Perfect fluid flows on $\mathbb {R}^d$ with growth/decay conditions at infinity

Abstract

We study the well-posedness and the spatial behavior at infinity of perfect fluid flows on $\mathbb {R}^d$ with initial velocity in a scale of weighted Sobolev spaces that allow spatial growth/decay at infinity as $|x|^\beta$ with $\beta <1/2$. Moreover, for initial velocity with sufficient spatial decay, we show that the solution of the Euler equation generically develops an asymptotic expansion at infinity with non-vanishing asymptotic terms that depend analytically on time and the initial data. For initial data in the Schwartz space, we identify the evolution space of the fluid velocity with a certain space of symbols.

Appendices

Auxiliary results

Most of the results in this Appendix are only generalizations to $\gamma +d/p>-1$ of results in [23] that assumed $\gamma +d/p>0$. Rather than repeat the detailed proofs given in [23], we will simply describe how to generalization them to the case $\gamma +d/p>-1$. In one instance, we generalize a statement from [14]. The following lemma follows directly from (3a).

Lemma A.1

If $w\in W^{m,p}_\gamma$ with $m>d/p$ and $\gamma +d/p>-1,$ then

$$\begin{aligned} C_1\langle x\rangle \le \big \langle x+w(x)\big \rangle \le C_2 \langle x\rangle \quad \hbox {for} \ x\in \mathbb {R}^d, \end{aligned}$$

(82)

where $C_1,C_2>0$ may be chosen locally uniformly for $w\in W^{m,p}_\gamma$.

Local uniformity means that for any $w_0\in W^{m,p}_\gamma$ there exists an open neighborhood U of $w_0$ in $W^{m,p}_\gamma$ such that the inequality (82) holds for any $w\in U$.

Our next result is analogous to Lemma 6.3 in [23]. Let $|\mathrm{d}\varphi (x)|$ denote the sum of the absolute values of the elements of the matrix $\mathrm{d}\varphi (x)$ for $x\in \mathbb {R}^d$.

Lemma A.2

If $\varphi =\mathrm{id}+w\in {{\mathcal {D}}}^{m,p}_\gamma$ with $m>1+d/p$ and $\gamma +d/p>-1,$ then

$$\begin{aligned} \big |\mathrm{d}\varphi (x)\big |\le C \quad \hbox {and} \quad 0<\varepsilon \le \det (\mathrm{d}\varphi (x)) \quad \hbox {for} \ x\in \mathbb {R}^d, \end{aligned}$$

(83)

where C may be chosen uniformly for $\Vert w\Vert _{W^{m,p}_\gamma }\le M$ and $\varepsilon$ may be chosen locally uniformly for $w\in W^{m,p}_\gamma$.

Proof of Lemma A.2

It follows from (3a) that for $m>1+d/p$ and $\delta +d/p\ge -1$,

$$\begin{aligned} W^{m-1,p}_{\delta +1}\subseteq L^\infty \end{aligned}$$

(84)

is bounded. Take $\varphi =\mathrm{id}+w\in D^{m,p}_\gamma$ with $m>1+d/p$ and $\gamma +d/p>-1$. Then, the first inequality in (83) follows from the boundedness of the inclusions

$$\begin{aligned} W^{m,p}_\gamma {\mathop {\hookrightarrow }\limits ^{\mathrm{d}}}W^{m-q,p}_{\gamma +1}\subseteq L^\infty \end{aligned}$$

and the fact that $\mathrm{d}\varphi =I+\mathrm{d}w$. The second inequality in (83) follows in a similar way from (84), $\mathrm{d}\varphi =I+\mathrm{d}w$, and (3a), since the latter implies that $|\mathrm{d}w|=C\,\Vert \mathrm{d}w\Vert _{W^{m-1,p}_{\delta +1}}/\langle x\rangle ^{(\delta +p/d)+1}$ with $(\delta +p/d)+1>0$. The estimates above a locally uniform for $\varphi \in D^{m,p}_\gamma$. $\square$

The following corollary follows from Lemma A.2 and Hadamard–Levy’s theorem.

Corollary A.1

If $\varphi _0=\mathrm{id}+w_0\in {{\mathcal {D}}}^{m,p}_\gamma$ where $m>1+d/p,$ $\gamma +d/p>-1,$ and ${\widetilde{w}}\in W^{m,p}_\gamma$ with $\Vert {\widetilde{w}}\Vert _{W^{m,p}_\gamma }$ sufficiently small,  then $\varphi _0+{\widetilde{w}}\in {{\mathcal {D}}}^{m,p}_\gamma$.

In particular, the set of maps ${{\mathcal {D}}}^{m,p}_\gamma$ can be identified with an open set in $W^{m,p}_\gamma$. Hence, ${{\mathcal {D}}}^{m,p}_\gamma$ is a Banach manifold modeled on $W^{m,p}_\gamma$.

For $\varphi \in {{\mathcal {D}}}^{m,p}_\gamma$, we know that $\varphi ^{-1}$ exists but we need estimates at infinity in order to conclude that $\varphi ^{-1}\in {{\mathcal {D}}}^{m,p}_\gamma$. The following is a first step and is analogous to Lemma 6.4 in [23].

