论文标题
在关键空间规律性下运输方程的弱解决方案的非唯一性
Nonuniqueness of weak solutions for the transport equation at critical space regularity
论文作者
论文摘要
我们考虑由尺寸不可压缩流驱动的线性传输方程$ d \ geq 3 $。对于不含差异的矢量字段$ u \ in l^1_t w^{1，q} $，著名的二型二型二叶式解决方案理论确定了$ l^\ infty_t l^p $ n时$ l^\ infty_t l^p $ $ \ frac \ frac {1} {p} {p} {p} + frac $ \ e}的独特性。对于此类矢量字段，我们表明，在制度$ \ frac {1} {p} + \ frac {1} {q}> 1 $中，弱解决方案在$ l^1_t l^p $的类中不是唯一的。证明的一种关键成分是在凸集成方案中使用时间间歇性和振荡。
We consider the linear transport equations driven by an incompressible flow in dimensions $d\geq 3$. For divergence-free vector fields $u \in L^1_t W^{1,q}$, the celebrated DiPerna-Lions theory of the renormalized solutions established the uniqueness of the weak solution in the class $L^\infty_t L^p$ when $\frac{1}{p} + \frac{1}{q} \leq 1$. For such vector fields, we show that in the regime $\frac{1}{p} + \frac{1}{q} > 1$, weak solutions are not unique in the class $ L^1_t L^p$. One crucial ingredient in the proof is the use of both temporal intermittency and oscillation in the convex integration scheme.
