Global weak solution of 3D-NSE with exponential damping

In this paper we prove the global existence of incompressible Navier-Stokes equations with damping $\alpha (e^{\beta |u|^2}-1)u$, where we use Friedrich method and some new tools. The delicate problem in the construction of a global solution, is the passage to the limit in exponential nonlinear term. To solve this problem, we use a polynomial approximation of the damping part and a new type of interpolation between $L^\infty(\mathbb{R}^+,L^2(\mathbb{R}^3))$ and the space of functions $f$ such that $(e^{\beta|f|^2}-1)|f|^2\in L^1(\mathbb{R}^3)$. Fourier analysis and standard techniques are used.