1 Introduction

Motivated by the study of the well-posedness of the inviscid primitive equations (PEs), we were led in earlier works, to study the well-posedness of the inviscid shallow water equations. Indeed, as shown in [9], the inviscid shallow water equations can be seen as a single mode of the PEs. Hence studying the SWE, besides being useful by itself, can be seen also as a step toward the study of the well-posedness of the PEs.

The inviscid shallow water equations are hyperbolic equations and there is a vast literature available concerning the initial and boundary value hyperbolic problems in a smooth domain in relation with the Kreiss–Lopatinskii conditions (see [11, 14]); many results in this direction can be found in [3]. However the usual framework for the Primitive Equations is to work in a rectangle in dimension two and in a cube in dimension three. Hence we encounter the difficulty of the well-posedness of hyperbolic initial and boundary value problems for non-smooth domains and the literature is rather scarce in this case; see the discussion in [9].

In [9], we presented the different set of boundary conditions that make the inviscid linearized SWE well-posed in a rectangle and classified these boundary conditions. For that reason, we made the equations as a linear evolution equation and used the semigroup theory. The classification of the flows correspond to different properties of the underlying linear (stationary) operator. In a same way as a stationary compressible flow can be subsonic (elliptic), transonic (parabolic), or supersonic (hyperbolic), we were led in [9] to classify the linearized shallow water flows in the fully hyperbolic case and the elliptic-hyperbolic case, where the fully hyperbolic case contains four sub-cases, the supercritical case, two mixed hyperbolic case, and the fully hyperbolic subcritical case.

Going from the linearized inviscid SWE to the full nonlinear inviscid SWE is not straight-forward, and truly new boundary conditions have to be derived. Besides, global existence of smooth solutions is not generally expected and we limit ourselves to the existence and uniqueness of smooth solutions for a limited time. However, we retain from [9] the classification of the (nonlinear) flows according to their initial values. When the flow is fully supercritical, the problem was studied in [8], we then considered in [10] the case where the flow was fully hyperbolic but took place in a channel \((0, 1)_x\times \mathbb {T}_y\) where \(\mathbb {T}_y\) is one dimensional torus. We now consider in this article the case of a fully hyperbolic (subcritical) flow taking place in a rectangle. We then encounter the difficulty of the compatibility of the initial and boundary values. When the datas are compatible, we are able to embed the flow in a channel flow (distinct from the case in [10]) and we are able to establish the existence and uniqueness of solutions.

The inviscid fully nonlinear shallow water equations (SWE) read

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t+uu_x + vu_y + g\phi _x -fv= 0, \\ v_t+uv_x + vv_y + g\phi _y + fu= 0, \\ \phi _t+u\phi _x + v\phi _y + \phi (u_x+v_y) = 0, \end{array}\right. } \end{aligned}$$
(1.1)

where u and v are the two horizontal components of the velocity and \(\phi \) is the height of the water; f and g are universal constants, standing for the Coriolis parameter and the gravitational acceleration, respectively. Setting \(U=(u,v, \phi )^t\), we write (1.1) in compact form

$$\begin{aligned} U_t + \mathcal {E}_1(U)U_x + \mathcal {E}_2(U)U_y + \ell (U) = 0. \end{aligned}$$
(1.2)

where \(\ell (U)=(-fv, fu, 0)^t\), and

$$\begin{aligned} \mathcal {E}_1( U)=\begin{pmatrix} u&{}\quad 0&{}\quad g\\ 0&{}\quad u&{}\quad 0\\ \phi &{}\quad 0&{}\quad u \end{pmatrix},\quad \mathcal {E}_2( U)=\begin{pmatrix} v&{}\quad 0&{}\quad 0\\ 0&{}\quad v&{}\quad g\\ 0&{}\quad \phi &{}\quad v \end{pmatrix}. \end{aligned}$$

The Assumptions

Our objective in this article is to study the initial and boundary value problem (IBVP) for the 2d nonlinear inviscid SWE (1.1) in the fully hyperbolic case in a rectangular domain. In [8, 10], we studied two types of problems for the 2d nonlinear inviscid SWE: the supercritical case in a rectangle and the subcritical case in a 2d channel with periodicity. For a classification of the 2d inviscid SWE, see [9, Sect. 5]. The fully hyperbolic case studied here corresponds to

$$\begin{aligned} u^2 + v^2 > g\phi , \end{aligned}$$
(1.3)

and the rectangular domain has to be properly chosen according to the characteristics of the 2d inviscid SWE, as explained in Sect. 4.2 below.

In order to show the idea for studying the IBVP of 2d nonlinear inviscid SWE, we assume that the domain \(\Omega \) is

$$\begin{aligned} \Omega = (0, 1)_x \times (0, 1)_y, \end{aligned}$$

and the 2d nonlinear SWE is supercritical in the direction (0, 1) (see Definition 4.1 below), that is

$$\begin{aligned} v^2 > g\phi . \end{aligned}$$
(1.4)

Note that condition (1.4) is stronger than (1.3), while after a suitable coordinate transformation, the condition (1.3) would become (1.4) (see Sect. 4.2 below). We can also assume that \(u,\; v \ge 0\) and the cases where u and, or v are negative can be treated in a similar manner.

Now, we have two sub-cases to consider according to the sign of \(u^2-g\phi \). The case when \(u^2 > g\phi \), that is the 2d nonlinear SWE is also supercritical in the direction (1, 0), is already studied in [8]. The remaining case when \(u^2 < g\phi \), that is the 2d nonlinear SWE is subcritical in the direction (1, 0), is the main goal of this article and we already termed it the mixed hyperbolic case in [9]. We now assume the enhanced mixed hyperbolic condition:

$$\begin{aligned} {\left\{ \begin{array}{ll} c_0\le u,\,v,\,\phi \le c_1,\\ u^2 + v^2 > g\phi ,\qquad u^2 - g\phi \le -c_2^2 ,\qquad v^2 - g\phi \ge c_2^2, \end{array}\right. } \end{aligned}$$
(1.5)

for some given positive constants \(c_0, c_1, c_2>0\).

This article is dedicated to the memory of Klaus Kirchgässner, a good friend and a gentle colleague, who has made deep and lasting contributions to the theory of bifurcation and partial differential equations, and has invested much time for services to the national and international mathematical communities.

