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Extended Crystal PDEs

In: Mathematics Without Boundaries

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  • Agostino Prástaro

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

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.

Suggested Citation

  • Agostino Prástaro, 2014. "Extended Crystal PDEs," Springer Books, in: Themistocles M. Rassias & Panos M. Pardalos (ed.), Mathematics Without Boundaries, edition 127, pages 415-481, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4939-1106-6_18
    DOI: 10.1007/978-1-4939-1106-6_18
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