Problems on Large Domains
Many theoretical problems in continuum fluid mechanics are formulated on unbounded physical domains, most frequently on the whole Euclidean space ℝ3. Although, arguably, any physical but also numerical space is necessarily bounded, the concept of an unbounded domain offers a useful approximation in situations when the influence of the boundary on the behavior of the system can be neglected. The acoustic waves examined in the previous chapters are often ignored in meteorological models, where the underlying ambient space is large when compared with the characteristic speed of the fluid as well as the speed of sound. However, as we have seen in Chapter 5, the way the acoustic waves “disappear” in the asymptotic limit may include fast oscillations in the time variable that may produce undesirable numerical instabilities. In this chapter, we examine the incompressible limit of the Navier-Stokes-Fourier System in the situation when the spatial domain is large with respect to the characteristic speed of sound in the fluid. Remarkably, although very large, our physical space is still bounded exactly in the spirit of the leading idea of this book that the notions of “large” and “small” depend on the chosen scale.
KeywordsAcoustic Wave Uniform Estimate Large Domain Ambient Space Uniform Bound
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