2 The Boundary Conditions

2.1 Failure of the (Linearized) Boundary Conditions

In [9, Sect. 5], the suitable boundary conditions for the linearized SWE in a rectangle have been proposed for the well-posedness and it is natural to consider these (modified) boundary conditions for the nonlinear SWE (1.1) in a rectangle. However, as we will see below, those boundary conditions are not suitable for the nonlinear problem (1.1) in the mixed hyperbolic case since we can not derive the suitable nonlinear boundary conditions from them. The arguments are as follows. Recall from [9, Sect. 5] that the boundary conditions for the linearized SWE around the state \((u_0, v_0, \phi _0)\) in the mixed hyperbolic case (1.5) are

$$\begin{aligned} {\left\{ \begin{array}{ll} v_0u - u_0v + \kappa _0 \phi = u_0u + v_0v + g\phi =0,\text { on } \{x=0\}, \\ v_0u - u_0v - \kappa _0 \phi = 0, \text { on } \{x=1\}, \\ u = v = \phi = 0, \text { on } \{y=0\}, \end{array}\right. } \end{aligned}$$
(2.1)

where \(\kappa _0=\sqrt{g(u_0^2 + v_0^2 - g\phi _0)/\phi _0}\). If we could derive a set of nonlinear boundary conditions from (2.1), then considering the boundary conditions at \(x=1\), there must exist two non-zero functions \(\Phi (u,v,\phi )\) and \(\Psi (u,v,\phi )\) such that

$$\begin{aligned} (\Phi _u(u,v,\phi ),\; \Phi _v(u,v,\phi ),\; \Phi _\phi (u,v,\phi ) )=(v,\; -u,\; -\kappa )\Psi (u,v,\phi ), \end{aligned}$$
(2.2)

where \(\kappa =\sqrt{g(u^2 + v^2 - g\phi )/\phi }\). We now infer from (2.2) the following identities

$$\begin{aligned} {\left\{ \begin{array}{ll} \Psi + v\Psi _v = \Phi _{uv} = -\Psi - u\Psi _u,\\ v\Psi _\phi = \Phi _{u\phi } = -\kappa _u \Psi - \kappa \Psi _u,\\ -u\Psi _\phi = \Phi _{v\phi } = -\kappa _v \Psi - \kappa \Psi _v. \end{array}\right. } \end{aligned}$$
(2.3)

Multiplying (2.3)\(_2\) by u and (2.3)\(_3\) by v, and adding the resulting identities together, we find

$$\begin{aligned} 0=(u\kappa _u + v\kappa _v)\Psi + \kappa (u\Psi _u + v\Psi _v), \end{aligned}$$

which, together with (2.3)\(_1\), shows that

$$\begin{aligned} (u\kappa _u + v\kappa _v)\Psi - 2\kappa \Psi = 0. \end{aligned}$$
(2.4)

Now, we directly calculate

$$\begin{aligned} \kappa _u = \frac{g u}{\phi \kappa },\quad \kappa _v = \frac{g v}{\phi \kappa }, \end{aligned}$$

and then deduce from (2.4) that

$$\begin{aligned} \big (gu^2/\phi + gv^2 /\phi - 2\kappa ^2 \big )\Psi =0, \end{aligned}$$

that is equivalent to

$$\begin{aligned} \big (u^2 + v^2 - 2g\phi \big )\Psi = 0, \end{aligned}$$

which is impossible for non-zero \(\Psi \). Therefore, we conclude that the (linearized) boundary conditions proposed in [9] are not suitable for the nonlinear SWE (1.1). This fact is rather general of course.

2.2 The Nonlinear Boundary Conditions

Under the mixed hyperbolic condition (1.5), all the eigenvalues of the matrix \(\mathcal {E}_2\) are positive and hence the y-direction could be viewed as a time-like direction, and we only need to impose the boundary conditions at \(y=0\), that is

$$\begin{aligned} (u,v,\phi )=(g_1, g_2, g_3),\quad \text { on }\quad y=0. \end{aligned}$$
(2.5)

For the boundary conditions in the x-direction, as in [10] where we studied the channel domain \((0, 1)_x\times \mathbb {T}\) with periodicity in the y-direction, we can take the following nonlinear boundary conditions:

$$\begin{aligned} (u+2\sqrt{g\phi }, v) = (\pi _1,\pi _2),\quad \text { on }\quad x=0,\quad u-2\sqrt{g\phi } = \pi _3,\quad \text { on }\quad x=1. \end{aligned}$$
(2.6)

Here, \((g_1, g_2, g_3)\) and \((\pi _1, \pi _2, \pi _3)\) are given boundary data.

We remark that although the linearized form of the boundary conditions (2.5)–(2.6) may not lead to the \(L^2\)-well-posedness of the linearized SWE in the rectangle \(\Omega \), these boundary conditions (2.5)–(2.6) will yield local well-posedness of the nonlinear SWE since we consider smooth solutions for the nonlinear problem (see Theorem 3.1).

3 The Fully Nonlinear Shallow Water System

In this section, we aim to investigate the well-posedness for Eq. (1.1) in the rectangular domain \(\Omega =(0, 1)_x\times (0,1)_y\) associated with initial and boundary conditions. The fully nonlinear shallow water system reads in compact form

$$\begin{aligned} U_t + \mathcal {E}_1(U)U_x + \mathcal {E}_2(U)U_y + \ell (U) = 0. \end{aligned}$$
(3.1)

3.1 Stationary Solutions

We want to study system (3.1) near a stationary solution, and we start by constructing such a stationary solution \(U=U_s\), that is \((u,v,\phi )=(u_s, v_s, \phi _s)\). These functions are independent of time and satisfy

$$\begin{aligned} \mathcal {E}_1(U)U_x + \mathcal {E}_2(U)U_y + \ell (U) = 0. \end{aligned}$$
(3.2)

In the following, we construct a y-independent stationary solution \(U_s\) of (3.2) satisfying the subcritical conditions (1.5). Thus \(U_s\) satisfies (see Subsec. 2.1 in [7] for a stationary solution in the supercritical case):

$$\begin{aligned} {\left\{ \begin{array}{ll} uu_x + g\phi _x - fv=0, \\ uv_x + fu =0, \\ (u\phi )_x = 0. \end{array}\right. } \end{aligned}$$
(3.3)

We infer from (3.3) that

$$\begin{aligned} {\left\{ \begin{array}{ll} u\phi = \kappa _1, \\ v = -fx + \kappa _2,\\ u^2 + 2g\phi = -f^2x^2 +2f\kappa _2 x+\kappa _3, \end{array}\right. } \end{aligned}$$

where \(\kappa _1,\kappa _2,\kappa _3\) are constants. We first choose \(\kappa _1 =1\), \(\kappa _2=2g+f\), and then we have \(\phi =u^{-1}\), \(v = 2g + f -fx\), and

$$\begin{aligned} u^2+\frac{2g}{u} = -f^2x^2 +{ 2f(2g+f) x }+\kappa _3,\quad x\in (0,1). \end{aligned}$$
(3.4)

We then set \(\Psi (u)=u^2+\frac{2g}{u}\) and \(\psi (x) = -f^2x^2 + 2f(2g+f) x +\kappa _3\) and we can easily deduce that

$$\begin{aligned} \kappa _3 \le \psi (x) \le \kappa _3 + 4 g f + f^2,\qquad \forall \, x\in (0, 1). \end{aligned}$$

