In 1.1.1 of 'The Mathematical Analysis of the Incompressible Euler and Navier-Stokes Equations, Jacob Bedrossian & Vlad Vicol', I don't understand the following estimate
$$\sup_{a\in\mathbb{R}^d}|\nabla_aX(t,a)|\leq \exp(\int_0^t||\nabla u(s)||_{L^{\infty}}ds).$$
$X(\cdot,t):\mathbb{R}^d\to\mathbb{R}^d$ is the flow map, which has the following property
$$\partial_t X(a,t)=u(t,X(a,t)),$$ where $u(t,x)$ is the velocity field. I have proved the following identity $$\partial_t \nabla_a X(t,a)=\nabla_x u(t,x)\cdot\nabla_aX(t,a).$$ It's definitely like the Gronwall's inequality of the scalar function, which is
Assume $u(x),v(x)$ is continue function and $$u^{'}(x)\leq v(x)u(x), \forall x\in [a,b]$$ then we have $$u(x)\leq u(a)\exp(\int_a^x v(t)dt), \forall x\in [a,b].$$ But in the matrix case, I can not understand how to deduce the result in the book.