Meandering Rivers: How Important is Lateral Variability for Species Persistence?

Abstract

Models for population dynamics in rivers and streams have highlighted the importance of spatial and temporal variations for population persistence. We present a novel model that considers the longitudinal variation as introduced by the sinuosity of a meandering river where a main channel is laterally extended to point bars in bends. These regions offer different habitat conditions for aquatic populations and therefore may enhance population persistence. Our model is a nonstandard reaction–advection–diffusion model where the domain of definition consists of the real line (representing the main channel) with periodically added intervals (representing the point bars). We give an existence and uniqueness proof for solutions of the equations. We then study population persistence as the (in-) stability of the trivial solution and population spread as the minimal wave speed of traveling periodic waves. We conduct a sensitivity analysis to highlight the importance of each parameter on the model outcome. We find that sinuosity can enhance species persistence.

Appendix A: Existence and Uniqueness of Solutions

In this appendix, we prove the existence and uniqueness of model (6) over a bounded periodic domain [0, nL] under the interface conditions (7) and (9), hostile upstream boundary condition and free-flow downstream boundary condition, i.e.,

$$\begin{aligned} u(0,t)=w(0,t)=u_x(0,t)=w_x(0,t)=0,\,\,u_x(nL,t)=0. \end{aligned}$$

(50)

Without loss of generality, we provide the proof over a two-period domain $\Omega =[0,2L] = \Omega _1 \cup \Omega _2$, where $\Omega _1 = [0, L_1] \cup [L, L+L_1]$ and $\Omega _2 = [L_1, L] \cup [L+L_1, 2L]$.

We briefly summarize the key steps of the Galerkin approximation and energy method that we use for the proof. A detailed process can be found in Larios and Pei (2016) and references therein.

Given a PDE system

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} u(x, t) = \mathcal {F}(u, \nabla u, \Delta u), \end{aligned}$$

for some smooth function $\mathcal {F}$, we consider a sequence of approximated systems

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} P_{\sigma }^{(m)}u(x, t) = \mathcal {F}(P_{\sigma }^{(m)}u, \nabla P_{\sigma }^{(m)}u, \Delta P_{\sigma }^{(m)}u), \end{aligned}$$

for $m = 1, 2, \dots$ after formally applying certain projections $P_{\sigma }^{(m)}$ onto a finite-dimensional subspace of the domain of the differential operator in the equation. Then, we take the inner product of the projected system with the eigenfunctions $\{\psi _{i}(x)\}_{i=1}^{m}$ of the differential operator, which is positive-definitive, self-adjoint, and compact, and obtain a sequence of ordinary differential equations (ODE). Each of the ODE system has a unique solution $\{\alpha (t)\}_{i=1}^{M}$ from which we obtain $P_{\sigma }^{(m)}u(x, t) = \sum _{i=1}^{m}\alpha _{i}(t)\psi _{i}(x)$ for $m = 1, 2, \dots$.

Then, we obtain the energy estimates for $P_{\sigma }^{(m)}u(x, t)$ in some Banach space Y, say, $Y = L^2(\Omega )$, as well as the boundedness of those solutions in some space X, say, $X = H^1(\Omega )$, which is compactly embedded in Y. Hence, we can extract a convergent subsequence of the above solutions as a sequence in Y, provided that the time derivative of the aforementioned sequence also remains bounded in Z, say, the dual space of X, and Y is continuously embedded in Z. The limit of the above convergent subsequence indeed satisfies the original PDE.

A critical result for the process to work is the Aubin–Lions Compactness Lemma below [see  remarks after Lemma 8.2 in Chapter Eight of Constantin and Foias (1988) as well as Lemma 2.1 in Sect. 2, Chapter 3 of Temam (2001)].

