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• W2935879897 abstract "This paper examines the uniqueness of weak solutions to the d-dimensional magnetohydrodynamic (MHD) equations with the fractional dissipation $(-Delta)^alpha u$ and without the magnetic diffusion. Important progress has been made on the standard Laplacian dissipation case $alpha=1$. This paper discovers that there are new phenomena with the case $alpha<1$. The approach for $alpha=1$ can not be directly extended to $alpha<1$. We establish that, for $alpha<1$, any initial data $(u_0, b_0)$ in the inhomogeneous Besov space $B^sigma_{2,infty}(mathbb R^d)$ with $sigma> 1+frac{d}{2}-alpha$ leads to a unique local solution. For the case $alphage 1$, $u_0$ in the homogeneous Besov space $mathring B^{1+frac{d}{2}-2alpha}_{2,1}(mathbb R^d)$ and $b_0$ in $ mathring B^{1+frac{d}{2}-alpha}_{2,1}(mathbb R^d)$ guarantees the existence and uniqueness. These regularity requirements appear to be optimal." @default.
• W2935879897 created "2019-04-25" @default.
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• W2935879897 date "2019-04-11" @default.
• W2935879897 modified "2023-09-26" @default.
• W2935879897 title "Unique weak solutions of the non-resistive magnetohydrodynamic equations with fractional dissipation" @default.
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• W2935879897 doi "https://doi.org/10.48550/arxiv.1904.06006" @default.
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