Lemma A.3

If $\varphi =\mathrm{id}+w\in {{\mathcal {D}}}^{m,p}_\gamma$ where $m>1+d/p$ and $\gamma +d/p>-1,$ then

$$\begin{aligned} \big |\mathrm{d}(\varphi ^{-1})(x)\big |\le C \quad \hbox {and} \quad 0<\varepsilon \le \det \,\big (\mathrm{d}(\varphi ^{-1})(x)\big ) \quad \hbox {for} \ x\in \mathbb {R}^d, \end{aligned}$$

where C and $\varepsilon$ may be chosen locally uniformly for $w\in W^{m,p}_\gamma$.

Proof of Lemma A.3

The lemma follows from Lemma A.2 and the formula

$$\begin{aligned} \mathrm{d}\big (\varphi ^{-1}\big )=\big [(\mathrm{d}\varphi )\circ \varphi ^{-1}\big ]^{-1} \end{aligned}$$

for the Jacobian matrix of $\varphi ^{-1}$. $\square$

Now let us consider compositions of maps as in Theorem 2.1. We begin with an estimate.

Lemma A.4

Suppose $m>1+d/p,$ $\gamma +d/p>-1$ and $\varphi \in {{\mathcal {D}}}^{m,p}_\gamma$. Then for every $0\le k\le m$ and $\delta \in \mathbb {R}$, we have

$$\begin{aligned} \Vert w\circ \varphi \Vert _{W^{k,p}_\delta } \le C\,\Vert w\Vert _{W^{k,p}_\delta }\quad \hbox {for all}\ f\in W^{k,p}_\delta , \end{aligned}$$

where C may be taken locally uniformly for $\varphi \in {{\mathcal {D}}}^{m,p}_\gamma$.

Proof of Lemma A.4

This is proved by induction using Lemma A.3 for a change of integration variable when $k=0$ and Proposition 2.2 in [23] to handle products in the induction step. For details see the proof of Lemma 6.5 in [23]. $\square$

Lemma A.5

Assume $m>1+d/p,$ $\gamma +d/p>-1,$ $\delta \in \mathbb {R},$ and $f\in C_c^\infty (\mathbb {R}^d)$. If $\varphi , \varphi _k\in {{\mathcal {D}}}^{m,p}_\gamma$ with $\varphi _k\rightarrow \varphi$ in ${{\mathcal {D}}}^{m,p}_\gamma$ as $k\rightarrow \infty ,$ then $f\circ \varphi _k\rightarrow f\circ \varphi$ in $W^{m,p}_\delta (\mathbb {R}^d)$.

Proof of Lemma A.5

The only difference from the proof of Lemma 6.6 in [23] is that $w_k(x)$ and w(x) in $\varphi _k(x)=x+w_k(x)$ and $\varphi (x)=x+w(x)$ are of order $O(|x|^{1-\varepsilon })$ for some $\varepsilon >0$ instead of just being bounded. But the rest of the proof in [23] can be used here without change. $\square$

Lemmas A.4 and A.5 can be combined to obtain the continuity of the composition.

Corollary A.2

Suppose $m>1+d/p,$ $\gamma +d/p>-1$ and $\delta \in \mathbb {R}$. Then the composition $(f,\varphi )\mapsto f\circ \varphi ,$ $W^{m,p}_\delta \times {{\mathcal {D}}}^{m,p}_\gamma \rightarrow W^{m,p}_\delta ,$ is continuous.

Proof of Corollary A.2

The details are the same as in the proof of Corollary 6.1 in [23]. $\square$

Next we investigate when the composition is $C^1$.

Lemma A.6

Assume $m>1+d/p,$ $\gamma +d/p>-1,$ $\delta \in \mathbb {R},$ and take $\varphi _0\in {{\mathcal {D}}}^{m,p}_\gamma$. For $f\in W^{m+1,p}_\delta$ and $\varphi \in {{\mathcal {D}}}^{m,p}_\gamma$ sufficiently close to $\varphi _0,$ we have

$$\begin{aligned} \Vert f\circ \varphi -f\circ \varphi _0\Vert _{W^{m,p}_{\delta +1}}\le C\,\Vert f\Vert _{W^{m+1,p}_\delta }\Vert \varphi -\varphi _0\Vert _{W^{m,p}_\gamma }, \end{aligned}$$

where $C>0$ can be taken uniformly for all $\varphi$ in an open neighborhood of $\varphi _0$.

Proof of Lemma A.6

The details are the same as in the proof of Lemma 6.7 in [23]. $\square$

Corollary A.3

Suppose $m>1+d/p,$ $\gamma +d/p>-1$ and $\delta \in \mathbb {R}$. Then the composition $(f,\varphi )\mapsto f\circ \varphi ,$ $W^{m+1,p}_\delta \times {{\mathcal {D}}}^{m,p}_\gamma \rightarrow W^{m,p}_\delta ,$ is $C^1$.

Proof of Corollary A.3

Using the above lemmas, Corollary A.3 can be proved following the proof of Proposition 5.1 in [23] (cf. also Appendix B in [32]). $\square$

This completes the proof of Theorem 2.1(a). Let us now prove Theorem 2.1(b).

Lemma A.7

If $\varphi \in {{\mathcal {D}}}^{m,p}_\gamma$ where $m>1+d/p$ and $\gamma +d/p>-1,$ then $\varphi ^{-1}\in {{\mathcal {D}}}^{m,p}_\gamma$.

