Then $(h, u) \in C([0, T^*); L^\infty((-1, 0); H^{s+2}(\mathbb{R}^2))$ the maximal-in-time solution to (3.1)-(3.2) emerging from initial data $(h, u)|_{t=0} = (h^0, u^0)$ is defined on the time interval $[0, T]$ and we have for almost all $r \in (-1, 0)$,

$$ \|(h_{\text{ref}} - h, u_{\text{ref}} - u)(\cdot, r)\|_{L^\infty_T H_x^s} \le \left( \|(h_{\text{ref}} - h, u_{\text{ref}} - u)(t = 0, \cdot, r)\|_{H_x^s} + \|(h_{\text{ref}} - h, u_{\text{ref}} - u)(t = 0, \cdot)\|_{L^1_r H_x^{s+1}} \right. \\ \left. + \left| (\underline{\rho}_{\text{ref}} - \underline{\rho}, \underline{u}_{\text{ref}} - \underline{u})(r) \right| + \|(\underline{\rho}_{\text{ref}} - \underline{\rho}, \underline{u}_{\text{ref}} - \underline{u})(r)\|_{L^1_r} \right) \times C \exp(CT/\kappa). \quad (3.10) $$

*Proof.* We first recall that Proposition 3.7 provides an estimate on the difference between the reference solution $(h_{\text{ref}}, u_{\text{ref}}) \in C([0, T]; L^\infty((-1, 0); H^{s+2}(\mathbb{R}^2))$ and the corresponding approximate solution defined as the solution to the system (3.7) with initial data $(h_{\text{app}}, u_{\text{app}})|_{t=0} = (h^0, u^0)$, $(h_{\text{app}}, u_{\text{app}}) \in C([0, T]; L^\infty((-1, 0); H_x^{s+2}(\mathbb{R}^2))$, whose existence and control on the time interval $[0, T]$ (lessening $T$ if necessary) is provided by Proposition 3.6. Specifically we have for almost any $r \in (-1, 0)$

$$ \|(h_{\text{ref}} - h_{\text{app}}, u_{\text{ref}} - u_{\text{app}})(\cdot, r)\|_{L^\infty_T H_x^{s+1}} \le \left( \|(h_{\text{ref}} - h_{\text{app}}, u_{\text{ref}} - u_{\text{app}})(t = 0, \cdot, r)\|_{H_x^{s+1}} \right. \\ \left. + C\kappa \left( \left| (\underline{\rho}_{\text{ref}} - \underline{\rho}, \underline{u}_{\text{ref}} - \underline{u})(r) \right| + \|(\underline{\rho}_{\text{ref}} - \underline{\rho})(r)\|_{L^1_r} \right) \right) \exp(CT/\kappa), $$

where $C$ depends only on $s, \zeta, \underline{M}, M_{\text{ref}}$ and $M_0$.

We then consider the difference between the exact solution $(h, u) \in C([0, T^*]; L^\infty((-1, 0); H_x^{s+2}(\mathbb{R}^2))$ — whose existence is provided by Proposition 3.2— and the approximate solution. By means of the consistency result, Proposition 3.5, we can adapt the proof of Proposition 3.3 and we find that under the assumptions that

$$ \|(h, u)\|_{L^\infty_T L^\infty_r H_x^s} + \kappa^{1/2} \|\partial_x h\|_{L^\infty_r L^2_T H_x^s} \le M \quad \text{and} \quad \underset{(t,x,r) \in [0,T] \times \mathbb{R} \times (-1,0)}{\operatorname{ess\,inf}} \quad 1 + h(t,x,r) \ge \zeta \quad (3.11) $$

there exists $C > 0$ depending only on $s, \zeta, \underline{M}, M_{\text{ref}}, M$ such that for all $\kappa \in (0, 1]$ and $t \in (0, \min(\{T, T^*\}))$ such that $Ct \le \kappa$ and for almost any $r \in (-1, 0)$ one has

$$ \max(\{\|(h - h_{\text{app}}, u - u_{\text{app}})(\cdot, r)\|_{L^\infty_t H_x^s}, \kappa^{1/2} \|\partial_x(h - h_{\text{app}})(\cdot, r)\|_{L^2_r H_x^s}\}) \\ \le 2 \|(h - h_{\text{app}}, u - u_{\text{app}})(t = 0, \cdot, r)\|_{L^\infty_r H_x^s} + Ct \|(h_{\text{ref}} - h_{\text{app}})(t, \cdot)\|_{L^1_r H_x^{s+1}}. $$

