  Combining experimental data, numerical transport calculations, and
theoretical analysis, we study the temperature-dependent resistivity of
high-mobility 2D Si MOSFETs to search for signatures of weak localization
induced quantum corrections in the effective metallic regime above the critical
density of the so-called two-dimensional metal-insulator transition (2D MIT).
The goal is to look for the effect of logarithmic insulating localization
correction to the metallic temperature dependence in the 2D conductivity so as
to distinguish between the 2D MIT being a true quantum phase transition versus
being a finite-temperature crossover. We use the Boltzmann theory of
resistivity including the temperature dependent screening effect on charged
impurities in the system to fit the data. We analyze weak perpendicluar field
magnetoresistance data taken in the vicinity of the transition and show that
they are consistent with weak localization behavior in the strongly disordered
regime $k_F\ell\gtrsim1$. Therefore we supplement the Botzmann transport theory
with a logarithmic in temperature quantum weak localization correction and
analyze the competition of the insulating temperature dependence of this
correction with the metallic temperature dependence of the Boltzmann
conductivity. Using this minimal theoretical model we find that the logarithmic
insulating correction is masked by the metallic temperature dependence of the
Botzmann resistivity and therefore the insulating $\log T$ behavior may be
apparent only at very low temperatures which are often beyond the range of
temperatures accessible experimentally. Analyzing the low-$T$ experimental Si
MOSFET transport data we identify signatures of the putative insulating
behavior at low temperature and density in the effective metallic phase.