Proof of Lemma A.7

Let $\varphi =\mathrm{id}+w$ and $\varphi ^{-1}=\mathrm{id}+u.$ To show $\varphi ^{-1}\in {{\mathcal {D}}}^{m,p}_\gamma$, we need to show $\partial ^\alpha u\in L^p_{\gamma +|\alpha |}$ for all $|\alpha |\le m$. For $\alpha =0$ we use the change of variables $x=\varphi (y)$ to compute

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^d}\langle x\rangle ^{\gamma p}|u(x)|^p\,dx&=\int _{\mathbb {R}^d} \langle x\rangle ^{\gamma p}|\varphi ^{-1}(x)-x|^p\,dx =\int _{\mathbb {R}^d} \langle \varphi (y)\rangle ^{\gamma p}|w(y)|^p\,\det (\mathrm{d}\varphi (y))\,dy \\&\le C\,\int \langle y\rangle ^{\gamma p}|w(y)|^p\,dy <\infty , \end{aligned} \end{aligned}$$

where we have used (82), (83), and $w=u\circ \varphi \in W^{m,p}_\gamma$ (cf. Lemma A.4). For $1\le |\alpha |\le m$, we can proceed as in (28) in [14] to show

$$\begin{aligned} \partial ^\alpha (\varphi ^{-1}-\mathrm{id})=F^{(\alpha )}\circ \varphi ^{-1}, \end{aligned}$$

(85)

where $F^{(\alpha )} : \mathbb {R}^d\rightarrow \mathbb {R}^d$ is in $W^{m-|\alpha |,p}_{\gamma +|\alpha |}$. Then, by (85), for any $\alpha$ with $0\le |\alpha |\le m$,

$$\begin{aligned} \begin{aligned} \int \langle x\rangle ^{(\gamma +|\alpha |)p}\,|\partial ^\alpha (\varphi ^{-1}(x)-x)|^p\,dx&= \int \langle \varphi (y)\rangle ^{(\gamma +|\alpha |)p}\, |F^{(\alpha )}(y)|^p\,\det (\mathrm{d}\varphi (y))\,dy \\&\le C\,\int \langle y\rangle ^{(\gamma +|\alpha |)p}\,|F^{(\alpha )}(y)|^p\,dy <\infty . \end{aligned} \end{aligned}$$

Hence $\varphi ^{-1}-\mathrm{id}\in W^{m,p}_\gamma$. This implies that $\varphi \in {{\mathcal {D}}}^{m,p}_\gamma$. $\square$

The continuity of the map $\varphi \mapsto \varphi ^{-1}$, ${{\mathcal {D}}}^{m,p}_\delta \rightarrow {{\mathcal {D}}}^{m,p}_\delta$, now follows from Corollary A.2 and Theorem 2 in [26].¹¹ The last statement in Theorem 2.1 (b) can be proved by following the proof of Proposition 2.13 in [14].

Remark A.1

Note that the regularity assumption $m>1+d/p$ in Lemma A.7 above is weaker than the regularity assumption $m>3+d/p$ in the analogous [23, Lemma 7.2] and [30, Lemma 2.5]. This happens since in [23, 30] one deals with the asymptotic expansion of $\varphi ^{-1}$, that complicates the situation. Note however, that the results in [23, 30] can be extended to the case when $m>1+d/p$ by expanding their proofs.

Inverting the Laplace operator

In this Appendix we present, in an extended form, a basic result about the inversion of the Laplace operator in weighted Sobolev spaces (see Lemma A.3 in [24]). Denote by $\mathcal {S}'$ the space of tempered distributions in $\mathbb {R}^d$.

Proposition B.1

Assume that $d\ge 3$ and $m\ge 0$ with $1<p<\infty$.¹² Then,  for any $g\in W^{m.p}_{\delta +2}(\mathbb {R}^d,\mathbb {R})$ with weight $\delta \in \mathbb {R}$ such that $\delta +d/p>0,$ $\delta +d/p\notin \mathbb {Z},$ there exists a unique (up to adding a constant term) solution u in $\mathcal {S}'\cap L^\infty$ of the Poisson equation

$$\begin{aligned} \Delta u=g \end{aligned}$$

such that

$$\begin{aligned} u:=\Delta ^{-1}g=\chi (r)\sum _{d-2\le k<\delta +d/p}\frac{a_k(\theta )}{r^k}+f,\quad f\in W^{m+2,p}_{\delta }, \end{aligned}$$

(86)

where $a_k(\theta )$ is an eigenfunction of the Laplace operator $-\Delta _S$ on the unit sphere $S^{d-1}$ in $\mathbb {R}^d$ with eigenvalue $\lambda _{k-d+2}=k(k-d+2)$. If we fix for any $k':=k-d+2\ge 0$ with $d-2\le k<\delta +d/p$ an orthonormal basis

$$\begin{aligned} \big \{h_{k';l}(\theta )\,\big |\, 1\le l\le \nu (k')\big \},\quad \nu (k'):=\dim \mathcal {H}_{k'}, \end{aligned}$$

of the eigenspace $\mathcal {H}_{k'}$ of $-\Delta _S$ with eigenvalue $\lambda _{k'}=k'(k'+d-2)$ and expand

$$\begin{aligned} a_k(\theta )=\sum _{l=1}^{\nu (k')}\widehat{a}_{k';l}\,h_{k';l}(\theta ) \end{aligned}$$

in the Fourier modes, then the Fourier coefficient $\widehat{a}_{k';l}$ equals

$$\begin{aligned} \widehat{a}_{k';l}=-\frac{1}{2k'+d-2}\int _{\mathbb {R}^d}g(x) H_{k';l}(x)\,\mathrm{d}x,\quad 1\le l\le \nu (k'), \end{aligned}$$

