Unique Continuation Property for the Coupled Ostrovsky System

In this paper, we establish a result of unique continuation for the coupled Ostrovsky system that models the evolution of long water waves with small amplitude in the presence of surface tension. More precisely, we will show that if (u,v)=(u(x,t),v(x,t))$(u,v) =(u(x, t), v(x, t))$ is a solution of the nonlinear system, in a suitable function space, and (u,v)$(u,v)$ vanishes on an open subset Ω$\Omega$ of R×[−T,T]$\mathbb {R}\times [-T, T]$, then (u,v)≡0$(u,v)\equiv 0$ in the horizontal component of Ω$\Omega$. To state such property, we use a Carleman‐type estimate for a differential operator L$\mathcal {L}$ related to the system. We prove the Carleman estimate using a particular version of the well‐known Treves' inequality.