On the Rigid-Lid Approximation for Two Shallow Layers of Immiscible Fluids with Small Density Contrast

Abstract

The rigid-lid approximation is a commonly used simplification in the study of density-stratified fluids in oceanography. Roughly speaking, one assumes that the displacements of the surface are negligible compared with interface displacements. In this paper, we offer a rigorous justification of this approximation in the case of two shallow layers of immiscible fluids with constant and quasi-equal mass density. More precisely, we control the difference between the solutions of the Cauchy problem predicted by the shallow-water (Saint-Venant) system in the rigid-lid and free-surface configuration. We show that in the limit of a small density contrast, the flow may be accurately described as the superposition of a baroclinic (or slow) mode, which is well predicted by the rigid-lid approximation, and a barotropic (or fast) mode, whose initial smallness persists for large time. We also describe explicitly the first-order behavior of the deformation of the surface and discuss the case of a nonsmall initial barotropic mode.

Appendix: Proof of Proposition 2.2

In this section, we detail the proof of Proposition 2.2, which follows the classical theory concerning Friedrichs-symmetrizable quasilinear systems. The proof is based on a priori energy estimates, for which the key ingredients are product and commutator estimates in Sobolev spaces. We first recall such results and refer the reader to, e.g., Alinhac and Gérard (1991), Lannes (2013) for the proof of Lemmata 5.1 and 5.3.

Lemma 5.1

(Product estimates) Let $s\ge 0$. For all $f,g\in H^s(\mathbb {R})\bigcap L^\infty (\mathbb {R})$, one has

$$\begin{aligned} \big \vert \ f \ g\ \big \vert _{H^s} \ \lesssim \ \big \vert \ f \ \big \vert _{L^\infty } \big \vert \ g\ \big \vert _{H^s}+\big \vert \ f \ \big \vert _{H^s} \big \vert \ g\ \big \vert _{L^\infty }. \end{aligned}$$

If $s\ge s_0>1/2$, then one deduces, thanks to a continuous embedding of Sobolev spaces,

$$\begin{aligned} \big \vert \ f \ g\ \big \vert _{H^s} \ \lesssim \ \big \vert \ f \ \big \vert _{H^s} \big \vert \ g\ \big \vert _{H^s}. \end{aligned}$$

Let $F\in C^\infty (\mathbb {R})$ such that $F(0)=0$. If $g\in H^s(\mathbb {R})\bigcap L^\infty (\mathbb {R})$ with $s\ge 0$, then one has $F(g)\in H^s(\mathbb {R})$ and

$$\begin{aligned} \big \vert \ F(g) \ \big \vert _{H^s} \ \le \ C(\big \vert \ g\ \big \vert _{L^\infty },\big \vert \ F\ \big \vert _{C^\infty })\big \vert \ g \ \big \vert _{H^s}. \end{aligned}$$

Throughout the paper, we repeatedly make use of the following corollary.

Corollary 5.2

Let $f,\zeta \in L^\infty \bigcap H^{ s}$, with $s\ge 0$ and $h(\zeta )\equiv 1-\zeta$, with $h(\zeta )\ge h_0>0$ for any $x\in \mathbb {R}$. Then one has

$$\begin{aligned} \big \vert \frac{1}{h(\zeta )} f \big \vert _{H^{s}} \&\le \ C(h_0^{-1},\big \vert \zeta \big \vert _{L^{\infty }})\big ( \big \vert f\big \vert _{H^{s}}+ \big \vert \zeta \big \vert _{H^{s}}\big \vert f\big \vert _{L^\infty }\big )\\ \big \vert f-\frac{1}{h(\zeta )} f \big \vert _{H^{s}} \&\le \ \ C(h_0^{-1},\big \vert \zeta \big \vert _{L^{\infty }})\big ( \big \vert \zeta \big \vert _{L^{\infty }}\big \vert f\big \vert _{H^{s}}+ \big \vert \zeta \big \vert _{H^{s}}\big \vert f\big \vert _{L^\infty }\big ). \end{aligned}$$

