  We are concerned with a two-dimensional ($2$-D) Riemann problem for
compressible flows modeled by the pressure gradient system that is a $2$-D
hyperbolic system of conservation laws. The Riemann initial data consist of
four constant states in four sectorial regions such that two shock waves and
two vortex sheets are generated between the adjacent states. This Riemann
problem can be reduced to a boundary value problem in the self-similar
coordinates with the Riemann initial data as its asymptotic boundary data,
along with two sonic circles determined by the Riemann initial data, for a
nonlinear system of mixed-composite type. The solutions keep the four constant
states and four planar waves outside the outer sonic circle. The two shocks
keep planar until they meet the outer sonic circle at two different points and
then generate a diffracted shock to be expected to connect these two points,
whose exact location is {\it apriori} unknown which is regarded as a free
boundary. Then the $2$-D Riemann problem can be reformulated as a free boundary
problem, in which the diffracted transonic shock is the one-phase free boundary
to connect the two points, while the other part of the outer sonic circle forms
the part of the fixed boundary of the problem. We establish the global
existence of a solution of the free boundary problem, as well as the
$C^{0,1}$--regularity of both the diffracted shock across the two points and
the solution across the outer sonic boundary which is optimal. One of the key
observations here is that the diffracted transonic shock can not intersect with
the inner sonic circle in the self-similar coordinates. As a result, this $2$-D
Riemann problem is solved globally, whose solution contains two vortex sheets
and one global $2$-D shock connecting the two original shocks generated by the
Riemann data.
