A Liouville Type Theorem For The Steady Water Waves

We prove some Liouville type theorems for steady water waves. For periodic waves, if the aggregate pressure in a period is large relative the L1, L2 norms of the water wave function, there is no wave. In other words, we use the L1 and L2norms of the water wave function to give upper bounds for the aggregate pressure and the total energy of vertical motion. For solitary waves, we also use these norms to bound the aggregate di?erence of pressure and the pressure at in?nity. Hence, we use the observable water waves to get properties of variables that are di?cult to be observed. To get the results, we apply Chae’s method in proving the Liouville theorem for the incompressible Euler equations. As the water wave is part of the boundary, the boundary terms from the integration by parts in the reasoning provide the L1 and L2norms of the water. The aggregate pressure and vertical energy appearing in the same formula can thus be estimated by them.