Proof

We will use the identity

$$\begin{aligned} \frac{1}{h(\zeta )} f \ = \ \frac{1}{1-\zeta }f \ = \ f \ + \ \frac{\zeta }{1-\zeta }f. \end{aligned}$$

By Lemma 5.1, we deduce

$$\begin{aligned} \big \vert \frac{1}{h(\zeta )} f \big \vert _{H^{s}} \le \ \big \vert f \big \vert _{H^{s}} \ + \ \big \vert \frac{\zeta }{1-\zeta } f \big \vert _{H^{s}}\\&\lesssim \ \big \vert f \big \vert _{H^{s}} \ + \ \big \vert \frac{\zeta }{1-\zeta } \big \vert _{L^\infty }\big \vert f \big \vert _{H^{s}} \ + \ \big \vert \frac{\zeta }{1-\zeta }\big \vert _{H^{s}}\big \vert f \big \vert _{L^\infty }. \end{aligned}$$

The only nontrivial term to estimate is now $\big \vert \frac{\zeta }{1-\zeta }\big \vert _{H^{s}}$. Using that $h(\zeta )=1-\zeta \ge h_0>0$, we introduce a function $F\in C^\infty (\mathbb {R})$ such that

$$\begin{aligned} F(X) \ = \ {\left\{ \begin{array}{ll} \frac{X}{1-X} &{} \text { if } 1-X\ge h>0,\\ 0 &{} \text { if } 1-X\le 0. \end{array}\right. } \end{aligned}$$

The function $F$ satisfies the hypotheses of Lemma 5.1, and we have

$$\begin{aligned} \big \vert \frac{\zeta }{1-\zeta }\big \vert _{H^{s}} \ = \ \big \vert F(\zeta )\big \vert _{H^{s}} \ \le \ C(\big \vert \zeta \big \vert _{L^\infty },h_0^{-1})\big \vert \zeta \big \vert _{H^{s}}. \end{aligned}$$

The first estimate of the lemma is proved. The second estimate is obtained in the same way using

$$\begin{aligned} f-\frac{1}{h(\zeta )} f \ =\ - \frac{\zeta }{1-\zeta }f. \end{aligned}$$

The corollary is proved. $\square$

The following lemma presents a generalization of the Kato–Ponce (Kato and Ponce 1988) commutator estimates due to Lannes Lannes (2006) (one has $\big \vert f\big \vert _{H^{s}}$ instead of $\big \vert \partial _x f\big \vert _{H^{s-1}}$ in the standard Kato–Ponce estimate).

Lemma 5.3

(Commutator estimates) For any $s\ge 0$ and $\partial _x f, g\in L^\infty (\mathbb {R})\bigcap H^{s-1}(\mathbb {R})$ we have

$$\begin{aligned} \big \vert \ [\Lambda ^s,f]g\ \big \vert _{L^2}\ \lesssim \ \big \vert \ \partial _x f\ \big \vert _{H^{s-1}}\big \vert \ g\ \big \vert _{L^\infty }+\big \vert \ \partial _x f\ \big \vert _{L^\infty }\big \vert \ g\ \big \vert _{H^{s-1}}. \end{aligned}$$

Thanks to the continuous embedding of Sobolev spaces, we have for $s\ge s_0+1, \ s_0>\frac{1}{2}$,

$$\begin{aligned} \big \vert \ [\Lambda ^s,f]g\ \big \vert _{L^2}\ \lesssim \ \big \vert \ \partial _x f\ \big \vert _{H^{s-1}}\big \vert \ g\ \big \vert _{H^{s-1}}. \end{aligned}$$

Let us now continue with the proof of Proposition 2.2. System (1.1) is quasilinear. In what follows we prove that it is Friedrichs-symmetrizable under conditions (2.2). We present below the symmetrizer of the system and compute the necessary energy estimates in Lemmata 5.5 and 5.6.

