Extended Crystal PDEs

DOI10.1007/978-1-4939-1106-6_18zbMATH Open1329.58027arXiv0811.3693OpenAlexW1901325078MaRDI QIDQ5265271

Publication date: 23 July 2015

Published in: Mathematics Without Boundaries (Search for Journal in Brave)

Abstract: In this paper we show that between PDE's and crystallographic groups there is an unforeseen relation. In fact we prove that integral bordism groups of PDE's can be considered extensions of crystallographic subgroups. In this respect we can consider PDE's as {em extended crystals}. Then an algebraic-topological obstruction ({em 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 that solve some well-known fundamental problems. These results, are also extended to singular PDE's, introducing ({em extended crystal singular PDE's}). An application to singular MHD-PDE's, is given following some our previous results on such equations, and showing existence of (finite times stable smooth) global solutions crossing critical nuclear energy production zone.

Full work available at URL: https://arxiv.org/abs/0811.3693

zbMATH Keywords

conservation laws Navier-Stokes equation crystallographic groups singular PDEs Ricci flow equation existence of local and global solutions in PDEs integral (co)bordism groups in PDEs

Mathematics Subject Classification ID

PDEs in connection with biology, chemistry and other natural sciences (35Q92) Navier-Stokes equations (35Q30) Applications of PDEs on manifolds (58J90)