Iterating the stability estimate on $T_n := \min(\{nT_0, T\})$ we deduce that (augmenting $C$ if necessary)

$$ \max(\{\|(h - h_{\text{app}}, u - u_{\text{app}})\|_{L^\infty_t L^\infty_r H_x^s}, \kappa^{1/2} \|\partial_x(h - h_{\text{app}})\|_{L^\infty_r L^2_t H_x^s}\}) \\ \le \exp(CT/\kappa) \|(h - h_{\text{app}}, u - u_{\text{app}})(t = 0, \cdot)\|_{L^\infty_r H_x^s} + C\kappa \exp(CT/\kappa) \|h_{\text{ref}} - h_{\text{app}}\|_{L^\infty_T L^1_r H_x^{s+1}}. $$

Since by construction $(h_{\text{app}}, u_{\text{app}})|_{t=0} = (h, u)|_{t=0}$ and using the above control on $h_{\text{ref}} - h_{\text{app}}$ we infer (augmenting $C$ if necessary)

$$ \max(\{\|(h - h_{\text{app}}, u - u_{\text{app}})\|_{L^\infty_t L^\infty_r H_x^s}, \kappa^{1/2} \|\partial_x(h - h_{\text{app}})\|_{L^\infty_r L^2_t H_x^s}\}) \\ \le C\kappa \exp(CT/\kappa) \|(h_{\text{ref}} - h, u_{\text{ref}} - u)(t = 0, \cdot)\|_{L^1_r H_x^{s+1}} + C(\kappa \exp(CT/\kappa))^2 \|(\underline{\rho}_{\text{ref}} - \underline{\rho}, \underline{u}_{\text{ref}} - \underline{u})(r)\|_{L^1_r} $$

Notice that this estimate implies by triangle inequality an upper bound on $\|(h, u)\|_{L^\infty_t L^\infty_r H_x^s} + \kappa^{1/2} \|\partial_x h\|_{L^\infty_r L^2_t H_x^s}$ which (choosing $M$ sufficiently large and $\delta_0$ sufficiently small) enforces strictly the condition (3.11), so that by the standard continuity argument the above holds without restriction on $t \in (0, \min(\{T, T^*\}))$. By the persistence of regularity stated in Proposition 3.2 we infer that $T^* > T$ and the above holds for any $t \in (0, T]$. The estimate (3.10) then immediately follows from the triangle inequality. $\square$
# 4 Conclusion

Thanks to Proposition 2.7 (concerning the well-posedness and control of solutions to the bilayer system) on one hand and Proposition 3.8 (concerning the control of the deviations of nearby solutions to some given reference solutions) on the other hand, one infers immediately the announced rigorous justification of the propagation in time of the columnar motion and sharp stratification assumptions in near-bilayer situations (within the hydrostatic framework). Specifically, we have the following result.

**Theorem 4.1.** Let $s \ge s_0 > 3/2$, $\underline{M} > 0$, $M_0 > 0$, $\zeta \in (0, 1)$, and $\kappa \in (0, 1]$. Then there exist $C > 0, T > 0$ independent of $\kappa$ and $\delta_0 > 0$ (depending on $\kappa$) such that the following holds.