System symmetrizer. Recall that (1.1) reads $\partial _t U \ + \ A[U]\partial _x U \ = \ 0$, with

$$\begin{aligned} A[U] \ \equiv \ \begin{pmatrix} u_1 &{}\quad \frac{ u_2-u_1}{\varrho } &{}\quad \frac{h_1}{\varrho } &{}\quad \frac{h_2}{\varrho } \\ 0 &{}\quad u_2 &{}\quad 0 &{}\quad h_2 \\ \frac{1}{\varrho } &{}\quad 0 &{}\quad u_1 &{}\quad 0 \\ \frac{\gamma }{\varrho } &{}\quad \delta +\gamma &{}\quad 0 &{}\quad u_2 \end{pmatrix}, \end{aligned}$$

(5.1)

where we use the notation $h_1\equiv 1+\varrho \zeta _1-\zeta _2$ and $h_2\equiv \delta ^{-1}+\zeta _2$. Define

$$\begin{aligned} S[U]\equiv \begin{pmatrix} \gamma &{}\quad 0&{}\quad 0 &{}\quad 0 \\ 0&{}\quad \gamma +\delta &{}\quad 0 &{}\quad u_2-u_1\\ 0 &{}\quad 0 &{}\quad \gamma h_1 &{}\quad 0 \\ 0&{}\quad u_2-u_1 &{}\quad 0 &{}\quad h_2 \end{pmatrix}. \end{aligned}$$

(5.2)

We can easily check that $S[U]A[U]\equiv \Sigma [U]$ and $S[U]$ are symmetric. More precisely, we have

$$\begin{aligned}&\Sigma [U]\nonumber \\&\quad \equiv \begin{pmatrix} \gamma u_1 &{}\quad \frac{\gamma (u_2-u_1)}{\varrho }&{}\quad \frac{\gamma h_1}{ \varrho }&{}\quad \frac{\gamma h_2}{ \varrho } \\ \frac{\gamma (u_2-u_1)}{\varrho } &{}\quad 2 (\gamma +\delta ) (2u_2-u_1)&{}\quad 0 &{}\quad (\gamma +\delta )h_2+ u_2(u_2-u_1)\\ \frac{\gamma h_1}{ \varrho }&{}\quad 0 &{}\quad \gamma h_1u_1 &{}\quad 0 \\ \frac{\gamma h_2}{\varrho } &{}\quad (\gamma +\delta )h_2+ u_2(u_2-u_1)&{}\quad 0 &{}\quad h_2 (2u_2-u_1) \end{pmatrix}.\nonumber \\ \end{aligned}$$

(5.3)

We can easily check that $S[U]$ is positive definite provided that the following holds:

$$\begin{aligned} \gamma >0 \quad ; \quad \gamma +\delta >0\quad ; \quad h_1 > 0 \quad ; \quad h_2-\frac{|u_2-u_1|^2}{\gamma +\delta }> 0, \end{aligned}$$

which is guaranteed by condition (2.2).

Energy of our system. The natural energy of our system is

$$\begin{aligned} E^s(U)&\equiv \big ( S[\underline{U}] \Lambda ^s U,\Lambda ^s U\big )\nonumber \\&= \gamma \big \vert \zeta _1\big \vert _{H^s}^2 + (\gamma +\delta )\big \vert \zeta _2\big \vert _{H^s}^2+\gamma \int _\mathbb {R}\underline{h}_1 \big \vert \Lambda ^s u_1\big \vert ^2\nonumber \\&+\int _\mathbb {R}\underline{h}_2 \big \vert \Lambda ^s u_2\big \vert ^2 +2\int _\mathbb {R}(\underline{u}_2- \underline{u}_1) \big \{\Lambda ^su_2\big \}\big \{\Lambda ^s\zeta _2\big \}, \end{aligned}$$

(5.4)

with $\underline{h}_1\equiv 1+ \varrho \underline{\zeta }_1-\underline{\zeta }_2$ and $\underline{h}_2\equiv \delta ^{-1}+\underline{\zeta }_2$.

