Co)Bordism Groups in PDEs

Abstract

We introduce a geometric theory of PDEs, by obtaining existence theorems of smooth and singular solutions. Within this framework, following our previous results on (co)bordisms in PDEs, we give characterizations of quantum and integral (co)bordism groups and relate them to the formal integrability of PDEs. An explicit proof that the usual Thom–Pontryagin construction in (co)bordism theory can be generalized also to a singular integral (co)bordism on the category of differential equations is given. In fact, we prove the existence of a spectrum that characterizes the singular integral (co)bordism groups in PDEs. Moreover, a general method that associates, in a natural way, Hopf algebras (fullp-Hopf algebras, 0 ≤ p ≤ n − 1), to any PDE, recently introduced, is further studied. Applications to particular important classes of PDEs are considered. In particular, we carefully consider the Navier–Stokes equation (NS) and explicitly calculate their quantum and integral bordism groups. An existence theorem of solutions of (NS) with a change in sectional topology is obtained. Relations between integral bordism groups and causal integral manifolds, causal tunnel effects, and the full p-Hopf algebras, 0 ≤ p ≤ 3, for the Navier–Stokes equation are determined.