Lemma A.1

Let $T>0$, fix some $p \in (1, \infty ),$ and let $\{f_{n}(\cdot , t)\}_{n=1}^{\infty }$ be a bounded sequence of functions in $L_{t}^{p}(Y;[0, T]),$ where Y is a Banach space. If $\{f_{n}\}_{n=1}^{\infty }$ is also bounded in $L_{t}^{p}(X;[0, T])$, where X is compactly imbedded in Y,  and $\{\partial f_{n}/\partial t\}_{n=1}^{\infty }$ is uniformly bounded in $L_{t}^{p}(Z;[0, T]),$ where Y is continuously imbedded in Z, then $\{f_{n}\}_{n=1}^{\infty }$ is relatively compact in $L_{t}^{p}(Y;[0, T])$.

For simplicity, we provide only the key a priori energy estimates under the interface conditions that lead to the boundedness results required in this lemma. We omit the details for constructing solutions via the standard Galerkin approximation; see  Constantin and Foias (1988), Temam (2001).

Throughout this section, we use $L^2$ and $H^{k} = W^{k,2}=\{u\in L^2: D^\alpha u\in L^2, \forall \alpha \text{ with } |\alpha |\le k\}$, $k=1, 2$, to denote Lebesgue and Sobolev spaces, respectively, while we denote by $C_{0}^{\infty }$ the space of compactly supported smooth functions in space and time.

First, we give the definition of weak solutions to (6).

Definition A.1

Let $T>0$ and suppose that $u_0, w_0 \in L^2$ with $u_0\ge 0$ and $w_0\ge 0$. We say that the pair (u, w) is a weak solution to (6), if $u, w \in L^{\infty }(L^2;[0, T]) \cap L^{2}(H^1;[0, T])$. Furthermore, for test functions $\phi ^{(1)} \in C_{0}^{\infty }([0, L_1] \times [0, T])$, $\phi ^{(2)} \in C_{0}^{\infty }([L, L+L_1] \times [0, T])$, and $\psi ^{(1)} \in C_{0}^{\infty }( [L_1, L]\times [0, T])$, $\psi ^{(2)} \in C_{0}^{\infty }([L+L_1, 2L] \times [0, T])$, (6) holds in the weak sense, i.e., we have

