A Unique Continuation Result for a 2D System of Nonlinear Equations for Surface Waves

Abstract

AbstractIn this paper, we establish a result of unique continuation for a special two-dimensional nonlinear 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 $$(\eta ,\Phi ) = (\eta (x,y, t),\Phi (x,y, t))$$ ( η , Φ ) = ( η ( x , y , t ) , Φ ( x , y , t ) ) is a solution of the nonlinear system, in a suitable function space, and $$(\eta ,\Phi )$$ ( η , Φ ) vanishes on an open subset $$\Omega $$ Ω of $$\mathbb {R}^2 \times [-T,T],$$ R 2 × [ - T , T ] , then $$(\eta ,\Phi )\equiv 0$$ ( η , Φ ) ≡ 0 in the horizontal component of $$\Omega .$$ Ω . To state such property, we use a Carleman-type estimate for a differential operator $$\mathcal {L}$$ L related to the system. We prove the Carleman estimate using a particular version of the well known Treves’ inequality.