(87)

where $H_{k';l}(x):=h_{k';l}(\theta ) r^{k'}$ is the homogeneous harmonic polynomial (of degree $k'\ge 0$) that corresponds to the Fourier mode $h_{k';l}(\theta )$.¹³ If we coordinatize the linear space $\mathcal {I}^{m,p}_\delta$ of functions of the form (86) by the Fourier coefficients of $a_k(\theta ),$ $d-2\le k<\delta +d/p,$ and the reminder $f\in W^{m,p}_\delta$ then the map

$$\begin{aligned} \Delta ^{-1} : W^{m+2,p}_{\delta +2}\rightarrow \mathcal {I}^{m,p}_\delta \end{aligned}$$

is an isomorphism of Banach spaces.

Remark B.1

Note that the eigenvalues of the Laplace operator on the unit sphere $S^{d-1}$ in $\mathbb {R}^d$ are highly degenerate: we have that $\dim \mathcal {H}_0=1$ (these are all constants), $\dim \mathcal {H}_1=d$ (all linear polynomials restricted to $S^{d-1}$), and

$$\begin{aligned} \dim \mathcal {H}_{k'}=\left( {\begin{array}{c}d-1+k'\\ d-1\end{array}}\right) -\left( {\begin{array}{c}d-3+k'\\ d-1\end{array}}\right) ,\quad k'\ge 2, \end{aligned}$$

where the second binomial coefficient above vanishes when $d=2$.

Proof of Proposition B.1

The fact that there exists a unique solution $u\in \mathcal {S}'\cap L^\infty$ of the form (86) follows from [24, Lemma A.3(b)] (cf. also [22]) and the fact that there is a bijective correspondence between the eigenfunctions of the Laplace operator $-\Delta _S$ on the unit sphere $S^{d-1}$ in $\mathbb {R}^d$ with eigenvalue $\lambda _{k'}=k'(k'+d-2)$ and the restriction to $S^{d-1}$ of harmonic polynomials of degree $k'\ge 0$ ( [31, §22.2]. Let us now prove the integral relation (87). To this end, take $d-2\le k<\delta +d/p$, $d-2\le n<\delta +d/p$, $1\le l_1\le \nu (k')$, $1\le l_2\le d(n')$ with $n':=n-d+2$, and consider the $k'$-th asymptotic term $A_{k';l}:=\chi h_{k';l}/r^k$ in (86). Then, it follows from the second Green’s identity and the fact that the eigenspaces $\mathcal {H}_{k'}$ and $\mathcal {H}_{n'}$ are $L^2$-orthogonal on $S^{d-2}$ for $k'\ne n'$, that

$$\begin{aligned}&\int _{\mathbb {R}^d}\Delta \big (A_{k';l_1}\big ) H_{n';l_2}\,\mathrm{d}x=\lim _{R\rightarrow \infty }\int _{S^{d-1}_R} \Big (\frac{\partial A_{k'l_1}}{\partial r}H_{n';l_2}-A_{k';l_1}\frac{\partial H_{n';l_2}}{\partial r}\Big ) \,d\sigma _R\nonumber \\&\quad =C(k,n)\lim _{R\rightarrow \infty }R^{n'-k'}\int _{S^{d-1}}h_{k';l_1} h_{n';l_2}\,d\sigma _1 =C(k,n)\,\delta _{kn} \end{aligned}$$

(88)

where $C(k,n):=-(k'+n'+d-2)\ne 0$, $S^{d-1}_R$ is the sphere of radius R, and $\delta _{kn}$ is the Kronecker delta. Now, consider the remainder $f\in W^{m+2,p}_\delta$ of the solution u in (86). For any $d-2\le n<\delta +d/p$ and $1\le l\le d(n')$ we have that $\Delta \big (f\big ) H_{n';l}\in W^{m,p}_{\delta +2-n'}$. This, together with the estimate (3a) implies that $\Delta \big (f\big ) H_{n';l}=O\big (1/r^{\delta +(d/p)+2-n'}\big )$, and hence $\Delta \big (f\big ) H_{n';l}\in L^1(\mathbb {R}^d)$ by the estimate $\delta +(d/p)+2-n'>d$. By arguing in the same way as above, we also have

$$\begin{aligned}&\int _{\mathbb {R}^d}\Delta \big (f\big ) H_{n';l}\,\mathrm{d}x=\lim _{R\rightarrow \infty } \int _{S^{d-1}_R}\Big (\frac{\partial f}{\partial r}\,H_{n';l}-f\,\frac{\partial H_{n';l}}{\partial r}\Big )\,d\sigma _R\nonumber \\&\quad =\lim _{R\rightarrow \infty }\int _{S^{d-1}_R}O\big (1/R^{\delta +(d/p)-n'+1}\big )\,d\sigma _R= \lim _{R\rightarrow \infty }O\big (1/R^{\delta +(d/p)-n}\big )=0, \end{aligned}$$

(89)

where we used that $n<\delta +d/p$ and the estimate (3a) on the decay of f at infinity. Finally, the integral formula (87) follows from (88), (89), and (86). $\square$

