Well-posedness of the initial value problem for the Ostrovsky-Hunter equation with spatially dependent flux

Abstract

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 (specified later), 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.