Existence of smooth solutions of the Navier-Stokes equations in three-dimensional Euclidean space (2507.18063v1)

Abstract: Based on the essential connection of the parabolic inertia Lam\'{e} equations and Navier-Stokes equations, we prove the existence of smooth solutions of the incompressible Navier-Stokes equations in three-dimensional Euclidean space $\mathbb{R}^3$ by showing the existence and uniqueness of smooth solutions of the parabolic inertia Lam\'{e} equations and by letting a Lam\'{e} constant $\lambda$ tends to infinity (the other Lam\'{e} constant $\mu>0$ is fixed).

• The paper establishes global existence and uniqueness of smooth solutions to the 3D Navier-Stokes equations using a penalization method via the inertia Lamé equations.
• It leverages explicit kernel representations, uniform energy and Sobolev estimates, and compactness arguments to effectively control nonlinear terms.
• The approach enables practical numerical schemes and paves the way for future research on bounded domains and inviscid limits.

Existence of Smooth Solutions to the 3D Navier-Stokes Equations: A Rigorous Analytical Approach

Introduction and Context

The existence and smoothness of solutions to the three-dimensional incompressible Navier-Stokes equations (NSE) in $\mathbb{R}^3$ is a central open problem in mathematical fluid dynamics. The paper under review addresses this problem by establishing the global existence and uniqueness of smooth solutions for the Cauchy problem associated with the 3D incompressible NSE, given smooth, divergence-free initial data with suitable decay at infinity. The approach is based on a novel connection between the NSE and the parabolic inertia Lamé equations, leveraging the latter's well-posedness and analytic properties to construct solutions to the former via a limiting process.

Analytical Framework and Main Results

Problem Formulation

The incompressible NSE in $\mathbb{R}^3$ are given by: $$\begin{aligned} &\frac{\partial \mathbf{u}}{\partial t} - \mu \Delta \mathbf{u} + \nabla p + (\mathbf{u} \cdot \nabla) \mathbf{u} = 0, \ &\nabla \cdot \mathbf{u} = 0, \ &\mathbf{u}(\mathbf{x}, 0) = \boldsymbol{\phi}(\mathbf{x}), \end{aligned}$$ where $\mathbf{u}$ is the velocity field, $p$ is the pressure, $\mu > 0$ is the viscosity, and $\boldsymbol{\phi}$ is a smooth, divergence-free initial datum with rapid decay.

The main theorem asserts that for any such initial datum, there exists a unique global-in-time smooth solution $(\mathbf{u}, p)$ to the above system, with bounded energy for all $t \geq 0$.

Methodological Innovation: The Inertia Lamé Equations

The core of the analysis is the introduction of the parabolic inertia Lamé equations: $$\frac{\partial \mathbf{u}}{\partial t} - \mu \Delta \mathbf{u} - (\lambda + \mu) \nabla \nabla \cdot \mathbf{u} + (\mathbf{u} \cdot \nabla) \mathbf{u} = 0,$$ where $\lambda$ is a Lamé parameter. For fixed $\mu > 0$ and $\lambda + \mu \geq 0$, this system is a nonlinear parabolic system with strong ellipticity, for which global existence and uniqueness of smooth solutions can be established using semigroup theory, energy estimates, and contraction mapping arguments.

The key insight is that as $\lambda \to +\infty$, the divergence term enforces incompressibility in the limit, and the solutions to the inertia Lamé equations converge (in appropriate function spaces) to solutions of the incompressible NSE.

Technical Highlights

• Explicit Fundamental Solutions: The paper constructs explicit fundamental solutions for the linearized Lamé and Stokes systems using Fourier analysis and semigroup theory, providing precise kernel representations and sharp Gaussian-type bounds.
• Uniform Estimates: Uniform-in-$\lambda$ sup-norm and Sobolev norm estimates are derived for the solutions of the inertia Lamé equations, ensuring control over the nonlinear terms and precluding finite-time blowup.
• Compactness and Weak Convergence: By establishing uniform bounds in $L^\infty_t H^1_x \cap L^2_t H^2_x$ and time-derivative spaces, the author applies weak compactness arguments to extract convergent subsequences as $\lambda \to \infty$.
• Passage to the Limit: The limiting function is shown to satisfy the incompressible NSE in the sense of distributions, with strong regularity inherited from the uniform estimates and the regularity theory for parabolic systems.
• Energy Inequality: The limiting solution satisfies the classical energy inequality, ensuring boundedness of kinetic energy for all time.

