On the Convergence of the Spectral Viscosity Method for the Two-Dimensional Incompressible Euler Equations with Rough Initial Data

Abstract

We propose a spectral viscosity method to approximate the twodimensional Euler equations with rough initial data and prove that the method converges to a weak solution for a large class of initial data, including when the initial vorticity is in the so-called Delort class i.e. it is a sum of a signed measure and an integrable function. This provides the first convergence proof for a numerical method approximating the Euler equations with such rough initial data and closes the gap between the available existence theory and rigorous convergence results for numerical methods. We also present numerical experiments, including computations of vortex sheets and confined eddies, to illustrate the proposed method. arXiv:1903.12361v2 [math.NA] 24 Oct 2019 providing a priori control on various norms of ω, such as L p -norms [33].Global existence and uniqueness results for the two-dimensional incompressible Euler equations with smooth initial data are classical [33,7]. For non-smooth initial conditions, the work by Yudovich [49] has established existence and uniqueness for bounded initial vorticity, i.e. ω 0 ∈ L ∞ . The uniqueness result of [49] has later been extended to vorticities belonging to slightly more general spaces [47,48,50]. An existence result for vorticity ω 0 ∈ L p , 1 < p < ∞ has been obtained by Diperna and Majda [7]. It is shown in [7] that the sequence obtained by solving the Euler equations for mollified initial data is strongly compact in L 2 . The existence of a weak solution for ω 0 ∈ L p is then established by passing to the limit. Further extensions of the result of Diperna and Majda can be obtained by compensated compactness methods for initial vorticity ω 0 belonging to e.g. Orlicz spaces such as ω 0 ∈ L log(L) α , α 1/2, which are compactly embedded in H −1 [34, 3, 2, 30]. These methods break down for ω 0 ∈ L 1 .In his celebrated work [6], Delort has shown the existence of solutions to the Euler equations with initial vorticity ω 0 = ω 0 + ω 0 , where ω 0 is a finite, nonnegative Radon measure belonging also to H −1 , and ω 0 ∈ L p , for some p > 1. These initial data correspond to the interesting case of vortex sheets i.e. vorticity concentrated on curves in the two-dimensional spatial domain [33]. In [6], it is remarked that the proof can be extended to allow for ω 0 ∈ L 1 . A detailed proof of this claim has subsequently been provided by Vecchi and Wu [46]. The results of Delort [6], and Vecchi and Wu [46], remain the most general existence results for the incompressible Euler equations in two dimensions. The question of existence of solutions beyond this Delort class, for instance, when ω 0 is an arbitrary signed bounded measure, remains open. The uniqueness question also remains open, even for vorticities ω 0 ∈ L p , p < ∞.1.2. Numerical schemes. It is not possible to represent solutions of the incompressible Euler equations in terms of analytical solution formulas, even in two space dimensions. Hence, numerical approximation of (1.1) is a necessary and key ingredient in the study of the inco...

Statistical solutions of the incompressible Euler equations

“…As discussed in Ref. 26 numerical approximations for sinusoidal vortex sheet initial data experience vortex sheet roll-up at ever smaller length-scales at increasing resolution Δ → 0 (and at low diffusivity). These small-scale Kelvin-Helmholtz instabilities slow down, and at even smaller values of Δ ultimately prevent the strong convergence of the numerical approximants to a limiting solution.…”

“…However, as noted in Ref. 26, this convergence can be very slow as the flow breaks down into smaller and smaller vortices. In fact, this phenomenon is also seen from Fig.…”

“…34 and 35, the authors showed convergence of vortex point and vortex blob methods in two space dimensions as long as the initial vorticity was a signed Radon measure. Finally in a recent paper, 26 the authors showed convergence of a spectral viscosity method for initial data in the Delort class.…”

“…Even when the numerical approximations converge (such as in Ref. 26), the inherent instabilities of the Euler equations lead to a very slow convergence rate (see also Fig. 8(left)) and render the computation of weak solutions of (1.1) prohibitively expensive.…”

“…the initial vorticity is concentrated on a sine curve. This test case was extensively studied in a recent paper 26 in the context of the numerical approximation of weak solutions (in Delort class) of the two-dimensional incompressible Euler equations. Whereas Ref.…”