论文标题
3D高度脱落的Navier-Stokes方程的急剧非唯一性：狮子指数上方
Sharp non-uniqueness for the 3D hyperdissipative Navier-Stokes equations: above the Lions exponent
论文作者
论文摘要
我们研究了圆环上的3D高度放电的Navier-Stokes方程，其中粘度指数$α$可能大于狮子指数$ 5/4 $。众所周知，由于狮子[55]，对于任何$ l^2 $ divergence的初始数据，当$α\ geq 5/4 $时，存在独特的光滑leray-hopf解决方案。我们证明，即使在这种高耗散性方面，唯一性也会在超临界空间中失败$ l^γ_tw^{s，p} _x $，鉴于广义的Ladyženskaja-prodi-serrin条件。非唯一性在很强的意义上被证明，尤其是在两个端点$（3/p+1-2α，\ infty，p）$和$（2α/γ+1-2α，γ，\ infty）$的情况下产生清晰度。 Moreover, the constructed solutions are allowed to coincide with the unique Leray-Hopf solutions near the initial time and, more delicately, admit the partial regularity outside a fractal set of singular times with zero Hausdorff $\mathcal{H}^{η_*}$ measure, where $η_*>0$ is any given small positive constant.这些结果还提供了超临界的Lebesgue和Besov空间的急剧非唯一性。此外，对于高脱离的Navier-Stokes方程，获得了强大的消失粘度结果。
We study the 3D hyperdissipative Navier-Stokes equations on the torus, where the viscosity exponent $α$ can be larger than the Lions exponent $5/4$. It is well-known that, due to Lions [55], for any $L^2$ divergence-free initial data, there exist unique smooth Leray-Hopf solutions when $α\geq 5/4$. We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces $L^γ_tW^{s,p}_x$, in view of the generalized Ladyženskaja-Prodi-Serrin condition. The non-uniqueness is proved in the strong sense and, in particular, yields the sharpness at two endpoints $(3/p+1-2α, \infty, p)$ and $(2α/γ+1-2α, γ, \infty)$. Moreover, the constructed solutions are allowed to coincide with the unique Leray-Hopf solutions near the initial time and, more delicately, admit the partial regularity outside a fractal set of singular times with zero Hausdorff $\mathcal{H}^{η_*}$ measure, where $η_*>0$ is any given small positive constant. These results also provide the sharp non-uniqueness in the supercritical Lebesgue and Besov spaces. Furthermore, the strong vanishing viscosity result is obtained for the hyperdissipative Navier-Stokes equations.