Let $(\underline{\rho}, \underline{u}) \in L^\infty((-1, 0))^2$ and $(h^0, u^0) \in L^\infty((-1, 0); H^{s+2}(\mathbb{R})^2)$ be such that

$$ \|(\underline{\rho}, \frac{1}{\underline{\rho}})\|_{L_r^\infty} \le \underline{M}, \quad \|(h^0, \kappa^{1/2} \partial_x h^0)\|_{L_r^\infty H_x^{s+1}} + \|u^0\|_{L_r^\infty H_x^{s+2}} \le M_0 $$

and there exists $(\rho_s, \rho_b, \underline{H}_s, \underline{H}_b, \underline{U}_s, \underline{U}_b) \in \mathbb{R}^6$ such that $\underline{H}_s + \underline{H}_b = 1$ and $\underline{U}_s + \underline{U}_b = 0$ as well as $(H_s^0, H_b^0, U_s^0, U_b^0) \in H^{s+4}(\mathbb{R})^2 \times H^{s+3}(\mathbb{R})^2$ such that the hyperbolicity condition holds:

$$ \forall x \in \mathbb{R}, \quad (\rho_s, \rho_b, \underline{H}_s + H_s^0(x), \underline{H}_b + H_b^0(x), \underline{U}_s + U_s^0(x), \underline{U}_b + U_b^0(x)) \in \mathfrak{p}^\zeta, $$

where $\mathfrak{p}^\zeta$ is defined in (2.5), and

$$ \|(H_s^0, H_b^0, U_s^0, U_b^0, \kappa \partial_x H_s^0, \kappa \partial_x H_b^0)\|_{H^{s+3}} \le M_0 $$

and such that denoting $(\underline{\rho}_{\text{bl}}^0, \underline{u}_{\text{bl}}^0, h_{\text{bl}}^0, u_{\text{bl}}^0)$ through (1.4) we have

$$ \|(h_{\text{bl}}^0 - h^0, u_{\text{bl}}^0 - u^0)\|_{L_r^1 H_x^{s+1}} + \|(\underline{\rho}_{\text{bl}} - \underline{\rho}, \underline{u}_{\text{bl}} - \underline{u})\|_{L_r^1} \le \delta_0, $$

then

1. there exists $(H_s, H_b, U_s, U_b) \in C([0, T]; H^{s+3}(\mathbb{R})^4)$ solution to (1.3) emerging from the initial data

$$ (H_s, H_b, U_s, U_b)|_{t=0} = (H_s^0, H_b^0, U_s^0, U_b^0); $$

2. there exists $(h, u) \in C([0, T]; L^\infty((-1, 0); H^{s+2}(\mathbb{R})^2)$ solution to (1.1) emerging from initial data

$$ (h, u)|_{t=0} = (h^0, u^0); $$

3. denoting $(\underline{\rho}_{\text{bl}}, \underline{u}_{\text{bl}}, h_{\text{bl}}, u_{\text{bl}})$ through (1.4) we have for almost all $r \in (-1, 0)$,

$$ \|(h_{\text{bl}} - h, u_{\text{bl}} - u)(\cdot, r)\|_{L_T^\infty H_x^s} \le \left( \| (h_{\text{bl}}^0 - h^0, u_{\text{bl}}^0 - u^0)(\cdot, r)\|_{H_x^s} + \| (h_{\text{bl}}^0 - h^0, u_{\text{bl}}^0 - u^0)\|_{L_r^1 H_x^{s+1}} \right. \\ \left. + |(\underline{\rho}_{\text{bl}} - \underline{\rho}, \underline{u}_{\text{bl}} - \underline{u})(\cdot, r)| + \|(\underline{\rho}_{\text{bl}} - \underline{\rho}, \underline{u}_{\text{bl}} - \underline{u})(\cdot, r)\|_{L_r^1} \right) \times C \exp(Ct/\kappa). $$

**Remark 4.2.** Proposition 3.8 is not limited to the bilayer framework and applies to any suitably regular reference solution. Hence it can be combined with results analogous to Proposition 2.7 to provide results analogous to Theorem 4.1 in the one-layer and multilayer frameworks.

The result analogous to Proposition 2.7 in the one-layer framework (that is associated with the standard shallow water equation with thickness diffusivity that was discussed for instance in [24]) is stated and proved in [2, Sect. 2.2]. Notice it requires neither the discussion on the hyperbolicity domain of the non-diffusive equations (since the standard non-cavitation assumption guarantees hyperbolicity) nor the discussion on the parabolic regularization of the total velocity (since the natural symmetrizer behaves well with diffusivity contributions).

