Global Smooth Solutions for the 2D incompressibleNavier--Stokes--Landau--Lifshitz equations

The Cauchy problem for incompressible Navier--Stokes--Landau--Lifshitz equations in two-dimensional space is solved by the following assumptions:the initial density satisfies \rho_00,the initial energy \|\rho_0^\frac12\mathbfu_0\|_L^2^2+\|\nabla\mathbfd_0\|_L^2^2 \varepsilon_0 is suitably small,and the global existence and uniqueness of the strong solutions are proved by energy method.