  This article is devoted to the proof of the well-posedness of a model
describing waves propagating in shallow water in horizontal dimension $d=2$ and
in the presence of a fixed partially immersed object. We first show that this
wave-interaction problem reduces to an initial boundary value problem for the
nonlinear shallow water equations in an exterior domain, with boundary
conditions that are fully nonlinear and nonlocal in space and time. This
hyperbolic initial boundary value problem is characteristic, does not satisfy
the constant rank assumption on the boundary matrix, and the boundary
conditions do not satisfy any standard form of dissipativity. Our main result
is the well-posedness of this system for irrotational data and at the
quasilinear regularity threshold. In order to prove this, we introduce a new
notion of weak dissipativity, that holds only after integration in time and
space. This weak dissipativity allows high order energy estimates without
derivative loss; the analysis is carried out for a class of linear
non-characteristic hyperbolic systems, as well as for a class of characteristic
systems that satisfy an algebraic structural property that allows us to define
a generalized vorticity. We then show, using a change of {unknowns}, that {it}
is possible to transform the linearized wave-interaction {problem} into a
non-characteristic system, {which} satisfies this structural property and for
which the boundary conditions are weakly dissipative. We can therefore use our
general analysis to derive linear, and then nonlinear, a priori energy
estimates. Existence for the linearized problem is obtained by a regularization
procedure that makes the problem non-characteristic and strictly dissipative,
and by the approximation of the data by more regular data satisfying higher
order compatibility conditions for the regularized problem. Due to the fully
nonlinear nature of the boundary conditions, it is also necessary to implement
a quasilinearization procedure. Finally, we have to lower the standard
requirements on the regularity of the coefficients of the operator in the
linear estimates to be able to reach the quasilinear regularity threshold in
the nonlinear well-posedness result.
