  Motivated by a new kind of initial boundary value problem (IBVP) with a free
boundary arising in wave-structure interaction, we propose here a general
approach to one-dimensional IBVP as well as transmission problems. For general
strictly hyperbolic 2x2 quasilinear hyperbolic systems, we derive new sharp
linear estimates with refined dependence on the source term and control on the
traces of the solution at the boundary. These new estimates are used to obtain
sharp results for quasilinear IBVP and transmission problems, and for fixed,
moving, and free boundaries. In the latter case, two kinds of evolution
equations are considered. The first one is of "kinematic type" in the sense
that the velocity of the interface has the same regularity as the trace of the
solution. Several applications that fall into this category are considered: the
interaction of waves with a lateral piston, and a new version of the well-known
stability of shocks (classical and undercompressive) that improves the results
of the general theory by taking advantage of the specificities of the
one-dimensional case. We also consider "fully nonlinear" evolution equations
characterized by the fact that the velocity of the interface is one derivative
more singular than the trace of the solution. This configuration is the most
challenging; it is motivated by a free boundary problem arising in
wave-structure interaction, namely, the evolution of the contact line between a
floating object and the water. This problem is solved as an application of the
general theory developed here.