Main Theorem (Paraphrased)

Given any smooth, divergence-free initial data $\boldsymbol{\phi} \in \mathscr{S}(\mathbb{R}^3)$, there exists a unique global smooth solution $(\mathbf{u}, p)$ to the 3D incompressible Navier-Stokes equations in $\mathbb{R}^3 \times [0, \infty)$, with

$$\sup_{t \geq 0} \int_{\mathbb{R}^3} |\mathbf{u}(\mathbf{x}, t)|^2 \, d\mathbf{x} < \infty,$$

and $$\mathbf{u}, p \in C^\infty(\mathbb{R}^3 \times [0, \infty))$$.

Detailed Analytical Pathway

Existence and Regularity for the Inertia Lamé System

• The inertia Lamé system is recast as an abstract evolution equation in Sobolev spaces, with the nonlinear term $(\mathbf{u} \cdot \nabla) \mathbf{u}$ treated via standard product estimates and Sobolev embeddings.
• The linear part generates an analytic semigroup, and the Duhamel formula is used to write the solution as a fixed point of a contraction mapping in a suitable Banach space.
• Uniform-in-$\lambda$ energy and sup-norm estimates are established using Gronwall-type inequalities and bootstrapping, ensuring global existence and smoothness.

Limiting Procedure and Recovery of the Navier-Stokes Solution

• For divergence-free initial data, the sequence of solutions $\{\mathbf{u}_{\lambda, \mu}\}$ is shown to be uniformly bounded in strong norms.
• Weak compactness yields a limit $\mathbf{u}_\mu$ as $\lambda \to \infty$, which is shown to be divergence-free and to satisfy the NSE with the same initial data.
• The pressure is recovered as the weak limit of the penalization term $$-(\lambda + \mu) \nabla \cdot \mathbf{u}_{\lambda, \mu}$$.
• Uniqueness follows from energy estimates and the linearity of the limiting process.

Regularity and Energy Bounds

• The limiting solution inherits $C^\infty$ regularity from the uniform bounds and the regularity theory for parabolic systems.
• The energy inequality is preserved in the limit, ensuring that the solution has bounded kinetic energy for all time.

Implications and Discussion

Theoretical Implications

• Resolution of the 3D NSE Regularity Problem: The paper claims to resolve the global regularity problem for the 3D incompressible NSE with smooth, rapidly decaying initial data, a result that stands in contrast to the prevailing consensus in the field.
• Methodological Novelty: The approach of using the inertia Lamé system as a penalization of the incompressibility constraint, and passing to the limit, provides a new analytical pathway that circumvents the direct difficulties associated with the pressure term and the nonlinear structure of the NSE.
• Generality: The method is robust and extends to periodic boundary conditions and to the viscous Burgers equations as a special case.

Practical and Computational Implications

• Numerical Schemes: The penalization approach via the inertia Lamé system suggests a practical computational method for approximating solutions to the NSE by solving a family of strongly parabolic systems with large penalization parameters.
• Stability and Convergence: The uniform estimates and limiting arguments provide a theoretical foundation for the stability and convergence of such numerical schemes.

Future Directions

• Extension to Other Geometries: The methodology may be adapted to bounded domains, exterior domains, or manifolds, provided appropriate uniform estimates can be established.
• Analysis of the Inviscid Limit: The paper remarks on the possibility of considering the Euler equations by taking the vanishing viscosity limit after the penalization limit, though this leads to additional analytical challenges due to loss of parabolicity.
• Further Investigation of Uniqueness and Regularity: While the paper establishes uniqueness in the class of smooth solutions, the behavior for rougher initial data or in weaker function spaces remains an open direction.

Conclusion

This work presents a comprehensive analytical framework for establishing the global existence and uniqueness of smooth solutions to the 3D incompressible Navier-Stokes equations in $\mathbb{R}^3$, based on a limiting procedure from the parabolic inertia Lamé equations. The approach is grounded in explicit kernel representations, uniform energy and sup-norm estimates, and compactness arguments. The results, if validated, would have significant implications for both the mathematical theory of fluid dynamics and the development of robust numerical methods for simulating incompressible flows. The methodology also opens new avenues for the analysis of related nonlinear parabolic systems and their singular limits.