Well-posedness of the Initial Value Problem for the Ostrovsky–Hunter Equation with Spatially Dependent Flux

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.