A posteriori error estimates for self-similar solutions to the Euler equations

Alberto Bressan; Wen Shen

doi:10.3934/dcds.2020168

Article Contents

2021, Volume 41, Issue 1: 113-130. Doi: 10.3934/dcds.2020168

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A posteriori error estimates for self-similar solutions to the Euler equations

Alberto Bressan^, and
Wen Shen

Mathematics Department, Pennsylvania State University, University Park, PA 16802, USA

^* Corresponding author
^* Corresponding author

Received: December 31, 2019

Early access: March 2020

Published: January 2021

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Abstract

The main goal of this paper is to analyze a family of "simplest possible" initial data for which, as shown by numerical simulations, the incompressible Euler equations have multiple solutions. We take here a first step toward a rigorous validation of these numerical results. Namely, we consider the system of equations corresponding to a self-similar solution, restricted to a bounded domain with smooth boundary. Given an approximate solution obtained via a finite dimensional Galerkin method, we establish a posteriori error bounds on the distance between the numerical approximation and the exact solution having the same boundary data.

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Mathematics Subject Classification: Primary: 35L60, 35Q31; Secondary: 35Q35.

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Figure 2. The vorticity distribution at time $ t = 1 $, for a solution to (1.2) with initial vorticity $ \overline \omega_\varepsilon $

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Figure 3. The vorticity distribution at time $ t = 1 $, for a solution to (1.2) with initial vorticity $ \overline\omega_\varepsilon^\dagger $

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Figure 1. The supports of the initial vorticity considered in (1.3)-(1.5)

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Figure 4. Decomposing the plane $ {\mathbb R}^2 = {\mathcal D}^\sharp\cup{\mathcal D}^\natural\cup{\mathcal D}^\flat $ into an outer, a middle, and an inner domain. Left: the case of a single spiraling vortex, as in Fig. 2. Right: the case of two spiraling vortices, as in Fig. 3

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Figure 5. According to (A1), every characteristic starting at a point $ y\in \Sigma_1\cap {\rm Supp }(h) $ exits from the domain $ {\mathcal D} $ at some boundary point $ z = z(y)\in \Sigma_2 $, at a time $ T(y)\leq T^* $. The shaded region represents the subdomain $ {\mathcal D}^* $ in (2.10)

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Figure 6. Computing the perturbed solution $ \Omega^\varepsilon(y) $ in (3.4), by estimating the change in the characteristic through $ y $

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