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In light of the question of finite-time blow-up vs. global well-posedness ofsolutions to problems involving nonlinear partial differential equations, weprovide several cautionary examples which indicate that modifications to theboundary conditions or to the nonlinearity of the equations can effect whetherthe equations develop finite-time singularities. In particular, we aim tounderscore the idea that in analytical and computational investigations of theblow-up of three-dimensional Euler and Navier-Stokes equations, the boundaryconditions may need to be taken into greater account. We also examine aperturbation of the nonlinearity by dropping the advection term in theevolution of the derivative of the solutions to the viscous Burgers equation,which leads to the development of singularities not present in the originalequation, and indicates that there is a regularizing mechanism in part of thenonlinearity. This simple analytical example corroborates recent computationalobservations in the singularity formation of fluid equations.

Global Regularity vs. Finite-Time Singularities: Some Paradigms on the Effect of Boundary Conditions and Certain Perturbations