Blowup in Stagnation-point Form Solutions of the Inviscid 2d Boussinesq Equations

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 $\Omega=\{(x,y)\in[0,1]\times\mathbb{R}^+\}$, we consider velocities of the form $u=(f(t,x),-yf_x(t,x))$, with scalar temperature\, $\theta=y\rho(t,x)$. 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 $\rho(0,x)\geq0$ and $f(t,x_i^*)=\rho(0,x_i^*)=0$, while $-yf_{xx}(0,x_i^*)\neq0$. 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 $\left\|f_x(t,\cdot)\right\|_{L^\infty([0,1])}$ are also provided.Comment: Minor typos corrected and streamlined the presentatio

Blow-up of Solutions to the Generalized Inviscid Proudman-Johnson Equation

The generalized inviscid Proudman-Johnson equation serves as a model for n-dimensional incompressible Euler flow, gas dynamics, high-frequency waves in shallow waters, and orientation of waves in a massive director field of a nematic liquid crystal. Furthermore, the equation also serves as a tool for studying the role that the natural fluid processes of convection and stretching play in the formation of spontaneous singularities, or of their absence. In this work, we study blow-up, and blow-up properties, in solutions to the generalized, inviscid Proudman-Johnson equation endowed with periodic or Dirichlet boundary conditions. More particularly,regularity of solutions in an Lp setting will be measured via a direct approach which involves the derivation of representation formulae for solutions to the problem. For a real parameter lambda, several classes of initial data are considered. These include the class of smooth functions with either zero or nonzero mean, a family of piecewise constant functions, and a large class of initial data with a bounded derivative that is, at least, continuous almost everywhere and satisfies Holder-type estimates near particular locations in the domain. Amongst other results, our analysis will indicate that for appropriate values of the parameter, the curvature of the data in a neighborhood of these locations is responsible for an eventual breakdown of solutions, or their persistence for all time. Additionally, we will establish a nontrivial connection between the qualitative properties of L-infinity blow-up in ux, and its Lp regularity. Finally, for smooth and non-smooth initial data, a special emphasis is made on the study of regularity of stagnation point-form solutions to the two and three dimensional incompressible Euler equations subject to periodic or Dirichlet boundary conditions

Blow-up of Solutions to the Generalized Inviscid Proudman-Johnson Equation

The generalized inviscid Proudman-Johnson equation serves as a model for n-dimensional incompressible Euler flow, gas dynamics, high-frequency waves in shallow waters, and orientation of waves in a massive director field of a nematic liquid crystal. Furthermore, the equation also serves as a tool for studying the role that the natural fluid processes of convection and stretching play in the formation of spontaneous singularities, or of their absence. In this work, we study blow-up, and blow-up properties, in solutions to the generalized, inviscid Proudman-Johnson equation endowed with periodic or Dirichlet boundary conditions. More particularly,regularity of solutions in an Lp setting will be measured via a direct approach which involves the derivation of representation formulae for solutions to the problem. For a real parameter lambda, several classes of initial data are considered. These include the class of smooth functions with either zero or nonzero mean, a family of piecewise constant functions, and a large class of initial data with a bounded derivative that is, at least, continuous almost everywhere and satisfies Holder-type estimates near particular locations in the domain. Amongst other results, our analysis will indicate that for appropriate values of the parameter, the curvature of the data in a neighborhood of these locations is responsible for an eventual breakdown of solutions, or their persistence for all time. Additionally, we will establish a nontrivial connection between the qualitative properties of L-infinity blow-up in ux, and its Lp regularity. Finally, for smooth and non-smooth initial data, a special emphasis is made on the study of regularity of stagnation point-form solutions to the two and three dimensional incompressible Euler equations subject to periodic or Dirichlet boundary conditions