In Majda/Bertozzi book, Incompressible flows etc.. p.118,he uses Gronwall theorem on the following inequality:
$$|\nabla v(.,t)|_{L^{\infty}} \leq C\left( 1 + \int_0^t|v(.,s|_{L^{\infty}}ds \right)\left(1 + |w(.,t)|_{L^{\infty}} \right)$$
To get that : $|\nabla v(.,t)|_{L^{\infty}} \leq |\nabla v(.,0)|_{L^{\infty}} e^{\int_0^t |w(.,s)|_{L^{\infty}} ds}$
I cannot see how to apply gronwall theorem to get this result, does it make sense to anybody ?