Some Wellposedness Results for the Ostrovsky–Hunter Equation

Oleinik type estimates for the Ostrovsky–Hunter equation

“…The two reformulations (1.8) and (1.10) of (1.1) are not equivalent. Therefore, the well-posedness result of [5,8] does not also apply to (1.8). Finally, the Kruzkov doubling of variables works for (1.10) but not for (1.8).…”

“…Second, it gives an Oleinik type estimate. We show that we can replace the Kruzkov-type entropy inequalities used in [5,8] with an Oleinik-type estimate and we prove the uniqueness via a nonlocal adjoint problem. This implies that a shock wave in an entropy weak solution to the Ostrovsky-Hunter equation is admissible only if it jumps down in value (similar to the inviscid Burgers' equation).…”

“…In agreement with the framework of [3][4][5]8], we introduce the following definition of the entropy solution of the initial value problem (1.1) and (1.2). Here, the Kruzkov doubling of variables does not work, therefore we use a one-sided Oleinik type estimate to prove the uniqueness of the entropy solutions.…”

“…A complete analysis of the well-posedness in this framework can be found in [5,8] under the additional condition that P (t, 0) = 0, which is natural in the reformulation of boundary value problems for (1.1) (see [3,4]). The equation analyzed in [5,8] is…”

“…We are interested in the bounded solutions of (1.1) (those in [5,8] are only locally bounded). Indeed, we have (1.5), which is an assumption on the decay at infinity of the initial condition u 0 .…”