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This article is devoted to the proof of the well-posedness of a modeldescribing waves propagating in shallow water in horizontal dimension $d=2$ andin the presence of a fixed partially immersed object. We first show that thiswave-interaction problem reduces to an initial boundary value problem for thenonlinear shallow water equations in an exterior domain, with boundaryconditions that are fully nonlinear and nonlocal in space and time. Thishyperbolic initial boundary value problem is characteristic, does not satisfythe constant rank assumption on the boundary matrix, and the boundaryconditions do not satisfy any standard form of dissipativity. Our main resultis the well-posedness of this system for irrotational data and at thequasilinear regularity threshold. In order to prove this, we introduce a newnotion of weak dissipativity, that holds only after integration in time andspace. This weak dissipativity allows high order energy estimates withoutderivative loss; the analysis is carried out for a class of linearnon-characteristic hyperbolic systems, as well as for a class of characteristicsystems that satisfy an algebraic structural property that allows us to definea generalized vorticity. We then show, using a change of {unknowns}, that {it}is possible to transform the linearized wave-interaction {problem} into anon-characteristic system, {which} satisfies this structural property and forwhich the boundary conditions are weakly dissipative. We can therefore use ourgeneral analysis to derive linear, and then nonlinear, a priori energyestimates. Existence for the linearized problem is obtained by a regularizationprocedure that makes the problem non-characteristic and strictly dissipative,and by the approximation of the data by more regular data satisfying higherorder compatibility conditions for the regularized problem. Due to the fullynonlinear nature of the boundary conditions, it is also necessary to implementa quasilinearization procedure. Finally, we have to lower the standardrequirements on the regularity of the coefficients of the operator in thelinear estimates to be able to reach the quasilinear regularity threshold inthe nonlinear well-posedness result.

The $2D$ nonlinear shallow water equations with a partially immersed obstacle