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

Abstract

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.

Small Scales and Singularity Formation in Fluid Dynamics

“…For the original HL model, finite time blow up has been proved in [12]. For the model including the additional term obtained from the boundary layer assumption into Biot-Savart law, finite time blow up has been proved in [25]. We now sketch a variant of the blow up proof for the HL model The initial data will be chosen as follows: ω 0 is odd, which together with periodicity implies that it is also odd with respect to x = L/2, and satisfy ω 0 (x) ≥ 0 if x ∈ [0, L/2].…”

Finite time blow up in the hyperbolic Boussinesq system

“…In [8], more information on the structure of blow up solutions has been obtained. Existence of finite time blow up in the original Hou-Luo model has been proved in [3], and a more general argument applying to a broader class of models was presented in [5]. Some infinite energy solutions of 2D Boussinesq system with simple structure and growing derivatives, inspired by Hou-Luo scenario, have been presented in [2].…”

Analysis of a singular Boussinesq model

“…A 1D model of the Hou-Luo scenario has been proposed already in [15]. Several works have analyzed this and a few other related models, in all cases proving finite time singularity formation [7,6,8,14]. All these models feature Biot-Savart laws u(x, t) = − ∞ 0 K(x, y)ω(y, t) dy with non-negative kernels K. This helps prove transport of vorticity and density towards the origin, accompanied by growth in ρ x 1 leading to growth of vorticity and thus to nonlinear feedback feedback loop driving blow up.…”

“…Here n is an integer. This space is well adapted to the mean field forcing term in (8). We also denote K ∞ = n≥1 K n .…”

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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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