[Paper Review] Wave breaking in a shallow water model

Vera Mikyoung Hur, Lizheng Tao|arXiv (Cornell University)|Aug 16, 2016

Advanced Mathematical Physics Problems10 references3 citations

TL;DR

This paper establishes wave breaking—where solutions remain bounded but their spatial derivatives become unbounded—in a bidirectional shallow water model that generalizes the Whitham equation by incorporating the full dispersion relation of water waves. Using rigorous analysis of a nonlocal, nonlinear PDE system, the authors prove finite-time blow-up of the solution gradient when the initial data's slope is sufficiently negative, confirming Whitham's conjecture on wave breaking in a physically accurate, dispersive shallow water framework.

ABSTRACT

We prove wave breaking --- bounded solutions with unbounded derivatives --- in the nonlinear nonlocal equations which combine the dispersion relation of water waves and the nonlinear shallow water equations, and which generalize the Whitham equation to permit bidirectional wave propagation, provided that the slope of the initial data is sufficiently negative.

Motivation & Objective

• To resolve the long-standing problem of wave breaking in water wave theory by developing a model that captures both nonlinearity and realistic dispersion.
• To extend the Whitham equation to allow bidirectional wave propagation while preserving the correct dispersion relation of water waves.
• To rigorously prove that solutions to this generalized model exhibit wave breaking—bounded solutions with unbounded derivatives—in finite time.
• To address the limitations of both the nonlinear shallow water equations (which always break) and the KdV equation (which never breaks) by introducing a balanced, physically accurate model.
• To confirm Whitham's conjecture that a suitable nonlocal, nonlinear equation with full dispersion can capture wave breaking phenomena.

Proposed method

• Formulates a bidirectional shallow water model by combining the nonlinear shallow water equations with the full dispersion relation of water waves, represented via a Fourier multiplier operator.
• Derives the governing equations as a system of nonlocal, nonlinear PDEs: ∂tη + ∂x(u(1 + aη)) = 0 and ∂t u + ∂xη + a u ∂x u = 0, where the dispersion is modeled by the Whitham-type operator M1/2 with symbol c(ξ) = tanh(ξ)/ξ.
• Applies energy estimates and a priori bounds to control solution behavior in Sobolev spaces, using the structure of the nonlocal kernel K(x) = (1/2π) ∫ e^{ixξ} c(ξ) dξ.
• Employs a contradiction argument based on the evolution of the maximum slope and velocity, tracking the behavior of v1(t;x) = ∂x u(t;x) and m(t) = max_x |∂x u(t;x)|.
• Uses integral estimates and logarithmic bounds on the kernel K to control the nonlocal term K1(t;x), proving that it remains small relative to the local nonlinear term.
• Applies Stirling’s inequality and combinatorial estimates to bound the growth of higher-order terms in the energy estimates, ensuring the blow-up condition is met.

Experimental results

Research questions

• RQ1 Can a bidirectional shallow water model with the full dispersion relation of water waves exhibit wave breaking, where solutions remain bounded but their derivatives blow up in finite time?
• RQ2 Does the inclusion of realistic dispersion—via the Whitham dispersion relation c(ξ) = tanh(ξ)/ξ—prevent the unphysical wave breaking predicted by the nonlinear shallow water equations?
• RQ3 Under what conditions on the initial data does wave breaking occur in this generalized Whitham-Boussinesq model?
• RQ4 Is Whitham's conjecture—that a nonlocal, nonlinear equation with full dispersion can capture wave breaking—rigorously provable?
• RQ5 How does the balance between nonlinearity and dispersion in this model affect the formation of singularities in the solution gradient?

Key findings

• Wave breaking occurs in the bidirectional shallow water model when the initial slope is sufficiently negative, leading to unbounded derivatives in finite time despite bounded solutions.
• The solution remains uniformly bounded in L∞, but its spatial derivative ∂x u becomes unbounded as t approaches a finite blow-up time T*.
• The blow-up is driven by the nonlinear term −u ∂x u, which dominates the nonlocal dispersion term when the slope steepens, leading to a finite-time gradient blow-up.
• The nonlocal kernel K(x) is shown to satisfy the bound |K_n(t;x)| ≤ C(ε) (δ^ε ‖ζ_n(t)‖_{L∞} + δ^{1−ε} ‖ζ_{n+1}(t)‖_{L∞}) for any ε > 0, ensuring control over the nonlocal term.
• The proof relies on a contradiction argument: if a point x1 exits the set Σ_γ(t) where the slope is large, it cannot re-enter, leading to a contradiction if wave breaking does not occur.
• The analysis confirms that the Whitham equation with full dispersion (c(ξ) = tanh(ξ)/ξ) supports wave breaking, validating Whitham’s long-standing conjecture.