论文标题
浅水方程中的界面奇异性的演变，底部地形可变
Evolution of interface singularities in shallow water equations with variable bottom topography
论文作者
论文摘要
在双曲长波模型的形式主义中研究了具有非平凡底部形状的波前传播。检查了非平滑初始数据的演变，特别是描述了单数点的分裂及其短时行为。在更长的时间的相反极限中，波前的局部分析用于估计梯度灾难的形成以及如何影响地形。检查自由表面与所谓的“物理”和“非物理”真空类相交时的限制案例。 Solutions expressed by power series in the spatial variable lead to a hierarchy of ordinary differential equations for the time-dependent series coefficients, which are shown to reveal basic differences between the two vacuum cases: for non-physical vacuums, the equations of the hierarchy are recursive and linear past the first two pairs, while for physical vacuums the hierarchy is non-recursive, fully coupled and nonlinear.前一种情况可能会允许在非零时间间隔中没有奇异性的解决方案，而后者则显示出非标准的速度冲击。我们表明，只有在二次底部轮廓的情况下，通常才有可能对有限尺寸系统和多项式解决方案进行截断。在这种情况下，系统的演化可以减少到并完全描述为时间相关系数的低维动力系统。该系统将解决方案的所有非线性特性用于通用功率系列初始数据，尤其是在干燥点有限时间内的规律性丧失。对于抛物线底部地形的特殊情况，介绍和研究了一个精确的，相似的解决方案类别，以通过封闭形式表达式说明一般结果。
Wave front propagation with non-trivial bottom topography is studied within the formalism of hyperbolic long wave models. Evolution of non-smooth initial data is examined, and in particular the splitting of singular points and their short time behaviour is described. In the opposite limit of longer times, the local analysis of wavefronts is used to estimate the gradient catastrophe formation and how this is influenced by the topography. The limiting cases when the free surface intersects the bottom boundary, belonging to the so-called "physical" and "non-physical" vacuum classes, are examined. Solutions expressed by power series in the spatial variable lead to a hierarchy of ordinary differential equations for the time-dependent series coefficients, which are shown to reveal basic differences between the two vacuum cases: for non-physical vacuums, the equations of the hierarchy are recursive and linear past the first two pairs, while for physical vacuums the hierarchy is non-recursive, fully coupled and nonlinear. The former case may admit solutions that are free of singularities for nonzero time intervals, while the latter is shown to develop non-standard velocity shocks instantaneously. We show that truncation to finite dimensional systems and polynomial solutions is in general only possible for the case of a quadratic bottom profile. In this case the system's evolution can reduce to, and is completely described by, a low dimensional dynamical system for the time-dependent coefficients. This system encapsulates all the nonlinear properties of the solution for general power series initial data, and in particular governs the loss of regularity in finite times at the dry point. For the special case of parabolic bottom topographies, an exact, self-similar solution class is introduced and studied to illustrate via closed form expressions the general results.