Note that the Coriolis parameter \(f\ll 1\) and the gravitational constant \(g \approx 9.8\). Hence, \(\Psi (\frac{1}{2g}) - \Psi (\frac{1}{g}) > g^2\) and we can choose \(\kappa _3\) such that

$$\begin{aligned} \Psi \Big (\frac{1}{2g}\Big ) > \kappa _3 + 4gf + f^2\ge \psi (x) \ge \kappa _3 > \Psi \Big (\frac{1}{g}\Big ). \end{aligned}$$

Then for any \(x\in (0,1)\) one solution (in u) of (3.4) is between 1 / 2g and 1 / g. We choose such a solution u, and therefore \(\phi =u^{-1}\) satisfies \(g\le \phi \le 2g\), and furthermore \(u^2-g\phi \le 1/g^2 - g^2<0\) and by the implicit function theorem, such a solution u is unique and smooth. Since \(x\in (0,1)\), we have \(v\ge 2g\) and hence \(v^2 > g\phi \). All these calculations mean that we can choose the stationary solution \(u_s,v_s,\phi _s\) satisfying the mixed hyperbolic conditions

$$\begin{aligned} u,\,v,\,\phi >0,\qquad u^2-g\phi <0,\qquad v^2 - g\phi >0. \end{aligned}$$
(3.5)

Therefore, we choose \(\kappa _{0,1}, \kappa _{0,2}, \kappa _{0,3}>0\) and \(\delta >0\) such that

$$\begin{aligned} {\left\{ \begin{array}{ll} c_0\le \kappa _{0,1} \pm c_3 \delta < c_1,\quad c_0\le \kappa _{0,2} \pm c_3 \delta < c_1,\quad c_0\le \kappa _{0,3} \pm c_3 \delta < c_1\\ (\kappa _{0,1} + c_3\delta )^2 - g(\kappa _{0,3} - c_3\delta ) \le -c_2^2,\quad (\kappa _{0,2} + c_3\delta )^2 - g(\kappa _{0,3} - c_3\delta ) \ge c_2^2, \end{array}\right. } \end{aligned}$$
(3.6)

where \(c_0,c_1,c_2>0\) are given positive constants and \(c_0<c_1\), and \(c_3\) is given by Lemma 3.2 below.

Note that the stationary solution we constructed for (3.2) is independent of y as described above, or saying in other way, exists for all \(y\in \mathbb {R}_y\). More generally, we assume that a stationary solution \(U_s(x,y)\) exists for all \((x,y)\in (0,1)_x\times \mathbb {R}_y\) and satisfies

$$\begin{aligned} \mathcal {E}_1(U_s)U_{s,x} + \mathcal {E}_2(U_s)U_{s,y} + \ell (U_s) = 0,\qquad \forall \,(x,y)\in (0, 1)_x\times \mathbb {R}_y. \end{aligned}$$
(3.7)

The reason why we assume \(U_s\) exists for all \(y\in \mathbb {R}_y\) instead of \(y\in (0,1)_y\) is that we are going to use the extension method below by extending the problem into the channel domain \((0, 1)_x\times \mathbb {R}_y\) and the assumption that \(U_s\) exists for all \(y\in \mathbb {R}_y\) will simplify our presentation.

In what follows, we think of the stationary solution \(U_s\) in a more general form (i.e. \(U_s\) depends on both x and y), and we choose \(U_s=(u_s,v_s,\phi _s)\) such that

$$\begin{aligned} |u_s - \kappa _{0,1} | \le \delta /4,\quad |v_s - \kappa _{0,2} |\le \delta /4,\quad |\phi _s - \kappa _{0,3} |\le \delta /4, \end{aligned}$$
(3.8)

and by (3.6), \(U_s\) satisfies the mixed hyperbolic condition (1.5). For convenience, we write

$$\begin{aligned} |U_s - \varvec{\kappa }_0 | \le \delta /4,\qquad \forall \,(x,y)\in (0, 1)_x\times \mathbb {R}_y. \end{aligned}$$
(3.9)

to stand for (3.8), where \(\varvec{\kappa }_0=(\kappa _{0,1}, \kappa _{0,2}, \kappa _{0,3} )\), and the \(\kappa _{0,i}\) (\(i=1,2,3\)) are positive constants satisfying (3.6).

We set \(U = U_s + \widetilde{U}\) and substitute these values into (3.1); we obtain a new system for \(\widetilde{U}\), and dropping the tildes, our new system reads:

$$\begin{aligned} L_{U_s+U}\, U = -L_{U_s+U}\,U_s, \end{aligned}$$
(3.10)

where the operator L is defined by

$$\begin{aligned} L_{ W }\, U = U_t + \mathcal {E}_1( W )U_x + \mathcal {E}_2( W )U_y + \ell (U). \end{aligned}$$

We supplement (3.10) with the following initial and boundary conditions (see (2.5)–(2.6)):

$$\begin{aligned} U=U_0(x,y),\text { on }t=0, \quad U=\mathsf {G}(x,t),\text { on }y=0,\quad b(U_s+U)=\Pi (y,t), \end{aligned}$$
(3.11)

where

$$\begin{aligned} b(U_s+U)\!=\!{\left\{ \begin{array}{ll} u+u_s + 2\sqrt{g(\phi +\phi _s)}{=}\pi _1(y,t),&{}\text { on }\;x=0,\\ v+v_s=\pi _2(y,t),&{}\text { on }\;x=0,\\ u+u_s - 2\sqrt{g(\phi +\phi _s)} {=}\pi _3(y,t),&{}\text { on }\;x=1,\\ \end{array}\right. } \quad \Pi =\begin{pmatrix} \pi _1\\ \pi _2 \\ \pi _3 \end{pmatrix},\;\, {\mathsf {G}}{=}\begin{pmatrix} g_1\\ g_2 \\ g_3 \end{pmatrix}, \end{aligned}$$

We regard the initial condition \(U_0=U_s+\widetilde{U_0}\) as a small perturbation of the stationary solution, and after dropping the tilde, we choose the small perturbation \(U_0\) satisfying

$$\begin{aligned} |U_0 | \le \epsilon \delta , \end{aligned}$$
(3.12)

for some \(0<\epsilon <1\) small enough.