Asymptotic spaces ${{\mathcal {A}}}^{m,p}_{n,N;\ell }$

In this section we will discuss the asymptotic spaces with log terms that are used in the proof of Theorem 1.2. For integers $m>d/p$, $0\le n\le N$, and $\ell \ge -n$, let ${{\mathcal {A}}}^{m,p}_{n,N;\ell }$ denote functions u on $\mathbb {R}^d$ of the form

$$\begin{aligned} \chi (r)\left( \frac{a_n^0(\theta )+\cdots +a_n^{n+\ell }(\theta )(\log r)^{n+\ell }}{r^n}+\cdots + \frac{a_N^0(\theta )+\cdots +a_N^{N+\ell }(\theta )(\log r)^{N+\ell }}{r^N}\right) +f(x),\nonumber \\ \end{aligned}$$

(90a)

where $a_k^j\in H^{m+1+N-k,p}(\mathrm{S}^{d-1},\mathbb {R}^d)$ for $0\le j\le k+\ell$, $0\le n\le k\le N$, and $f\in W_{\gamma _N}^{m,p}$ with $N\le \gamma _N+d/p<N+1$ so, by (3b), $f(x)=o\big (r^{-N}\big )$ as $r\rightarrow \infty$. This is a Banach space with norm

$$\begin{aligned} \Vert u\Vert _{{{\mathcal {A}}}^{m,p}_{n,N;\ell }}=\sum _{0\le j\le k+\ell , \, n\le k\le N} \Vert a_k^j\Vert _{H^{m+1+N-k,p}} \ + \ \Vert f\Vert _{W^{m,p}_{\gamma _N}}. \end{aligned}$$

(90b)

Note that $W_{\gamma _N}^{m,p}$ is a closed subspace of ${{\mathcal {A}}}^{m,p}_{n,N;\ell }$. It is easy to confirm that the following inclusions are bounded for $N\le \gamma _N+d/p<N=1$:

$$\begin{aligned} {{\mathcal {A}}}^{m,p}_{n_1,N_1;\ell _1} \subseteq {{\mathcal {A}}}^{m,p}_{n,N;\ell } \quad \hbox {if}\ n_1\ge n,\ N_1\ge N,\ \ell \ge \ell _1\ge -n, \end{aligned}$$

(91a)

$$\begin{aligned} \partial _{x_J} : {{\mathcal {A}}}^{m,p}_{n,N;\ell } \rightarrow {{\mathcal {A}}}^{m-1,p}_{n+1,N+1;\ell -1} \quad \hbox {if}\ m>1+d/p,\quad 1\le j\le d. \end{aligned}$$

(91b)

Moreover, if $n=n_1+n_2$, $\ell =\ell _1+\ell _2$, and $N<\min (N_1+n_2,N_2+n_1)$, then pointwise multiplication $(u,v)\mapsto uv$,

$$\begin{aligned} {{\mathcal {A}}}^{m,p}_{n_1,N_1;\ell _1} \times {{\mathcal {A}}}^{m,p}_{n_2,N_2;\ell _2} \rightarrow {{\mathcal {A}}}^{m,p}_{n,N;\ell }, \end{aligned}$$

(91c)

is a bounded bilinear map. When $\ell _i=-n_i$ there are no log terms in the leading asymptotic and we have the sharper version with $N=\min (N_1+n_2,N_2+n_1)$, $n=n_1+n_1$:

$$\begin{aligned} {{\mathcal {A}}}^{m,p}_{n_1,N_1;-n_1} \times {{\mathcal {A}}}^{m,p}_{n_2,N_2;-n_2} \rightarrow {{\mathcal {A}}}^{m,p}_{n,N;\ell }. \end{aligned}$$

(91d)

These may be combined to conclude that

$$\begin{aligned} {{\mathcal {A}}}^{m,p}_{n,N;\ell } \ \hbox {is a Banach algebra} \end{aligned}$$

(91e)

in the case when $n\ge 1$, or when $\ell =-n$. For more details, see Appendix B in [23] (for the case when $N<\gamma _N<N+1$).

Analogous to (14), for $m>1+d/p$ we introduce asymptotic diffeomorphisms

$$\begin{aligned} {{{\mathcal {A}}}{\mathcal {D}}}^{m,p}_{n,N;\ell }:=\big \{\varphi : \mathbb {R}^d\rightarrow \mathbb {R}^d\,\big |\,\varphi =\mathrm{id}+w,\,w\in {{\mathcal {A}}}^{m,p}_{n,N;\ell }\, \text { and}\,\det (\mathrm{d}\varphi )>0\big \}. \end{aligned}$$

(92)

Analogous to Theorem 4.1 above, Theorem 6.1 in [24] shows for $m>3+d/p$ and $0\le n\le \min (d+1,N)$ that the Euler vector field

$$\begin{aligned} {{\mathcal {E}}}: {{{\mathcal {A}}}{\mathcal {D}}}^{m,p}_{n,N;0} \times {{{\mathcal {A}}}}^{m,p}_{n,N;0} \rightarrow {{{\mathcal {A}}}{\mathcal {D}}}^{m,p}_{n,N;0} \times {{{\mathcal {A}}}}^{m,p}_{n,N;0} \end{aligned}$$

(93)

is smooth. Theorem 1.1 in [24] uses this vector field as we did in Sect. 4 to show the existence of a unique solution as in (43). However, all results in [24] are under the assumption $N<\gamma _N+d/p<N+1$.

