Global regularity for the Navier-Stokes equations with application to global solvability for the Euler equations

This paper establishes the global regularity of Leray-Hopf weak solutions to the $d$-dimensional Navier-Stokes equations for initial data in $H^s$ with $s \geq -1 + d/2$ by constructing a novel supercritical space and deriving viscosity-independent energy estimates through a re-scaling argument, thereby implying global solvability for the Euler equations.

The Gist

Imagine you are trying to predict how a giant, invisible fluid (like air or water) will move forever. In the world of physics, this is described by two famous sets of rules: the Navier-Stokes equations and the Euler equations.

Think of the Navier-Stokes equations as describing a fluid that has a little bit of "stickiness" or friction (viscosity), like honey or thick oil. The Euler equations describe a "perfect" fluid with absolutely no friction, like a ghost moving through space.

For decades, mathematicians have been stuck on a massive puzzle: Can we guarantee that these fluids will keep moving smoothly forever, or will they suddenly explode into chaos (a "singularity")?

This paper by Myong-Hwan Ri claims to solve this puzzle for fluids in 3D (and higher dimensions), provided the fluid starts off in a certain "smooth enough" condition. Here is how the author did it, explained through simple analogies.

1. The Problem: The "Friction" Trap

Usually, when mathematicians try to prove that a fluid won't explode, they rely on the fluid's friction (viscosity) to smooth things out. It's like using a brake pedal to stop a car from crashing.

• The issue: If you want to use these results to understand the "perfect" fluid (Euler equations), you have to imagine the friction disappearing completely (turning the brake pedal off).
• The danger: If your proof depends on the brake pedal working, it falls apart the moment you remove it. The author needed a way to prove the fluid stays smooth even if the friction is tiny or zero.

2. The Solution: A New "Safety Net"

The author invented a new mathematical "safety net" (called a supercritical space) to catch the fluid's energy before it gets too wild.

• The Old Net: Previous nets were too tight. They only caught the fluid if it was already very calm. If the fluid got a little rowdy, the net snapped.
• The New Net: The author built a net with a very specific, strange pattern. Imagine a fishing net where the holes are mostly tiny, but every now and then, there is a massive, gaping hole.
  • This net is designed to catch the "high-frequency" ripples (the tiny, fast vibrations in the fluid).
  • The "gaping holes" are placed so cleverly that they don't let the dangerous energy escape, but they are loose enough to let the math work even when the friction (viscosity) is almost zero.

3. The Trick: The "Zoom and Shrink" Camera

To prove this new net works, the author used a clever camera trick called re-scaling.

• Imagine you are watching a stormy ocean. It looks chaotic and huge.
• The author says, "Let's zoom in on a tiny drop of water and shrink the whole ocean down to the size of a bathtub."
• When you do this mathematically, the "friction" of the water changes. By zooming in enough, the author showed that the fluid's behavior becomes so predictable that it fits inside the new safety net.
• Because the net works in this "shrunken" world, and the math rules are the same, it proves the fluid is safe in the "real" world too, regardless of how much friction it has.

4. The Result: No More Explosions

By using this new net and the zooming trick, the author proved:

1. For Sticky Fluids (Navier-Stokes): If the fluid starts smooth enough, it will stay smooth forever. It will never explode into chaos.
2. For Perfect Fluids (Euler): Because the proof didn't rely on the friction being strong, it works even when the friction is zero. This means we can now guarantee that perfect fluids also stay smooth forever, provided they start in the right condition.

Summary

Think of the fluid as a wild horse.

• Old Math: "We can keep the horse calm if we have a strong rope (friction). But if the rope breaks, we don't know what happens."
• This Paper: "We built a magical fence (the supercritical space) that holds the horse calm even if the rope is cut. We proved this by imagining the horse shrunk down to the size of a mouse, where it's easier to see that it won't run wild."

The Bottom Line: The author has shown that for a wide range of starting conditions, these fluids will never suddenly break down or explode. They will flow smoothly for all time, whether they are sticky or perfectly frictionless.

Technical Summary

Technical Summary: Global Regularity for the Navier-Stokes Equations with Application to Global Solvability for the Euler Equations

Problem Statement
The paper addresses the long-standing problem of global regularity for the $d$-dimensional incompressible Navier-Stokes equations (where $d \ge 3$). Specifically, it investigates whether Leray-Hopf weak solutions, which are known to exist globally in time, remain smooth (regular) for all time when the initial data $u_0$ belongs to the Sobolev space $H^s(\mathbb{R}^d)$ with $s \ge -1 + d/2$. While global existence of weak solutions is established, their uniqueness and regularity in dimensions $d \ge 3$ remain open. Furthermore, the paper seeks to apply these regularity results to the vanishing viscosity limit to establish global existence for the incompressible Euler equations under similar initial data conditions.

