论文标题
磁性流动力方程的尖锐而强大的非唯一性
Sharp and strong non-uniqueness for the magneto-hydrodynamic equations
论文作者
论文摘要
在本文中，我们证明了三维磁磁动力（MHD）系统的一类弱解（MHD）的敏感解决方案是一种敏锐而强大的唯一性。更准确地说，我们表明，任何弱解决方案$（v，b）\ in l^p_tl^{\ infty} _x $在$ l^p_tl^{\ inftl^{\ infty} _x $中，带有$ 1 \ le p <2 $，揭示了强大的非唯一性，以及在Classical of Classical of Clastical of Crapical cr的敏锐度中，并以此为目的。 $（2，\ infty）$。此外，对于任何$ 1 \ le p <2 $和$ε> 0 $，我们在$ l^p_tl^{\ infty} _x \ cap l^1_tc^{1-ε} $中构造非leray-hopf弱解决方案。 \ cite {1cheskidov}中Navier-Stokes方程的结果暗示了MHD系统具有琐碎的磁场$ B $的急剧非唯一性。我们的结果表明，任何弱解决方案$（V，b）$（包括非平凡磁场$ b $）的非唯一性。
In this paper, we prove a sharp and strong non-uniqueness for a class of weak solutions to the three-dimensional magneto-hydrodynamic (MHD) system. More precisely, we show that any weak solution $(v,b)\in L^p_tL^{\infty}_x$ is non-unique in $L^p_tL^{\infty}_x$ with $1\le p<2$, which reveals the strong non-uniqueness, and the sharpness in terms of the classical Ladyzhenskaya-Prodi-Serrin criteria at endpoint $(2, \infty)$. Moreover, for any $1\le p<2$ and $ε>0$, we construct non-Leray-Hopf weak solutions in $L^p_tL^{\infty}_x\cap L^1_tC^{1-ε}$. The results of Navier-Stokes equations in \cite{1Cheskidov} imply the sharp non-uniqueness of MHD system with trivial magnetic field $b$. Our result shows the non-uniqueness for any weak solution $(v,b)$ including non-trivial magnetic field $b$.