We now show that this solvability of the Euler equations in ${{\mathcal {A}}}^{m,p}_{n,N;0}$ also holds for $N\le \gamma _N+d/p<N+1$, at least when $n=1$.

Proposition C.1

Assume $m>3+d/p,$ $N\ge 1,$ and $N\le \gamma _N<N+1$. Then,  for any given $\rho >0$ there exists $\tau >0$ such that for any with $\Vert u_0\Vert _{{{\mathcal {A}}}^{m,p}_{1,N;0}}<\rho$ there exists a unique solution of the Euler equation

(94)

that depends continuously on the initial data $u_0$.

Proof of Proposition C.1

As previously stated, this was proved in [24] when $N<\gamma _N+d/p<N+1$, so we need only consider the case when $\gamma _N+d/p=N$. To do this, let us change notation within this proof and denote by $\mathfrak {A}^{m,p}_{n,N;0}$ the asymptotic space ${{\mathcal {A}}}^{m,p}_{n,N;\ell }$ with $\gamma _N+d/p=N$; we reserve the notation ${{\mathcal {A}}}^{m,p}_{n,N;0}$ for an asymptotic space with a remainder $f\in W^{m,p}_{{\tilde{\gamma }}_N}$ with $N<{\tilde{\gamma }}_N<N+1$. In particular, referring to (91a), we have the following bounded inclusions for any $\ell \ge -n$:

$$\begin{aligned} {{\mathcal {A}}}^{m,p}_{n,N;\ell }\subseteq \mathfrak {A}^{m,p}_{n,N;\ell }\subseteq {{\mathcal {A}}}^{m,p}_{n,N-1;\ell }. \end{aligned}$$

(95)

We will need to take the square of elements in $\mathfrak {A}^{m,p}_{n,N;\ell }$. For $n+\ell >0$, we know from (91c) and (91d) that the log terms in the leading asymptotic prevent us from keeping the same order of decay; but decreasing this order of decay by one enables us to also embed in a space with $0<\gamma _0+d/p<1$:

$$\begin{aligned} u\mapsto u^2 \quad \hbox {is smooth} \quad \mathfrak {A}^{m,p}_{n,N;\ell } \rightarrow {{\mathcal {A}}}^{m,p}_{2n,N+n-1;2\ell }. \end{aligned}$$

(96)

Using (91b), for $u\in \mathfrak {A}^{m,p}_{1,N;0}$ we have $\mathrm{d}u\in \mathfrak {A}^{m-1,p}_{2,N+1;-1}$ so (96) implies that $Q(u)\equiv \mathrm{tr}\,\big (\mathrm{d}u\big )^2$ defines a bounded quadratic polynomial map $Q :\mathfrak {A}^{m,p}_{1,N;0}\rightarrow {{\mathcal {A}}}^{m-1,p}_{4,N+2;-2}\subseteq {{\mathcal {A}}}^{m-1,p}_{2,N+2;-2}$. Consequently, the maps

$$\begin{aligned} Q : \mathfrak {A}^{m,p}_{1,N;0}\rightarrow {{\mathcal {A}}}^{m-1,p}_{2,N+2;-2}\quad \text { and}\quad \nabla \circ Q : \mathfrak {A}^{m,p}_{1,N;0}\rightarrow {{\mathcal {A}}}^{m-2,p}_{3,N+3;-3} \end{aligned}$$

(97)

are smooth. We can now apply Proposition 3.1 and (17b) in [24] to conclude that the linear map $\Delta ^{-1}: {{\mathcal {A}}}^{m-2,p}_{3,N+3;-3}\rightarrow {{\mathcal {A}}}^{m,p}_{1,N+1;0}$ is bounded and injective. Combined with (97) and the embeddings ${{\mathcal {A}}}^{m,p}_{1,N+1;0}\subseteq {{\mathcal {A}}}^{m,p}_{1,N;0} \subseteq \mathfrak {A}^{m,p}_{1,N;0}$ we see that

$$\begin{aligned} \Delta ^{-1}\circ \nabla \circ Q : \mathfrak {A}^{m,p}_{1,N;0}\rightarrow \mathfrak {A}^{m,p}_{1,N;0} \end{aligned}$$

is smooth. The arguments in [24, Section 4, 5, and 6] (cf. Lemma 5.2, Proposition 5.1, Lemma 6.1, and Theorem 6.1 in [24]) then show that the associated conjugate map

$$\begin{aligned}&(\varphi ,f)\mapsto \big (R_\varphi \circ \Delta ^{-1}\circ R_{\varphi ^{-1}}\big )\circ \big (R_{\varphi }\circ \nabla \circ Q\circ R_{\varphi ^{-1}}\big )(f),\\&\quad \mathfrak {A}{{\mathcal {D}}}^{m,p}_{1,N;0}\times \mathfrak {A}^{m,p}_{1,N;0}\rightarrow \mathfrak {A}{{\mathcal {D}}}^{m,p}_{1,N;0}\times \mathfrak {A}^{m,p}_{1,N;0}, \end{aligned}$$

is smooth. This implies that the Euler vector field $\mathcal {E}$ is smooth as a map

$$\begin{aligned} \mathcal {E} : {\mathcal {A}D}^{m,p}_{1,N;0}\times {{\mathcal {A}}}^{m,p}_{1,N;0}\rightarrow {\mathcal {A}D}^{m,p}_{1,N;0}\times {{\mathcal {A}}}^{m,p}_{1,N;0}. \end{aligned}$$