3.2 Compatibility Conditions on the Data

In order to be able to solve the system (3.10) we need to introduce some technical conditions (see [3, Sect. 11.1.2]). First we require that \(U=0\) is a solution of the special IBVP (3.10) with zero initial data and boundary data \(\Pi (y, t=0)\) and \(\mathsf {G}(x, t=0)\), which amounts to asking that the following compatibility conditions are satisfied by \(U_s\):

$$\begin{aligned} b(U_s)={\left\{ \begin{array}{ll} u_s + 2\sqrt{g\phi _s} = \pi _1(y, 0),&{}\text { on }x=0,\\ v_s = \pi _2(y,0),&{}\text { on }x=0,\\ u_s - 2\sqrt{ g\phi _s } =\pi _3(y, 0),&{}\text { on }x=1, \end{array}\right. } \quad U_s=\mathsf {G}(x, 0),\text { on }y=0. \end{aligned}$$
(3.13)

The second condition is that the initial and boundary data should satisfy some additional compatibility conditions and these conditions are very natural for smooth solutions, which we are looking for. Let us first rewrite (3.10) as

$$\begin{aligned} U_t = H(U+U_s) - \mathcal {E}_1(U+U_s)U_x - \mathcal {E}_2(U+U_s)U_y - \ell (U), \end{aligned}$$
(3.14)

where we denote by \(H(U+U_s)\) the right-hand side of (3.10), that is \(-L_{U_s+U}U_s\). Now, if U is continuous, then necessarily at \(t=0\), there should holds

$$\begin{aligned} b(U_s+U_0)=\Pi (y, 0),\qquad \mathsf {G}(x,0)=U_0|_{y=0}; \end{aligned}$$
(3.15)

and if U is \(\mathcal {C}^1\) up to the boundary, then at \(t=0\),

$$\begin{aligned} \begin{aligned} \partial _t \Pi (y,0)&={\mathrm {d}} b(U_s+U_0)\cdot \partial _t U(x,0)\\&={\mathrm {d}} b(U_s+U_0)\cdot \big ( H(U_0+U_s) \,{-}\, \mathcal {E}_1(U_0+U_s)U_{0,x} - \mathcal {E}_2(U_0+U_s)U_{0,y} {-} \ell (U_0) \big ),\\ \partial _t \mathsf {G}(x,0)&=\partial _t U(x,0)= H(U_0+U_s) - \mathcal {E}_1(U_0+U_s)U_{0,x} - \mathcal {E}_2(U_0+U_s)U_{0,y} - \ell (U_0), \end{aligned} \end{aligned}$$

where \({\mathrm {d}} b(U_s+U)\) is a matrix-valued function, the gradient of the function \(b(U_s+U)\) with respect to the variable U. More generally, if U is \(\mathcal {C}^{m-1}\) up to the boundary, then at \(t=0\),

$$\begin{aligned} \begin{aligned} \partial _t^p \Pi (y, 0)&={\mathsf {C}}_{p, U_0}(V_0,\ldots ,V_p),\qquad \forall p\in \left\{ 1,\ldots ,m-1 \right\} ,\\ \partial _t^p \mathsf {G}(x, 0)&=V_p\big |_{y=0},\qquad \qquad \quad \;\,\forall p\in \left\{ 1,\ldots ,m-1 \right\} , \end{aligned} \end{aligned}$$
(3.16)

where the complicated nonlinear function \({\mathsf {C}}_{p, U_0}\) is given by

$$\begin{aligned} {\mathsf {C}}_{p, U_0}(V_0,\ldots ,V_p)=\sum _{k=1}^p\sum _{j_1+\cdots +j_k=p}c_{j_1,\ldots ,j_k}{\mathrm {d}}^kb(U_s+U_0)\cdot (V_{j_1},\ldots ,V_{j_k}), \end{aligned}$$

and the functions \(V_i\) (\(i=0,\ldots ,m\)) are defined by induction by (with U being replaced by \(U_0\))

$$\begin{aligned} \begin{aligned} V_0&= U,\\ V_1&= \partial _t U = H(U+U_s) - \mathcal {E}_1(U+U_s)U_x - \mathcal {E}_2(U+U_s)U_y - \ell (U),\\ \end{aligned} \end{aligned}$$
(3.17)

and for all \(i=1,\ldots ,m-1\),

$$\begin{aligned} \begin{aligned} V_{i+1}=\partial _t^{i+1}U&=\sum _{k=1}^{i}\sum _{j_1+\cdots +j_k=i}c_{j_1,\ldots ,j_k}({\mathrm {d}}^kH(U+U_s))\cdot (V_{j_1},\ldots ,V_{j_k}) \\&\quad -\sum _{l=1}^{i}\left( {\begin{array}{c}i\\ l\end{array}}\right) \sum _{k=1}^l\sum _{j_1+\cdots +j_k=l}c_{j_1,\ldots ,j_k}({\mathrm {d}}^k\mathcal {E}_1(U+U_s))\cdot (V_{j_1},\ldots ,V_{j_k})V_{i-l,x} \\&\quad -\sum _{l=1}^{i}\left( {\begin{array}{c}i\\ l\end{array}}\right) \sum _{k=1}^l\sum _{j_1+\cdots +j_k=l}c_{j_1,\ldots ,j_k}({\mathrm {d}}^k\mathcal {E}_2(U+U_s))\cdot (V_{j_1},\ldots ,V_{j_k})V_{i-l,y} \\&\quad -\mathcal {E}_1(U+U_s)V_{i,x}-\mathcal {E}_2(U+U_s)V_{i,y} -\ell (V_{i}). \\ \end{aligned} \end{aligned}$$
(3.18)

Here the coefficients \(c_{j_1,\ldots ,j_k}\) are derived from the Fa\(\grave{\text {a}}\) di Bruno’s formula, see [4, 5]. The conditions (3.15)–(3.16) express the classical compatibility conditions which are necessary for the solution U of (3.10) to be \(\mathcal {C}^{m-1}\) near \(t=0\); see e.g. [15, 17, 19].

We also need to express the compatibility conditions at \(y=0\). For this reason, we rewrite (3.10) as

$$\begin{aligned} \begin{aligned} U_y&= \mathcal {E}_2(U+U_s)^{-1}\big (H(U+U_s) - \mathcal {E}_1(U+U_s)U_x - U_t - \ell (U)\big )\\&=\widetilde{H}(U+U_s) - \widetilde{\mathcal {E}}_1(U+U_s)U_x - \mathcal {E}_2(U+U_s)^{-1}(U_t -\ell (U)), \end{aligned} \end{aligned}$$
(3.19)

where

$$\begin{aligned} \widetilde{H}(U+U_s)=\mathcal {E}_2(U+U_s)^{-1}H(U+U_s),\qquad \widetilde{\mathcal {E}}_1(U+U_s)=\mathcal {E}_2(U+U_s)^{-1} \mathcal {E}_1(U+U_s). \end{aligned}$$