$$\begin{aligned} \left\{ \begin{aligned}&- \int _{0}^{T}\int _{0}^{L_1} u(x, t) \frac{\partial \phi ^{(1)}}{\partial t}\,\mathrm{d}x\,\mathrm{d}t -\int _{0}^{T}\int _{L}^{L+L_1} u(x, t) \frac{\partial \phi ^{(2)}}{\partial t}\,\mathrm{d}x\,\mathrm{d}t \\&\qquad - D_1\int _{0}^{T}\int _{0}^{L_1} u(x, t) \frac{\partial ^2 \phi ^{(1)}}{\partial x^2}\,\mathrm{d}x\,\mathrm{d}t - D_1\int _{0}^{T}\int _{L}^{L+L_1} u(x, t) \frac{\partial ^2 \phi ^{(2)}}{\partial x^2}\,\mathrm{d}x\,\mathrm{d}t \\&= v_1\int _{0}^{T}\int _{0}^{L_1} u(x, t) \frac{\partial \phi ^{(1)}}{\partial x}\,\mathrm{d}x\,\mathrm{d}t + v_1\int _{0}^{T}\int _{L}^{L+L_1} u(x, t) \frac{\partial \phi ^{(2)}}{\partial x}\,\mathrm{d}x\,\mathrm{d}t \\&\qquad + \alpha _{u}\int _{0}^{T}\int _{0}^{L_1} (w(x, t)-u(x, t)) \phi ^{(1)}\,\mathrm{d}x\,\mathrm{d}t + \alpha _{u}\int _{0}^{T}\int _{L}^{L+L_1} (w(x, t)-u(x, t)) \phi ^{(2)}\,\mathrm{d}x\,\mathrm{d}t \\&\qquad + \int _{0}^{T}\int _{0}^{L_1} f_1(u(x, t))u(x, t) \phi ^{(1)}\,\mathrm{d}x\,\mathrm{d}t + \int _{0}^{T}\int _{L}^{L+L_1} f_1(u(x, t))u(x, t) \phi ^{(2)}\,\mathrm{d}x\,\mathrm{d}t, \\&- \int _{0}^{T}\int _{0}^{L_1} w(x, t) \frac{\partial \phi ^{(1)}}{d t}\,\mathrm{d}x\,\mathrm{d}t - \int _{0}^{T}\int _{L}^{L+L_1} w(x, t) \frac{\partial \phi ^{(2)}}{d t}\,\mathrm{d}x\,\mathrm{d}t \\&\qquad - D_{w}\int _{0}^{T}\int _{0}^{L_1} w(x, t) \frac{\partial ^2 \phi ^{(1)}}{\partial x^2}\,\mathrm{d}x\,\mathrm{d}t - D_{w}\int _{0}^{T}\int _{L}^{L+L_1} w(x, t) \frac{\partial ^2 \phi ^{(2)}}{\partial x^2}\,\mathrm{d}x\,\mathrm{d}t \\&= v_{w}\int _{0}^{T}\int _{0}^{L_1} w(x, t)\frac{\partial \phi ^{(1)}}{\partial x}\,\mathrm{d}x\,\mathrm{d}t + v_{w}\int _{0}^{T}\int _{L}^{L+L_1} w(x, t) \frac{\partial \phi ^{(2)}}{\partial x}\,\mathrm{d}x\,\mathrm{d}t \\&\qquad + \alpha _{w}\int _{0}^{T}\int _{0}^{L_1} (u(x, t)-w(x, t)) \phi ^{(1)}\,\mathrm{d}x\,\mathrm{d}t + \alpha _{w}\int _{0}^{T}\int _{L}^{L+L_1} (u(x, t)-w(x, t)) \phi ^{(2)}\,\mathrm{d}x\,\mathrm{d}t \\&\qquad + \int _{0}^{T}\int _{0}^{L_1} g(w(x, t))w(x, t) \phi ^{(1)}\,\mathrm{d}x\,\mathrm{d}t + \int _{0}^{T}\int _{L}^{L+L_1} g(w(x, t))w(x, t) \phi ^{(2)}\,\mathrm{d}x\,\mathrm{d}t, \\&- \int _{0}^{T}\int _{L_1}^{L} u(x, t) \frac{\partial \psi ^{(1)}}{d t}\,\mathrm{d}x\,\mathrm{d}t - \int _{0}^{T}\int _{L+L_1}^{2L} u(x, t) \frac{\partial \psi ^{(2)}}{d t}\,\mathrm{d}x\,\mathrm{d}t \\&\qquad - D_2\int _{0}^{T}\int _{L_1}^{L} u(x, t) \frac{\partial ^2 \psi ^{(1)}}{\partial x^2}\,\mathrm{d}x\,\mathrm{d}t - D_2\int _{0}^{T}\int _{L+L_1}^{2L} u(x, t) \frac{\partial ^2 \psi ^{(2)}}{\partial x^2}\,\mathrm{d}x\,\mathrm{d}t \\&= v_2\int _{0}^{T}\int _{L_1}^{L} u(x, t) \frac{\partial \psi ^{(1)}}{\partial x}\,\mathrm{d}x\,\mathrm{d}t + v_2\int _{0}^{T}\int _{L+L_1}^{2L} u(x, t) \frac{\partial \psi ^{(2)}}{\partial x}\,\mathrm{d}x\,\mathrm{d}t. \end{aligned} \right. \end{aligned}$$

(51)

In the following proof of the well-posedness of system (6), for the sake of simplicity, we provide only the a priori $L^2$-energy and $H^1$-enstrophy estimates. We denote $u(\cdot ) = u(\cdot , t)$ and similarly for the derivatives of u in space and time.

