In this talk, we study the stationary linear and nonlinear Navier-Stokes equations on the hyperbolic space. This study includes discussing the existence of weak solutions to the linear stationary equation in two and three dimensions, proving the uniqueness of the weak solutions to the linear stationary problem in three dimensions, and examining the nonuniqueness in two dimensions. We also prove the existence of weak solutions to the nonlinear stationary equation. We then outline the construction of a coercive estimate for the linear stationary Navier-Stokes equation in two dimensions. Finally, we show how using the Hodge Laplacian in two dimensions requires us to look for weak solutions in a different function space, and we discuss an exterior domain problem in this case. This is a thesis defense.
Foundations of the steady Navier-Stokes equation on the hyperbolic space
Residuated lattices were introduced by Ward and Dilworth as tools in the study of ideal lattices of rings. Residuated lattices have a monoid and a lattice reduct, as well as division-like operations; examples include Boolean algebras, lattice-ordered groups and relation algebras. Also, they form algebraic semantics for substructural logics and are connected to mathematical linguistics and computer science (for example pointer management and memory allocation). We focus on a class of residuated lattices that have an idempotent multiplication and all elements are comparable to the monoid identity; these are related to algebraic models of relevance logic. After establishing a decomposition result for this class, we show that it has the strong amalgamation property, and extend the result to the variety generated by this class; this implies that the corresponding logic has the interpolation property and Beth definability.
Amalgamation for certain conic idempotent residuated lattices
Apr. 06, 2021 4pm (zoom: 994…
Thesis Defenses
Carlos E. Pinilla (CU Boulder)
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The Hodge decomposition is well-known for compact manifolds. The result has been extended by Kodaira to non-compact complete manifolds in the space of forms. We showed this decomposition with Chan and Czubak in the Sobolev space on a space form of a nonpositive sectional curvature. In this thesis we extend the decomposition to the Sobolev space for integers We also prove that this decomposition holds in the strong sense, depending on the dimension of the space and the degree of the differential form. This is a thesis defense.
Hodge decomposition for the Sobolev space on a space form of nonpositive sectional curvature
Foundations of the steady Navier-Stokes equation on the hyperbolic space