Zero-Dissipation and Zero-Dispersion Limits Arising in Fluid Mechanics

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The Principal Investigators will study several related asymptotic limits of nonlinear partial differential equations arising in fluid mechanics. Interest is focused on how solutions behave in limits where certain terms in the equations become increasingly negligible. For the situations in view here, these terms correspond to the physical effects of dissipation and dispersion. It is planned to work at two levels of complexity. The first, and easier level is that of model equations for wave propagation where nonlinear, dispersive and dissipative effects are all present. Both qualitative and quantitative information will be sought. The information obtained will yield information helpful to modelling near-shore zone processes. At a more complex level, it is planned to investigate various limits of the Navier-Stokes equations including the inviscid limit for the Navier-Stokes equations in bounded domains with fixed boundaries, and for statistical solutions with periodic boundary conditions (or in all of space). It is also intended to investigate how well the Navier-Stokes equations posed in a channel are modelled by dissipative nonlinear wave equations. The present award will support research on several interesting and important asymptotic limits for mathematical models. The kind of limits under consideration here arise in various areas of physics, mechanics, oceanography, materials science, biology, and elsewhere when partial differential equations are used as models. The problems considered here derive principally from fluid mechanics, but in so far as we are successful in our program, there is a broader implied scope. The zero-limits under study are those associated with dissipation and dispersion in fluid motion. In various modeling situations, one or the other of these effects may be ignored. The question then arises whether or not such approximations are justified and, if so, under what flow conditions and over what time scales. These questions are of theoretical and practical importance since the approximating equations are often easier to use. This award will support work that aims to address issues including fundamental points and aspects that arise in the use of these equations as descriptions of real phenomena.