Proof of Theorem 2.1

Step 1: $L^2$ -estimates Multiply the three equations in (6) by $A_1u$, $A_{w}w$, and $A_2 u$, respectively, integrate by parts over $\Omega _1$ and $\Omega _2$ accordingly, and add, so we obtain

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left( A_1\Vert u\Vert _{L^2(\Omega _1)}^2 + A_2 \Vert u\Vert _{L^2(\Omega _2)}^2 + A_{w}\Vert w\Vert _{L^2(\Omega _1)}^2\right) \nonumber \\&\qquad + A_1 D_1 \Vert u_x\Vert _{L^2(\Omega _1)}^2 + A_2 D_2 \Vert u_x\Vert _{L^2(\Omega _2)}^2 + A_{w} D_{w} \Vert u_x\Vert _{L^2(\Omega _1)}^2 +\mathbf{(B_1)} \nonumber \\&= \underbrace{ - A_1 v_1 \int _{\Omega _1}u\,u_{x}\,\mathrm{d}x - A_2 v_2 \int _{\Omega _2}u\,u_{x}\,\mathrm{d}x - A_{w} v_{w} \int _{\Omega _1}w\,w_{x}\,\mathrm{d}x}_{I} \nonumber \\&\qquad \underbrace{ + A_1\alpha _1\int _{\Omega _1}(w-u)u\,\mathrm{d}x + A_{w}\alpha _{w}\int _{\Omega _1}(u-w)w\,\mathrm{d}x}_{II} \nonumber \\&\qquad \underbrace{ + A_1\int _{\Omega _1}f_1(u)u^2\,\mathrm{d}x + A_2\int _{\Omega _2}f_2(u)u^2\,\mathrm{d}x + A_{w}\int _{\Omega _1}g(w)w^2\,\mathrm{d}x}_{III}, \end{aligned}$$

(52)

where $\mathbf{(B_1)}$ on the left side of the equation represents the boundary terms, which from integration by parts can be written as

$$\begin{aligned} \mathbf{(B_1)}&= A_1D_1u_{x}(0)u(0) - A_1D_1u_{x}(L_1)u(L_1) - A_1D_1u_{x}(L+L_1)u(L+L_1) \nonumber \\&\quad +\,A_1D_1u_{x}(L)u(L) A_{w}D_{w}w_{x}(0)w(0)-A_{w}D_{w}w_{x}(L_1)w(L_1) \nonumber \\&\quad -\,A_{w}D_{w}w_{x}(L+L_1)w(L+L_1) + A_{w}D_{w}w_{x}(L)w(L) - A_2D_2u_{x}(L)u(L) \nonumber \\&\quad +\, A_2D_2u_{x}(L_1)u(L_1) - A_2D_2u_{x}(2L)u(2L) + A_2D_2u_{x}(L+L_1)u(L+L_1) \nonumber \\&= A_1D_1u_{x}(0)u(0)+A_{w}D_{w}w_{x}(0)w(0)- A_2D_2u_{x}(2L)u(2L), \end{aligned}$$

(53)

by using the interface conditions (9).

In order to estimate the three parts I, II, and III on the right side of (52), we first notice that due to the hydrological condition (1) and integration by parts, we have

$$\begin{aligned} I&= \frac{1}{2} A_1v_1u^2(0) - \frac{1}{2} A_1v_1u^2(L_1) - \frac{1}{2}A_1v_1u^{2}(L+L_1) + \frac{1}{2}A_1v_1u^2(L) \\&\qquad \frac{1}{2}A_{w}v_{w}w^2(0) - \frac{1}{2}A_{w}v_{w}w^2(L_1) - \frac{1}{2}A_{w}v_{w}w^2(L+L_1) + \frac{1}{2}A_{w}v_{w}w^2(L) \\&\qquad - \frac{1}{2}A_{2}v_{2}u^2(L) + \frac{1}{2}A_{2}v_{2}u^2(L_1) - \frac{1}{2}A_{2}v_{2}u^2(2L) + \frac{1}{2}A_{2}v_{2}u^2(L+L_1) \\&= \frac{1}{2} A_1v_1u^2(0)+\frac{1}{2}A_{w}v_{w}w^2(0) - \frac{1}{2}A_{2}v_{2}u^2(2L). \end{aligned}$$