Similar to the definition of the \(V_i\)’s, we now define the functions \(W_i\) (\(i=0,\ldots ,m\)) by induction by setting (with U being replaced by \(\mathsf {G}\))

$$\begin{aligned} \begin{aligned} W_0&=U,\\ W_1&=\partial _y U=\widetilde{H}(U+U_s) - \widetilde{\mathcal {E}}_1(U+U_s)U_x - \mathcal {E}_2(U+U_s)^{-1}U_t -\mathcal {E}_2(U+U_s)^{-1}\ell (U),\\ \end{aligned} \end{aligned}$$
(3.20)

and for all \(i=1,\ldots ,m-1\),

$$\begin{aligned} \begin{aligned} W_{i+1}&=\partial _y^{i+1}U=\sum _{k=1}^{i}\sum _{j_1+\cdots +j_k=i}c_{j_1,\ldots ,j_k}({\mathrm {d}}^k\widetilde{H}(U+U_s))\cdot (W_{j_1},\ldots ,W_{j_k}) \\&\quad -\sum _{l=1}^{i}\left( {\begin{array}{c}i\\ l\end{array}}\right) \sum _{k=1}^l\sum _{j_1+\cdots +j_k=l}c_{j_1,\ldots ,j_k}({\mathrm {d}}^k\widetilde{\mathcal {E}}_1(U+U_s))\cdot (W_{j_1},\ldots ,W_{j_k})W_{i-l,x} \\&\quad -\sum _{l=1}^{i}\left( {\begin{array}{c}i\\ l\end{array}}\right) \sum _{k=1}^l\sum _{j_1+\cdots +j_k=l}c_{j_1,\ldots ,j_k}({\mathrm {d}}^k\mathcal {E}_2(U+U_s)^{-1})\cdot (W_{j_1},\ldots ,W_{j_k})(W_{i-l,t}-\ell (W_{i-l}))\\&\quad -\widetilde{\mathcal {E}}_1(U+U_s)W_{i,x}-\mathcal {E}_2(U+U_s)^{-1}(W_{i,t} -\ell (W_{i})). \\ \end{aligned} \end{aligned}$$
(3.21)

Now, if U is continuous, then necessarily at \(y=0\), there should holds

$$\begin{aligned} b(U_s+\mathsf {G})=\Pi (0, t),\qquad U_0|_{y=0}=\mathsf {G}(x,0). \end{aligned}$$
(3.22)

More generally, if U is \(\mathcal {C}^{m-1}\) up to the boundary, then at \(y=0\),

$$\begin{aligned} \begin{aligned} \partial _y^p \Pi (0, t)&={\mathsf {C}}_{p, \mathsf {G}}(W_0 + U_s, \ldots , W_p + \partial _y^pU_s),\qquad \forall p\in \left\{ 1,\ldots ,m-1 \right\} ,\\ \partial _y^p U_0|_{y=0}&=W_p|_{t=0}, \end{aligned} \end{aligned}$$
(3.23)

where \({\mathsf {C}}_{p, \mathsf {G}}\) is defined in the same fashion as \({\mathsf {C}}_{p, U_0}\), and the reason why we have the additional term \(\partial _y^pU_s\) in \({\mathsf {C}}_{p, \mathsf {G}}\) is because \(U_s\) is independent of t but generally depends on y. The conditions (3.22)–(3.23) express the classical compatibility conditions which are necessary for the solution U of (3.10) to be \(\mathcal {C}^{m-1}\) near \(y=0\).

We remark that the compatibility conditions between the boundary data \(\mathsf {G}\) at \(y=0\) and the initial data \(U_0\) which are expressed either near \(t=0\) or near \(y=0\) are equivalent.

3.3 Approximate Solutions

The disadvantage of this new formulation (3.10)–(3.11) is that the boundary conditions in both the x-and y-directions and the initial condition \(U_0\) are generally non-zero. To overcome these difficulties, we will use two approximate solutions lifting the boundary data \(\mathsf {G}\) at \(y=0\) and the initial data \(U_0\) at \(x=0\). The approximate lifting solutions \(U_{\mathsf {g}}\) of the boundary data \(\mathsf {G}\) is given by the following lemma.

Lemma 3.1

We are given \(m\ge 3\), the stationary solution \(U_s\in H^{m+1}(\Omega )\), the initial data \(U_0=(u_0,v_0,\phi _0)\) belonging to \(H^{m+1/2}(\Omega )\), the boundary data \(\mathsf {G}=(g_1, g_2, g_3)\) belonging to \(H^{m+1/2}((0,1)_x\times (0, T))\) and \(\Pi =(\pi _1,\pi _2,\pi _3)\) belonging to \(H^{m+1/2}((0,1)_y\times (0,T))\). Then there exists a function \(U_{\mathsf {g}}\in H^{m+1}(\Omega \times (0,T))\) such that \(U_{\mathsf {g}}|_{y=0}=\mathsf {G}\), and

$$\begin{aligned} \Vert U_{\mathsf {g}} \Vert _{ H^{m+1}(\Omega \times (0, T)) } \le C \Vert \mathsf {G} \Vert _{H^{m+1/2}((0, 1)_x\times (0, T)) } \Vert U_s \Vert _{ H^{m+1}(\Omega \times (0, T)) }, \end{aligned}$$
(3.24)

for some constant \(C>0\) depending on m and \(\Omega \), independent of \(\mathsf {G}\) and \(U_s\).

If we let \(U_0^0=U_0-U_{\mathsf {g}}|_{t=0}\), \(\Pi ^0=-b(U_{\mathsf {g}} + U_s)+\Pi \), and

$$\begin{aligned} \widetilde{F}^0=-\partial _yU_{\mathsf {g}} + \mathcal {E}_2(U_{\mathsf {g}}+U_s)^{-1}\big (H(U_{\mathsf {g}}+U_s) - \mathcal {E}_1(U_{\mathsf {g}}+U_s)U_{\mathsf {g}, x} - U_{\mathsf {g}, t} - \ell (U_{\mathsf {g}})\big ), \end{aligned}$$

then \(U_0^0\in H^{m+1/2}(\Omega )\), \(\widetilde{F}^0\in H^m(\Omega \times (0, T))\), \(\Pi ^0\in H^{m+1/2}((0,1)_y\times (0,T))\), and

$$\begin{aligned} \partial _y^j \widetilde{F}^0=0,\quad \partial _y^jU_0^0=0,\quad \partial _y^j \Pi ^0 = 0,\text { on }y=0,\qquad \forall \,j\in \left\{ 0,\ldots ,m-1 \right\} . \end{aligned}$$
(3.25)

Proof

Similar to [3, Lemma 11.1], we can construct \(\{W_i=W_i|_{y=0}\}_{i=0,\ldots ,m-1 }\) with \(U|_{y=0}=\mathsf {G}\) satisfying (3.20)–(3.21) and

$$\begin{aligned} W_i\in H^{m+1/2-i}((0, 1)_x\times (0, T)),\qquad \forall \,i\in \left\{ 0,\ldots ,m-1 \right\} . \end{aligned}$$

Then, by the lifting result in Proposition 5.3, we find \(U_{\mathsf {g}}\in H^{m+1}(\Omega \times (0, T))\) such that

$$\begin{aligned} (\partial _y^i U_{\mathsf {g}})|_{y=0}=W_i,\qquad \forall \,j\in \left\{ 0,\ldots ,m-1 \right\} , \end{aligned}$$

and by the classical inequalities for the Sobolev spaces (see for example [18, Chapter 13], [3, Appendix C], or [10, Lemma B.1]):

$$\begin{aligned} \begin{aligned} \Vert U_{\mathsf {g}} \Vert _{ H^{m+1}(\Omega \times (0, T)) } \lesssim \sum _{i=0}^m\Vert W_i \Vert _{H^{m+1/2-i}((0, 1)_x\times (0, T))}\\ \lesssim \Vert \mathsf {G} \Vert _{H^{m+1/2}((0, 1)_x\times (0, T)) } \Vert U_s \Vert _{ H^{m+1}(\Omega \times (0, T)) }, \end{aligned} \end{aligned}$$
(3.26)

where \(\lesssim \) means \(\le \) up to a multiplicative absolute constant C.