In what follows, we specify the equivalence between our energy and the norm $X^s$ offered by the well-posedness of the symmetrizer. Recall that $X^s$ denotes the space $H^s(\mathbb {R})^4$, endowed with the following norm:

$$\begin{aligned} \big \vert U\big \vert _{X^s}^2 \ = \ \gamma \big \vert \zeta _1\big \vert _{H^s}^2 \ + \ \big \vert \zeta _2\big \vert _{H^s}^2+\gamma \big \vert u_1\big \vert _{H^s}^2+\big \vert u_2\big \vert _{H^s}^2. \end{aligned}$$

Lemma 5.4

Let $s\ge 0$ and $\underline{\zeta }\in L^{\infty }(\mathbb {R})$, satisfying (2.2). Then $E^s(U)$ is uniformly equivalent to the $\vert \cdot \vert _{X^s}$-norm. More precisely, there exists positive constants $C_2=C(h_{0}^{-1},\delta _{\min }^{-1})>0$, and $C_1=C(\big \vert \underline{h}_1\big \vert _{L^\infty },\big \vert \underline{h}_2\big \vert _{L^\infty },\delta _{\max })>0$ such that

$$\begin{aligned} \frac{1}{C_1}E^s(U) \ \le \ \big \vert U \big \vert _{X^s}^2 \ \le \ C_2 E^s(U). \end{aligned}$$

Proof

The fact that $E^s(U) \ \le \ C_1 \big \vert U \big \vert _{X^s}$ is a simple consequence of the Cauchy–Schwarz inequality, applied to (5.4), where we use that (2.2) yields $\vert \underline{u}_2-\underline{u}_1 \vert ^2 < (\gamma +\delta ) \underline{h}_2$.

The other inequality follows directly from (2.2). More precisely, we have

$$\begin{aligned} E^s(U) \ge&\,\, \gamma \big \vert \zeta _1\big \vert _{H^s}^2+\gamma h_0 \int _\mathbb {R}\big \vert \Lambda ^s u_1\big \vert ^2 \\&+ (\gamma +\delta )\big \vert \zeta _2\big \vert _{H^s}^2\!+\!\int _\mathbb {R}\underline{h}_2 \big \vert \Lambda ^s u_2\big \vert ^2 \!-2\int _\mathbb {R}\sqrt{(\underline{h}_2-h_0)(\gamma +\delta )} \big \{\Lambda ^su_2\big \}\big \{\Lambda ^s\zeta _2\big \}, \end{aligned}$$

and the result is now clear. Lemma 5.4 is proved. $\square$

We now highlight energy estimates with respect to the linearized system from (1.1), namely

$$\begin{aligned} \partial _t U \ + \ A[\underline{U}]\partial _x U \ = \ \mathcal {R}\ , \end{aligned}$$

(5.5)

with given $\underline{U},\mathcal {R}$.

Lemma 5.5

($L^2$ energy estimate) Set $T,M>0$. Let $U\in L^\infty ( [0,T];X^0)$ satisfy (5.5), with given $\mathcal {R}\in L^1([0,T];X^0)$, and $\underline{U}$ satisfying (2.2), with $h_0>0$ (for any $t\in [0,T]$) as well as

$$\begin{aligned} \big \Vert \underline{U} \big \Vert _{L^\infty ([0,T]\times \mathbb {R})^4}+\big \Vert \partial _x\underline{U} \big \Vert _{L^\infty ([0,T]\times \mathbb {R})^4}+\varrho \big \Vert \partial _t \underline{U} \big \Vert _{L^\infty ([0,T]\times \mathbb {R})^4} \ \le \ M. \end{aligned}$$

Then there exists $C_0\equiv C(M,h_0^{-1},\delta _{\min }^{-1},\delta _{\max })$ such that $\forall t\in [0,T],$

$$\begin{aligned} E^0(U)(t)\le e^{C_0M\varrho ^{-1} t}E^0(U\left| _{\scriptstyle t=0}\right. )+\,\, C_0 \int ^{t}_{0} e^{C_0M\varrho ^{-1}( t-t')}\big \vert \mathcal {R}(t',\cdot )\big \vert _{X^s}\ dt'. \end{aligned}$$

(5.6)