Therefore, by the boundary conditions (50), we have

$$\begin{aligned} I-\mathbf{(B_1)}&= -\left[ A_1D_1u_{x}(0)+A_{w}D_{w}w_{x}(0)- A_1v_1u(0)-A_{w}v_{w}w(0)\right] u(0)\nonumber \\&-\frac{1}{2}A_1v_1u^2(0)-\frac{1}{2}A_wv_ww^2(0)+\left[ A_2D_2u_{x}(2L) - \frac{1}{2}A_{2}v_{2}u(2L)\right] u(2L)\nonumber \\&\le 0. \end{aligned}$$

(54)

Next, by denoting $A = \max \{A_1, A_2, A_{w}\}$ and $\alpha = \max \{\alpha _{u}, \alpha _{w}\}$, we estimate II as

$$\begin{aligned} II&\le A\alpha \left( \Vert u\Vert _{L^2(\Omega _1)}^2 + \Vert u\Vert _{L^2(\Omega _2)}^2 + \Vert w\Vert _{L^2(\Omega _1)}^2 \right) \end{aligned}$$

where we used Cauchy–Schwarz inequality. Regarding the remaining integrals, we use the assumption that $f_1, f_2, g \in L^{\infty }(\Omega )$ and proceed as

$$\begin{aligned} III&\le A\left( \Vert f_1\Vert _{L^{\infty }(\Omega _1)} \Vert u\Vert _{L^2(\Omega _1)}^2 + \Vert f_2\Vert _{L^{\infty }(\Omega _2)} \Vert u\Vert _{L^2(\Omega _2)}^2 + \Vert g\Vert _{L^{\infty }(\Omega _1)} \Vert w\Vert _{L^2(\Omega _1)}^2 \right) \end{aligned}$$

(55)

Thus, by denoting

$$\begin{aligned} X^2(t) = A_1\Vert u\Vert _{L^2(\Omega _1)}^2 + A_2\Vert u\Vert _{L^2(\Omega _2)}^2 + A_{w}\Vert w\Vert _{L^2(\Omega _1)}^2, \end{aligned}$$

all the above estimates imply that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}X^2(t) + A_1 D_1 \Vert u_x\Vert _{L^2(\Omega _1)}^2 + A_2 D_2 \Vert u_x\Vert _{L^2(\Omega _2)}^2 + A_{w} D_{w} \Vert w_x\Vert _{L^2(\Omega _1)}^2 \nonumber \\&\quad \le C X^2(t), \end{aligned}$$

(56)

where for $j=1,2$, the constant C depends on $A_{j}$, $A_{w}$, $\alpha _{u}$, $\alpha _{w}$, and $\Vert f_{j}\Vert _{L^{\infty }(\Omega )}$, $\Vert g\Vert _{L^{\infty }(\Omega )}$. Hence, we conclude that $u, w \in L^{\infty }(L^2;[0, T]) \cap L^{2}(H^1;[0, T])$.

Step 2: Uniqueness Suppose there are two solutions to the system (6), say, $(u^{(1)}, w^{(1)})$ and $(u^{(2)}, w^{(2)})$, possessing the same initial data and satisfying the identical hostile boundary and interior interface conditions. We denote the difference $\tilde{u} = u^{(1)} - u^{(2)}$ and $\tilde{w} = w^{(1)} - w^{(2)}$ and subtract the systems satisfied by the two solutions. Then, we perform the same energy estimates to the system for $(\tilde{u}, \tilde{w})$ as in Step 1, and obtain

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t} \left( A_1\Vert \tilde{u}\Vert _{L^2(\Omega _1)}^2 + A_2\Vert \tilde{u}\Vert _{L^2(\Omega _2)}^2 + A_{w}\Vert \tilde{w}\Vert _{L^2(\Omega _1)}^2\right) \nonumber \\&\qquad + A_1 D_1 \Vert \tilde{u}_x\Vert _{L^2(\Omega _1)}^2 + A_2 D_2 \Vert \tilde{u}_x\Vert _{L^2(\Omega _2)}^2 + A_{w} D_{w} \Vert \tilde{w}_x\Vert _{L^2(\Omega _1)}^2 \nonumber \\&\quad \le \tilde{C} \left( A_1\Vert \tilde{u}\Vert _{L^2(\Omega _1)}^2 + A_2\Vert \tilde{u}\Vert _{L^2(\Omega _2)}^2 + A_{w}\Vert \tilde{w}\Vert _{L^2(\Omega _1)}^2\right) , \end{aligned}$$

