On the Discretely Self-similar Solutions to the Euler Equations in ${\mathbb {R}}^3$

Abstract

We remove $(\alpha , \lambda )$-discretely self-similar blow up for solutions to the Euler equations for $\alpha \ge \frac{3}{2}$, for which we allow sublinear growth for the profile. More precisely, we show that there are only spatial constant $(\alpha , \lambda )$-discretely self-similar solutions $v=c(t)$ having the sublinear growth at the infinity. For the proof, we establish a new a priori $L^2_{\textrm{loc}} ({\mathbb {R}}^3)$ estimate for the 3D Euler equations.

Appendix A: Auxiliary lemmas

Lemma A.1

Let $v\in C^1({\mathbb {R}}^{3})$ with $\nabla \cdot v=0$ in ${\mathbb {R}}^{3}$, such that $\nabla \times v\in L^{ p}({\mathbb {R}}^{3})$, $1< p< \infty$. Then there exist $u\in {\dot{W}}^{1,\,p}({\mathbb {R}}^{3} )$, and a harmonic function $\pi : {\mathbb {R}}^{3} \rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} v = u + \nabla \pi \quad \text { in}\quad {\mathbb {R}}^{3}. \end{aligned}$$

(A.1)

Proof

Let us set

$$\begin{aligned} u_i= \sum _{j,k=1}^{3} \varepsilon _{ ijk} \Delta ^{ -1} (\partial _j ((\nabla \times v)_k)) = (\Delta ^{ -1}\nabla \times (\nabla \times v))_i, \quad i=1,2,3. \end{aligned}$$

where $\Delta ^{ -1} \partial _j$ is well defined linear bounded operator from $L^p({\mathbb {R}}^{3} )$ into the space $X_p$, where

$$\begin{aligned} X_p={\left\{ \begin{array}{ll} L^{ \frac{3p}{3-p}}({\mathbb {R}}^{3} ) &{}\quad \text {if}\quad 1 \le p<3,\\ C^{ 1- \frac{3}{p}}({\mathbb {R}}^{3} ) &{}\quad \text {if}\quad 3< p <+\infty ,\\ BMO &{}\quad \text {if}\quad p =3, \end{array}\right. } \end{aligned}$$

which is a consequence of Sobolev’s embedding theorem together with Calderón-Zygmund inequality. In particular, we find $\Vert \nabla u\Vert _{p} \le C \Vert \nabla \times v\Vert _{p} <+\infty ,$ which shows $u\in {\dot{W}}^{1,p} ({\mathbb {R}}^3)$. We observe

$$\begin{aligned} \nabla \times (v-u)= \nabla \times v- \Delta ^{-1} \nabla \times \nabla \times (\nabla \times v) = \nabla \times v- \nabla \times v=0. \end{aligned}$$

Therefore, by the Poincare’s lemma $v-u= \nabla \pi$ on ${\mathbb {R}}^3$ for some scalar function $\pi$. Since $\Delta \pi = \nabla \cdot v- \nabla \cdot u=0$, the claim is proved. $\square$

Lemma A.2

(Iteration lemma) Let $\beta _m: [a, b] \rightarrow {\mathbb {R}}$, $m\in {\mathbb {N}}\cup \{ 0\}$ be a sequences of continuous functions. Furthermore let $\alpha \in L^1(a, b)$ with $\alpha \ge 0$. We assume that the following recursive integral inequality holds true for a constant $C>0$

$$\begin{aligned} \beta _{ m}(t) \le C+ \int \limits _{a}^{t} \alpha (s) \beta _{ m+1}(s) \textrm{d}s, \quad m\in {\mathbb {N}}\cup \{ 0\}. \end{aligned}$$

(A.2)

Furthermore, suppose that there exists $K>0$ such that

$$\begin{aligned} \max _{ t\in [a, b]}| \beta _m(t)| \le K^m\quad \forall \,m\in {\mathbb {N}}. \end{aligned}$$

(A.3)

Then, the following inequality holds true for all $t\in [a,b]$

$$\begin{aligned} \beta _0(t) \le { C } { } e^{\int \limits _{a}^{t} \alpha (\tau ) d\tau }. \end{aligned}$$

(A.4)

Proof

Iterating (A.2) m-times $(m \in {\mathbb {N}}, m \ge 3)$, and arguing similar as in the proof of (Chae and Wolf 2019, Lemma B.1), we find

$$\begin{aligned} \beta _{0}&\le C \Bigg \{1+ \int \limits _{t_{0}}^{t} \alpha (s)\textrm{d}s + \sum _{k= 2}^{m-1}\int \limits _{t_{0}}^{t} \int \limits _{t_{0}}^{s_{1}} \ldots \int \limits _{t_{0}}^{s_{k-1}} \alpha (s_{1}) \ldots \alpha (s_{k}) \textrm{d}s_{k} \ldots \textrm{d}s_{1} \Bigg \}\\&\qquad \qquad + \int \limits _{t_{0}}^{t} \int \limits _{t_{0}}^{s_{1}} \ldots \int \limits _{t_{0}}^{s_{m-1}} \beta _{m} (s_{m})\alpha (s_{1}) \ldots \alpha (s_{m}) \textrm{d}s_{m} \ldots d s_{1}\\&\le C e ^{ \int \limits _{t_{0}}^{t} \alpha (s) \textrm{d}s} + \frac{K^{m}}{ m!} \bigg (\int \limits _{t_{0}}^{t} \alpha (s) \textrm{d}s\bigg )^{m}. \end{aligned}$$

Since the last term on the right-hand side tends to zero as $m \rightarrow + \infty$ we get (A.4)