator embodying the nonlinearities in fluid dynamics. The core objective is to determine whether solutions to these equations remain smooth for all time or if singularities can develop, which is known as the global regularity problem.

Key Construction and Strategies

The paper introduces a systemic modification $\partial_t u = \Delta u + \tilde{B}(u,u)$, where $\tilde{B}$ is an averaged bilinear operator of the original $B$. Here, $\tilde{B}$ involves rotations and order-zero Fourier multipliers designed to obey the same cancellation condition $\langle \tilde{B}(u,u), u \rangle = 0$, maintaining the energy identity. The averaging effectively makes $\tilde{B}$ not stronger from a harmonic analysis perspective but sufficiently generalized to reveal potential singularities otherwise hidden in the original problem setup.

Finite Time Blowup Construct

By analyzing systems akin to a dyadic Navier-Stokes model, a specific construction demonstrates a smooth solution showing finite time blowup. The paper establishes an instance where the solution loses regularity abruptly, illustrating the failure to maintain global regularity under the averaged system. This approach suggests that existing harmonic analytical methods and use of energy identity, though effective in other contexts, fall short in ruling out this blowup.

Theoretical Implications

The results indicate that any solution to the Navier-Stokes regularity problem will need to leverage more specific structures of the equation’s nonlinearity beyond these generalized harmonic analytical estimates or cancellation properties. While blowup in an averaged context does not directly refute the global regularity of the true Navier-Stokes equations, it underlines key limitations and presents concrete challenges that require addressing the finer, potentially undiscovered elements of the equation.

Future Directions and Conclusion

Tao proposes further probing into the Navier-Stokes structure that may provide insights overlooked by current mathematical frameworks. The analysis highlights a crucial gap in understanding supercritical regimes and encourages investigations into modelling causalities for systems exhibiting such blowup, potentially by examining certain intricate symmetries or conservation laws.

Understanding finite time blowups in averaged variants is a step towards addressing global regularity comprehensively, as it identifies what specific tools must be developed or improved in order to achieve a robust resolution of the Navier-Stokes problem. This paper, by describing a failure mode in the averaged context, advances constructive paths that intertwine rigorous mathematical deduction with insightful explorative constructs in fluid dynamics research.