论文标题
在几乎尖锐的liouville型定理上，用于分数Navier-Stokes方程
On an almost sharp Liouville type theorem for fractional Navier-Stokes equations
论文作者
论文摘要
我们研究了3D固定和不可压缩的分数Navier-Stokes方程的存在，liouville型定理和规律性结果：在这种情况下，通常的laplacian被其分数功率$（ - δ）^{\fracα{2}}}替换为$ 0 <α<2 $。 By applying a fixed point argument, weak solutions can be obtained in the Sobolev space $\dot{H}^{\fracα{2}}(\mathbb{R})$ and if we add an extra integrability condition, stated in terms of Lebesgue spaces, then we can prove for some values of $α$ that the zero function is the unique smooth solution.额外的集成性条件几乎是鲜明的，$ 3/5 <α<5/3 $。此外，如果在某些条件下建立了$ 1 <α<2 $ $ 1 <α<2 $，但是对制度的规律性研究$ 0 <α\ leq 1 $，目前似乎是一个空旷的问题。
We investigate existence, Liouville type theorems and regularity results for the 3D stationary and incompressible fractional Navier-Stokes equations: in this setting the usual Laplacian is replaced by its fractional power $(-Δ)^{\fracα{2}}$ with $0<α<2$. By applying a fixed point argument, weak solutions can be obtained in the Sobolev space $\dot{H}^{\fracα{2}}(\mathbb{R})$ and if we add an extra integrability condition, stated in terms of Lebesgue spaces, then we can prove for some values of $α$ that the zero function is the unique smooth solution. The additional integrability condition is almost sharp for $3/5<α<5/3$. Moreover, in the case $1<α<2$ a gain of regularity is established under some conditions, however the study of regularity in the regime $0<α\leq 1$ seems for the moment to be an open problem.