Proof

Let us consider the $L^2$ inner product of (5.5) and $S[\underline{U}] U$:

$$\begin{aligned} \big (\partial _t U,S[\underline{U}] U\big ) \ + \ \big (A[\underline{U}]\partial _x U,S[\underline{U}] U\big ) \ = \ \big ( \mathcal {R},S[\underline{U}] U\big ). \end{aligned}$$

From the symmetry property of $S[\underline{U}],\Sigma [\underline{U}]$, and using the definition of $E^0(U)$, we deduce

$$\begin{aligned} \frac{1}{2} \frac{d}{dt}E^0(U) = \frac{1}{2}\big ( U,\big [\partial _t, S[\underline{U}]\big ] U\big )-\big (\Sigma [\underline{U}]\partial _xU, U\big ) + \big (\mathcal {R},S[\underline{U}] U\big )\nonumber \\&= \frac{1}{2}\big ( U,\big [\partial _t, S[\underline{U}]\big ] U\big )+\frac{1}{2}\big (\big [\partial _x,\Sigma [\underline{U}]\big ] U, U\big ) + \big (\mathcal {R},S[\underline{U}] U\big ). \end{aligned}$$

(5.7)

We now estimate each of the terms on the right-hand side of (5.7).

Estimate of $\big ( U,\big [\partial _t, S[\underline{U}]\big ] U\big )$. We have $\big ( U,\big [\partial _t, S[\underline{U}]\big ] U\big ) \ = \ \big ( U, \mathrm{d}S[\partial _t \underline{U}] U\big )$, with

$$\begin{aligned} \mathrm{d}S[\partial _t\underline{U}] \equiv \begin{pmatrix} 0 &{}0&{} 0 &{} 0 \\ 0&{} 0&{} 0 &{} \partial _t(\underline{u}_2-\underline{u}_1)\\ 0 &{} 0 &{}\gamma \partial _t (\varrho \underline{\zeta }_1-\underline{\zeta }_2) &{} 0 \\ 0&{} \partial _t(\underline{u}_2-\underline{u}_1) &{} 0 &{}\partial _t \underline{\zeta }_2 \end{pmatrix}. \end{aligned}$$

Using the Cauchy–Schwarz inequality and Lemma 5.4 we have straightforwardly

$$\begin{aligned} \big | \big ( U,\big [\partial _t, S[\underline{U}]\big ] U\big ) \big | \ \le \ C_0\big \vert \partial _t \underline{U}\big \vert _{L^\infty } C_2^{-1}\ \big \vert U \big \vert _{X^0}^2 \ \le \ C_0\ M\ \varrho ^{-1}\ E^0(U) , \end{aligned}$$

(5.8)

with $C_0=C(h_0^{-1},\delta _{\min }^{-1},\delta _{\max })$.

Estimate of $\big (\big [\partial _x,\Sigma [\underline{U}]\big ] U, U\big )$. We have $\big (\big [\partial _x,\Sigma [\underline{U}]\big ] U, U\big ) =\big ( U, \mathrm{d}\Sigma [\underline{U}] U\big )$, with

$$\begin{aligned} \mathrm{d}\Sigma [\underline{U}]\equiv \begin{pmatrix} \gamma \partial _x \underline{u}_1 &{}\frac{\gamma \partial _x(\underline{u}_2-\underline{u}_1)}{\varrho }&{} \frac{\gamma \partial _x(\varrho \underline{\zeta }_1-\underline{\zeta }_2)}{ \varrho }&{} \frac{\gamma \partial _x \underline{\zeta }_2}{ \varrho } \\ \frac{\gamma \partial _x(\underline{u}_2-\underline{u}_1)}{\varrho } &{} (\gamma +\delta )\partial _x(2 \underline{u}_2-\underline{u}_1)&{} 0 &{}\partial _x \big ((\gamma +\delta ) \underline{\zeta }_2+\underline{u}_2(\underline{u}_2-\underline{u}_1)\big )\\ \frac{\gamma \partial _x(\varrho \underline{\zeta }_1-\underline{\zeta }_2)}{ \varrho } &{} 0 &{} \gamma \partial _x (\underline{h}_1\underline{u}_1) &{} 0 \\ \frac{\gamma \partial _x \underline{\zeta }_2}{ \varrho } &{}\partial _x \big ((\gamma +\delta ) \underline{\zeta }_2+\underline{u}_2(\underline{u}_2-\underline{u}_1)\big ) &{} 0 &{}2\partial _x\big (\underline{h}_2(2 \underline{u}_2-\underline{u}_1)\big ) \end{pmatrix}. \end{aligned}$$