(57)

where we assumed that $f_1$, $f_2$, and g are Lipschitz continuous. Hence, the uniqueness of the above solution follows from Grönwall’s inequality applied to the analogues of (57) with the initial value $\tilde{u}_0 = \tilde{w}_0 = 0$. For simplicity, we skip the details here.

Step 3: Higher regularity ($H^1$-estimates) Assume $u_0, w_0 \in H^1$ with $u_0\ge 0$ and $w_0\ge 0$ and $T>0$. We prove a priori that the solution (u, w) is actually regular on $\Omega \times [0, T]$. Multiply (6) by $-A_1D_1u_{xx}$, $-A_{w}D_{w}w_{xx}$, and $-A_2D_2u_{xx}$, respectively, integrate by parts over $\Omega _1$ and $\Omega _2$ accordingly, and add, so that we have

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left( A_1D_1\Vert u_{x}\Vert _{L^2(\Omega _1)}^2 + A_2D_2\Vert u_{x}\Vert _{L^2(\Omega _2)}^2 + A_{w}D_{w}\Vert w_{x}\Vert _{L^2(\Omega _1)}^2 \right) \nonumber \\&\qquad + A_1D_1^2\Vert u_{xx}\Vert _{L^2(\Omega _1)}^2 + A_2D_2^2\Vert u_{xx}\Vert _{L^2(\Omega _1)}^2 + A_{w}D_{w}^2\Vert w_{xx}\Vert _{L^2(\Omega _1)}^2 + \mathbf{(B_2)} \nonumber \\&\quad = \underbrace{ A_1D_1v_1\int _{\Omega _1}u_{x}u_{xx}\,dx + A_2D_2v_2\int _{\Omega _2}u_{x}u_{xx}\,\mathrm{d}x + A_{w}D_{w}v_{w}\int _{\Omega _1}w_{x}w_{xx}\,\mathrm{d}x}_{IV} \nonumber \\&\qquad \underbrace{ - A_1D_1\alpha _{u}\int _{\Omega _1}(w - u)u_{xx}\,\mathrm{d}x - A_{w}D_{w}\alpha _{w}\int _{\Omega _1}(u - w)w_{xx}\,\mathrm{d}x}_{V} \nonumber \\&\qquad \underbrace{ - A_1D_1\int _{\Omega _1}f_1(u)uu_{xx}\,\mathrm{d}x - A_2D_2\int _{\Omega _2}f_2(u)uu_{xx}\,\mathrm{d}x - A_{w}D_{w}\int _{\Omega _1}g(w)ww_{xx}\,\mathrm{d}x}_{VI}, \end{aligned}$$

(58)

where $\mathbf{(B_2)}$ represents the boundary terms from the left side of (58), which by the interface conditions, add up to 0, i.e.,

