\documentclass[JEP,XML,SOM,Unicode]{cedram}
\datereceived{2026-07-01}
\dateaccepted{2026-08-03}
\dateepreuves{2028-08-18}
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\begin{document}
\title{Corrigendum: The Cauchy problem for quasi-linear parabolic systems revisited}

\author[\initial{I.} \lastname{Gallagher}]{\firstname{Isabelle} \lastname{Gallagher}}
\address{DMA, École normale supérieure, CNRS, PSL University,\\
75005 Paris, France}
\address{\& UFR de mathématiques, Université Paris Cité,\\
75013 Paris, France}
\email{isabelle.gallagher@ens.fr}
\urladdr{https://webusers.imj-prg.fr/~isabelle.gallagher/}

\author[\initial{A.} \lastname{Moussa}]{\firstname{Ayman} \lastname{Moussa}}
\address{DMA, École normale supérieure, CNRS, PSL University,\\
75005 Paris, France}
\address{\& LJLL, Sorbonne Université, Université Paris Cité,\\
75005 Paris, France}
\email{ayman.moussa@sorbonne-universite.fr}
\urladdr{https://www.ljll.fr/~moussa/}

\begin{abstract}
In \cite[p.\,1666]{galmou}, the proof of Theorem 6 had a misprint, which turned into a wrong argument that we correct here.
\end{abstract}

\subjclass{35K40, 35K45, 35K59, 35K57}

\keywords{Quasi-linear parabolic systems, Petrovskii's condition, Littlewood-Paley, paraproduct}

\altkeywords{Systèmes parabolique quasi-linéaires, condition de Petrovskii, Littlewood-Paley, paraproduit}

\alttitle{Corrigendum: Le problème de Cauchy pour les systèmes paraboliques quasi-linéaires: une nouvelle approche}

\begin{altabstract}
Dans \cite[p.\,1666]{galmou}, la démonstration du théorème 6 comportait une coquille qui a donné lieu à un raisonnement erroné, que nous corrigeons ici.
\end{altabstract}

\maketitle

Let us first recall a few useful notations. We use here the Besov spaces $B^s_{p,1}$ defined in \cite[App.\,B]{galmou}. For any given~$T>0$ and~$p \in [1,\infty)$, the analogue of the solution space~$E_T^s$ in the Besov setting is:
\[
\E^s_T:= \mathcal C^0 ([0,T];B^s_{p,1}) \cap \L^1 (0,T;B^{s+2}_{p,1}),
\]
while the analogue of the exterior force space~$Y_T^s$ is:
\[
{\mathbb Y}_T^{s}:= \L^1 (0,T;B^{s }_{p,1}).
\]

Section~7.4 of \cite{galmou} intended to define a fixed-point procedure leveraging on \cite[Th.\,7]{galmou}. We propose below a complete rewriting of this subsection.

\setcounter{section}{7}
\setcounter{subsection}{3}
\subsection{Conclusion}
To conclude the proof we shall use the linear estimate provided by \cite[Th.\,7]{galmou} to implement a fixed point argument. Let us set
\[
\mathbb G^p_T:={\mathbb E}^\sfrac dp_T \cap \mathcal C^0(Q_T).
\]
Given $(U^0,F)\in B^\sfrac dp_{p,1}\cap {\mathbb Y}^\sfrac dp_T$, we consider the following map
\begin{align*}
\Theta : {\mathbb G}^\sfrac dp_T &\to {\mathbb E}^\sfrac dp_T \\
U &\mto U^\star,
\end{align*}
where $U^\star$ is the only element of~${\mathbb E}^\sfrac dp_T$ solving $L_{A(U)} U^\star = F$ with~$U(0) = U^0$. We~define as before the reference solution $U_F:=\Theta(0)$.

