On the Convergence of the Spectral Viscosity Method for the Two-Dimensional Incompressible Euler Equations with Rough In
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On the Convergence of the Spectral Viscosity Method for the Two-Dimensional Incompressible Euler Equations with Rough Initial Data Samuel Lanthaler1 · Siddhartha Mishra1 Received: 16 April 2019 / Revised: 10 October 2019 / Accepted: 14 October 2019 © SFoCM 2019
Abstract We propose a spectral viscosity method to approximate the two-dimensional 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. Keywords Incompressible Euler · Spectral viscosity · Vortex sheet · Convergence · Compensated compactness Mathematics Subject Classification 65M12 · 65M70
1 Introduction Flow of incompressible fluids at (very) high Reynolds numbers is often approximated by the incompressible Euler equations that model the motion of an ideal (incompressible and inviscid) fluid, [33] and references therein. The incompressible Euler equations are nonlinear partial differential equations of the form,
Communicated by Eitan Tadmor.
B 1
Samuel Lanthaler [email protected] Seminar for Applied Mathematics, Department of Mathematics, ETH Zurich, Rämistrasse 101, 8092 Zurich, Switzerland
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Foundations of Computational Mathematics
⎧ ⎪ ⎨ ∂t u + u · ∇u + ∇ p = 0, div(u) = 0, ⎪ ⎩ u|t=0 = u0 .
(1.1)
Here, the velocity field is denoted by u ∈ Rd (for d = 2, 3), and the pressure is denoted by p ∈ R+ . The pressure acts as a Lagrange multiplier to enforce the divergence-free constraint. The equations need to be supplemented with suitable boundary conditions. For simplicity, we will only consider the case of periodic boundary conditions in this paper. 1.1 Mathematical Results Although short-time (or small data) well-posedness results are classical [26], the questions of well-posedness, i.e., existence, uniqueness, stability and regularity, of global solutions of the three-dimensional Euler equations are largely open. Notable exceptions are provided by the striking results of [22,23,35,36], where it is established that weak solutions, even Hölder continuous ones (with Hölder exponent < 13 ), are not necessarily unique. On the other hand, the analysis of the Euler equations (1.1) in two space dimensions is significantly more mature. This is mainly due to the fact that, in two dimensions, the vorticity ω = curl(u) of a solution u to the PDE (1.1) satisfies a transport equation ∂t ω + u · ∇ω = 0,
(1.2)
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 Eu
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