$$\begin{aligned} \mathbf{(B_2)}&= A_1D_1u_{t}(0)u_{x}(0) - A_1D_1u_{t}(L_1)u_{x}(L_1) + A_1D_1u_{t}(L)u_{x}(L)\nonumber \\&\quad -\, A_1D_1u_{t}(L+L_1)u_{x}(L+L_1) \nonumber \\&\quad +\, A_{w}D_{w}w_{t}(0)w_{x}(0) - A_{w}D_{w}w_{t}(L_1)w_{x}(L_1) + A_{w}D_{w}w_{t}(L)w_{x}(L) \nonumber \\&\quad -\, A_{w}D_{w}w_{t}(L+L_1)w_{x}(L+L_1) + A_2D_2u_{t}(L_1)u_{x}(L_1) - A_2D_2u_{t}(L)u_{x}(L) \nonumber \\&\quad +\, A_2D_2u_{t}(L+L_1)u_{x}(L+L_1) - A_2D_2u_{t}(2L)u_{x}(2L) \nonumber \\&= u_{t}(0)[A_1D_1u_{x}(0)+A_wD_ww_x(0)]+u_t(L_1)[-A_1D_1u_x(L_1)\nonumber \\&\qquad -A_wD_ww_x(L_1)+A_2D_2u_x(L_1)] \nonumber \\&\quad +\,u_t(L)[A_1D_1u_x(L)+A_wD_ww_x(L)-A_2D_2u_x(L)] \nonumber \\&\quad +\,u_t(L+L_1)[-A_1D_1u_x(L+L_1)-A_wD_ww_x(L+L_1)+A_2D_2u_x(L+L_1)] \nonumber \\&\quad -\, A_2D_2u_{t}(2L)u_{x}(2L)\nonumber \\&=u_{t}(0)[A_1D_1u_{x}(0)+A_wD_ww_x(0)]- A_2D_2u_{t}(2L)u_{x}(2L)\nonumber \\&=0, \end{aligned}$$

(59)

where we used the condition $u_{t}=w_{t}$ at $x=0, L, L_1$, and $L+L_1$, and boundary conditions (50). In order to estimate IV, we use Cauchy–Schwarz inequality and proceed as

$$\begin{aligned} IV&\le C_1\left( \Vert u_{x}\Vert _{L^2(\Omega _1)}^2 + \Vert u_{x}\Vert _{L^2(\Omega _2)}^2 + \Vert w_{x}\Vert _{L^2(\Omega _1)}^2 \right) \\ {}&\qquad + \frac{A_1D_1^2}{16}\Vert u_{xx}\Vert _{L^2(\Omega _1)}^2 + \frac{A_2D_2^2}{16}\Vert u_{xx}\Vert _{L^2(\Omega _2)}^2 + \frac{A_{w}D_{w}^2}{16}\Vert w_{xx}\Vert _{L^2(\Omega _1)}^2, \end{aligned}$$

where for $j=1, 2$, the constant $C_1$ depends on $A_{j}$, $v_{j}$, and $A_{w}$, $v_{w}$. Next, we integrate by parts and bound V as

$$\begin{aligned} V&= A_1D_1v_1\int _{\Omega _1}u_{x}w_{x}\,\mathrm{d}x - A_1D_1v_1\int _{\Omega _1}u_{x}^2\,\mathrm{d}x \\&\qquad + A_{w}D_{w}v_{w}\int _{\Omega _1}u_{x}w_{x}\,\mathrm{d}x - A_{w}D_{w}v_{w}\int _{\Omega _1}w_{x}^2\,\mathrm{d}x \\&\le C_2\left( \Vert u_{x}\Vert _{L^2(\Omega _1)}^2 + \Vert u_{x}\Vert _{L^2(\Omega _2)}^2 + \Vert w_{x}\Vert _{L^2(\Omega _1)}^2 \right) , \end{aligned}$$

where we used Cauchy–Schwarz inequality. Here the constant $C_2$ depends on all parameters including $D_1$, $D_2$ and $D_{w}$. Note that the boundary terms from the above integration by parts add up to 0 due to the continuity condition (7). Regarding VI, we first apply Hölder’s inequality and get

