A Regularized Shallow Water System

Technical Summary: A Regularized Shallow Water System

Problem Statement
The classical shallow-water system (Equation 1.1) is a standard hyperbolic model for long waves in shallow water. While widely used in coastal engineering and river flow modeling, the system suffers from a fundamental mathematical limitation: for a large class of initial data, solutions develop steep gradients leading to shock formation (infinite derivatives) in finite time. This phenomenon, known as the "long-wave paradox," invalidates the model's underlying assumption of slowly varying flow conditions once singularities appear. While shocks can be physically interpreted as bores, their mathematical appearance restricts the regime of validity for smooth solutions. Existing regularizations often modify linear terms or introduce dissipation, which may alter the conservative structure of the system.

Methodology
The authors introduce a regularized shallow-water system (Equation 1.3) where the nonlinear flux terms are modified by a bounded operator $K$. The system is defined as:
$$\begin{aligned} \eta_t + h_0 u_x + K(\eta u)_x &= 0, \\ u_t + g \eta_x + K(u^2/2)_x &= 0, \end{aligned}$$
where $K$ is a regularizing operator chosen such that $K\partial_x$ is bounded in $L^2(\mathbb{R})$ while $K \approx 1$ in the shallow-water regime. A specific example provided is the Fourier multiplier $K = \frac{\tanh(h_0 D)}{h_0 D}$, where $D = -i\partial_x$.

The authors establish that this system possesses a Hamiltonian structure with a conserved energy functional $H(\eta, u)$ (Equation 1.5). The analysis proceeds in two main stages:

1. Local Well-posedness: Using the space $X^s = H^s \times H^s$ for $s > 1/2$, the Cauchy problem is reformulated as an integral equation. The authors employ Picard iteration and the contraction mapping principle to prove the existence of a unique local solution. They further demonstrate that the flow map is real analytic and that the solution is classical ($C^1$ in time).
2. Global Well-posedness (Small Data): To extend the local result globally, the authors utilize the conservation of the energy functional $H$. By establishing a relationship between the energy norm and the $H^{1/2}$ norm, they show that for sufficiently small initial data, the solution remains bounded in $H^{1/2}$. They then apply the Brezis-Gallouet-Wainger inequality to bound the $L^\infty$ norm of the solution in terms of the $H^s$ norm and logarithmic growth. Combining this with a Gronwall-type inequality allows them to prove that the $H^s$ norm does not blow up in finite time, ensuring global existence for small data.

Key Results

• Local Well-posedness: The system admits a unique solution in $C([0, T]; H^s \times H^s)$ for $s > 1/2$, with the solution depending real analytically on the initial data.
• Energy Conservation: The energy functional $H$ is conserved for solutions in the appropriate Sobolev spaces. The proof involves a regularization argument using frequency truncation to handle the non-smoothness of the energy functional in lower regularity spaces.
• Global Well-posedness: For initial data with sufficiently small $H^{1/2}$ norm, the solution exists globally in time. This contrasts with the classical system, where global smooth solutions do not exist for general initial data due to shock formation.
• Numerical Simulations:
  • Shock Suppression: Numerical comparisons between the standard system (1.2), the regularized system (1.3), and a fully dispersive model (4.1) show that while the standard system develops a shock, the regularized system maintains smooth profiles without visible shock formation.
  • Solitary Waves: The authors numerically compute solitary-wave solutions for the regularized system using the Petviashvili iteration method. The amplitude-speed relation of these solitary waves is found to be very close to that of the fully dispersive model and the full Euler equations, particularly for moderate speeds.
  • Accuracy: For high and fast waves, the regularized system exhibits accuracy comparable to the fully dispersive model when approximating full Euler solitary waves.

Significance and Claims
The paper claims that the proposed regularized system offers a mathematically rigorous alternative to the standard shallow-water equations that avoids the problem of singularity formation while remaining consistent with the long-wave regime. The primary contribution is the establishment of local and small-data global well-posedness in Sobolev spaces, thereby providing a framework where smooth solutions can be guaranteed to exist globally.

The authors note that while numerical simulations strongly suggest the system admits solitary waves and suppresses shocks, a rigorous analytical proof of global existence for arbitrary (large) data is not addressed in this work and remains a hypothesis. Similarly, a rigorous proof of the existence of solitary waves is left for future analytical work. The paper positions this model as a potential substitute for the standard system in practical applications, particularly where maintaining smooth solutions is critical, without resorting to dissipative schemes that might alter the physical nature of the wave propagation.