理想MHD中的磁螺旋度和子解
Magnetic helicity and subsolutions in ideal MHD
摘要：
我们证明了理想的二维MHD方程不存在具有时间紧支集和非平凡磁场的弱解（甚至子解）。我们还表明，$Λ$凸包的理想MHD有空的内部在2D和3D，这是通过找到合适的$Λ$ -凸函数。因此，我们证明了二维均方磁势在物理自然能量空间$L^\infty_t L^2_x$中通过解的子解和弱极限是守恒的，三维磁螺旋度在物理自然能量空间中通过$L^3$ -可积子解和弱极限是守恒的.然而，在3D的$Λ$ -凸包被证明是足够大的，非平凡的光滑，compensated严格的子解存在。
英文摘要：
We show that ideal 2D MHD does not possess weak solutions (or even subsolutions) with compact support in time and non-trivial magnetic field. We also show that the $Λ$-convex hull of ideal MHD has empty interior in both 2D and 3D; this is seen by finding suitable $Λ$-convex functions. As a consequence we show that mean-square magnetic potential is conserved in 2D by subsolutions and weak limits of solutions in the physically natural energy space $L^\infty_t L^2_x$, and in 3D we show the conservation of magnetic helicity by $L^3$-integrable subsolutions and weak limits of solutions. However, in 3D the $Λ$-convex hull is shown to be large enough that nontrivial smooth, compactly supported strict subsolutions exist.
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