Methodology
The author employs a novel approach centered on the construction of a specific "supercritical" function space, denoted as $X_1$, and utilizes a superposition method of energy norms for high-frequency components of the solution.

1. Construction of a Supercritical Space ($X_1$):
   The core innovation is the definition of a space $X_1$ that is slightly weaker than the critical homogeneous Sobolev space $\dot{H}^{-1+d/2}(\mathbb{R}^d)$. The norm in $X_1$ incorporates a "very sparse inverse logarithmic weight" in the frequency domain. Specifically, the space is defined via Littlewood-Paley decomposition $\Delta_j$, where the norm involves a sequence $a(j)$ that grows logarithmically only on a sparse set of indices (specifically where $j = i + 2^{2k}$). This sparsity is crucial for satisfying specific averaging conditions required for the estimates.

2. High-Frequency Energy Estimates:
   Instead of treating the convection term $(u \cdot \nabla)u$ merely as a quadratic nonlinearity, the proof utilizes the cancellation property $((u \cdot \nabla)u_k, u_k) = 0$ by testing the momentum equation against high-frequency parts $u_k$ (where $u_k$ represents frequencies $|\xi| \ge k$). This yields a differential inequality for the $L^2$ norm of $u_k$. The author derives an estimate where the growth of high-frequency energy is controlled by the norm of the solution in the supercritical space $X_1$ and a factor $c(k)$ dependent on the topology of $X_1$.

3. Superposition and Rescaling:
   The author sums these high-frequency estimates over all $k \in \mathbb{N}$. A key technical lemma (Lemma 3.1 and 3.2) demonstrates that the sequence of coefficients $b(j)$ arising from the summation satisfies an averaging condition ($\sum_{k=1}^n c(k) \lesssim n$). This allows the superposition of the energy norms to recover estimates for the critical and subcritical norms.
   Crucially, the proof employs a rescaling argument. By rescaling the solution $u_\lambda(t, x) = \lambda u(\lambda^2 t, \lambda x)$, the author shows that for sufficiently large scaling parameters $\lambda$, the norm $\|u_\lambda\|_{X_1}$ becomes uniformly small. This smallness allows the absorption of the nonlinear term into the dissipative term in the energy inequality, independent of the viscosity coefficient $\nu$.

Key Results
The paper establishes the following main theorems:

• Theorem 1.1 (Global Regularity for Navier-Stokes): For initial data $u_0 \in H^s(\mathbb{R}^d)$ with $d \ge 3$ and $s \ge -1 + d/2$, any Leray-Hopf weak solution to the Navier-Stokes equations belongs to $L^\infty(0, \infty; H^s(\mathbb{R}^d)) \cap L^2(0, \infty; H^{s+1}(\mathbb{R}^d))$. The solution is globally regular and unique.

  • Significance of the Estimate: The derived energy estimate (1.3) is uniform with respect to the viscosity coefficient $\nu$. This independence is a pivotal feature of the proof.

• Theorem 4.2 (Global Solvability for Euler Equations): As a direct corollary of the uniform regularity estimates for the Navier-Stokes equations, the paper proves the global existence of a weak solution to the incompressible Euler equations for initial data $u_0 \in H^s(\mathbb{R}^d)$ with $s \ge -1 + d/2$.

  • Uniqueness: The solution is unique if $s > 1 + d/2$.
  • Method: The result is obtained via the vanishing viscosity limit, relying on the compactness of the sequence of Navier-Stokes solutions and the uniform bounds established in Theorem 1.1.

Significance and Claims
The paper claims that the primary contribution lies in circumventing the traditional limitations of regularity criteria, which often require smallness conditions on critical norms or additional scaling-invariant constraints on the solution itself. By constructing the specific supercritical space $X_1$ with its sparse logarithmic weights, the author demonstrates that the "averaging" of high-frequency energy dissipation is sufficient to control the nonlinearity globally.

The author emphasizes that the independence of the regularity constants from the viscosity coefficient $\nu$ is the critical factor enabling the transition to the Euler equations. This provides a pathway to global existence for the Euler equations in dimensions $d \ge 3$ for initial data in $H^s$ with $s \ge -1 + d/2$, a result that the paper notes has not been generally established under such regularity assumptions in previous literature. The work extends previous results on critical spaces (such as $\dot{B}^{-1}_{\infty, \infty}$) by providing a global regularity result for a broader class of initial data without requiring the solution to remain in a critical space for all time, but rather by proving the solution stays in the subcritical space $H^s$.