(98)

The arguments in the proof of Theorem 1.1 above then complete the proof of the proposition (cf. also Section 7 in [24]). $\square$

Global existence in the case when $d=2$

In this section we generalize Theorem 1.1 in [30] and prove that for $d=2$ the solution of the Euler equation (1) has a unique global in time solution in the asymptotic space ${{\mathcal {Z}}}^{m,p}_N$ with weight $\gamma _N$ such that $\gamma _N+d/p>0$ is integer.¹⁴ For a given $a\in \mathbb {R}$ denote by $\lfloor a \rfloor$ the integer part of a. We will follow the notation introduced in [30, Section 2].

For a given $1<p<\infty$, $m>2/p$, and $\delta +2/p>0$ we set $N:=\lfloor \delta +2/p \rfloor$, $\gamma _N:=\delta$, and consider the space of complex valued functions of $z\in \mathbb {C}$,

$$\begin{aligned} \mathcal {Z}^{m,p}_{n,N}:=\Big \{\chi \sum _{n\le k+l\le N}\frac{a_{kl}}{z^k{{\bar{z}}}^l}+f\,\Big |\,f\in W^{m,p}_{\gamma _N}, a_{kl}\in \mathbb {C}\Big \}, \end{aligned}$$

(99)

where $0\le n\le N+1$ and where we omit the summation term if $n=N+1$ and set $\mathcal {Z}^{m,p}_{n,N}\equiv W^{m,p}_{\gamma _N}$. We also set $\mathcal {Z}^{m,p}_N\equiv \mathcal {Z}^{m,p}_{0,N}$. The space (99) is a closed subspace in the asymptotic space ${{\mathcal {A}}}^{m,p}_N$ of vector fields on $\mathbb {R}^2$ that satisfies Proposition 2.1 and 2.2 in [30]. Note however, that Proposition 3.3 and Theorem 3.2 in [30] does not hold for integer $\delta +2/p$. As a consequence, the proof of the global well-posedness of the Euler equation for $d=2$ in [30, Section 5] does not apply for integer values of $\delta +2/p$. Following [30, Section 2] we denote the group of diffeomorphisms of $\mathbb {R}^2$ modeled on $\mathcal {Z}^{m,p}_N$ by ${{\mathcal {Z}}}D^{m,p}_N$. First, we prove the following lemma.

Lemma D.1

Take $m>3+2/p,$ a non-integer $\delta +2/p>0,$ and let ${\hat{\delta }}$ be the lowest integer ${\hat{\delta }}>\delta$ such that ${\hat{\delta }}+2/p\in \mathbb {Z}$. Then,  for a given volume preserving $\varphi \in {{\mathcal {Z}}}D^{m,p}_M$ (with $\gamma _M:=\delta ,$ $M:=\lfloor \delta +2/p \rfloor )$ and $u_0\in \mathcal {Z}^{m,p}_N$ (with $\gamma _N:={\hat{\delta }},$ $N:={{\hat{\delta }}}+2/p)$ we have that

$$\begin{aligned} \big (R_\varphi \circ \partial _z^{-1}\circ R_{\varphi ^{-1}}\big )(\partial _z u_0)=u_0+\mathcal {R}(\varphi ,u_0),\quad \mathcal {R}(\varphi ,u_0)\in \mathcal {Z}^{m,p}_{1,M+1}, \end{aligned}$$

(100)

where the map $\mathcal {R} : {{\mathcal {Z}}}D^{m,p}_M\times {{\mathcal {Z}}}^{m,p}_N\rightarrow {{\mathcal {Z}}}^{m,p}_{M+1}$ is analytic and $\partial _z$ denotes the Cauchy operator $\partial _z : \mathcal {Z}^{m,p}_{1,M}\rightarrow \widetilde{\mathcal {Z}}^{m-1,p}_{M+1}$.¹⁵

Proof of Lemma D.1

Since $u_0\in \mathcal {Z}^{m,p}_{N}$ (with $\gamma _N={\hat{\delta }}$) we have that $\partial _z u_0\in \widetilde{\mathcal {Z}}^{m-1,p}_{1,M+1}$ and, by Proposition 3.4 in [30], $\big (R_\varphi \circ \partial _z^{-1}\circ R_{\varphi ^{-1}}\big )(\partial _z u_0)$ is well defined and belongs to $\mathcal {Z}^{m,p}_{1,M}$. By setting $w:=\big (R_\varphi \circ \partial _z^{-1}\circ R_{\varphi ^{-1}}\big )(\partial _z u_0)$ we then obtain from Lemma 2.4 in [30] that

$$\begin{aligned} \big (R_\varphi \circ \partial _z\circ R_{\varphi ^{-1}}\big )(w)=\partial _z u_0. \end{aligned}$$

