Extended Crystal PDE's

TL;DR

This paper reveals a novel connection between PDEs and crystallographic groups, introducing the concept of extended crystals and a crystal obstruction to determine the existence of global solutions, with applications to fundamental equations like Ricci-flow and Navier-Stokes.

Contribution

It introduces the concept of extended crystals linking PDEs and crystallography, and develops a new obstruction theory for global solutions, with applications to key physical PDEs.

Findings

Extended crystal theory applies to Ricci-flow and Navier-Stokes equations.

Crystal obstruction characterizes the existence of smooth solutions.

Applications include solutions crossing critical nuclear energy zones.

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.