Article Contents
Article Contents

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

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• 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.

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

 Citation:

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

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

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

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

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)

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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