Blowup with vorticity control for a 2D model of the Boussinesq equations

Abstract

We propose a system of equations with nonlocal flux in two space dimensions which is closely modeled after the 2D Boussinesq equations in a hyperbolic flow scenario. Our equations involve a simplified vorticity stretching term and Biot-Savart law and provide insight into the underlying intrinsic mechanisms of singularity formation. We prove stable, controlled finite time blowup involving upper and lower bounds on the vorticity up to the time of blowup for a wide class of initial data.

Small Scales and Singularity Formation in Fluid Dynamics

“…There are a few more recent papers that have contributed towards understanding the hyperbolic point blow up scenario. Two-dimensional simplified models of the 2D Boussinesq system have been considered in [40] and in [53]. In both cases, the derivative forcing term in (2.2) is replaced by a simpler sign-definite approximation θ x 1 , and the Biot-Savart law is replaced by a simpler version u = (−x 1 Ω(x, t), x 2 Ω(x, t)).…”

“…In both cases, the derivative forcing term in (2.2) is replaced by a simpler sign-definite approximation θ x 1 , and the Biot-Savart law is replaced by a simpler version u = (−x 1 Ω(x, t), x 2 Ω(x, t)). In [40], Ω takes form similar to the 2D Euler example (3.6). In [53], Ω is closely related but is also chosen to keep u incompressible.…”

Analysis of a singular Boussinesq model

“…The first two-dimensional models of the Hou-Luo scenario have been considered in [12,18]. Both models are set in the first quadrant of the plane (implicitly assuming odd symmetry of the solution) and are given by…”

“…In [18] a slightly different integration domain D = {(y 1 , y 2 ) : y 1 y 2 ≥ x 1 x 2 } is chosen in the definition of Ω. The choice of the Ω in [18] leads to incompressible fluid velocity, while the velocity in [12] is not incompressible but is closer in form to the velocity representation for the 2D Euler solutions established in [17]. Also, both models use simplified mean field forcing term ρ/x 1 , which ensures that vorticity has fixed sign.…”

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Recently, a new singularity formation scenario for the 3D axi-symmetric Euler equation and the 2D inviscid Boussinesq system has been proposed by Hou and Luo based on extensive numerical simulations [15,16]. As the first step to understand the scenario, models with simplified sign-definite Biot-Savart law and forcing have recently been studied in [7,6,8,12,14,18]. In this paper, we aim to bring back one of the complications encountered in the original equation -the sign changing kernel in the Biot-Savart law.

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