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

Advances in Mathematics

Tan

2018

Abstract. In recent work of Luo and Hou [10], a new scenario for finite time blow up in solutions of 3D Euler equation has been proposed. The scenario involves a ring of hyperbolic points of the flow located at the boundary of a cylinder. In this paper, we propose a two dimensional model that we call "hyperbolic Boussinesq system". This model is designed to provide insight into the hyperbolic point blow up scenario. The model features an incompressible velocity vector field, a simplified Biot-Savart law, and a simplified term modeling buoyancy. We prove that finite time blow up happens for a natural class of initial data.

Section: Introductionmentioning

confidence: 99%

Finite time blow up in the hyperbolic Boussinesq system

Advances in Mathematics

Tan

2018

“…[13]). Indeed, if in (1), (2), (3), (4) we re-label ω θ /r ≡ ω, ru θ ≡ θ, r = y, z = x, and set r = 1 in the coefficients, we obtain (10). Since in the Hou-Luo scenario, the fastest growth of vorticity is observed at the boundary of the cylinder r = 1, and in particular away from the axis, the analogy should apply.…”

Section: Derivation Of the Model Equationsmentioning

confidence: 99%

“…They established finite time blow up for a broad class of initial data. Hou and Liu [10] have described the blow up solutions in the CKY model in more detail, and showed that these solutions possess self-similar structure.…”

Section: Introductionmentioning

confidence: 99%

Stability of Blowup for a 1D Model of Axisymmetric 3D Euler Equation

2016

J Nonlinear Sci

Abstract. The question of the global regularity vs finite time blow up in solutions of the 3D incompressible Euler equation is a major open problem of modern applied analysis. In this paper, we study a class of one-dimensional models of the axisymmetric hyperbolic boundary blow up scenario for the 3D Euler equation proposed by Hou and Luo [12] based on extensive numerical simulations. These models generalize the 1D Hou-Luo model suggested in [12], for which finite time blow up has been established in [1]. The main new aspects of this work are twofold. First, we establish finite time blow up for a model that is a closer approximation of the three dimensional case than the original Hou-Luo model, in the sense that it contains relevant lower order terms in the Biot-Savart law that have been discarded in [12], [1]. Secondly, we show that the blow up mechanism is quite robust, by considering a broader family of models with the same main term as in the Hou-Luo model. Such blow up stability result may be useful in further work on understanding the 3D hyperbolic blow up scenario.

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mentioning

confidence: 99%

Analysis of a singular Boussinesq model

Yang

2018

Res Math Sci

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. This makes analysis harder, as there are two competing terms in the fluid velocity integral whose balance determines the regularity properties of the solution. The equation we study here is based on the CKY model introduced in [6]. We prove that finite time blow up persists in a certain range of parameters.