Nonlinear Singular Perturbation Phenomena: Theory and Applications

Springer Science & Business Media
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Our purpose in writing this monograph is twofold. On the one hand, we want to collect in one place many of the recent results on the exist ence and asymptotic behavior of solutions of certain classes of singularly perturbed nonlinear boundary value problems. On the other, we hope to raise along the way a number of questions for further study, mostly ques tions we ourselves are unable to answer. The presentation involves a study of both scalar and vector boundary value problems for ordinary dif ferential equations, by means of the consistent use of differential in equality techniques. Our results for scalar boundary value problems obeying some type of maximum principle are fairly complete; however, we have been unable to treat, under any circumstances, problems involving "resonant" behavior. The linear theory for such problems is incredibly complicated already, and at the present time there appears to be little hope for any kind of general nonlinear theory. Our results for vector boundary value problems, even those admitting higher dimensional maximum principles in the form of invariant regions, are also far from complete. We offer them with some trepidation, in the hope that they may stimulate further work in this challenging and important area of differential equa tions. The research summarized here has been made possible by the support over the years of the National Science Foundation and the National Science and Engineering Research Council.
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Publisher
Springer Science & Business Media
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Published on
Dec 6, 2012
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Pages
180
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ISBN
9781461211143
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Best For
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Language
English
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Genres
Mathematics / Calculus
Mathematics / Mathematical Analysis
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Content Protection
This content is DRM protected.
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Turbulence is the lIDst natural nDde of fluid lIDtion, and has been the subject of scientific study for all!Dst a century. During this period, various ideas and techniques have evolved to nDdel turbulence. Following Saffman, these theoretical approaches can be broadly divided into four overlapping categories -- (1) analytical lIDdelling, (2) physical lIDdelling, (3) phenomenologicalllDdelling, and (4) nurerical lIDdelling. With the purpose of stmtnarizing our =ent understanding of these theoretical approaches to turbulence, recognized leaders (fluid dynamicists, mathematicians and physicists) in the field were invited to participate in a formal workshop during October 10-12, 1984, sponsored by The Institute for CooIputer Applications in Science and Engineering and NASA Langley Research Center. Kraiciman, McCcxnb, Pouquet and Spiegel represented the category of analytical nDdelling, while Landahl and Saffman represented physical lIDdelling. The contributions of Latmder and Spalding were in the category of phenanenological lIDdelling, and those of Ferziger and Reynolds in the area of nurericalllDdelling. Aref, Cholet, Lumley, Moin, Pope and Temam served on the panel discussions. With the care and cooperation of the participants, the workshop achieved its purpose, and we believe that its proceedings published in this vol\. llre has lasting scientific value. The tone of the workshop was set by two introductory talks by Bushnell and ChaImm. Buslmell presented the engineering viewpoint while Chapman reviewed from a historical perspective developments in the study of turbulence. The remaining talks dealt with specific aspects of the theoretical approaches to fluid turbulence.
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