Continuous dependence on the density for stratified steady water waves

Monday, September 19, 2016 - 15:00
Thackeray 703
Speaker Information
University of Pittsburgh

Abstract or Additional Information

There are two distinct regimes commonly used to model traveling waves in  stratified water: continuous stratification, where the density is smooth throughout the fluid, and layer-wise continuous stratification, where the fluid consists of multiple immiscible strata. The former is the more physically accurate description, but the latter is frequently more amenable to analysis and computation. By the conservation of mass, the density is constant along the streamlines of the flow; the stratification can therefore be specified by prescribing the value of the density on each streamline. We call this the streamline density function. Intuitively speaking, one expects that it is possible to use layer-wise smooth waves (for which the streamline density function is piecewise smooth) to approximate smoothly stratified waves (for which the streamline density function is of course smooth).

In this talk, we will discuss some recent work in this direction. Our main result states that, for every smoothly stratified periodic traveling wave in a certain small-amplitude regime, there is an $L^\infty$ neighborhood of its streamline density function such that, for any piecewise smooth streamline density function in that neighborhood, there is a corresponding traveling wave solution. Moreover, the mapping from streamline density function to wave is Lipschitz continuous in a certain function space framework. As this neighborhood includes piecewise smooth densities with arbitrarily many jump discontinues, this theorem provides a rigorous justification for the ubiquitous practice of approximating a smoothly stratified wave by a layered one.

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