  It is well-known that shock will form in finite time for hyperbolic
conservation laws from initial nonlinear compression no matter how small and
smooth the data are. Classical results, including Lax [14], Liu [22],
Li-Zhou-Kong [16], confirm that when initial data are small smooth
perturbations near constant states, blowup in gradient of solutions occurs in
finite time if and only if intial data contain any compression in some truly
nonlinear characteristic field. A natural puzzle is that: Will this picture
keep true for large data problem of physical systems such as compressible Euler
equations? One of the key issues is how to find an effective way to obtain
sharp enough control on density lower bound. For isentropic flow, we offer a
complete picture on the finite time shock formation from smooth initial data
away from vacuum, which is consistent with small data theory. For adiabatic
flow, we show a striking observation that initial weak compressions do not
necessarily develop singularity in finite time, in a sharp contrast to the
small data theory. Furthermore, we find the critical strength of nonlinear
compression, and prove that if the compression is stronger than this critical
value, then singularity develops in finite time, and otherwise there are a
class of initial data admitting global smooth solutions with maximum strength
of compression equals to this critical value.
