Forward self-similar solutions to the 2D Navier--Stokes equations

TL;DR

The paper proves the existence of forward self-similar solutions to the 2D Navier–Stokes equations for -1-homogeneous initial data, showing the profile $U$ exists, is smooth, and decays as $|U(y)-(e^{\Delta}u_0)(y)|\lesssim \langle y\rangle^{-1-\alpha}$ for any $\alpha\in(0,1)$. The construction uses a Leray-Schauder fixed-point argument in a weighted function space, with a decomposition $U=a+V$ where $a=e^{\Delta}u_0$, and a carefully derived a priori control via a self-similar Bernoulli-type pressure $\Phi$. The work highlights significant 2D-specific challenges due to infinite energy of -1-homogeneous data, addresses them through weighted elliptic estimates and parabolic bootstrapping, and provides numerical evidence of non-uniqueness through a pitchfork bifurcation in the self-similar regime. This advances understanding of non-uniqueness mechanisms in the Navier–Stokes equations within the infinite-energy, self-similar framework and connects rigorous analysis with computational bifurcation evidence."

Abstract

We construct self-similar solutions to the 2D Navier--Stokes equations evolving from arbitrarily large $-1$--homogeneous initial data and present numerical evidence for their non-uniqueness.

Figures (4)

• Figure 1.1: Symmetric and non-symmetric solutions with $\sigma = 80$.
• Figure 6.1: Representations of the solution $\widetilde{U}_\sigma$ of \ref{['eq:LerayEquationsBall']} for $\sigma\in\{0,20,80\}$: all solutions are $\mathcal{R}$-symmetric. The first column is on the full computational domain and represents in color the magnitude $|\widetilde{y}||\widetilde{U}(\widetilde{y})|$ and the streamlines of $\widetilde{U}$. In particular, for $\sigma=0$, the solution is in addition symmetric with respect to the vertical axis because it solves the self-similar Stokes equations. The second column represent the solution near the origin with the color being the magnitude of $|\widetilde{U}(\widetilde{y})|$.
• Figure 6.2: Eigenvalues of the linearized operator around $\widetilde{U}_\sigma$ found numerically. The color of the lines represents the absolute value of the imaginary part of the eigenvalues. The eigenvalue $\lambda=1$ is associated to the projection given by the mean of the eigenvectors. Near $\sigma_0\approx39.2$, one single real eigenvalue is crossing the imaginary axis.
• Figure 6.3: Representation of the symmetric solution $\widetilde{U}_\sigma$ on the left and of the non-symmetric solution $\widetilde{U}_\sigma + \widetilde{V}_\sigma$ on the right for $\sigma\in\{40,60,80\}$. The representation is done near the origin but slightly shifted to the left since the interesting part is in that direction. One visually sees that the two solutions are different.

Theorems & Definitions (22)

• Theorem 1.1
• Lemma 2.1
• proof
• Proposition 2.2
• Proposition 2.3
• proof : Proof of Proposition \ref{['prop:K']}
• Lemma 3.1
• proof
• Proposition 4.1
• Lemma 4.2