[Paper Review] Are the incompressible 3d Navier-Stokes equations locally ill-posed in the natural energy space?

Hao Jia, Vladimír Šverák|arXiv (Cornell University)|Jun 10, 2013

Navier-Stokes equation solutions8 citations

TL;DR

This paper investigates the local ill-posedness of the 3D incompressible Navier-Stokes equations in the natural energy space by analyzing spectral properties of a linear operator associated with scale-invariant solutions. It shows that if the linearized operator has an eigenvalue with positive real part and suitable decay, non-uniqueness and ill-posedness emerge even for compactly supported, $L^2$-initial data at the borderline of classical perturbation theory, providing a spectral criterion for non-uniqueness of Leray-Hopf solutions.

ABSTRACT

An important open problem in the theory of the Navier-Stokes equations is the uniqueness of the Leray-Hopf weak solutions with $L^2$ initial data. In this paper we give sufficient conditions for non-uniqueness in terms of spectral properties of a natural linear operator associated to scale-invariant solutions recently constructed in \cite{JiaSverak}. If the spectral conditions are satisfied, non-uniqueness and ill-posedness can appear for quite benign compactly supported data, just at the borderline of applicability of the classical perturbation theory. The verification of the spectral conditions seems to be approachable by relatively straightforward numerical simulations which involve only smooth functions.

Motivation & Objective

• To determine whether the 3D incompressible Navier-Stokes equations are locally ill-posed in the natural energy space $L^2$.
• To establish sufficient spectral conditions on a linearized operator $σ$ for non-uniqueness and ill-posedness of solutions with $L^2$-initial data.
• To demonstrate that non-uniqueness can occur for compactly supported, smooth initial data at the threshold of classical perturbation theory.
• To link spectral properties of the linearized operator $σ$ to the existence of multiple Leray-Hopf weak solutions.
• To provide a framework for verifying non-uniqueness via numerical simulations of smooth functions in Banach spaces.

Proposed method

• The authors analyze scale-invariant solutions $u_\sigma(x,t) = \frac{1}{\sqrt{t}} U_\sigma(\frac{x}{\sqrt{t}})$ to the 3D Navier-Stokes equations with $-1$-homogeneous initial data $\sigma u_0$.
• They linearize the Navier-Stokes equation around $u_\sigma$, leading to a time-dependent equation for perturbations $\phi$ governed by the operator $\mathcal{L}_\sigma$.
• The time evolution is transformed to $\phi_t = \mathcal{L}_\sigma \phi$ on $(-\infty, 0)$, where $\mathcal{L}_\sigma$ is a linear operator involving $U_\sigma$, gradients, and pressure correction.
• The spectral properties of $\mathcal{L}_\sigma$ are analyzed in the Banach space $X = L^2 \cap L^4$ with divergence-free vector fields, with decay conditions at infinity.
• Non-uniqueness is linked to the existence of eigenvalues of $\mathcal{L}_\sigma$ with positive real part and $o(|x|^{-1})$ decay at infinity.
• The construction is localized to compactly supported initial data by decomposing the solution into a scale-invariant part and a perturbation, using a priori estimates and regularity theory.

Experimental results

Research questions

• RQ1 Under what spectral conditions on the linearized operator $\mathcal{L}_\sigma$ does the 3D Navier-Stokes system exhibit non-uniqueness and ill-posedness in the natural energy space?
• RQ2 Can non-uniqueness of Leray-Hopf weak solutions arise for compactly supported, $L^2$-initial data at the threshold of classical perturbation theory?
• RQ3 How do bifurcations in the spectrum of $\mathcal{L}_\sigma$ under variation of $\sigma$ lead to multiple solutions with the same initial data?
• RQ4 To what extent can the spectral conditions for non-uniqueness be verified numerically using smooth functions and standard Banach space techniques?
• RQ5 Is the presence of an eigenvalue with positive real part in $\mathcal{L}_\sigma$ sufficient to imply ill-posedness for initial data in $L^2$?

Key findings

• Non-uniqueness and ill-posedness of the 3D Navier-Stokes equations can occur for compactly supported, divergence-free initial data in $L^2$ when the linearized operator $\mathcal{L}_\sigma$ has an eigenvalue with positive real part and the corresponding eigenfunction decays as $o(|x|^{-1})$ at infinity.
• The spectral condition (A) — existence of an eigenvalue of $\mathcal{L}_\sigma$ with positive real part and suitable decay — is sufficient to construct two distinct Leray-Hopf weak solutions with the same compactly supported initial data $v_0 \in C^\infty(\mathbb{R}^3 \setminus \{0\})$.
• For $\sigma$ sufficiently close to a critical value $\sigma_0$, the solution curve $U_\sigma$ undergoes a pitchfork bifurcation, leading to non-uniqueness of self-similar solutions and hence non-uniqueness of weak solutions.
• The two solutions constructed are smooth on $\mathbb{R}^3 \times (0,1)$ and uniformly bounded in $L^2(\mathbb{R}^3)$, confirming they are Leray-Hopf weak solutions.
• The $L^4$-norm of the difference between the two solutions diverges as $t \to 0^+$, proving they are not identical, even though they share the same initial data.
• The spectral conditions for non-uniqueness are amenable to verification via numerical simulations involving only smooth functions and standard $L^p$-based Banach spaces.