One page 5 in Landau & Lifshitz Fluid Mechanics (2nd edition), the authors pose the following problem:

The authors then go on to give their solutions and assumptions. Here are the important parts:

For the condition of mass conversation the authors arrive at (where [itex]ρ_0=ρ(a)[/itex] is the given initial density distribution):
[tex]
ρ\mathrm{d}x=ρ_0 \mathrm{d}a
[/tex]

or alternatively:
[tex]
ρ\left(\frac{∂x}{∂a}\right)_t=ρ_0
[/tex]

Now the authors go on to write out Euler's equation, where I start to miss something. With the velocity of the fluid particle [itex]v=\left(\frac{∂x}{∂t}\right)_a[/itex] and [itex]\left(\frac{∂v}{∂t}\right)_a[/itex] the rate of change of the velocity of the particle during its motion, they write for Euler's equation:

How are the authors arriving at that equation?

In particular, when looking at the full Euler's equation:
[tex]
\frac{∂v}{∂t}+(\mathbf{v}⋅\textbf{grad})\mathbf{v}=−1 ρ\, \textbf{grad}\, p
[/tex]

what happens with the second term on the LHS, [itex](\mathbf{v}⋅\textbf{grad})\mathbf{v}[/itex]? Why does it not appear in the authors' solution?

That's precisely the point of this exercise.
The derivation is simple: consider a bunch of particles in a range [a,a+da],
and apply Newtons law on this bunch of particles.
You will immediately get the result.
The point is that

The gradient term (convective term) is precisely the variation of velocity that comes from "following the fluid".
The Euler view, where a is taken constant, follows the fluid.
Therefore the "Euler derivative" incorporates the convective term: it is the change of speed when following the fluid.