Before starting the fixed point procedure, let us check that~$\Theta$ maps~$ \mathbb G^p_T$ onto itself. We recall that~$U^\star:=\Theta(U)$ solves
\[
\partial_tU^\star = \sum_k \partial_k \big(A(U)\partial_kU^\star\big) + F,
\]
and we know that~$F$ belongs to~$ {\mathbb Y}^\sfrac dp_T$. Let us prove that the same information holds for~$\partial_k \big(A(U)\big)\partial_kU^\star\big)$. We write
\[
\partial_k \big(A(U)\partial_kU^\star\big) = A(U)\partial^2_kU^\star + A'(U)\partial_kU \partial_kU^\star,
\]
and we know that smooth functions are continuous over~$B^\sfrac dp_{p,1}$, so since~$B^\sfrac dp_{p,1}$ is an algebra. It follows that
\[
\big\| A(U)\partial^2_kU^\star \big\| _{ {\mathbb Y}^\sfrac dp_T} \lesssim \big\| A(U) \big\| _{L^\infty_T(B^\sfrac dp_{p,1})} \|\partial^2_k U^\star\| _{ {\mathbb Y}^\sfrac dp_T} \lesssim \Phi( \| U \| _{L^\infty_T(B^\sfrac dp_{p,1})} )\|U^\star\| _{ {\mathbb Y}^{\Psfrac dp+2}_T}
\]
with~$\Phi$ smooth and increasing. Similarly,
\[
\big\| A'(U)\partial_kU \partial_kU^\star
\big\| _{ {\mathbb Y}^\sfrac dp_T} \lesssim \big\| A'(U) \big\| _{L^\infty_T(B^\sfrac dp_{p,1})} \|\partial_k U \| _{L^2_T(B^{\Psfrac dp+1}_{p,1})} \|\partial_k U^\star\| _{L^2_T(B^{\Psfrac dp+1}_{p,1})}.
\]
We thus find that~$ \partial_tU^\star$ belongs to~$\L^1(0,T;B^\sfrac dp_{p,1})$, which implies the expected result since~$B^\sfrac dp_{p,1}$ is embedded in the space of continuous functions. In the following, we~restrict our attention to the space~${\mathbb E}^\sfrac dp_T$, as the estimates can then be extended to the full space~$ \mathbb G^p_T$ by the same argument on the time derivative.

Consider~$U_1$ in the ${\mathbb E}^\sfrac dp_T$ closed ball of radius $r$ centered at $U_F$ and set~$V:=U_1^\star - U_F$. The same argument as in \cite[Cor.\,2.5]{galmou} allows to replace the continuity constant~$[A(U_F)]_0$ by~$\|U_F\|_{\mathbb G^p_T}$, so that
\[
\|V\|_{{\mathbb E}^\sfrac dp_T} \lesssim_{T,\|U_F\|_{\mathbb G^p_T},\omega_{U_F} } \| V (0)\|_{B^{\sfrac dp }_{p,1}} + \|L_{A(U_F)}V\|_{\mathbb E_T^\sfrac dp}.
\]
Since
\[
L_{A(U_F)} V = L_{A(U_1)} V + \sum_k \partial_k \big(A(U_1) - A(U_F)\big)\partial_k V,
\]
we find, thanks to \cite[Prop.\,B.2]{galmou},
\begin{multline*}
\|V\|_{{\mathbb E}^\sfrac dp_T}\lesssim_{T,\|U_F\|_{\mathbb G^p_T},\omega_{U_F}} \|L_{A(U_1)}V\|_{\mathbb E_T^\sfrac dp} + \|A(U_1)- A(U_F)\|_{\L^\infty(Q_T)} \|V\|_{{\mathbb Y}^{\Psfrac dp+2}_T} \\
+ \int_0^T
\big \|\big[A(U_1)- A(U_F)\big](t)\|_{B_{p,1}^{\Psfrac dp+2}} \| V (t)\|_{B^{\sfrac dp }_{p,1}} \, \dd t.
\end{multline*}
It follows that, if~$r$ is small enough (depending only on~$T$ and~$ U_F $), then
\begin{multline*}
\|V\|_{{\mathbb E}^\sfrac dp_T} \lesssim_{T, \|U_F\|_{\mathbb G^p_T} ,\omega_{U_F} } \|L_{A(U_1)}V\|_{\mathbb E_T^\sfrac dp}\\
+ \int_0^T
\big \|\big[A(U_1)- A(U_F)\big](t)\|_{B_{p,1}^{\Psfrac dp+2}} \| V (t)\|_{B^{\sfrac dp }_{p,1}} \, \dd t.
\end{multline*}
But
\[
L_{A(U_1)} V = \sum_k \partial_k \big(A(U_1) - A(0)\big)\partial_k {U_F},
\]
so again, thanks to \cite[Prop.\,B.2]{galmou},
\[
\begin{aligned}
\|V\|_{{\mathbb E}^\sfrac dp_T} & \lesssim_{T, \|U_F\|_{\mathbb G^p_T} ,\omega_{U_F}} \|A(U_1)- A(0)\|_{\L^\infty(Q_T)} \|U_F\|_{\mathbb Y_T^{\Psfrac dp+2}} \\
&\quad+ \int_0^T
\big \|\big[A(U_1)- A(0)\big](t)\|_{B_{p,1}^{\Psfrac dp+ 1 }} \| U_F (t)\|_{B^{\Psfrac dp +1 }_{p,1}} \, \dd t\\
&\quad+ \int_0^T
\big \|\big[A(U_1)- A(U_F)\big](t)\|_{B_{p,1}^{\Psfrac dp+2}} \| V (t)\|_{B^{\sfrac dp }_{p,1}} \, \dd t.
\end{aligned}
\]
In the right-hand side, the two first lines are small as soon as both $T$ and $r$ are small: for the first one we can split $A(U_1)-A(0)$ into $A(U_1)-A(U_F) + A(U_F)-A(0)$ and use the triangular inequality while for the second one, we use the embedding of ~${\mathbb E}^{d/p}_T$ into~$\L^2([0,T]; B^{d/p+1}_{p,1})$ by interpolation. Then the last line is handled, thanks to a Gronwall estimate, and we can find~$T$ small enough so that~$ \|V\|_{{\mathbb E}^{d/p}_T} \leq r$.

