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Sharp non-uniqueness of weak solutions to 3D magnetohydrodynamic equations | | Yachun Li
; Zirong Zeng
; Deng Zhang
; | | Date: |
1 Aug 2022 | | Abstract: | We prove the non-uniqueness of weak solutions to 3D hyper viscous and
resistive MHD in the class $L^gamma_tW^{s,p}_x$, where the exponents
$(s,gamma,p)$ lie in two supercritical regimes. The result reveals that the
scaling-invariant Ladyv{z}enskaja-Prodi-Serrin (LPS) condition is the right
criterion to detect non-uniqueness, even in the highly viscous and resistive
regime beyond the Lions exponent. In particular, for the classical viscous and
resistive MHD, the non-uniqueness is sharp near the endpoint $(0,2,infty)$ of
the LPS condition. Moreover, the constructed weak solutions admit the partial
regularity outside a small fractal singular set in time with zero
$mathcal{H}^{eta_*}$-Hausdorff dimension, where $eta_*$ can be any given
small positive constant. Furthermore, we prove the strong vanishing viscosity
and resistivity result, which yields the failure of Taylor’s conjecture along
some subsequence of weak solutions to the hyper viscous and resistive MHD
beyond the Lions exponent. | | Source: | arXiv, 2208.00624 | | Services: | Forum | Review | PDF | Favorites | |
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