On Ill-Posedness of Truncated Series Models for Water Waves

The evolution of surface gravity waves on a large body of water, such as an ocean or a lake, is reasonably well approximated by the Euler system for ideal, free–surface flow under the influence of gravity. The origins of this mathematical formulation of the water wave problem lie in the 18 century. The well-posedness theory for initialvalue problems for these equations, which has a long and distinguished history, reveals that solutions exist, are unique, and depend continuously upon initial data in various function–space contexts. This theory is subtle and the design of stable, accurate, numerical schemes is likewise challenging. Starting already in the 19 century, when concrete issues have arisen concerning wave propagation, attention has been turned to model equations which formally approximate the full, Euler system. This latter thrust, which also has a long history, has been a mainstay of developments in oceanography and theoretical fluid mechanics in the 20 century. Depending upon the wave regime in question, there are many different approximate models that can be formally derived from the Euler equations. As the Euler system is known to be well-posed, it seems appropriate that associated approximate models should also have this property, and indeed, certain approximations of the Euler equations are known to be well-posed. However, there are classes of weakly nonlinear models for which well-posedness has not been established. It is to this issue that the present essay is directed. Evidence is presented calling into question the well-posedness of a wellknown class of model equations which are widely used in simulations. A simplified version of these models is shown explicity to be ill-posed and numerical simulations of quadraticand cubic-order water-wave models, initiated with initial data predicted by