As previously, the Cauchy–Schwarz inequality and Lemmata 5.1 and 5.4 yield

$$\begin{aligned} \big |\big (\Sigma [\underline{U}]\partial _xU, U\big )\big | \ \le \ C_0\ M\ \varrho ^{-1}\ E^0(U) , \end{aligned}$$

(5.9)

with $C_0=C(M,h_0^{-1},\delta _{\min }^{-1},\delta _{\max })$.

Estimate of $\big (\mathcal {R},S[\underline{U}] U\big )$. By the Cauchy–Schwarz inequality and Lemmata 5.1 and 5.4,

$$\begin{aligned} \big | \big (\mathcal {R},S[\underline{U}] U\big ) \big | \ \le \ C_0\ \big \vert U \big \vert _{X^s} \big \vert \mathcal {R}\big \vert _{X^s} \ \le \ C_0' E^s(U)^{1/2} \big \vert \mathcal {R}\big \vert _{X^s} \ , \end{aligned}$$

(5.10)

with $C_0,C_0'=C(M,h_0^{-1},\delta _{\min }^{-1},\delta _{\max })$.

Estimate (5.6) is now a consequence of the Gronwall–Bihari inequality applied to the differential inequality obtained when plugging (5.8), (5.9), (5.10) into (5.7). $\square$

Lemma 5.6

($H^s$ energy estimate) Set $M,T>0$ and $s\ge s_0+1, s_0>1/2$. Let $U\in L^\infty ([0,T];X^s)$ satisfy (5.5), with $\mathcal {R}\in L^1([0,T];X^s)$, and $\underline{U}\in L^\infty ([0,T];X^s)$ satisfying (2.2) as well as

$$\begin{aligned} \big \Vert \underline{U} \big \Vert _{L^\infty ([0,T];X^s)}+\varrho \big \Vert \partial _t \underline{U} \big \Vert _{L^\infty ([0,T];X^{s-1} )} \ \le \ M. \end{aligned}$$

Then there exists $C_0\equiv C(M,h_0^{-1},\delta _{\min }^{-1},\delta _{\max })$ such that, for all $t \in [0, T]$,

$$\begin{aligned}&E^s(U)(t)\le e^{C_0M\varrho ^{-1}}E^s(U\left| _{\scriptstyle t=0}\right. ) + C_0 \int ^{t}_{0} e^{C_0M\varrho ^{-1}( t-t')}\big \vert \mathcal {R}(t',\cdot )\big \vert _{X^s}\ dt'.\quad \quad \quad \end{aligned}$$

(5.11)

Proof

As previously, we deduce from (5.5) the identity

$$\begin{aligned} \big (\Lambda ^s\partial _t U,S[\underline{U}] \Lambda ^s U\big ) \ + \ \big (\Lambda ^s A[\underline{U}]\partial _x U,S[\underline{U}] \Lambda ^s U\big ) \ = \ \big ( \Lambda ^s \mathcal {R},S[\underline{U}] \Lambda ^s U\big ) \ , \end{aligned}$$

where we recall the notation $\Lambda \equiv ({{\mathrm{Id}}}-\partial _x^2)^{1/2}$. It follows that

$$\begin{aligned} & \frac{1}{2} \frac{d}{dt}E^s(U) = \frac{1}{2}\big ( \Lambda ^s U,\big [\partial _t, S[\underline{U}]\big ] \Lambda ^s U\big )-\big (S[\underline{U}]\Lambda ^sA[\underline{U}]\partial _xU, \Lambda ^s U\big ) \nonumber \\&\qquad +\,\, \big ( \Lambda ^s \mathcal {R},S[\underline{U}] \Lambda ^s U\big )\nonumber \\&= \frac{1}{2}\big ( \Lambda ^s U,\big [\partial _t, S[\underline{U}]\big ] \Lambda ^s U\big )+\frac{1}{2}\big (\big [\partial _x,\Sigma [\underline{U}]\big ] \Lambda ^s U, \Lambda ^s U\big )\nonumber \\&\qquad +\,\, \big ( \Lambda ^s \mathcal {R},S[\underline{U}] \Lambda ^s U\big )\nonumber \\&\qquad -\big (S[\underline{U}]\big [\Lambda ^s,A[\underline{U}]\big ]\partial _xU, \Lambda ^s U\big ). \end{aligned}$$

