We consider hypothetical solutions of 3D Euler which blow up in finite time in a self-similar fashion. We prove that if the initial data has finite kinetic energy, then the similarity exponent $γ$ which governs the rate of zooming in must be larger than $2/5$. If a smooth globally self-similar blowup profile exists, and this profile satisfies an outgoing property, we prove that $γ\geq 1/2$. For axisymmetric solutions, we establish the bound $γ\geq 1/2$ in more general settings, including ones in which the outgoing property is not present.
Self-similarity [1] is a powerful idea: it says that things look the same at different scales, if you just know how to zoom in or out. It is a particularly useful tool for identifying singular solutions of nonlinear PDE via a reduction to ODE [39,23]. Self-similarity is found in compressible fluids and in many other physical systems when dynamics are determined by local interactions. Compressible Euler equations for instance exhibit self-similar explosions [38,45,43], implosions [29,33,35,3,41] and shocks [4,5].
Incompressible fluids are not local, the outside matters. This makes the link between possible singularities and local self-similar behavior rather tenuous. There have been a number of recent works where self-similar incompressible singularities have been reported, either as rigorous proofs or in computational studies. In all of them there is a remnant of compression due to either the presence of boundaries [12,13,14] and [30,46,47,48,49], or the lack of smoothness of vorticity [24,11,26,27,9,16]. The goal of this paper is to analyze constraints on putative self-similar singularities for the incompressible 3D Euler equations with smooth initial data, in the absence of boundaries.
We recall the vorticity formulation of incompressible 3D Euler equations:
The smooth and localized initial data ω 0 for the Cauchy problem associated to (1.1) is specified at time t = 0, and the equations are posed on the whole space (x ∈ R 3 ). It is well known that singularities of any kind cannot arise in finite time from smooth and localized initial data unless the vorticity becomes infinite, such that its maximum magnitude is not time integrable. This is the well-known Beale-Kato-Majda criterion [2]. We say that (1.1) exhibits a self-similar singularity if there exists a first time T * > 0, a space location x * ∈ R 3 , a similarity exponent γ > 0, and a vorticity profile Ω such that
The factor (T * -t) -1 appearing in (1.2) is unavoidable, and consistent with [2]. A more precise meaning of a self-similar singularity for (1.1) will be given later in the paper; see Section 2 for an asymptotically self-similar blowup, and Section 3 for a globally self-similar blowup.
If a true self-similar singularity forms in (1.1), the value of the exponent γ is an important and powerful dynamic property, and needs to be understood. Finding a profile Ω is a well known challenge. Because access to self-similar solutions is fraught with numerical and theoretical difficulties, it is useful to establish strict mathematical guardrails and to provide computational benchmarks.
The Euler equations have a great variety of different types of solutions. Blow up of smooth solutions with infinite kinetic energy has been proved [15,20,28,42]. In this paper we prove that if the initial data of (1.1) has finite kinetic energy, then a solution behaving as in (1.2) must have γ ≥ 2/5; see Theorem 2.1.
Beyond a general intrinsic interest in lower bounds for γ as a benchmark for ongoing computational and analytical studies, in the second half of the paper we focus on the hypothetical case when γ lies below the parabolic threshold of 1/2, relevant for incompressible 3D Navier-Stokes. At γ = 1/2, the advection operator ∂ t + u • ∇ and the dissipative operator -∆ are in exact balance when acting on vorticities in the form (1.2). Liouville theorems [34,44,40,8] rule out certain globally self-similar solutions for the 3D incompressible Navier-Stokes equations with γ = 1/2. Under reasonable assumptions, γ > 1 2 cannot be the exponent of an approximate self-similar blow up for 3D incompressible Navier-Stokes equations (see Proposition 3.1). For self-similar scaling with γ < 1/2, the viscous term yields a vanishingly small force competing with the Euler nonlinearity. If self-similar solutions of Euler equations with γ < 1 2 are found, they can be used to obtain 3D Navier-Stokes blow up as a perturbation of the 3D Euler one, allowing the inviscid blowup to be used to find a viscous one. This is indeed a rigorously proven fact in the realm of compressible flows: globally self-similar implosion singularities for 3D compressible isentropic Euler equations can be used to provide asymptotically self-similar implosion singularities for 3D compressible barotropic Navier-Stokes equations with constant viscosity coefficients; see [35,36] and [3,41]. In the process of establishing the blow up [36,3,41], in addition to having γ < 1/2 the authors prove and use that the inviscid similarity profile has infinite regularity (real-analyticity) in a neighborhood of all sonic points (stagnation points on the self-similar fast-acoustic characteristics).
An analogous approach to establish blow up for incompressible viscous flows hinges on finding self-similar blowup of (1.1) with γ < 1/2. This is advocated in [46] and [47].
The paper [25] identifies an outgoing property (cf. (3.34) with c * > 0) as being an essential ingredient for the existence of self-similar profiles. This
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