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The well-posedness of three-dimensional Navier-Stokes and magnetohydrodynamic equations with partial fractional dissipation

Qibo Ma, Li Li

TL;DR

The paper analyzes 3D Navier–Stokes and magnetohydrodynamic equations with partial fractional diffusion or diffusion in selectively weakened directions. It introduces sets of 'bad' indices that mark where dissipation is missing and proves global existence and regularity in $H^1$ for NS and MHD when index configurations are not in these sets, along with conditional uniqueness requiring extra regularity. The approach hinges on meticulous trilinear-term estimates and a systematic case analysis, including computer-assisted identification of all admissible index configurations (24 good indices). The results extend prior work (YJW2019M) to a broader anisotropic diffusion framework and include corollaries for modified directional operators, broadening the regime of well-posedness in anisotropic dissipative settings.

Abstract

It is well-known that if one replaces standard velocity and magnetic dissipation by $(-Δ)^αu$ and $(-Δ)^βb$ respectively, the magnetohydrodynamic equations are well-posed for $α\ge\frac{5}{4}$ and $α+ β\ge \frac{5}{2}$. This paper considers the 3D Navier-Stokes and magnetohydrodynamic equations with partial fractional hyper-dissipation. It is proved that when each component of the velocity and magnetic field lacks dissipation along some direction, the existence and conditional uniqueness of the solution still hold. This paper extends the previous results in (Yang, Jiu and Wu J. Differential Equations 266(1): 630-652, 2019) to a more general case.

The well-posedness of three-dimensional Navier-Stokes and magnetohydrodynamic equations with partial fractional dissipation

TL;DR

The paper analyzes 3D Navier–Stokes and magnetohydrodynamic equations with partial fractional diffusion or diffusion in selectively weakened directions. It introduces sets of 'bad' indices that mark where dissipation is missing and proves global existence and regularity in for NS and MHD when index configurations are not in these sets, along with conditional uniqueness requiring extra regularity. The approach hinges on meticulous trilinear-term estimates and a systematic case analysis, including computer-assisted identification of all admissible index configurations (24 good indices). The results extend prior work (YJW2019M) to a broader anisotropic diffusion framework and include corollaries for modified directional operators, broadening the regime of well-posedness in anisotropic dissipative settings.

Abstract

It is well-known that if one replaces standard velocity and magnetic dissipation by and respectively, the magnetohydrodynamic equations are well-posed for and . This paper considers the 3D Navier-Stokes and magnetohydrodynamic equations with partial fractional hyper-dissipation. It is proved that when each component of the velocity and magnetic field lacks dissipation along some direction, the existence and conditional uniqueness of the solution still hold. This paper extends the previous results in (Yang, Jiu and Wu J. Differential Equations 266(1): 630-652, 2019) to a more general case.
Paper Structure (9 sections, 11 theorems, 162 equations)

This paper contains 9 sections, 11 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Uniqueness for the Navier-Stokes equations
  3. Global existence and regularity for the MHD equations
  4. The estimates of $K_2$
  5. The estimates of $K_4$
  6. Uniqueness for the MHD equations
  7. The estimates of $L_2$
  8. The estimates of $L_3$
  9. The estimates of $L_4$

Key Result

Theorem 1.1

If the indices $(i_1, i_2, i_3) \notin \mathcal{I}$ and $\boldsymbol{u}_0 \in H^1( \mathbb{R}^3 )$, then there exists a global solution ${\bm u}$ of eq:general:NS satisfying In addition, the $H^1$-norm of ${\bm u}$ is bounded uniformly in time.

Theorems & Definitions (15)

  • Theorem 1.1: Existence
  • Theorem 1.2: Uniqueness
  • Theorem 1.3: Existence
  • Theorem 1.4: Uniqueness
  • Corollary 1.5
  • Corollary 1.6
  • Lemma A.1: Sobolev embedding
  • Lemma A.2
  • proof
  • Lemma A.3
  • ...and 5 more