A result analogous to Proposition 2.7 in the multilayer framework follows from combining the result of [22] with the analysis of Section 2.2. In the former, it is proved that assuming sufficiently small shear velocities is a sufficient condition for the (strict) hyperbolicity of the multilayer system in the stably stratified situation. Notice however that this smallness condition is implicit, and not uniform with respect to the number of layers.
# 4 Conclusion

Thanks to Proposition 2.7 (concerning the well-posedness and control of solutions to the bilayer system) on one hand and Proposition 3.8 (concerning the control of the deviations of nearby solutions to some given reference solutions) on the other hand, one infers immediately the announced rigorous justification of the propagation in time of the columnar motion and sharp stratification assumptions in near-bilayer situations (within the hydrostatic framework). Specifically, we have the following result.

**Theorem 4.1.** Let $s \ge s_0 > 3/2$, $\underline{M} > 0$, $M_0 > 0$, $\zeta \in (0, 1)$, and $\kappa \in (0, 1]$. Then there exist $C > 0, T > 0$ independent of $\kappa$ and $\delta_0 > 0$ (depending on $\kappa$) such that the following holds.

Let $(\underline{\rho}, \underline{u}) \in L^\infty((-1, 0))^2$ and $(h^0, u^0) \in L^\infty((-1, 0); H^{s+2}(\mathbb{R})^2)$ be such that

$$ \|(\underline{\rho}, \frac{1}{\underline{\rho}})\|_{L_r^\infty} \le \underline{M}, \quad \|(h^0, \kappa^{1/2} \partial_x h^0)\|_{L_r^\infty H_x^{s+1}} + \|u^0\|_{L_r^\infty H_x^{s+2}} \le M_0 $$

and there exists $(\rho_s, \rho_b, \underline{H}_s, \underline{H}_b, \underline{U}_s, \underline{U}_b) \in \mathbb{R}^6$ such that $\underline{H}_s + \underline{H}_b = 1$ and $\underline{U}_s + \underline{U}_b = 0$ as well as $(H_s^0, H_b^0, U_s^0, U_b^0) \in H^{s+4}(\mathbb{R})^2 \times H^{s+3}(\mathbb{R})^2$ such that the hyperbolicity condition holds:

$$ \forall x \in \mathbb{R}, \quad (\rho_s, \rho_b, \underline{H}_s + H_s^0(x), \underline{H}_b + H_b^0(x), \underline{U}_s + U_s^0(x), \underline{U}_b + U_b^0(x)) \in \mathfrak{p}^s, $$

where $\mathfrak{p}^s$ is defined in (2.5), and

$$ \|(H_s^0, H_b^0, U_s^0, U_b^0, \kappa \partial_x H_s^0, \kappa \partial_x H_b^0)\|_{H^{s+3}} \le M_0 $$

and such that denoting $(\underline{\rho}_{\text{bl}}^0, \underline{u}_{\text{bl}}^0, h_{\text{bl}}^0, u_{\text{bl}}^0)$ through (1.4) we have

$$ \|(h_{\text{bl}}^0 - h^0, u_{\text{bl}}^0 - u^0)\|_{L_r^1 H_x^{s+1}} + \|(\underline{\rho}_{\text{bl}} - \underline{\rho}, \underline{u}_{\text{bl}} - \underline{u})\|_{L_r^1} \le \delta_0, $$

then

1. there exists $(H_s, H_b, U_s, U_b) \in C([0, T]; H^{s+3}(\mathbb{R})^4)$ solution to (1.3) emerging from the initial data

$$ (H_s, H_b, U_s, U_b)|_{t=0} = (H_s^0, H_b^0, U_s^0, U_b^0); $$

2. there exists $(h, u) \in C([0, T]; L^\infty((-1, 0); H^{s+2}(\mathbb{R})^2)$ solution to (1.1) emerging from initial data

$$ (h, u)|_{t=0} = (h^0, u^0); $$

3. denoting $(\underline{\rho}_{\text{bl}}, \underline{u}_{\text{bl}}, h_{\text{bl}}, u_{\text{bl}})$ through (1.4) we have for almost all $r \in (-1, 0)$,

