关于拉格朗日流映射梯度的Gronwall不等式应用推导问询
Hey folks, I'm new around here so go easy on me if I slip up a bit with my questions!
I'm working through The Mathematical Analysis of the Incompressible Euler and Navier-Stokes Equations by Vicol and Bedrossian, and on page 3, they start from the core Eulerian-Lagrangian relation for the Lagrangian flow map $X$:
$$
\partial_t X(t,a)=u(t,X(t,a))
$$
Using the chain rule, I was able to work out the derivation for the gradient of the flow map without too much trouble, getting this equation:
$$
\partial_t\frac{\partial X_j}{\partial a_k}(t,a)=\frac{\partial u_j}{\partial x_i}(t,X(t,a))\frac{\partial X_i}{\partial a_k}(t,a), \quad \frac{\partial X_j}{\partial a_k}(a,0)=\delta_{jk} \tag{1}
$$
But then they throw out this inequality and say it's obtained by applying Gronwall's inequality to equation (1):
$$
\sup_{a\in\mathbb{R}d}|\nabla_aX(t,a)|\leq\exp\bigg(\int_0t\lVert\nabla u(s) \rVert_{L^\infty}ds\bigg)
$$
I'm totally stuck on this step. I get how Gronwall's inequality works in general, but I can't see how to bridge the gap between equation (1) and this specific bound. Could someone break down this application for me step by step?
备注：内容来源于stack exchange，提问作者Hyun Jun Kim
