Sobolev空间中轴对称霍尔磁流体动力学的低正则性范数膨胀
Low Regularity Norm Inflation of Axisymmetric Hall Magnetohydrodynamics in Sobolev Space
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中文总结 AI 辅助
本文研究轴对称霍尔磁流体动力学，证明其光滑解在$1<σ<7/2$时于对应Sobolev空间中存在范数膨胀，且若光滑解唯一则该结论可推广至一般霍尔磁流体动力学。
中文摘要 AI 辅助
本文研究满足$u = u^re_r+u^ze_z$且$B = B^θe_θ$的轴对称霍尔磁流体动力学。我们证明：若该系统存在光滑解，则对于$1<σ<7/2$，该解必然在$H^{σ-1}(\u211d^3) \times H^σ(\u211d^3)$中出现范数膨胀。若霍尔磁流体动力学的光滑解唯一，这也意味着霍尔磁流体动力学中存在范数膨胀。
英文摘要
In this paper, we study the axisymmetric Hall magnetohydrodynamics where $u = u^re_r+u^ze_z$ and $B = B^θe_θ$. We construct small, smooth, compactly supported, and axisymmetric initial data $(u_{0},B_{0})$ and prove that there is a unique smooth axisymmetric solution with norm inflation in $H^{σ-1}(\mathbb R^3) \times H^σ(\mathbb R^3)$ for $1<σ<7/2$. Moreover, we study the case with fractional dissipation $(-Δ)^βu$ and $(-Δ)^αB$, modifying the initial data slightly and proving that the solution inflates in $H^{σ-1}(\mathbb R^3) \times H^σ(\mathbb R^3)$ for $1<σ<7/2-2\max\{α,β\}$ and $ 0 \le α, β< 5/4$.
发表机构
- University of Illinois Chicago（伊利诺伊大学芝加哥分校）
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