A multivariable calculus exercise in the weak formulation of the Navier Stokes Equations The following is an excerpt from Robinson's An introduction to the classical theory of the
Navier–Stokes equations.

Here (1.1):

Using integration by parts I can get the $(\nabla u,\nabla\varphi)$ term from the second term of (1.1). But I don't get $\int_0^T(u,\varphi_t)$ and $-(u_0,\varphi)$. Also, the signs in (1.8) are confusing. Would anyone help me to derive (1.8) from (1.1)?

The sign is wrong in the pressure term.
The integration by parts in time gives
$$
\int_0^T (u_t,\varphi)=(u(T),\varphi(T))-(u(0),\varphi(0))-\int_0^T (u,\varphi_t)=
-(u(0),\varphi(0))-\int_0^T (u,\varphi_t).
$$
The correct formula should be
$$
\int_0^T[(u,\varphi_t)-(\nabla u,\nabla\varphi)-((\nabla\cdot u)u,\varphi)-(\nabla p,\varphi)]dt=-(u(0),\varphi(0))
$$
