Stationary wave solutions to two dimensional viscous shallow water equations: theory of small and large solutions

TL;DR

The paper develops a comprehensive theory of stationary waves for a forced, viscous shallow water system with bathymetry in two dimensions. It advances both periodic and solitary settings by first proving small-data well-posedness via a real-analytic implicit function framework and weighted-space analysis, then constructing large-data solution curves using an analytic global implicit function theorem and detailed a priori estimates. The work reveals two complementary regimes: periodic waves and localized solitary waves, each supporting small solutions and global continuation with explicit blowup scenarios that tie directly to physical constraints such as bottom contact and unbounded forcing. The results represent the first general construction of stationary large- and small-amplitude viscous shallow water waves with bathymetry, providing a rigorous foundation for understanding stationary viscous free-boundary flows in shallow regimes with forcing.

Abstract

We study a system of forced viscous shallow water equations with nontrivial bathymetry in two spatial dimensions. We develop a well-posedness theory for small but arbitrary forcing data, as well as for a fixed data profile but large amplitude. In the latter case, solutions may actually fail to exist for large amplitude, but in this case we prove that one of three physically meaningful breakdown scenarios occurs. Through the use of implicit function theorem techniques and a priori estimates, we construct both spatially periodic and solitary (non-periodic but spatially localized) solutions. The solitary case is substantially more complicated, requiring a delicate analysis in weighted Sobolev spaces. To the best of our knowledge, these results constitute the first general construction of stationary wave solutions, large or otherwise, to the viscous shallow water equations and the first general analysis of large solitary wave solutions to any viscous free boundary fluid model.

Figures (2)

• Figure 1: Depicted here are the height data from numerical simulations of periodic stationary solutions to \ref{['time-dependent variable-batheymetry shallow water equations']}, in contour plot form (top) and graph form (bottom), for three values of applied forcing strength ${\upkappa}\in\{{20000,90000,280000}\}$. The bathymetry $b$ (shown in orange on the bottom) is a fixed half-ellipse with base at height $-2$ and maximum at height $0$. The forcing profile $F$ is a component of gravity in the $e_1$ direction, i.e. $\varphi=0$, $\Xi=0$, $\xi=0$, and $f=e_1$. The surface satisfies $h=1+h_0$ for an average zero perturbation $h_0$. The other physical parameters are $\alpha=0.65$, $g=0.5$, $\mu=0.3$, and $\sigma=1.1$. This is meant to depict stationary shallow water flow over an inclined bathymetry.
• Figure 2: Shown here are are numerical simulations of the divergence (top row) and curl (bottom row) of the tangential velocity $v$ of gravity driven shallow water flow over a fixed, but randomly generated bathymetry $b$ (shown on the left). These correspond to stationary solutions to system \ref{['time-dependent variable-batheymetry shallow water equations']} with forcing $F$ determined by $f=e_1$, $\varphi=0$, $\Xi=0$, and $\xi=0$. The forcing strength varies in the set ${\upkappa}\in\{{400000,1000000,2700000}\}$ while the other parameters $\alpha$, $g$, $\mu$, $\sigma$, and equilibrium height are all set to $1$.

Theorems & Definitions (119)

• Theorem 1: Proved in Theorem \ref{['thm on theory of small solutions']}
• Theorem 2: Proved in Theorem \ref{['thm on theory of large solutions']}
• Theorem 3: Proved in Theorem \ref{['thm on theory of small solutions']}
• Theorem 4: Proved in Theorem \ref{['thm on theory of large solutions']}
• Definition 2.1: Weighted Sobolev spaces
• Remark 2.2: Completeness
• Proposition 2.3: Equivalent norm on the weighted Sobolev spaces
• proof
• Definition 2.4: A choice of Sobolev inner product
• Remark 2.5: Pseudolocality