That \(\widetilde{F}^0\) is in \(H^m\) follows in a classical way from the inequalities in [10, Lemma B.1] (see also [18, Chapter 13] and [3, Appendix C]) and that \(U_0^0\) and \(\Pi ^0\) are in \(H^{m+1/2}\) follows from the trace theorem (see Proposition 5.3). The vanishing of \(\partial _y^j \widetilde{F}^0\) at \(y=0\) follows from the construction of \(U_{\mathsf {g}}\) and the vanishing of \(\partial _y^j \Pi ^0\) and \(U_0^0\) at \(y=0\) are consequences of the compatibility conditions in (3.22)–(3.23). \(\square \)

If we let \(F^0 = \mathcal {E}_2(U_{\mathsf {g}}+U_s) \widetilde{F}^0\), then \(F^0=-L_{U_{\mathsf {g}}+U_s}(U_{\mathsf {g}}+U_s)\) and \(F^0\) has the same properties as \(\widetilde{F}^0\), that is

$$\begin{aligned} F^0\in H^m(\Omega \times (0, T)),\qquad \partial _y^j F^0=0,\text { on }y=0,\qquad \forall \,j\in \left\{ 0,\ldots ,m-1 \right\} . \end{aligned}$$
(3.27)

We recall that we have set \(U=U_s+\widetilde{U}\) and that we have dropped the tilde in the above. Now let us reintroduce the tilde and set \(\widetilde{U}=U_{\mathsf {g}} + \overline{U}\), so that \(U=U_s+U_{\mathsf {g}}+\overline{U}\). We then substitute this expression into the system (3.10) and in (3.12) (where \(U_0=\widetilde{U_0}\)) and drop the bars. Then the new system for \(U=\overline{U}\) becomes the following initial and boundary value problem (IBVP):

$$\begin{aligned} {\left\{ \begin{array}{ll} L_{U_{\mathsf {g}} + U_s + U} U= - L_{U_{\mathsf {g}} + U_s +U } (U_{\mathsf {g}} + U_s) , \\ U|_{t=0}= U_0- U_{\mathsf {g}}, \\ U|_{y=0}=0,\\ b(U_{\mathsf {g}}+U_s+U)= \Pi .\\ \end{array}\right. } \end{aligned}$$
(3.28)

If U is a solution of the IBVP (3.28), then by use of the induction method we can show that

$$\begin{aligned} \partial _y^j U|_{y=0} = 0,\quad \forall \, j\in \{0,\ldots ,m-1 \}. \end{aligned}$$
(3.29)

Indeed, applying \(\partial _y^j\) to the equation (3.19) with \(U_s\) replaced by \(U_s+U_{\mathsf {g}}\) and \(H(U+U_s)\) replaced by \(- L_{U_{\mathsf {g}} + U_s +U } (U_{\mathsf {g}} + U_s)\), we find

$$\begin{aligned} \begin{aligned} \partial _y^{j+1}U&= \sum _{k=0}^j\partial _y^{j-k}(\mathcal {E}_2(U+U_s+U_{\mathsf {g}})^{-1} \big [ \partial _y^k (- L_{U_{\mathsf {g}}+U_s +U } (U_{\mathsf {g}} + U_s) ) - \partial _y^k U_t - \partial _y^k \ell (U)\big ]\\&\quad +\sum _{k=0}^j\partial _y^{j-k}(\mathcal {E}_2(U+U_s+U_{\mathsf {g}})^{-1}\mathcal {E}_1(U+U_s+U_{\mathsf {g}} )) \partial _y^kU_x. \end{aligned} \end{aligned}$$
(3.30)

The result (3.29) then follows from the above identity and \(U|_{y=0}=0\). The vanishing property (3.29) points to the fact that we may extend the system to the smooth domain \((0, 1)_x\times \mathbb {R}_y\), which we will now do. We point out that the assumption that the 2d nonlinear SWE is supercritical in the direction (0, 1) enables us to prescribe all the boundary conditions at \(y=0\) in the y-direction and hence by the lifting Lemma 3.1, the boundary conditions at \(y=0\) are lifted to 0, which yields the vanishing property (3.29) at \(y=0\).

3.4 The Extension Problem

We now aim to extend the problem (3.28) to the channel (smooth) domain \(\mathcal {Q}:=(0, 1)_x\times \mathbb {R}_y\) and for this reason, we need the following extension result.

Lemma 3.2

(Extension Theorem) There exists a continuous linear operator \(P=P_m\) from \(H^m(\Omega \times [0,T])\) into \(H^m(\mathcal {Q}\times [0, T])\) such that for all \(u\in H^m(\Omega \times [0,T])\), the restriction of Pu to \(\Omega \times [0,T]\) is u itself, i.e.

$$\begin{aligned} Pu|_{\Omega \times [0,T]} = u, \end{aligned}$$

and furthermore Pu has compact support in the y-direction (i.e. in \(\mathcal {Q}\times [0,T]\)) and satisfies the estimate

$$\begin{aligned} \Vert Pu \Vert _{L^\infty (\mathcal {Q}\times [0,T])} \le c_3 \Vert u \Vert _{L^\infty (\Omega \times [0,T])},\quad \Vert Pu \Vert _{H^m(\mathcal {Q}\times [0,T])} \le c_4\Vert u \Vert _{H^m(\Omega \times [0,T])}. \end{aligned}$$

where \(c_3>1,c_4>1\) only depend on m, and are independent of u.

Furthermore, we have the following vanishing properties:

  1. (1)

    if u vanishes on \(x=0\) (resp. \(x=1\)), then Pu also vanishes on \(x=0\) (resp. \(x=1\));

  2. (2)

    if \(\partial _t^p u\) vanishes on \(t=0\) for all \(p=0,\ldots ,m-1\), then \(\partial _t^p Pu\) also vanishes on \(t=0\) for all \(p=0,\ldots ,m-1\).

See [12, Chapter 2] for a detailed proof of Lemma 3.2, and using the Babitch extension procedure, the \(L^\infty \)-estimate and the vanishing properties come from the reflection formula (4.8) in [12, Chapter 2]. See also [1, 6].

