By a semi-Lagrangian change of coordinates, the hydrostatic Euler equations describing free-surface sheared flows is rewritten as a system of quasilinear equations, where stability conditions can be determined by the analysis of its hyperbolic structure. The system one obtains can be written as a quasi linear system in time and horizontal variables and involves no more vertical derivatives. However, the coefficients in front of the horizontal derivatives include an integral operator acting on the new vertical variable. The spectrum of these operators is studied in detail, in particular it includes a continuous part. Riemann invariants are then determined as conserved quantities along the characteristic curves. Examples of solutions are provided, in particular stationary solutions and solutions blowing-up in finite time. Eventually, we propose an exact multilayer $\mathbb{P}_0$-discretization, which could be used to solve numerically this semi-Lagrangian system, and analyze the eigenvalues of the corresponding discretized operator to investigate the hyperbolic nature of the approximated system.

翻译：通过半拉格朗日坐标变换，描述自由表面剪切流的静水欧拉方程组被重写为拟线性方程组，其稳定性条件可通过分析该系统的双曲结构来确定。所得方程组可表示为关于时间变量和水平变量的拟线性系统，且不再包含垂直方向导数项。然而，水平导数前的系数包含作用于新垂直变量的积分算子。本文详细研究了这些算子的谱特性，特别指出其包含连续谱部分。随后确定了沿特征曲线守恒的黎曼不变量。文中给出了若干解例，特别是稳态解和有限时间内爆破的解。最后，我们提出了一种精确的多层$\mathbb{P}_0$离散格式，该格式可用于数值求解此半拉格朗日系统，并通过分析相应离散化算子的特征值来研究近似系统的双曲性质。