I have two related questions.
First what is a Lagrangian map? I've searched online but I have not been able to find an explanation that I understand.
Second, for a solenoidal field $u(x,t)$ how do you calculate said Lagrangian map $x\mapsto X(t;x)$?
I apologize if this question is vague, but I honestly don't understand the basics of finding a Lagrangian map.
In the Lagrangian specification of the flow field you track the position of fluid particles through time as they follow pathlines.
Let $\mathbb{X}(t,\mathbb{x}_0)$ denote the position at time $t$ of a fluid particle that occupied the intial position $\mathbb{x}_0$ at time $t=0$. As in basic mechanics, the time derivative of the position vector of a particle is the velocity.
For a given velocity field $\mathbb{u}(\mathbb{x},t)$ the map $\mathbb{x}_0 \mapsto\mathbb{X}(t,\mathbb{x}_0)$ is obtained as the solution to the initial value problem
$$\frac{\partial}{\partial t}\mathbb{X}(t,\mathbb{x}_0)= \mathbb{u}(\mathbb{X}(t,\mathbb{x}_0),t), \\ \mathbb{X}(0,\mathbb{x}_0)= \mathbb{x}_0.$$
This is a general result regardless of whether the flow is compressible or incompressible (solenoidal).
Why is this useful?
It would be an impractical and pointless exercise to track fluid particles in this way as they are moving through a continuum. Furthermore, the Eulerian velocity field $\mathbb{u}$ is typically the function that must be determined in solving a fluid flow problem.
However, the theoretical construct of the Lagrangian map is useful for proving theorems that allow us to derive the equations of fluid motion. An important example is the Reynolds Transport Theorem. This pertains to the time derivative of the integral of a scalar-, vector- or tensor-valued function over a region $V(t)$ that is moving and deforming through time with the flow. An application of this theorem to density $\rho$ results in
$$\frac{d}{dt}\int_{V(t)} \rho(\xi,t) \, d\xi = \int_{V(t)} \left[\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbb{u}) \right] \, d\xi. $$
Since mass is conserved, we have
$$\frac{d}{dt}\int_{V(t)} \rho(\xi,t) \, d\xi = 0.$$
Using the transformation $\xi = \mathbb{X}(t, \mathbb{x}_0)$ we obtain
$$\frac{d}{dt}\int_{V(t)} \rho(\xi,t) \, d\xi = \frac{d}{dt}\int_{V(0)} \rho(\mathbb{X}(t, \mathbb{x}_0),t) \frac{\partial \xi}{\partial \mathbb{x}_0} \, d\mathbb{x}_0, $$
where $\frac{\partial \xi}{\partial \mathbb{x}_0}$ is the Jacobian determinant.
By transforming to a region that is not changing in time we can interchange the derivative and the integral and ultimately obtain
$$\int_{V(t)} \left[\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbb{u}) \right] \, d\xi = 0,$$
eventually leading to the continuity equation.