Applying Gronwall lemma in Majda-Bertozzi book. I am studying Majda-Bertozzi book about incompressible flows. I have applied Gronwall lemma several times, but I do not know how to do in the following case:
We have
$$
|\nabla v(\cdot,t)|_{L^{\infty}}\le C\left(1+\int_0^t|\nabla v(\cdot,s)|_{L^{\infty}}\right)\left(1+|w(\cdot,t)|_{L^{\infty}}\right)
$$
and we must use Gronwall lemma to get:
$$
|\nabla v(\cdot,t)|_{L^{\infty}}\le|\nabla v_0|_0 \exp\left(\int_0^t|w(\cdot,s)|_{L^{\infty}}ds\right).
$$
Thanks a lot!
 A: I can't quite obtain that inequality, but I can get something that is enough to finish the proof of the theorem. Majda-Bertozzi quotes lemma 3.1 as Gronwall,
$$q(t) \le c(t) + \int_0^t u(s) q(s) ds\implies q(t) \le c(0) \exp\left(\int_0^t u(s) ds\right) + \int_0^t c'(s)\left(\exp\int_s^t u(\tau)d\tau \right)ds$$
To put it in this form, you can set 
$$ q(t) = \frac{|\nabla v(t)|_{L^\infty}}{1+|\omega(t)|_{L^\infty}}, u(t) = C(1+|\omega(t)|_{L^\infty}), c(t) = C$$ Then we have
$$ \frac{|\nabla v(t)|_{L^\infty}}{1+|\omega(t)|_{L^\infty}} \le C\exp\left(C\int _0^t 1+ |\omega(s)|_{L^\infty}ds \right) $$
i.e.
$$ |\nabla v(t)|_{L^\infty} \le C (1+|\omega(t)|_{L^\infty})\exp\left(C\int _0^t 1+ |\omega(s)|_{L^\infty}ds \right) = \frac{d}{dt} \exp \left(C\int _0^t 1+ |\omega(s)|_{L^\infty}ds \right)$$
And hence
$$\int_0^t |\nabla v(s)|_{L^\infty} ds \le C\exp \left(C\int _0^t 1+ |\omega(s)|_{L^\infty}ds \right) $$
which you can plug into the $H^m$ energy estimate (3.79),
$$ \|u(T)\|_m \le \|u_0\|_m \exp\left(c_m\int_0^T|\nabla v(t)|_{L^\infty} dt \right)$$
to get the required a priori estimate.
