The Inviscid Voigt-Regularization for Hydrodynamic Models: Global Regularity, Boundary Conditions, and Blow-Up Phenomena

We prove global-wellposedness results for Voigt-regularized three-dimensional Euler equations for ideal incompressible fluids. We also prove higher-order and a Gevrey class (spatial analytic) regularity of solutions to these equations.;We establish the global well-posedness of two Voigt-regularizations of the three-dimensional inviscid Magnetohydrodynamic (MHD) equations. Weak solutions for one of these regularized models are also considered, and proven to exist globally in time, but the question of uniqueness for weak solutions is still open.;We prove global well-posedness theorems for the two-dimensional non-diffusive Boussinesq system with viscosity only in the horizontal direction, which arises in Ocean dynamics. We improve the best-known result in this direction, without needing the tools of harmonic analysis, used by other authors. We also study an inviscid Voigt-regularization for the two-dimensional inviscid, non-diffusive Boussinesq system of equations, which we call the Boussinesq-Voigt equations. Global regularity of this system is established.;Moreover, for each of the Voigt-regularized equations mentioned above, we show that the solutions of the Voigt-regularized system converge, as the regularization parameter alpha → 0, to strong solutions of the original equations, on the corresponding time interval of existence of the latter. Furthermore, we derive a criterion for finite-time blow-up of each system based on this inviscid regularization.;We provide some paradigms on the question of global well-posedness of nonlinear PDEs as related to their boundary conditions. In particular we present nonlinear PDE which are well-posed under periodic boundary conditions, but which experience finite-time blow-up of solutions under Dirichlet boundary conditions. The examples used here involve the viscous Hamilton-Jacobi equations and the Kuramoto-Sivashinsky equation.;Finally, we discuss some recent literature on the Navier-Stokes and Euler equations in which it is claimed, based on numerical evidence, that a specific alteration to the nonlinearity in the axi-symmetric Navier-Stokes equations leads the modified equations to develop a singularity on a finite interval of time. We compare this with the viscous Burgers equation to show analytically that a similar phenomenon occurs in this simpler case.