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Global dissipative solutions of the 3D Naiver-Stokes and MHD equations

Alexey Cheskidov, Zirong Zeng, Deng Zhang

TL;DR

The paper establishes the global existence and nonuniqueness of dissipative weak solutions to the 3D Navier–Stokes and MHD equations for divergence-free initial data in the critical space $H_x^{1/2}$, while also obtaining continuous-energy solutions for rough $L_x^2$ data. It introduces a telescoping convex integration framework built around a $ vLambda(t)$-dependent (the $ vLambda$-MHD) approximation, a frequency-truncation scheme, and heat correctors that leverage dissipation to achieve energy decay along prescribed channels. By constructing backward/forward time sequences and carefully balancing energy and cross helicity profiles, the method produces infinitely many solutions with distinct energy trajectories, including energy-decreasing ones, thereby violating uniqueness in the dissipative class. The approach extends to stochastic settings and provides a detailed, technically involved pathway to control Reynolds and magnetic stresses through a layered perturbation scheme with precise estimates for perturbations, heat corrections, and various error terms. The results have implications for the understanding of turbulence-like behaviors and nonuniqueness phenomena in viscous, dissipative fluid and MHD models, highlighting the nuanced roles of dissipation and frequency interactions in weak solution theory.

Abstract

For any divergence free initial data in $H^\frac12$, we prove the existence of infinitely many dissipative solutions to both the 3D Navier-Stokes and MHD equations, whose energy profiles are continuous and decreasing on $[0,\infty)$. If the initial data is only $L^2$, our construction yields infinitely many solutions with continuous energy, but not necessarily decreasing. Our theorem does not hold in the case of zero viscosity as this would violate the weak-strong uniqueness principle due to Lions. This was achieved by designing a convex integration scheme that takes advantage of the dissipative term.

Global dissipative solutions of the 3D Naiver-Stokes and MHD equations

TL;DR

The paper establishes the global existence and nonuniqueness of dissipative weak solutions to the 3D Navier–Stokes and MHD equations for divergence-free initial data in the critical space , while also obtaining continuous-energy solutions for rough data. It introduces a telescoping convex integration framework built around a -dependent (the -MHD) approximation, a frequency-truncation scheme, and heat correctors that leverage dissipation to achieve energy decay along prescribed channels. By constructing backward/forward time sequences and carefully balancing energy and cross helicity profiles, the method produces infinitely many solutions with distinct energy trajectories, including energy-decreasing ones, thereby violating uniqueness in the dissipative class. The approach extends to stochastic settings and provides a detailed, technically involved pathway to control Reynolds and magnetic stresses through a layered perturbation scheme with precise estimates for perturbations, heat corrections, and various error terms. The results have implications for the understanding of turbulence-like behaviors and nonuniqueness phenomena in viscous, dissipative fluid and MHD models, highlighting the nuanced roles of dissipation and frequency interactions in weak solution theory.

Abstract

For any divergence free initial data in , we prove the existence of infinitely many dissipative solutions to both the 3D Navier-Stokes and MHD equations, whose energy profiles are continuous and decreasing on . If the initial data is only , our construction yields infinitely many solutions with continuous energy, but not necessarily decreasing. Our theorem does not hold in the case of zero viscosity as this would violate the weak-strong uniqueness principle due to Lions. This was achieved by designing a convex integration scheme that takes advantage of the dissipative term.

Paper Structure

This paper contains 68 sections, 19 theorems, 404 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Review of related literature
  3. Convex integration for the NSE: intermittent schemes
  4. Convex integration for the MHD
  5. $L^\infty$-scheme for ideal MHD
  6. Intermittent scheme for MHD
  7. $\Lambda$-MHD approximation and telescoping energy levels
  8. Heat correctors for the 3-D MHD
  9. Telescoping convex integration scheme
  10. Initialization: the $\Lambda$-MHD system
  11. Selection of backward and forward times
  12. Selection of energy and cross helicity profiles
  13. Main iterative estimates
  14. The $\Lambda$-MHD system
  15. Step $1$: Global existence.
  16. ...and 53 more sections

Key Result

Theorem 1.2

Given any divergence-free initial data in $H_x^{\frac{1}{2}}(\mathbb{T}^3)$, there exist infinitely many dissipative solutions to 3D NSE equa-NSE on $[0,\infty)$.

Figures (2)

  • Figure 1: Wavenumber, initial Reynolds and magnetic errors
  • Figure 2: Upper and lower energy perturbation profiles

Theorems & Definitions (35)

  • Definition 1.1: Dissipative solution
  • Theorem 1.2: Dissipative solutions to NSE
  • Remark 1.3
  • Theorem 1.4: Dissipative solutions to MHD
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 2.1
  • Remark 2.2
  • ...and 25 more