Similarly, we can prove that~$\Theta$ is Lipschitz on the ball of~${\mathbb E}^\sfrac dp_T$ centered at~$U_F$ and of radius~$r$ with a Lipschitz constant smaller than one (for~$r$ and~$T$ small enough). This follows the same lines as the above computations: we fix~$U_1$ and~$U_2$ such that~$\| U_i-\nobreak U_F\|_{{\mathbb E}^\sfrac dp_T} \leq r$ and we note that thanks to \cite[Prop.\,B.2]{galmou}
\begin{multline*}
\|U_1^\star-U_2^\star\|_{{\mathbb E}^\sfrac dp_T} \lesssim_{T, \|U_1\|_{\mathbb G^p_T} ,\omega_{U_1} } \int_0^T \big \|(A(U_1)- A(U_2)(t)\big)\|_{B_{p,1}^{\Psfrac dp+1}} \|U_2^\star(t)\|_{B_{p,1}^{\Psfrac dp+1}} \, \dd t\\
\shoveright{+ \int_0^T \big \|(A(U_1)- A(U_2)(t)\big)\|_{\L^\infty} \|U_2^\star(t)\|_{B_{p,1}^{\Psfrac dp+2}} \, \dd t} \\
\lesssim_{T, r,\|U_F\|_{\mathbb G^p_T} ,\omega_{U_F} } \|U_1 -U_2 \|_{{\mathbb E}^\sfrac dp_T} (r+ \|U_F\|_{L^1_TB_{p,1}^{\Psfrac dp+2}} )\\
+
\|U_1 -U_2 \|_{{\mathbb E}^\sfrac dp_T}(r+ \|U_F\|_{L^2_TB_{p,1}^{\Psfrac dp+1}} ),
\end{multline*}
where we have used that, since~$\|U_1-U_F\|_{\mathbb{E}_T^{d/p}} \leq r$, then in particular for any~$(t,x) $ and~$ (t',x') $ in~$Q_T$ there holds
\[
|U_1(t,x) - U_1(t',x')| \leq 2r + |U_F(t,x)-U_F(t',x')|,
\]
whence
\[
\omega_{U_1} \leq r + \omega_{U_F}.
\]
The result follows, choosing~$r$ and~$T$ small enough.
\qed

\backmatter
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