Unique local weak solutions of the non-resistive MHD equations in homogeneous Besov space

ABSTRACT

In this paper, the local existence and uniqueness of weak solutions to a d-dimensional non-resistive MHD equations in homogeneous Besov spaces are studied. Specifically we obtain the local existence of a weak solution $(u,b)$ of the non-resistive MHD equations for the initial data $u_0\in {\dot{B}}_{p,1}^{\frac{d}{p}-1}({\mathbb{R}}^d)$ and $b_0\in {\dot{B}}_{p,1}^{\frac{d}{p}}({\mathbb{R}}^d)$ with $1\le p\le \mathrm{\infty }$, and the uniqueness of the weak solution when $1\le p\le 2d$. Compared with the previous results for the non-resistive MHD equations, in the local existence part, the range of p extends to $1\le p\le \mathrm{\infty }$ from $1\le p\le 2d$, but the uniqueness of the solution requires $1\le p\le 2d$ yet.

KEYWORDS:

• Non-resistive MHD equations
• homogeneous Besov space
• uniqueness
• weak solution

MATHEMATIC SUBJECT CLASSIFICATIONS (2000):

• 35Q35
• 76D03
• 76W05

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Additional information

Funding

The work of B. Yuan was partially supported by the Innovative Research Team of Henan Polytechnic University [grant number T2022-7], and double first-class discipline project [grant number AQ20230775].