We now describe how to extend the initial data \(U_0\) and the boundary data \(\Pi \) to make sure that the extended data still satisfy the compatibility condition stated in Sect. 3.2. We first extend the initial data \(U_0\) and we find from (3.13) and (3.15) that

$$\begin{aligned} b(U_s) = \Pi (y,0) = b(U_s + U_0),\quad \text { on }\quad t=0; \end{aligned}$$

specifically, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} u_s + 2\sqrt{g\phi _s} = \pi _1(y, 0) = u_s+u_0 + 2\sqrt{g(\phi _s + \phi _0)},&{}\text { on }\; x=0,\\ v_s = \pi _2(y,0) = v_s + v_0,&{}\text { on }\;x=0,\\ u_s - 2\sqrt{ g\phi _s } =\pi _3(y, 0)=u_s+u_0 - 2\sqrt{g(\phi _s + \phi _0)},&{}\text { on }\; x=1, \end{array}\right. } \end{aligned}$$
(3.31)

which is equivalent to

$$\begin{aligned} {\left\{ \begin{array}{ll} u_0 + 2\sqrt{g(\phi _s + \phi _0)} - 2\sqrt{g\phi _s} =0,&{}\text { on }\;x=0,\\ v_0 =0,&{}\text { on }\;x=0,\\ u_0 - 2\sqrt{g(\phi _s + \phi _0)} + 2\sqrt{g\phi _s} =0,&{}\text { on }\; x=1. \end{array}\right. } \end{aligned}$$
(3.32)

We now set

$$\begin{aligned} {\left\{ \begin{array}{ll} \xi = u_0 + 2\sqrt{g(\phi _s + \phi _0)} - 2\sqrt{g\phi _s} ,\\ \eta = v_0,\\ \zeta = u_0 - 2\sqrt{g(\phi _s + \phi _0)} + 2\sqrt{g\phi _s}, \end{array}\right. } \end{aligned}$$

and we have for all \(y\in (0, 1)\):

$$\begin{aligned} \xi = \eta = 0,\text { on }x=0,\qquad \zeta \text { on }x=1. \end{aligned}$$

Using Lemma 3.2, we can extend \((\xi ,\, \eta ,\, \zeta )\) to \((\hat{\xi },\, \hat{\eta },\, \hat{\zeta })\) in the channel domain \(\mathcal {Q}\) such that for all \(y\in \mathbb {R}\):

$$\begin{aligned} \hat{\xi } = \hat{\eta } = 0,\text { on }x=0,\qquad \hat{\zeta } \text { on }x=1. \end{aligned}$$

Note that the stationary solution \(U_s\) exists in the channel domain \((0, 1)_x\times \mathbb {R}_y\), then the extended initial data \(\widehat{U}_0 = (\hat{u}_0,\, \hat{v}_0,\, \hat{\phi }_0)\) are now given by the following equations

$$\begin{aligned} {\left\{ \begin{array}{ll} \hat{u}_0 + 2\sqrt{g(\phi _s + \hat{\phi }_0)} - 2\sqrt{g \phi _s} = \hat{\xi },\\ \hat{v}_0 = \hat{\eta },\\ \hat{u}_0 - 2\sqrt{g(\phi _s + \hat{\phi }_0)} + 2\sqrt{g \phi _s} = \hat{\zeta }. \end{array}\right. } \end{aligned}$$

We remark that as long as \(U_0\) is small enough in the sense of the \(L^\infty \)-norm, then \((\xi ,\, \eta ,\, \zeta )\) will also be small enough and hence the extended data \((\hat{\xi },\, \hat{\eta },\, \hat{\zeta })\) and the extended initial data \(\widehat{U}_0\).

We are now going to describe how to extend the boundary data \(\Pi \). We first construct \(\{ \widehat{V}_i\}|_{i=0,\ldots ,m-1}\) with \(U = \widehat{U}_0\) by (3.17) and (3.18). We now set

$$\begin{aligned} \Xi (y,t) = \Pi (y,t) - b(\widehat{U}_0 + U_s) - \sum _{p=1}^{m-1} \frac{t^p}{p!} {\mathsf {C}}_{p, \widehat{U}_0}(\widehat{V}_0,\ldots , \widehat{V}_p),\qquad \forall \,y\in (0, 1), \end{aligned}$$

and we find from the compatibility conditions (3.15) and (3.16) that

$$\begin{aligned} \partial _t^p\Xi (y, t=0) = 0,\qquad \forall \, p =0,\cdots ,m-1,\quad \forall \,y\in (0, 1). \end{aligned}$$

Using Lemma 3.2, we can extend \(\Xi \) to \(\widehat{\Xi }\) in the domain \(\mathbb {R}_y\times (0, T)\) such that

$$\begin{aligned} \partial _t^p\widehat{\Xi }(y, t=0) = 0,\quad \forall \, p =0,\ldots ,m-1,\quad \forall \,y\in \mathbb {R}_y. \end{aligned}$$

The extended boundary data \(\widehat{\Pi }\) are now given by

$$\begin{aligned} \widehat{\Pi }(y,t) = \widehat{\Xi }(y,t) + b(\widehat{U}_0 + U_s) + \sum _{p=1}^{m-1} \frac{t^p}{p!} {\mathsf {C}}_{p, \widehat{U}_0}(\widehat{V}_0,\ldots , \widehat{V}_p),\qquad \forall \,y\in \mathbb {R}_y. \end{aligned}$$

By the construction of \(\widehat{U}_0\) and \(\widehat{\Pi }\), we can see that \(\widehat{U}_0\) and \(\widehat{\Pi }\) satisfy the compatibility conditions (3.15) and (3.16) for all \(y\in \mathbb {R}_y\).

Finally, we also use Lemma 3.2 to extend \(U_{\mathsf {g}}\) to \(\widehat{U_{\mathsf {g}}}\) in the domain \(\mathcal {Q}\times (0,T)\).

Now, to solve the IBVP (3.28), we first consider the following extension problem, that is we look for a solution \(\widehat{U}\) satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} L_{\widehat{U_{\mathsf {g}}} +{ U_s} + \widehat{U}} \widehat{U}= - L_{\widehat{U_{\mathsf {g}}} +{ U_s} +\widehat{U}} ( \widehat{U_{\mathsf {g}}} +{ U_s} ) , \quad (x,y)\in \mathcal {Q}=(0, 1)_x\times \mathbb {R}_y, \\ \widehat{U}|_{t=0}= \widehat{{U}_0} - \widehat{{U}_{\mathsf {g}}}, \\ b(\widehat{U_{\mathsf {g}}} +{ U_s} +\widehat{U})=\widehat{\Pi }.\\ \end{array}\right. } \end{aligned}$$
(3.33)

We are going to apply [3, Theorem 11.1] to the IBVP (3.33) and the use of [3] is legitimate since the domain \((0,1)_x\times \mathbb {R}_y\) is smooth. In order to exactly fit the statements in [3, Theorem 11.1], we search for a solution \(V:=\widehat{{U}_{\mathsf {g}}} +\widehat{U}\) satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} L_{ V +{ U_s} } V= - L_{ V+{ U_s} } { U_s} , \quad (x,y)\in \mathcal {Q}=(0, 1)_x\times \mathbb {R}_y, \\ V|_{t=0}= \widehat{{U}_0}, \\ b( V +{ U_s} )=\widehat{\Pi }.\\ \end{array}\right. } \end{aligned}$$
(3.34)

