Numerical investigations of non-uniqueness for the Navier-Stokes initial value problem in borderline spaces

TL;DR

This work studies potential non-uniqueness of the Navier–Stokes Cauchy problem in borderline spaces by numerically analyzing scale-invariant, axi-symmetric data and their self-similar profiles $u(t,x)=t^{-1/2}U(x/t^{1/2})$. Using axisymmetric reduction, continuation in the scaling parameter $\sigma$, spectral analysis of the linearization $\mathcal{L}(U_\sigma)$, and bifurcation theory, the authors identify a real eigenvalue crossing at $\sigma_0\approx 292$ that triggers a supercritical pitchfork bifurcation, producing symmetry-broken self-similar branches. The results imply non-uniqueness for large data in critical-like spaces (e.g., $L^{3,\infty}$ and ${\mathrm{BMO}}^{-1}$) and motivate localization of scale-invariant solutions to finite-energy Leray–Hopf solutions, suggesting near-optimal local well-posedness limits. The study also lays groundwork for computer-assisted proofs and highlights the intimate link between spectral crossings and non-unique Navier–Stokes evolutions in borderline function spaces.

Abstract

We consider the Cauchy problem for the incompressible Navier-Stokes equations in $\mathbb{R}^3$ for a one-parameter family of explicit scale-invariant axi-symmetric initial data, which is smooth away from the origin and invariant under the reflection with respect to the $xy$-plane. Working in the class of axi-symmetric fields, we calculate numerically scale-invariant solutions of the Cauchy problem in terms of their profile functions, which are smooth. The solutions are necessarily unique for small data, but for large data we observe a breaking of the reflection symmetry of the initial data through a pitchfork-type bifurcation. By a variation of previous results by Jia & Šverák (2013) it is known rigorously that if the behavior seen here numerically can be proved, optimal non-uniqueness examples for the Cauchy problem can be established, and two different solutions can exists for the same initial datum which is divergence-free, smooth away from the origin, compactly supported, and locally $(-1)$-homogeneous near the origin. In particular, assuming our (finite-dimensional) numerics represents faithfully the behavior of the full (infinite-dimensional) system, the problem of uniqueness of the Leray-Hopf solutions (with non-smooth initial data) has a negative answer and, in addition, the perturbative arguments such those by Kato (1984) and Koch & Tataru (2001), or the weak-strong uniqueness results by Leray, Prodi, Serrin, Ladyzhenskaya and others, already give essentially optimal results. There are no singularities involved in the numerics, as we work only with smooth profile functions. It is conceivable that our calculations could be upgraded to a computer-assisted proof, although this would involve a substantial amount of additional work and calculations, including a much more detailed analysis of the asymptotic expansions of the solutions at large distances.

Figures (19)

• Figure 1: Different subclasses of the space $L^{p}(0,T;L^{q}(\mathbb{R}^{3}))$. The Leray–Hopf solutions belong to the blue region characterized by \ref{['region-leray']}. The Serrin criterion for uniqueness and regularity is given by the green region defined by \ref{['region-serrin']}. The corrector used in the proof of localization to localize a self-similar solution belongs to the space $X_{T}$ defined by \ref{['def-X_T']}, hence to the red region characterized by \ref{['reqion-w']}. The localization of the numerical solutions found belong to the complement of the green region, hence showing that the Serrin uniqueness criterion is sharp in these spaces.
• Figure 2: Construction of the discretization $\Omega(R,n)$ for $R=20$ and $n=8$. First the domain $\Omega(R)$ is discretized into $2n^{2}$ squares and then refined near the origin.
• Figure 3: Azimuthal component of the numerical solution $\boldsymbol{U}_{\!\sigma}$ multiplied by $|\boldsymbol{x}|$ on the whole computational domain $\Omega(R_{\sigma},600)$ for various $\sigma$. One can see that the choice of $\varkappa_{\sigma}$ is made such that the solution remains $(-1)$-homogeneous in most of the computational domain except near the origin.
• Figure 4: Norm of the radial and vertical component of the numerical solution $\boldsymbol{U}_{\!\sigma}$ multiplied by $|\boldsymbol{x}|$. As expect since the boundary condition $\sigma\boldsymbol{a}_{0}$ is pure swirl, these two components decays like $|\boldsymbol{x}|^{-3}$.
• Figure 5: Decay of the function $\Gamma(s)=\sigma^{-1}\sup_{|\boldsymbol{x}|=\varkappa_{\sigma}s}|\boldsymbol{x}||\boldsymbol{U}_{\!\sigma}(\boldsymbol{x})|$ in term of $s\in(0,20)$ for various values of $\sigma$. The function $\Gamma(s)$ is almost flat for $s\geq10$, so this means that the computational domain is large enough, since the numerical solution is already $(-1)$-homogeneous in a large region.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Proposition 1
• proof
• proof : Proof of spectrum-L.
• proof : Proof of continuation-bifurcation.
• Theorem 5
• proof : Proof of localization.