Correction to: Wellposedness and Convergence of Solutions to a Class of Forced Non-diffusive Equations with Applications

1 Correction to: J. Math. Fluid Mech. (2019) 21:50 https://doi.org/10.1007/s00021-019-0454-1

In this erratum we revise the hypothesis and statement of [1, Theorem 2.3] to prove local in time existence of analytic, rather than Gevrey class solutions. As a consequence, we also revise the results in [1, Theorem 2.5] for the convergence of analytic solutions as $\nu$ goes to zero.

To begin with, we first recall that the abstract active scalar equations which are given by

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t\theta ^{\nu }+u^\nu \cdot \nabla \theta ^{\nu }=S, \\ u_j^{\nu }=\partial _{x_i} T_{ij}^{\nu }[\theta ^{\nu }],\theta ^{\nu }(x,0)=\theta _0(x) \end{array}\right. \end{aligned}$$

(1.1)

where ${\mathbb {T}}^d\times (0,\infty )=[0,2\pi ]^d\times (0,\infty )$ with $d\ge 2$. We assume that

$$\begin{aligned} \displaystyle \int _{{\mathbb {T}}^d}\theta ^\nu (x,t)dx=\int _{{\mathbb {T}}^d}S(x)=0 \text{ for } \text{ all } t\ge 0. \end{aligned}$$

The symbols $\{T_{ij}^{\nu }\}_{\nu \ge 0}$ refer to a sequence of operators which satisfy:

• A1 $\partial _i\partial _j T^{\nu }_{ij}f=0$ for any smooth functions f for all $\nu \ge 0$.

• A2 $T_{ij}^\nu :L^\infty ({\mathbb {T}}^d)\rightarrow BMO({\mathbb {T}}^d)$ are bounded for all $\nu \ge 0$.

• A3 For each $\nu >0$, there exists a constant $C_{\nu }>0$ such that for all $1\le i,j\le d$,

  $$\begin{aligned} |{\widehat{T}}^{\nu }_{ij}(k)|\le C_{\nu }|k|^{-3}, \forall k\in {\mathbb {Z}}^d. \end{aligned}$$

• A4 For each $1\le i,j\le d$,

  $$\begin{aligned} \lim _{\nu \rightarrow 0}\sum _{k\in {\mathbb {Z}}^d}|\widehat{T^{\nu }_{ij}}(k)-\widehat{T^0_{ij}}(k)|^2|\widehat{g}(k)|^2=0 \end{aligned}$$

  for all $g\in L^2$.

Moreover, we further assume that $\{T_{ij}^{\nu }\}_{\nu \ge 0}$ satisfy either one of the following assumptions:

$\hbox {A5}_{{1}}$:

There exists a constant $C_{0}>0$ independent of $\nu$, such that for all $1\le i,j\le d$,

$$\begin{aligned} \sup _{\nu \in (0,1]}\sup _{\{k\in {\mathbb {Z}}^d\}}|{\widehat{T}}^{\nu }_{ij}(k)|\le C_{0}; \end{aligned}$$

$$\begin{aligned} \sup _{\{k\in {\mathbb {Z}}^d\}}|{\widehat{T}}^0_{ij}(k)|\le C_{0}. \end{aligned}$$

(1.2)

$\hbox {A5}_{2}$:

There exists a constant $C_{0}>0$ independent of $\nu$, such that for all $1\le i,j\le d$,

$$\begin{aligned} \sup _{\nu \in (0,1]}\sup _{\{k\in {\mathbb {Z}}^d\}}|k_i{\widehat{T}}^{\nu }_{ij}(k)|\le C_{0}; \end{aligned}$$

$$\begin{aligned} \sup _{\{k\in {\mathbb {Z}}^d\}}|k_i{\widehat{T}}^0_{ij}(k)|\le C_{0}. \end{aligned}$$

(1.3)

We now give the revised statement for Theorem 2.3 given in [1]:

Theorem 2.3

(Analytic local wellposedness in the case $\nu =0$). Fix $r>\frac{d}{2}+\frac{3}{2}$ and $K_0>0$. Let $\theta ^0(\cdot ,0)=\theta _0$ and S be analytic functions with radius of convergence $\tau _0>0$ and satisfy

$$\begin{aligned} \Vert \Lambda ^re^{\tau _0\Lambda }\theta ^0(\cdot ,0)\Vert _{L^2}\le K_0,\qquad \Vert \Lambda ^re^{\tau _0\Lambda }S\Vert _{L^2}\le K_0. \end{aligned}$$

(2.4)

For $\nu =0$, under the assumptions A1–A2 and A5$_1$, there exists ${\bar{T}},{\bar{\tau }}>0$ and a unique analytic solution $\theta ^0$ to (1.1) defined on ${\mathbb {T}}^d\times [0,{\bar{T}}]$ with radius of convergence at least ${\bar{\tau }}$. In particular, there exists a constant $C=C(K_0)>0$ such that for all $t\in [0,{\bar{T}}]$,

$$\begin{aligned} \Vert \Lambda ^re^{{\bar{\tau }}\Lambda }\theta ^0(\cdot ,t)\Vert _{L^2}\le C. \end{aligned}$$

(2.5)

Moreover, if the assumption A3 holds as well, then we have

$$\begin{aligned} \Vert \Lambda ^re^{{\bar{\tau }}\Lambda }\theta ^\nu (\cdot ,t)\Vert _{L^2}\le C,\,\forall \nu >0, \end{aligned}$$

(2.6)

where $\theta ^\nu$ are analytic solutions to (1.1) for $\nu >0$ as described in [1, Theorem 2.2].

Proof of Theorem 2.3

The results follow by the similar argument given by Friedlander and Vicol [2] for the unforced system with $S\equiv 0$ in (1.1). $\square$

We also revise the statement of [1, Theorem 2.5] for the convergence of analytic solutions to (1.1).

Theorem 2.5

(Convergence of solutions as $\nu \rightarrow 0$). Depending on the assumptions A$5_1$ and A$5_2$, we have the following cases:

• Assume that the hypotheses and notations of Theorem 2.3 are in force. Under the assumptions A3–A4, if $\theta ^\nu$ and $\theta ^0$ are analytic solutions to (1.1) for $\nu >0$ and $\nu =0$ respectively with initial datum $\theta _0$ on ${\mathbb {T}}^d\times [0,{\bar{T}}]$ with radius of convergence at least ${\bar{\tau }}$ as described in Theorem 2.3, then there exists $T<{\bar{T}}$ and $\tau =\tau (t)<{\bar{\tau }}$ such that, for $t\in [0,T]$, we have

  $$\begin{aligned} \lim _{\nu \rightarrow 0}\Vert (\Lambda ^re^{\tau \Lambda }\theta ^{\nu }-\Lambda ^re^{\tau \Lambda }\theta ^0)(\cdot ,t)\Vert _{L^2}=0. \end{aligned}$$

  (2.7)

• Assume that the hypotheses and notations of [1, Theorem 2.4] are in force. Under the assumptions A3–A4, for $d\ge 2$ and $s>\frac{d}{2}+1$ and $t\in [0,T]$, we have

  $$\begin{aligned} \lim _{\nu \rightarrow 0}\Vert (\theta ^{\nu }-\theta ^0)(\cdot ,t)\Vert _{H^{s-1}}=0. \end{aligned}$$

  (2.8)

Remark 2.6

The proof for the convergence result (2.7) follows by the same argument given in [1, pp. 16–18] by taking $s=1$.

Remark 2.7

The applications to the magnetostrophic equations given in [1, Sect. 6] now hold under the revised statements of Theorems 2.3 and 2.5.