In this paper we study the Ostrovsky–Hunter equation for the case where the flux function f(x, u) may depend on the spatial variable with certain smoothness. Our main results are that if the flux function is smooth enough (namely f_x(x, u) is uniformly Lipschitz locally in u and f_u(x, u) is uniformly bounded), then there exists a unique entropy solution. To show the existence, after proving some a priori estimates we have used the method of compensated compactness and to prove the uniqueness we have employed the method of doubling of variables.

中文翻译：

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具有空间相关通量的Ostrovsky-Hunter方程初值问题的适定性

在本文中，对于通量函数f（x，u）可能依赖于具有一定平滑度的空间变量的情况，我们研究了Ostrovsky-Hunter方程。我们的主要结果是，如果通量函数足够平滑（即f _x（x，u）在u中均匀地局部Lipschitz且f _u（x，u）被均匀定界），则存在唯一的熵解。为了证明其存在，在证明了一些先验估计之后，我们使用了补偿紧致度方法，并且为了证明唯一性，我们采用了加倍变量的方法。