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

TL;DR

This paper proves the well-posedness of 3D Navier-Stokes and MHD equations with partial fractional dissipation, showing solutions exist and are unique even when dissipation is missing in some directions.

Contribution

It extends previous results by establishing existence and uniqueness of solutions for equations with partial fractional hyper-dissipation in more general cases.

Findings

Solutions exist under partial dissipation conditions

Conditional uniqueness holds with missing dissipation directions

Generalizes previous well-posedness results

Abstract

It is well-known that if one replaces standard velocity and magnetic dissipation by $(-\Delta)^\alpha u$ and $(-\Delta)^\beta b$ respectively, the magnetohydrodynamic equations are well-posed for $\alpha\ge\frac{5}{4}$ and $\alpha + \beta \ge \frac{5}{2}$. 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.