论文标题
对随机Navier-Stokes方程的解决方案的急剧非唯一性
Sharp non-uniqueness of solutions to stochastic Navier-Stokes equations
论文作者
论文摘要
在本文中，我们为随机$ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d \ geq2 $）建立了不可压缩的Navier-Stokes方程。首先，对于$ l^2 $中的每一个无差的初始条件，我们显示了无限的存在，许多全球范围内的概率强度强，并且在$ l^α\ big（ω，l^p_tl^\ infty \ big）$ 1 \ leq p <2，leq p <2，α\ geq1 $中的$ l^α\ big（ω，l^p_tl^\ infty \ big）中的存在。其次，我们证明了上述结果是明显的，因为在[2，\ infty]中的某些$ p \的$ l^p_tl^q $中保持着唯一性，q \ in（2，\ infty] $，因此$ \ frac2 {p}+\ frac}+\ frac {p} ladyzhenskaya-serrodi-serlin标准，与随机的欧拉方程式存在，与\ cite {hzz19，hzz21a is a new new new nek a nek a nek a nek nek a nek Insection cite cite相比我们在凸集成方案中介绍了期望，并在整个时间间隔$ [0，\ infty）$上直接构建解决方案。
In this paper we establish a sharp non-uniqueness result for stochastic $d$-dimensional ($d\geq2$) incompressible Navier-Stokes equations. First, for every divergence free initial condition in $L^2$ we show existence of infinite many global in time probabilistically strong and analytically weak solutions in the class $L^α\big(Ω,L^p_tL^\infty\big)$ for any $1\leq p<2,α\geq1$. Second, we prove the above result is sharp in the sense that pathwise uniqueness holds in the class of $L^p_tL^q$ for some $p\in[2,\infty],q\in(2,\infty]$ such that $\frac2{p}+\frac{d}{q}\leq1$, which is a stochastic version of Ladyzhenskaya-Prodi-Serrin criteria. Moreover, for stochastic $d$-dimensional incompressible Euler equation, existence of infinitely many global in time probabilistically strong and analytically weak solutions is obtained. Compared to the stopping time argument used in \cite{HZZ19, HZZ21a}, we developed a new stochastic version of the convex integration. More precisely, we introduce expectation during convex integration scheme and construct directly solutions on the whole time interval $[0,\infty)$.
