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 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.

A: Miscellaneous Results

We shall need some estimates for trigonometric polynomials $f_M(x) {=}\sum _{|\varvec{k}|\le M} \widehat{f}_{\varvec{k}} \mathrm{e}^{i\varvec{k}\cdot \varvec{x}}$. We denote by ${\mathcal {P}}_N$ the projection onto this space. We take them from [15] (though they may have appeared elsewhere).

Theorem A.1

Let $1<p\le q<\infty$, or $1<p<q\le \infty$. Then

$$\begin{aligned} \Vert {\mathcal {P}}_N f \Vert _q \le C_p N^{d\left( \frac{1}{p} - \frac{1}{q}\right) } \Vert f \Vert _p. \end{aligned}$$

and

Theorem A.2

Let $s\ge 0$. Then,

$$\begin{aligned} \Vert |\nabla |^s {\mathcal {P}}_N f\Vert _p \le N^s C_p \Vert f\Vert _p. \end{aligned}$$

Let us furthermore state a multidimensional version of the one-dimensional Bernstein inequality. We first recall the one-dimensional case:

Theorem A.3

(Bernstein) Let $f_N$ be a trigonometric polynomial on $\mathbb {T}$, of order N. Then, we have the following $L^p$ inequality ($1\le p \le \infty$) for its derivative

$$\begin{aligned} \Vert f_N' \Vert _{L^p} \le N \Vert f_N \Vert _{L^p}. \end{aligned}$$

We will require the following (multidimensional) inequality for the $L^p$-norm of the Laplacian.

Theorem A.4

Let $f_N: \mathbb {T}^d \rightarrow \mathbb {C}$ be a trigonometric polynomial of degree at most N. Then, for any $1\le p \le \infty$:

$$\begin{aligned} \Vert \Delta f_N \Vert _{L^p} \le N^2d \Vert f_N \Vert _{L^p}. \end{aligned}$$

Proof

Since the constant $N^2d$ in this estimate is independent of p, it will suffice to consider $p<\infty$. The result for the $L^\infty$-norm then follows by letting $p\rightarrow \infty$. From the one-dimensional inequality applied to the trigonometric polynomial

$$\begin{aligned} x_i \mapsto f_N(x_1,\ldots ,x_i,\ldots ,x_d), \end{aligned}$$

where the other variables $x_j$, $j\ne i$ are frozen, we immediately obtain

$$\begin{aligned} \int \left| \frac{\partial ^2 f_N}{\partial x_i^2}\right| ^p \, \mathrm{d}x_i \le N^{2p} \int \left| f_N\right| ^p \, \mathrm{d}x_i. \end{aligned}$$

Integrating over $x_1, \ldots ,x_{i-1},x_{i+1},\ldots , x_d$, it then follows that

$$\begin{aligned} \int \left| \frac{\partial ^2 f_N}{\partial x_i^2}\right| ^p \, \mathrm{d}x \le N^{2p} \int \left| f_N\right| ^p \, \mathrm{d}x, \end{aligned}$$

and therefore

$$\begin{aligned} \left( \int \left| \Delta f_N\right| ^p \, \mathrm{d}x\right) ^{1/p}&\le \sum _{i=1}^d \left( \int \left| \frac{\partial ^2 f_N}{\partial x_i^2}\right| ^p \, \mathrm{d}x\right) ^{1/p} \\&\le \sum _{i=1}^d N^2 \left( \int \left| f_N\right| ^p \, \mathrm{d}x\right) ^{1/p} \\&= N^2 d \left( \int \left| f_N\right| ^p \, \mathrm{d}x\right) ^{1/p}. \end{aligned}$$

$\square$

We also recall the following characterization of weakly compact subsets of $L^1([0,T]\times \mathbb {T}^2)$, due to Dunford–Pettis theorem (for a proof, see [9]).

Theorem A.5

(Dunford–Pettis) A subset $K \subset L^1([0,T]\times \mathbb {T}^2)$ is weakly compact, if and only if

• K is bounded in the $L^1$-norm,

• for every $\epsilon >0$, there exists a $\delta >0$ such that

  $$\begin{aligned} |A|< \delta \implies \int _A f \, \mathrm{d}x \, \mathrm{d}t < {\epsilon }, \quad \text {for all }f\in K. \end{aligned}$$

We shall also need the following “Aubin–Lions lemma.” For a proof and thorough discussion of compactness in spaces $L^p([0,T];B)$ with B a Banach space, we refer to [41] and references therein.

Theorem A.6

[41, Thm. 5] Fix $T>0$. Let $X\subset B\subset Y$ be Banach spaces, with compact embedding $X\rightarrow B$. If $1\le p \le \infty$ and

• $F \subset L^p([0,T];X)$ is bounded,

• $\Vert f(\cdot +h) - f(\cdot ) \Vert _{L^p([0,T];Y)} \rightarrow 0$ as $h\rightarrow 0$, uniformly for $f\in F$.

Then, F is relatively compact in $L^p([0,T];B)$ (and in C([0, T]; B) if $p=\infty$).