Extended Crystal PDEs

Agostino Prástaro ()
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Agostino Prástaro: University of Roma La Sapienza, Department SBAI-Mathematics

A chapter in Mathematics Without Boundaries, 2014, pp 415-481 from Springer

Abstract: Abstract In this paper we show that between partial differential equations (PDEs) and crystallographic groups there is an unforeseen relation. In fact we prove that integral bordism groups of PDEs can be considered extensions of crystallographic subgroups. In this respect we can consider PDEs as extended crystals. Then an algebraic topological obstruction (crystal obstruction), characterizing existence of global smooth solutions for smooth boundary value problems, is obtained. Applications of this new theory to the Ricci flow equation and Navier–Stokes equation are given, which solve some well-known fundamental problems. These results are also extended to singular PDEs (introducing extended crystal singular PDEs). An application to singular magnetohydrodynamics partial differential equations (MHD-PDEs) is given following some our previous results on such equations, and showing existence of (finite times stable smooth) global solutions crossing the critical nuclear energy production zone.

Keywords: Integral (co)bordism groups in PDEs; Existence of local and global solutions in PDEs; Conservation laws; Crystallographic groups; Singular PDEs (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4939-1106-6_18

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DOI: 10.1007/978-1-4939-1106-6_18

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