Finite Time Blow Up for a 1D Model of 2D Boussinesq System

Abstract

Abstract. The 2D conservative Boussinesq system describes inviscid, incompressible, buoyant fluid flow in gravity field. The possibility of finite time blow up for solutions of this system is a classical problem of mathematical hydrodynamics. We consider a 1D model of 2D Boussinesq system motivated by a particular finite time blow up scenario. We prove that finite time blow up is possible for the solutions to the model system.

On the Finite‐Time Blowup of a One‐Dimensional Model for the Three‐Dimensional Axisymmetric Euler Equations

“…finite-time blowup when a < 0, Castro and Córdoba (2010) Córdoba et al (2005), SQG analogy u = Hω ω t + uω x = 0 finite-time blow-up from smooth data HL-model, Hou and Luo (2013), 2d Boussinesq/3d axi-symmetric Euler analogy u x = Hω ω t + uω x = θ x θ t + uθ x = 0 finite time blow-up from smooth data, the main new result of this paper CKY-model, Choi et al (2015), simplified HL-model…”

“…In this subsection, we use an approach different from Choi et al (2015) to show the finite time blow up of the CKY model. Namely, we will prove that the entropy functional I(t) blows up in finite time, implying that the solution must have a finite time singularity as well.…”

Self-similar singularity of a 1D model for the 3D axisymmetric Euler equations

“…Inspired by the work of [15] and [17], Choi, Kiselev, and Yao proposed the following 1D model (we call it the CKY model for short) [7] on [ 0, 1]: This 1D model can be viewed as a simplified approximation to the 1D model proposed by Hou and Luo in [15], and its finite-time singularity from smooth initial data has been proved in [7]. Like the 1D model of Hou and Luo, the CKY model approximates the 3D axisymmetric Euler equations (1.1) on the boundary of the cylinder r = 1 with θ ∼ u The positivity of θ x (x, t) near the origin creates a compressive flow which is responsible for the finite-time singularity of this model (1.2), and we will use this fact in our construction in the next section.…”

One-dimensional model equations for hyperbolic fluid flow

“…Equations (1) and (2) are the vorticity form of 2D Euler equation. When we take u = ∇ ⊥ (−∆) − 1 2 ω, (1) becomes the surface quasi-geostrophic (SQG) equation, which has important applications in geophysics, or can be regarded as a toy model for the 3D-Euler equations.…”

“…This was begun in [2,1], where one-dimensional models for the 2D-Boussinesq and 3D axisymmetric Euler equations were introduced and blowup was proven.…”