具有部分分数阶扩散项的三维Navier-Stokes方程的适定性

众所周知,具有超耗散(-Δ)^5/4的Navier-Stokes方程是适定的.许多学者研究了具有弱超耗散的问题.去除Navier-Stokes方程中一些不同方向的超耗散分量,在一些额外的条件下,可以证明解的存在性和唯一性.本文推导出从u₂和u₃的方程中去除沿x₃方向的超耗散问题解的存在性和唯一性.需要指出的是,如果从所有方程中去除沿x₃方向的超耗散,则本方法无法获得此问题的适定性结果.

Well-posedness of three dimensional Navier-Stokes equations with partial fractional diffusion terms

It is a well-known fact that the Navier-Stokes equation with hyperdissipation is well-posed. There are many scholars committing themselves to solving the problem with weaker hyperdissipation. When some components of hyperdissipation in different spatial directions are removed from the Navier-Stokes equations, the existence as well as uniqueness under some extra conditions is thus proved. In this paper, we derive the existence and uniqueness of the problem with hyperdissipation in x3 direction removed from the equations of u2 and u3. It is worth mentioning that the well-posedness result shall not be obtained by the present method if the hyperdissipation in x3 direction is removed from all three equations.