A priori estimates for the moving contact line problem for the 2D nonlinear shallow water equations with a partially immersed obstacle

We consider the initial value problem for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>D</mml:mi> </mml:mrow> </mml:math> nonlinear shallow water model in the presence of a fixed partially immersed solid body. In this problem, we have a contact line where the solid body, the water, and the air meet, and whose projection on the horizontal plane moves freely even if the solid body is fixed. This wave-structure interaction problem reduces to an initial boundary value problem for the nonlinear shallow water equations in an exterior domain with a free boundary, for which we prove a priori energy estimates locally in time for solutions at the quasilinear regularity threshold under irrotationality and subcriticality assumptions. We use the weak dissipativity of the system, second order Alinhac good unknowns associated with a regularizing diffeomorphism, and a new type of hidden boundary regularity.