Pure-swirl loss of boundedness under $L^1_tL^2_x$ forcing: exact mixed-norm ranges

Technical Summary: Pure-Swirl Loss of Boundedness Under $L^1_t L^2_x$ Forcing

Problem Statement
The paper addresses the regularity and boundedness properties of solutions to the forced incompressible Navier–Stokes equations in a three-dimensional circular cylinder $D = B_2(0, 1) \times (0, 1)$. Specifically, it investigates whether a classical solution can lose $L^\infty$ boundedness (blow up) at a finite time $T$ while the external force $f$ remains within specific mixed-norm spaces, particularly the energy-class space $L^1(0, T; L^2(D))$. This construction aims to isolate the effect of external forcing on blow-up mechanisms without relying on nonlinear amplification, as the convective term is absorbed by the pressure in the specific ansatz used.

Methodology
The authors construct an explicit "pure-swirl" solution, where the velocity field is purely azimuthal: $v(x, t) = W(r, t)e_\theta$. In this configuration, the nonlinear convection term $(v \cdot \nabla)v$ has no azimuthal component, and its radial component is exactly balanced by the pressure gradient. Consequently, the azimuthal component $W(r, t)$ satisfies a scalar parabolic equation driven by the azimuthal component of the force $F(r, t)$:
$$\partial_t W - \left(\partial_r^2 + \frac{1}{r}\partial_r - \frac{1}{r^2}\right)W = F.$$
The construction refines a previous profile by Zhang [23] by introducing an annular cancellation. The authors define a profile $\phi_0(\rho)$ based on a smooth, compactly supported function $A(\rho)$ on an annulus $(a, b)$. This specific choice ensures that the profile vanishes near the symmetry axis ($r=0$) and behaves like $1/r$ at the outer edge of the support. This cancellation is crucial for maintaining smoothness at the symmetry axis when the solution is multiplied by a non-integer power $r^\alpha$.

The solution is defined as $W(r, t) = r^\alpha \phi(r, t) + \beta r$, where $\phi(r, t)$ is a self-similar rescaling of $\phi_0$ with a time-dependent scale $R(t) = \sqrt{2(T-t)}$. The force $f$ is derived directly from this ansatz.

Key Contributions and Results

1. Exact Mixed-Norm Ranges:
   The paper establishes the precise mixed-norm conditions for both the force $f$ and the velocity $v$.

   • Force: The force belongs to $L^q(0, T; L^p(D))$ if and only if $\alpha > 3 - \frac{2}{p} - \frac{2}{q}$.
   • Velocity: The velocity belongs to $L^\sigma(0, T; L^m(D))$ if and only if $\alpha > 1 - \frac{2}{m} - \frac{2}{\sigma}$.
   • Endpoint Achievement: By choosing parameters such that $1/p + 1/q > 1$, the construction yields a force in $L^1(0, T; L^2(D))$. This is a significant refinement over previous results, as it reaches the energy-class forcing threshold.

2. Blow-up Rates:
   As $t \uparrow T$, the solution exhibits the following asymptotic behaviors:

   • Velocity Supremum: $\|v(t)\|_{L^\infty(D)} \asymp (T-t)^{-(1+\alpha)/2} \to \infty$.
   • Enstrophy: $\|\nabla v(t)\|_{L^2(D)}^2 \asymp (T-t)^{\alpha-1}$.
   • Force Norm: $\|f(t)\|_{L^p(D)} \asymp (T-t)^{-(3-\alpha)/2 + 1/p}$.

3. Regularity and Uniqueness:

   • The solution is classical on $[0, T)$ and smooth for $t < T$.
   • The solution extends strongly in $L^2(D)$ to a unique Leray–Hopf solution at time $T$.
   • The constructed solution satisfies the energy equality on $[0, T]$.
   • The loss of boundedness occurs strictly at $t=T$, while the force becomes unbounded only at this terminal time.

4. Stokes System Applicability:
   Due to the cancellation of the convective term by the pressure, the same example applies to the forced Stokes system, demonstrating that the blow-up mechanism is driven entirely by the forcing term rather than nonlinear interactions.

Significance and Claims
The paper claims to provide a "direct, self-contained sharpening" of the $k=1$ case from the authors' previous weighted construction [2]. Its primary significance lies in:

• Refining the Endpoint: It corrects the mixed-norm condition for the $k=1$ case, successfully reaching the $L^1_t L^2_x$ forcing regime, which was not fully captured in the previous general hierarchy argument.
• Isolating Forcing Effects: By using the pure-swirl ansatz, the construction isolates the mechanism of loss of boundedness to the external force, proving that even with energy-class forcing ($L^1_t L^2_x$) and a solution that satisfies the energy equality and uniqueness in the Leray–Hopf class, $L^\infty$ boundedness can fail.
• Smoothness at the Axis: The introduction of the annular cancellation allows for the use of non-integer exponents $\alpha$ while preserving smoothness at the symmetry axis, a technical hurdle in previous profile constructions.

The authors explicitly note that this result does not address the Millennium Prize problem regarding the global regularity of unforced Navier–Stokes equations with smooth data, as the blow-up is driven by a singular forcing term. However, it provides a rigorous counterexample to the intuition that energy-class forcing guarantees bounded solutions.