$$ \|(h_{\text{bl}} - h, u_{\text{bl}} - u)(\cdot, r)\|_{L_T^\infty H_x^s} \le \left( \| (h_{\text{bl}}^0 - h^0, u_{\text{bl}}^0 - u^0)(\cdot, r)\|_{H_x^s} + \| (h_{\text{bl}}^0 - h^0, u_{\text{bl}}^0 - u^0)\|_{L_r^1 H_x^{s+1}} \right. \\ \left. + |(\underline{\rho}_{\text{bl}} - \underline{\rho}, \underline{u}_{\text{bl}} - \underline{u})(\cdot, r)| + \|(\underline{\rho}_{\text{bl}} - \underline{\rho}, \underline{u}_{\text{bl}} - \underline{u})(\cdot, r)\|_{L_r^1} \right) \times C \exp(Ct/\kappa). $$

**Remark 4.2.** Proposition 3.8 is not limited to the bilayer framework and applies to any suitably regular reference solution. Hence it can be combined with results analogous to Proposition 2.7 to provide results analogous to Theorem 4.1 in the one-layer and multilayer frameworks.

The result analogous to Proposition 2.7 in the one-layer framework (that is associated with the standard shallow water equation with thickness diffusivity that was discussed for instance in [24]) is stated and proved in [2, Sect. 2.2]. Notice it requires neither the discussion on the hyperbolicity domain of the non-diffusive equations (since the standard non-cavitation assumption guarantees hyperbolicity) nor the discussion on the parabolic regularization of the total velocity (since the natural symmetrizer behaves well with diffusivity contributions).

A result analogous to Proposition 2.7 in the multilayer framework follows from combining the result of [22] with the analysis of Section 2.2. In the former, it is proved that assuming sufficiently small shear velocities is a sufficient condition for the (strict) hyperbolicity of the multilayer system in the stably stratified situation. Notice however that this smallness condition is implicit, and not uniform with respect to the number of layers.
**Acknowledgements** The authors thank the Institute for applied mathematics “Mauro Picone” (IAC) for its hospitality when this work was initiated, and Laboratoire Ypatia des Sciences Mathématiques LYSM CNRS-INdAM International Research Laboratory as well as Univ. Rennes funding for its financial support. MA and VD thank the program Centre Henri Lebesgue ANR-11-LABX-0020-0, for fostering an attractive mathematical environment. RB acknowledges fundings from the Italian Ministry of University and Research, PRIN 2022HSSYPN (Teseo).

**References**

[1] M. Adim. Approximating a continuously stratified hydrostatic system by the multi-layer shallow water system. arXiv preprint: 2307.11426.

[2] M. Adim. *Modèles continument stratifiés et systèmes multi-couches pour les écoulements géophysiques*. PhD thesis, Univ. Rennes, 2024.

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**Acknowledgements** The authors thank the Institute for applied mathematics “Mauro Picone” (IAC) for its hospitality when this work was initiated, and Laboratoire Ypatia des Sciences Mathématiques LYSM CNRS-INdAM International Research Laboratory as well as Univ. Rennes funding for its financial support. MA and VD thank the program Centre Henri Lebesgue ANR-11-LABX-0020-0, for fostering an attractive mathematical environment. RB acknowledges fundings from the Italian Ministry of University and Research, PRIN 2022HSSYPN (Teseo).

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[17] D. Bresch and P. Noble. Mathematical justification of a shallow water model. *Methods Appl. Anal.*, 14(2):87–117, 2007.

[18] R. Camassa and R. Tiron. Optimal two-layer approximation for continuous density stratification. *J. Fluid Mech.*, 669:32–54, 2011.

[19] B. Desjardins, D. Lannes, and J.-C. Saut. Normal mode decomposition and dispersive and nonlinear mixing in stratified fluids. *Water Waves*, 3(1):153–192, 2021.

[20] V. Duchêne. Many Models for Water Waves. Open Math Notes, OMN:202109.111309, 2021.

[21] V. Duchêne. Asymptotic shallow water models for internal waves in a two-fluid system with a free surface. *SIAM J. Math. Anal.*, 42(5):2229–2260, 2010.

[22] V. Duchêne. A note on the well-posedness of the one-dimensional multilayer shallow water model. HaL preprint: 00922045, 2013.

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