In this work, we investigate the Cauchy problem for a generalized Riemann-type hydrodynamical equation. The local well-posedness of the equation in Besov spaces is derived by using Littlewood–Paley decomposition and transport equation theory. Then, we show that a finite maximal life span for a solution necessarily implies wave breaking for this solution and give a condition on the initial data to ensure wave breaking for this equation by making use of the method of characteristics; otherwise, the equation has a global smooth solution. In addition, we establish persistence results for solutions of the equation in weighted L^p spaces for a large class of moderate weights.

中文翻译：

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广义Riemann型水动力学方程的Cauchy问题。

在这项工作中，我们研究了广义Riemann型流体动力学方程的Cauchy问题。利用Littlewood–Paley分解和输运方程理论推导了Besov空间中方程的局部适定性。然后，我们证明一个有限的最大使用寿命将必然意味着该解决方案发生了波浪破裂，并通过使用特征方法为初始方程提供了一个条件，以确保该方程的波浪破裂。否则，该方程具有全局光滑解。此外，我们为一类中等权重的加权L ^p空间建立了方程解的持久性结果。