$$\begin{aligned} VI&\le A_1D_1\Vert f_1\Vert _{L^{\infty }(\Omega _1)} \Vert u\Vert _{L^{2}(\Omega _1)} \Vert u_{xx}\Vert _{L^2(\Omega _1)} + A_2D_2\Vert f_2\Vert _{L^{\infty }(\Omega _2)} \Vert u\Vert _{L^{2}(\Omega _2)} \Vert u_{xx}\Vert _{L^2(\Omega _2)} \\&\qquad + A_{w}D_{w}\Vert g\Vert _{L^{\infty }(\Omega _1)} \Vert w\Vert _{L^{2}(\Omega _1)} \Vert w_{xx}\Vert _{L^2(\Omega _1)} \\&\le C_3\left( \Vert u_{x}\Vert _{L^2(\Omega _1)}^2 + \Vert u_{x}\Vert _{L^2(\Omega _2)}^2 + \Vert w_{x}\Vert _{L^2(\Omega _1)}^2 \right) \\&\qquad + \frac{A_1D_1^2}{16}\Vert u_{xx}\Vert _{L^2(\Omega _1)}^2 + \frac{A_2D_2^2}{16}\Vert u_{xx}\Vert _{L^2(\Omega _2)}^2 + \frac{A_{w}D_{w}^2}{16}\Vert w_{xx}\Vert _{L^2(\Omega _1)}^2, \end{aligned}$$

where we also used Cauchy–Schwarz and Poincaré’s inequality in the last step. The constant $C_3$ in the above estimates depends on $A_{j}$, $A_{w}$, L, $L_{1}$, and $\Vert f_{j}\Vert _{L^{\infty }(\Omega _{j})}$, $\Vert g\Vert _{L^{\infty }(\Omega _1)}$, for $j=1, 2$. Summing up all the above estimates, and denoting

$$\begin{aligned} Y^{2}(t) = A_1D_1\Vert u_{x}\Vert _{L^2(\Omega _1)}^2 + A_2D_2\Vert u_{x}\Vert _{L^2(\Omega _2)}^2 + A_{w}D_{w}\Vert w_{x}\Vert _{L^2(\Omega _1)}^2, \end{aligned}$$

we get

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}Y^2(t) + \left( A_1D_1^2\Vert u_{xx}\Vert _{L^2(\Omega _1)}^2 + A_2D_2^2\Vert u_{xx}\Vert _{L^2(\Omega _1)}^2 + A_{w}D_{w}^2\Vert w_{xx}\Vert _{L^2(\Omega _1)}^2 \right)&\le CY^2(t), \end{aligned}$$

where C depends on $A_{j}$, $v_{j}$, $\alpha _{u}$, $\alpha _{w}$, and $A_{w}$, $v_{w}$, for $j=1, 2$. Thus, Grönwall’s inequality implies that $u, w \in L^{\infty }( H^1;[0, T]) \cap L^{2}(H^2;[0, T] )$. Hence, the proof is complete. $\square$

Remark A.1

From the above $L^2$ and $H^1$ estimates, we see that the time derivatives $\frac{\mathrm{d}u}{\mathrm{d}t}$ and $\frac{\mathrm{d}w}{\mathrm{d}t}$ are also uniformly bounded. Therefore, we obtain that u and w are in fact weakly continuous in time. Namely, $u, w \in C_{w}(H^1;[0, T])$ for all $T>0$, where the space $C_{w} \subset L^{\infty }$ here consists of all functions that are weakly continuous in time with values in $H^1$. In particular, the initial conditions are satisfied in the weak sense, i.e.,

$$\begin{aligned} \lim _{t \rightarrow 0}\int _{\Omega _{i}}u(\cdot , t)\xi \,\mathrm{d}x = \int _{\Omega _{i}}u(\cdot , 0)\xi \,\mathrm{d}x, \end{aligned}$$

for $i=1, 2$ and any test function $\xi$ being $\phi$ or $\psi$. Moreover, by standard arguments, e.g., Constantin and Foias (1988), Foias et al. (2001), we have that the solution is actually analytic in time.

Remark A.2

In view of Sobolev embedding $H^1 \hookrightarrow L^{\infty }$ in one-dimensional space, we conclude that u and w are in fact uniformly bounded for all time.

Remark A.3

Now with the $H^1$ estimates available, we can repeat the above arguments but testing the system by $\Delta ^2 u$ and $\Delta ^2 w$ over $\Omega _{j}$, $j=1, 2$, and obtain the $H^2$ boundedness of the solution (see e.g.,  Larios and Pei 2016). Similarly, we have that the solution is in fact smooth. For simplicity, we omit the details here.$\square$