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Non-uniqueness in the Leray-Hopf class for a dyadic Navier-Stokes model

Stan Palasek

Abstract

The uniqueness of Leray-Hopf solutions to the incompressible Navier-Stokes equations remains a significant open question in fluid mechanics. This paper proposes a potential mechanism for non-uniqueness, illustrated in a natural dyadic shell model. We show that, for the Obukhov model with $d>2$, there exist initial data at the critical regularity that give rise to two distinct Leray-Hopf solutions. These solutions exhibit an approximately discretely self-similar structure, with non-uniqueness resulting from a partial breaking of the scaling symmetry. The fundamental observation is that, in a certain scenario, the dynamics reduce to a sequence of weakly coupled finite-dimensional systems. Moreover, the predominant nonlinear interactions are identical to those arising in convex integration, suggesting the possibility of a similar construction in the full PDE setting.

Non-uniqueness in the Leray-Hopf class for a dyadic Navier-Stokes model

Abstract

The uniqueness of Leray-Hopf solutions to the incompressible Navier-Stokes equations remains a significant open question in fluid mechanics. This paper proposes a potential mechanism for non-uniqueness, illustrated in a natural dyadic shell model. We show that, for the Obukhov model with , there exist initial data at the critical regularity that give rise to two distinct Leray-Hopf solutions. These solutions exhibit an approximately discretely self-similar structure, with non-uniqueness resulting from a partial breaking of the scaling symmetry. The fundamental observation is that, in a certain scenario, the dynamics reduce to a sequence of weakly coupled finite-dimensional systems. Moreover, the predominant nonlinear interactions are identical to those arising in convex integration, suggesting the possibility of a similar construction in the full PDE setting.
Paper Structure (12 sections, 9 theorems, 143 equations, 3 figures)

This paper contains 12 sections, 9 theorems, 143 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Previous results for the Navier--Stokes equations
  3. Dyadic models
  4. Heuristics for the binary system
  5. Non-uniqueness and breaking the scaling symmetry
  6. Prospects for the Navier--Stokes equations
  7. Notation and definitions
  8. Plan of the paper
  9. Constructing the stable manifold
  10. Bootstrapping the trajectories
  11. Initial data and lack of uniqueness
  12. Stability in the delays

Key Result

Theorem 1.2

For any $\alpha\in(2,4)$ and all sufficiently large $\lambda$, there exists initial data $u^0\in\cap_{s<\alpha-2}H^s$ that gives rise to two distinct Leray--Hopf solutions of system.

Figures (3)

  • Figure 1: The approximate dynamics of the system for $(u_{k-1},u_k)$, obtained by setting $b_k=b_k(0)$ and $f_k=0$. Part of the stable manifold of $(0,0)$ is highlighted in red and is nearby the semicircle centered at $(\delta_k,0)$ with radius $\delta_k$. In this simplified picture, one should imagine the initial data being chosen on the stable manifold, close to the point $(2\delta_k,0)$, at an angle $\sim\lambda^{-\alpha+2}$ above the horizontal. The result is the $k-1$ and $k$th modes annihilate on the short time scale $N_k^{-2}$.
  • Figure 2: The data $u^0$ is such that any adjacent pair of modes $(u_{k-1},u_k)$ can annihilate (by which we mean approximately follow the dynamics in Figure \ref{['binarysystemphaseportraitfigure']}) on the time scale $N_{k}^{-2}$. This annihilation takes place in such a way that it is only weakly coupled to the rest of the system. There are two possible configurations which are approximately rescaled and shifted versions of each other: the solid interactions leading to $u$, and the dashed interactions leading to $v$.
  • Figure 3: A schematic view of the definition of $T(\mu)$. The values of $\mu_k$ for $k$ odd determine $u^0$, while the values of $\mu_k$ for $k$ even along with $u^0_0$ determine $v^0$. Fixed points of $T$ correspond to the desired outcome $u^0=v^0$.

Theorems & Definitions (27)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Proposition 2.1
  • proof
  • ...and 17 more