Global regularity of Leray-Hopf weak solutions to 3D Navier-Stokes equations

Abstract: We show that any Leray-Hopf weak solution to 3D Navier-Stokes equations with initial values u0 2 H1=2(R3) belong to L1(0; 1; H1=2(R3)) and thus it is regular. For the proof, flrst, we construct a supercritical space, the norm of which is compared to the homogeneous Sobolev H_ 1=2-norm in that it has inverse logarithmic weight very sparsely in the frequency domain. Then we obtain the energy estimates of high frequency parts of the solution which involve the supercritical norm on the right-hand side. Finally, we superpose the energy norm of high frequency parts of the solution to get estimates of the critical norms of the weak solution via the re-scaling argument.

• The paper proves that Leray–Hopf weak solutions starting from H1/2(R3) data remain globally regular without imposing smallness conditions.
• It introduces a novel supercritical frequency-weighted space, X1, to effectively control high-frequency components of the solution.
• The approach leverages high-frequency energy estimates and rescaling arguments to close energy inequalities globally in time.

Global Regularity of Leray-Hopf Weak Solutions to 3D Navier-Stokes Equations

Introduction and Context

The global regularity of solutions to the three-dimensional Navier-Stokes equations remains a central open problem in mathematical fluid dynamics. The existence of global-in-time weak solutions (Leray-Hopf solutions) is classical, but their regularity and uniqueness are unresolved for general initial data. This work addresses the regularity of Leray-Hopf weak solutions with initial data in the critical Sobolev space $H^{1/2}(\mathbb{R}^3)$, establishing that such solutions are globally regular and remain in $L^\infty(0,\infty; H^{1/2}(\mathbb{R}^3))$.

Main Result

The principal theorem asserts that if the initial velocity $u_0$ belongs to $H^{1/2}(\mathbb{R}^3)$, then the corresponding Leray-Hopf weak solution $u$ satisfies

$$u \in L^\infty(0,\infty; H^{1/2}(\mathbb{R}^3)) \cap L^2(0,\infty; H^{3/2}(\mathbb{R}^3)),$$

with the quantitative estimate

$$\|u(t)\|_{H^{1/2}}^2 + \nu \int_0^t \|u(\tau)\|_{H^{3/2}}^2\,d\tau \leq 2\|u_0\|_{H^{1/2}}^2, \quad \forall t > 0.$$

This result implies global regularity for such initial data, resolving the regularity question in this critical setting.

Methodological Innovations

Supercritical Frequency-Weighted Spaces

A key technical innovation is the construction of a supercritical function space $X_1$, defined via a frequency-weighted norm that is strictly weaker than the critical $\dot{H}^{1/2}$ norm but remains close to it. The $X_1$ norm is given by

$$\|v\|_{X_1} = \left( \sum_{j \in \mathbb{Z}} \left[ 2^{j/2} a^{-1}(j) \|\Delta_j v\|_2 \right]^2 \right)^{1/2},$$

where $a(j)$ is a sparsely growing sequence (inverse logarithmic weight) in the frequency domain. This construction allows for the control of high-frequency components of the solution in a manner that is almost critical, yet admits the necessary averaging properties for the subsequent energy estimates.

High-Frequency Energy Estimates and Superposition

The proof leverages the cancellation property of the nonlinear term in the Navier-Stokes equations, specifically when testing the momentum equation with high-frequency projections $u^k$. The analysis yields differential inequalities for the $L^2$ norm of $u^k$ and its gradient, with the right-hand side involving the $X_1$ norm. By summing over all high-frequency bands and exploiting the structure of $X_1$, the author derives a global-in-time bound for the critical $\dot{H}^{1/2}$ norm.

Rescaling and Uniform Smallness

A crucial step is the use of rescaling arguments to ensure the uniform smallness of the $X_1$ norm over time intervals. The scaling properties of the Navier-Stokes equations and the $X_1$ norm are carefully analyzed, showing that for sufficiently large rescaling factors, the $X_1$ norm of the rescaled solution can be made arbitrarily small. This enables the application of Gronwall-type arguments to close the energy estimates globally in time.

Technical Highlights

• The frequency decomposition is performed using dyadic blocks $\Delta_j$, and the equivalence of the $\dot{H}^s$ norm with the $\ell_2$ sum of $2^{sj}\|\Delta_j u\|_2$ is exploited.
• The sequence $a(j)$ is constructed to be $1$ except on a sparse set of indices where it grows logarithmically, ensuring that $X_1$ is supercritical but only marginally so.
• The energy inequality for high-frequency components is derived using the cancellation property $((u \cdot \nabla) u^k, u^k) = 0$.
• The summation over frequency bands is handled with careful combinatorial estimates, ensuring that the cumulative effect of the supercritical weights does not destroy the control over the critical norm.
• The rescaling argument is formalized via explicit estimates on the $X_1$ norm of rescaled functions, showing that for any $\varepsilon > 0$, there exists a scaling such that the $X_1$ norm is less than $\varepsilon$.

Implications and Discussion

Theoretical Implications

This result provides a positive answer to the global regularity problem for Leray-Hopf weak solutions with initial data in $H^{1/2}(\mathbb{R}^3)$. The approach demonstrates that the criticality barrier can be overcome by exploiting the structure of the nonlinearity and by working in a carefully constructed supercritical space. The method circumvents the need for smallness conditions on the initial data or additional regularity assumptions, which are common in previous works.

Comparison with Prior Work

Previous regularity criteria often required the solution to remain bounded in $L^3$ or in critical Besov spaces, or imposed smallness conditions. The present work extends the regularity theory to the endpoint Sobolev space $H^{1/2}$, which is critical for the Navier-Stokes scaling. The use of supercritical spaces with sparse frequency weights is novel and provides a new avenue for addressing regularity in borderline cases.

Practical Implications

While the result is primarily theoretical, it has implications for numerical simulations and the analysis of turbulent flows. The identification of $H^{1/2}$ as a sufficient space for global regularity suggests that numerical schemes preserving this regularity may be robust against singularity formation. Moreover, the techniques developed may inform the design of regularization strategies in computational fluid dynamics.

Future Directions

The methodology introduced may be applicable to other critical or supercritical PDEs, particularly those with similar scaling and cancellation properties. Further research could explore the extension of these techniques to bounded domains, to the case of nonzero external forces, or to other fluid models. Additionally, the sharpness of the $H^{1/2}$ threshold and the potential for further relaxation of regularity assumptions warrant investigation.

Conclusion

This work establishes the global regularity of Leray-Hopf weak solutions to the 3D Navier-Stokes equations for initial data in $H^{1/2}(\mathbb{R}^3)$. The proof combines a novel supercritical frequency-weighted space, high-frequency energy estimates, and rescaling arguments to control the critical norm globally in time. The result advances the understanding of regularity in critical spaces and introduces techniques with potential applicability to a broader class of nonlinear PDEs.