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Energy equality of the weak solutions to the fractional Navier-Stokes / MHD equations

Yi Feng, Weihua Wang

TL;DR

The paper studies energy equality for weak solutions to the 3D fractional Navier–Stokes and MHD equations with dissipation $(-\Delta)^\alpha$ and initial data in $L^2$-based spaces. It develops a symmetrization framework together with interpolation in Sobolev multiplier spaces to establish new sufficient conditions guaranteeing energy conservation, extending existing results to the fractional setting and linking to Onsager-type criteria. The results include Theorem 1.3 and related corollaries, generalizing prior multiplier-space criteria (e.g., CZ23, Wu24, ZY22) and recovering classical NS/MHD energy-equality results in the appropriate limits. The work contributes to the understanding of energy dissipation and potential uniqueness of weak solutions in fractional fluid models, with implications for turbulence theory and the Onsager conjecture in the fractional regime.

Abstract

In this paper, we study the problem of energy equality for weak solutions of the 3D incompressible fractional Navier-Stokes / MHD equations. With the help of the technique of symmetrization and interpolation method, we obtain some new sufficient conditions including the Sobolev multiplier spaces, which insures the validity of the energy equality of the weak solution to fractional MHD equations. Correspondingly, the results of fractional Navier-Stokes equations are obtained. And these energy equations are usually related to the uniqueness of solutions to the corresponding fractional Navier-Stokes / MHD equations.

Energy equality of the weak solutions to the fractional Navier-Stokes / MHD equations

TL;DR

The paper studies energy equality for weak solutions to the 3D fractional Navier–Stokes and MHD equations with dissipation and initial data in -based spaces. It develops a symmetrization framework together with interpolation in Sobolev multiplier spaces to establish new sufficient conditions guaranteeing energy conservation, extending existing results to the fractional setting and linking to Onsager-type criteria. The results include Theorem 1.3 and related corollaries, generalizing prior multiplier-space criteria (e.g., CZ23, Wu24, ZY22) and recovering classical NS/MHD energy-equality results in the appropriate limits. The work contributes to the understanding of energy dissipation and potential uniqueness of weak solutions in fractional fluid models, with implications for turbulence theory and the Onsager conjecture in the fractional regime.

Abstract

In this paper, we study the problem of energy equality for weak solutions of the 3D incompressible fractional Navier-Stokes / MHD equations. With the help of the technique of symmetrization and interpolation method, we obtain some new sufficient conditions including the Sobolev multiplier spaces, which insures the validity of the energy equality of the weak solution to fractional MHD equations. Correspondingly, the results of fractional Navier-Stokes equations are obtained. And these energy equations are usually related to the uniqueness of solutions to the corresponding fractional Navier-Stokes / MHD equations.

Paper Structure

This paper contains 4 sections, 10 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The proof of Theorem 1.3
  4. The proof of Theorem 1.6 and Theorem 1.9

Key Result

Theorem 1.3

Let $$ be a weak solutions of the fractional MHD equations eq1.2 on some time interval $[0,T]$ with $u_0,B_0\in L^{2}_{\sigma}(\mathbb{R}^{3})$ and $0<T\leq\infty$. If $u$ and $B$ satisfy $|u|^{\frac{2}{\beta+\theta}-2}u$ and $|B|^{\frac{2}{\beta+\theta}-2}B \in L^{\frac{2\alpha(\beta+\theta)}{\alph holds for any $\;0\le t<T$.

Theorems & Definitions (21)

  • Definition 1.1: Leray-Hopf Weak Solutions
  • Definition 1.2: Leray-Hopf Weak Solutions
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Remark 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 11 more