  Multi-valley spin relaxation in $n$-type GaAs quantum wells with in-plane
electric field is investigated at high temperature by means of kinetic spin
Bloch equation approach. The spin relaxation time first increases and then
decreases with electric field, especially when the temperature is relatively
low. We show that $L$ valleys play the role of a ``drain'' of the total spin
polarization due to the large spin-orbit coupling in $L$ valleys and the strong
$\Gamma$-$L$ inter-valley scattering, and thus can enhance spin relaxation of
the total system effectively when the in-plane electric field is high. Under
electric field, spin precession resulting from the electric-field-induced
magnetic field is observed. Meanwhile, due to the strong $\Gamma$-$L$
inter-valley scattering as well as the strong inhomogeneous broadening in $L$
valleys, electron spins in $L$ valleys possess almost the same damping rate and
precession frequency as those in $\Gamma$ valley. This feature still holds when
a finite static magnetic field is applied in Voigt configuration, despite that
the $g$-factor of $L$ valleys is much larger than that of $\Gamma$ valley.
Moreover, it is shown that the property of spin precession of the whole system
is dominated by electrons in $\Gamma$ valley. Temperature, magnetic field, and
impurity can affect spin relaxation in low electric field regime. However, they
are shown to have marginal influence in high electric field regime.