This, together with formula (54) in [30] and the fact that $\varphi =\mathrm{id}_\mathbb {C}+u\in {{\mathcal {Z}}}D^{m,p}_M$ is volume preserving, then implies that $\partial _z w+(\partial _z w)(\partial _{{\bar{z}}}{{\bar{u}}}_0)-(\partial _{{\bar{z}}}w)(\partial _z{{\bar{u}}}_0)=\partial _z u_0$, or equivalently,

$$\begin{aligned} w=u_0+\partial _z^{-1}\big [(\partial _{{\bar{z}}}w)(\partial _z{{\bar{u}}}_0)-(\partial _z w)(\partial _{{\bar{z}}}{{\bar{u}}}_0)\big ]. \end{aligned}$$

(101)

Since, by Lemma 3.5 in [30], $w\equiv \big (R_\varphi \circ \partial _z^{-1}\circ R_{\varphi ^{-1}}\big )(\partial _z u_0)\in \mathcal {Z}^{m,p}_{1,M}$ depends analytically on $(\varphi ,u_0)\in {{\mathcal {Z}}}D^{m,p}_M\times {{\mathcal {Z}}}D^{m,p}_N$, we obtain from Proposition 2.2 in [30] that

$$\begin{aligned} (\partial _{{\bar{z}}}w)(\partial _z{{\bar{u}}}_0)-(\partial _z w)(\partial _{{\bar{z}}}{{\bar{u}}}_0)\in \widetilde{\mathcal {Z}}^{m-1,p}_{M+2} \end{aligned}$$

and depends analytically on $(\varphi ,u_0)\in {{\mathcal {Z}}}D^{m,p}_M\times {{\mathcal {Z}}}^{m,p}_N$. By combining this with Theorem 3.2 in [30] we then see that

$$\begin{aligned} \mathcal {R}(\varphi ,u_0):=\partial _z^{-1}\big [(\partial _{{\bar{z}}}w)(\partial _z{{\bar{u}}}_0)-(\partial _z w)(\partial _{{\bar{z}}}{{\bar{u}}}_0)\big ] \in \mathcal {Z}^{m,p}_{1,M+1} \end{aligned}$$

and depends analytically on $(\varphi ,u_0)\in {{\mathcal {Z}}}D^{m,p}_M\times {{\mathcal {Z}}}^{m,p}_N$. This completes the proof of the lemma. $\square$

Now, we are ready to prove

Proposition D.1

Assume that $m>3+2/p,$ $\delta +2/p>0$ is an integer,  and $d=2$. Then,  for any (with $\gamma _N:=\delta$ and $N:=\delta +2/p)$ the Euler equation (1) has a unique global in time solution that depends continuously on the initial data (cf. [30, Theorem 1.1] for the case when $\gamma _N+2/p$ is not integer).

Proof of Proposition D.1

Assume that $\delta +2/p>0$ is an integer and choose $\delta ^-\in \mathbb {R}$ such that $0<\delta -\delta ^-<1$ and $\delta ^-+2/p>0$. Take $u_0\in {{\mathcal {Z}}}^{m,p}_N$ (with $\gamma _N=\delta$). Since ${{\mathcal {Z}}}^{m,p}_N$ is a subspace in ${{\mathcal {Z}}}^{m,p}_M$ (with $\gamma _M:=\delta ^-$ and $M:=\lfloor \delta ^-+2/p \rfloor =N-1$) and since $\delta ^-+2/p$ is not integer, we conclude from [30, Theorem 1.1] that there exists a unique solution of the Euler equation

$$\begin{aligned} u\in C\big ([0,\infty ),\mathcal {Z}^{m,p}_M\big )\cap C^1\big ([0,\infty ),\mathcal {Z}^{m-1,p}_M\big ) \end{aligned}$$

that depends continuously on the initial data $u_0\in \mathcal {Z}^{m,p}_N$. By [30, Proposition 4.2], $\varphi \in C^1\big ([0,\infty ),\mathcal {Z}D^{m,p}_M\big )$ where , $\varphi |_{t=0}=u_0$, and it depends continuously on the initial data $u_0\in \mathcal {Z}^{m,p}_N$. The preservation of vorticity (cf. formula (76) in [30]) and Lemma D.1 then imply that

$$\begin{aligned} u&=\partial _z^{-1}\big ((\partial _z u_0)\circ \varphi ^{-1}\big )= R_{\varphi ^{-1}}\circ \big (R_\varphi \circ \partial _z^{-1}\circ R_{\varphi ^{-1}}\big )(\partial _z u_0)\nonumber \\&=u_0\circ \varphi ^{-1}+\mathcal {R}(\varphi ,u_0)\circ \varphi ^{-1} \end{aligned}$$

(102)

where $\mathcal {R}(\varphi ,u_0)\in \mathcal {Z}^{m,p}_{1,M+1}$ and it depends analytically on $(\varphi ,u_0)\in {{\mathcal {Z}}}D^{m,p}_M\times {{\mathcal {Z}}}^{m,p}_N$. Since $\gamma _M+1>\delta$ we have that $\mathcal {Z}^{m,p}_{M+1}\subseteq \mathcal {Z}^{m,p}_N$. By Proposition 2.3 and Proposition 2.4 in [30] we then obtain that

$$\begin{aligned} u\in C\big ([0,\infty ),\mathcal {Z}^{m,p}_N\big )\cap C^1\big ([0,\infty ),\mathcal {Z}^{m-1,p}_N\big ) \end{aligned}$$

and it depends continuously on the initial data $u_0\in \mathcal {Z}^{m,p}_N$. This completes the proof of the proposition. $\square$