For arbitrary values of a parameter λ ∈ R, finite-time blowup of solutions to the generalized, inviscid Proudman-Johnson equation is studied via a direct approach which involves the derivation of representation formulae for solutions to the problem. Mathematics Subject Classification (2010). 35B44, 35B10, 35B65, 35Q35.

In [20], we derived representation formulae for spatially periodic solutions to the generalized, inviscid Proudman-Johnson equation and studied their regularity for several classes of initial data. The purpose of this paper is to extend these results to larger classes of functions including those having arbitrary local curvature near particular points in the domain. Mathematics Subject Classification (2010). 35B44, 35B10, 35B65, 35Q35. Lastly, let P C R (0, 1) denote the family of piecewise constant functions with zero mean in [0, 1]. Then, in [20] we proved the following:Theorem 1.5. For the initial boundary value problem (1.1)-(1.2), 1. Suppose u 0 (x) ∈ P C R (0, 1) and λ > 1/2. Then, there exist solutions and a finite t * > 0 for which ux undergoes a two-sided, everywhere blow-up as t ↑ t * . If λ < 0, a one-sided discrete blow-up may occur instead. In contrast, for λ ∈ [0, 1/2], solutions may persist globally in time. More particularly, these either vanish as t ↑ t * = +∞ if λ ∈ (0, 1/2), or converge to a nontrivial steady-state for λ = 1/2.

The 2d Boussinesq equations model large scale atmospheric and oceanic flows. Whether its solutions develop a singularity in finite-time remains a classical open problem in mathematical fluid dynamics. In this work, blowup from smooth nontrivial initial velocities in stagnation-point form solutions of this system is established. On an infinite strip = {(x, y) ∈ [0, 1] × R + }, we consider velocities of the form u = (f (t, x), −yf x (t, x)), with scalar temperature θ = yρ(t, x). Assuming f x (0, x) attains its global maximum only at points x * i located on the boundary of [0, 1], general criteria for finite-time blowup of the vorticity −yf xx (t, x * i ) and the time integral of f x (t, x * i ) are presented. Briefly, for blowup to occur it is sufficient that ρ(0, x) ≥ 0 and f (t, x * i ) = ρ(0, x * i ) = 0, while −yf xx (0, x * i ) = 0. To illustrate how vorticity may suppress blowup, we also construct a family of global exact solutions. A local-existence result and additional regularity criteria in terms of the time integral of f x (t, ·) L ∞ ([0,1]) are also provided.

The generalized, two-component Hunter-Saxton system comprises several well-known models of fluid dynamics and serves as a tool for the study of one-dimensional fluid convection and stretching. In this article a general representation formula for periodic solutions to the system, which is valid for arbitrary values of parameters (λ, κ) ∈ R × R, is derived. This allows us to examine in great detail qualitative properties of blow-up as well as the asymptotic behaviour of solutions, including convergence to steady states in finite or infinite time.

In this work, we present analytic formulas for calculating the critical buckling states of some plastic axial columns of constant crosssections. The associated critical buckling loads are calculated by Eulertype analytic formulas and the associated deformed shapes are presented in terms of generalized trigonometric functions. The plasticity of the material is defined by the Hollomon's power-law equation. This is an extension of the Euler critical buckling loads of perfect elastic columns to perfect plastic columns. In particular, critical loads for perfect straight plastic columns with circular and rectangular cross-sections are calculated for a list of commonly used metals. Connections and comparisons to the classical result of the Euler-Engesser reduced-modulus loads are also presented.