(5.12)

The first three terms are bounded exactly as previously when replacing $U$ with $\Lambda ^s U$. The only novelty lies in the use of continuous Sobolev embeddings, so that

$$\begin{aligned} \big \Vert \underline{U} \big \Vert _{L^\infty ([0,T]\times \mathbb {R})^4}+\big \Vert \partial _x\underline{U} \big \Vert _{L^\infty ([0,T]\times \mathbb {R})^4}\ \lesssim \ \big \Vert \underline{U}\big \Vert _{L^\infty ([0,T];X^{s})}. \end{aligned}$$

Similarly, we have

$$\begin{aligned} \varrho \big \Vert \partial _t \underline{U} \big \Vert _{L^\infty ([0,T]\times \mathbb {R})^4} \ \lesssim \ \varrho \big \Vert \partial _t \underline{U}\big \Vert _{L^\infty ([0,T];X^{s-1})}. \end{aligned}$$

The remaining term is estimated as follows. Using the commutator estimate in Lemma 5.3 we have

$$\begin{aligned} \big \vert \big [\Lambda ^s,A[\underline{U}]\big ]\partial _xU\big \vert _{L^2} \ \le \ C \big \vert \partial _x U\big \vert _{H^{s-1}} \big \vert \big [\partial _x, A[\underline{U}]\big ]\big \vert _{H^{s-1}}\ \le \ C_0\ M \ \varrho ^{-1}\ \big \vert U\big \vert _{X^{s}} , \end{aligned}$$

with $C_0=C(M,h_0^{-1},\delta _{\min }^{-1},\delta _{\max })$. Altogether, we deduce from (5.12)

$$\begin{aligned} \frac{1}{2} \frac{d}{dt}E^s(U) \ \le \ C_0M \varrho ^{-1} E^s(U) \ + \ C_0 E^s(U)^{1/2} \big \vert \mathcal {R}\big \vert _{X^s}. \end{aligned}$$

Estimate (5.11) is now a consequence of the Gronwall–Bihari inequality, and the lemma is proved. $\square$

Completion of Proof of Proposition 2.2

The well-posedness of system (1.1) is now a consequence of the energy estimates of Lemmata 5.5 and 5.6, following the standard strategy (we refer the reader to standard textbooks, e.g., Taylor 1997; Alinhac and Gérard 1991; Métivier 2008, for more details). More precisely, we first show that the linearized problem (5.5) is well posed, then the solution of the nonlinear problem (1.1) is obtained as the limit of an iterative scheme:

$$\begin{aligned} \partial _t U^{n+1} \ + \ A[U^n]\partial _x U^{n+1} \ = \ 0. \end{aligned}$$

The restriction on the time scale $t\in [0, T\varrho ]$ is necessary to guarantee that $(U^n)_{n\in \mathbb {N}}$ is a Cauchy sequence, and in particular that $U^n$ is uniformly bounded with respect to $n$, over a time domain which can be chosen independent of $n$. The desired estimate on $\big \vert U\big \vert _{X^s}$ follows directly from Lemma 5.6, with $\underline{U}=U$ and $R\equiv 0$, and the corresponding estimate on $\big \vert \partial _t U\big \vert _{X^s}$ is then deduced using (1.1). The uniqueness comes from a similar estimate on the difference between two solutions, and the blow-up criterion as $t\rightarrow T_{\max }$ if $T_{\max }<\infty$ follows from standard continuation arguments. This concludes the proof of Proposition 2.2. $\square$