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<p>In this paper, we investigate the long-time existence of solutions for a Boussinesq-like system modeling surface water waves in shallow water with <i>strong-bottom topography variations</i> (i.e., <i>b</i> = <i>O</i>(1), where <i>b</i> measures the bathymetry). The system is derived under the <i>long-wavelength, small-amplitude</i> regime, with <i>h</i> = 1 − <i>b</i> representing the mean water depth. The strong-bottom topography may exhibit <i>slow</i> (∇<i>h</i> = <i>O</i>(<i>ε</i>)) or <i>fast</i> (∇<i>h</i> = <i>O</i>(1)) oscillations, where the small parameter <i>ε</i> measures the comparable long-wave and weak-nonlinearity effects. We establish the long-time existence results in two typical regimes.<OrderedList> <ListItem> <ItemNumber>(1)</ItemNumber> <ItemContent> <p>Slow oscillation (∇<i>h</i> = <i>O</i>(<i>ε</i>)). For both 1<i>D</i> and 2<i>D</i> cases, we prove the local existence on the time scale <i>O</i>(1/<i>ε</i>) under some restrictions on the parameters ensuring the symmetry of the main dispersive terms. To overcome key challenges arising from <i>h</i>-dependent coefficients and the incomplete control of ∇<Emphasis Type="BoldItalic">V</Emphasis> in 2<i>D</i>, we design adapted energy functionals and exploit the cancellation of lower-order linear terms in 2<i>D</i>.</p> </ItemContent> </ListItem> <ListItem> <ItemNumber>(2)</ItemNumber> <ItemContent> <p>Fast oscillation (∇<i>h</i> = <i>O</i>(1)). In 1<i>D</i>, we extend the natural <i>O</i>(1) existence time to <i>O</i>(1/<i>ε</i>) for a special case, maintaining the symmetric principal dispersive structure. The proof uses a strategy that first establishes time regularity before recovering spatial derivatives, supported by carefully designed energy estimates and the cancellation of lower-order terms.</p> </ItemContent> </ListItem> </OrderedList></p><p>These results provide the first rigorous long-time existence theory for the Boussinesq-like system with strong topography variations in both 1<i>D</i> and 2<i>D</i>, significantly extending previous studies limited to fully symmetric cases.</p>

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Long-time existence for a Boussinesq-like system with strong topography variations

  • 期刊论文

  • 发表日期：

    2026年6月18日

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  • Qi Li,
  • Jean-Claude Saut,
  • Li Xu

摘要

In this paper, we investigate the long-time existence of solutions for a Boussinesq-like system modeling surface water waves in shallow water with strong-bottom topography variations (i.e., b = O(1), where b measures the bathymetry). The system is derived under the long-wavelength, small-amplitude regime, with h = 1 − b representing the mean water depth. The strong-bottom topography may exhibit slow (∇h = O(ε)) or fast (∇h = O(1)) oscillations, where the small parameter ε measures the comparable long-wave and weak-nonlinearity effects. We establish the long-time existence results in two typical regimes. (1)

Slow oscillation (∇h = O(ε)). For both 1D and 2D cases, we prove the local existence on the time scale O(1/ε) under some restrictions on the parameters ensuring the symmetry of the main dispersive terms. To overcome key challenges arising from h-dependent coefficients and the incomplete control of ∇V in 2D, we design adapted energy functionals and exploit the cancellation of lower-order linear terms in 2D.

(2)

Fast oscillation (∇h = O(1)). In 1D, we extend the natural O(1) existence time to O(1/ε) for a special case, maintaining the symmetric principal dispersive structure. The proof uses a strategy that first establishes time regularity before recovering spatial derivatives, supported by carefully designed energy estimates and the cancellation of lower-order terms.

These results provide the first rigorous long-time existence theory for the Boussinesq-like system with strong topography variations in both 1D and 2D, significantly extending previous studies limited to fully symmetric cases.