The corresponding functions h, b, and \(\underline{\mathrm{b}}\) in [3, Theorem 11.1] are as follows:

$$\begin{aligned} h(V) := -\ell (V) - L_{ V+{ U_s} } { U_s}, \qquad b(V):= b( V +{ U_s} ), \qquad \underline{\mathrm{b}}=\widehat{\Pi }, \end{aligned}$$
(3.35)

and we choose \(\mathscr {U}\) to be the open ball in the space \(H^m(\mathcal {Q}\times (0, T))\) with radius \(\delta /(4\nu _m)\), where \(\nu _m\) denotes the norm of the Sobolev embedding \(H^m(\mathcal {Q}\times (0, T))\hookrightarrow L^\infty (\mathcal {Q}\times (0, T))\). Hence, if \(V\in \mathscr {U}\), then

$$\begin{aligned} \Vert V \Vert _{L^\infty (\mathcal {Q}\times (0, T))}\le \delta /4, \end{aligned}$$

and then \({U_s}+V\) satisfies the mixed hyperbolic condition (1.5). Therefore, it is not hard to verify that the conditions (CH), (T), (NC \(_b\)), (N \(_b\)), (UKL \(_b\)) in [3, pp. 317–319] hold in this context. The compatibility conditions hold true for the extended data \(\widehat{U_0}\) and \(\widehat{\Pi }\) by our construction. We apply [3, Theorem 11.1] with \(\Omega =\mathcal {Q}\), and the condition \(h(0)=0\) follows from (3.7) and the condition \(b(0)\equiv \underline{\mathrm{b}}(\cdot , t=0)\) follows from (3.13) and the construction of \(\widehat{\Pi }\); we arrive at the local well-posedness of the system (3.33) if the initial data \(\widehat{U_0}\in \mathscr {U}\) belongs to \(H^{m+1/2}(\mathcal {Q})\).

Once we have a unique solution V for the system (3.34) and hence a unique solution \(\widehat{U} = V-\widehat{U_{\mathsf {g}}} \) for the extension IBVP (3.33), we now set \(U=\widehat{U}|_{\Omega \times (0, T)}\). Then U satisfies the IBVP (3.28) except that we need to verify that \(U|_{y=0}=0\). In order to show \(U|_{y=0}=0\), restricting the extension IBVP (3.33) to \(y=0\) yields

$$\begin{aligned} {\left\{ \begin{array}{ll} L_{\widehat{U_{\mathsf {g}}} +{ U_s} + \widehat{U}|_{y=0}} \widehat{U}|_{y=0}= - L_{\widehat{{U}_{\mathsf {g}}} +{ U_s} +\widehat{U}|_{y=0} } ( \widehat{{U}_{\mathsf {g}}} +{ U_s} ),\quad &{}\text { on }y=0, \\ \widehat{U}|_{t=0,y=0}= (\widehat{{U}_0} - \widehat{U_{\mathsf {g}}})|_{y=0},&{} \\ b(\widehat{U_{\mathsf {g}}} +{ U_s} +\widehat{U}|_{y=0})= \widehat{\Pi },\quad &{}\text { on }y=0.\\ \end{array}\right. } \end{aligned}$$
(3.36)

We observe from (3.22) and (3.27) that \(V=\widehat{U_{\mathsf {g}}}|_{y=0}=\mathsf {G}\) is a solution of the following system

$$\begin{aligned} {\left\{ \begin{array}{ll} L_{ V +{ U_s} |_{y=0} } V= - L_{ V+{ U_s}|_{y=0} } { U_s}|_{y=0} , \quad x\in (0, 1)_x, \\ V|_{t=0}= \widehat{U_0}|_{y=0} = U_0|_{y=0}, \\ b( V +{ U_s}|_{y=0} )=\widehat{\Pi }|_{y=0} = \Pi |_{y=0},\\ \end{array}\right. } \end{aligned}$$
(3.37)

and the uniqueness result in [3, Theorem 11.1] implies that \(V=\widehat{U_{\mathsf {g}}}|_{y=0}\) is the unique solution of (3.37), and hence we can conclude from (3.36) that \(\widehat{U}|_{y=0} = 0\). Therefore, \(U=\widehat{U}|_{\Omega \times (0, T)}\) satisfies the IBVP (3.28) and consequently, the system (3.10)–(3.11) admits a solution \(U_{\mathsf {g}} + U\).

3.5 The Main Result

We now conclude by stating the main result proved in the previous subsections.

Theorem 3.1

We are given a rectangular domain \(\Omega =(0,1)_x\times (0,1)_y\), a real number \(T>0\), an integer \(m\ge 3\) Footnote 1, the stationary solution \(U_s\in H^{m+1}(\Omega )\) satisfying (3.8) (i.e. the mixed hyperbolic condition (1.5)), the initial data \(U_0=(u_0,v_0,\phi _0)\) belonging to \(H^{m+1/2}(\Omega )\), the boundary data \(\mathsf {G}=(g_1, g_2, g_3)\) belonging to \(H^{m+1/2}((0,1)_x\times (0, T))\) and \(\Pi =(\pi _1,\pi _2,\pi _3)\) belonging to \(H^{m+1/2}((0,1)_y\times (0,T))\). We assume the condition (3.13) and the compatibility conditions (3.15)–(3.16) and (3.22)–(3.23), which are necessary to obtain a smooth solution in \(H^m(\Omega \times (0, T))\). We also assume that the initial data \(U_0\) is small enough in the space \(H^{m}(\Omega )\) so that the extended function \(\widehat{U_0}\) in Sect. 3.4 belongs to the ball \(\mathscr {U}\). Then there exists \(T^*>0\) ( \(T^*\le T\)) such that the system (3.10)–(3.11) admits a unique solution \(U\in H^m(\Omega \times (0, T^*))\).

Proof

The existence part is already proved in the previous subsections by considering the extension problem. We are now going to prove the uniqueness part. Suppose there are two solutions \(U_1\) and \(U_2\) belonging to \(H^m(\Omega \times (0, T^*))\) that satisfy the system (3.10)–(3.11), and set \(W=U_1 - U_2\). Then W satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} L_{U_s + U_1} W = ( L_{U_s + U_2} - L_{U_s + U_1}) (U_2 + U_s),\\ W = 0,\quad \text { on }\; t=0,\\ W = 0,\quad \text { on }\; y=0,\\ {\mathrm {d}} b(U_s+U_1)\cdot W=b(U_s + U_1)-b(U_s+U_2) -{\mathrm {d}} b(U_s+U_1)\cdot W,\quad \text { on }\; x=0,1.\\ \end{array}\right. } \end{aligned}$$
(3.38)

In order to obtain the \(L^2\)-estimate for W, as in [10], we set \(S_0={\mathrm {diag}}(1,1, g/(\phi _s+\phi _1)\), which is positive-definite; we denote by \(\langle \cdot , \cdot \rangle \) the \(L^2\)-inner product in \(L^2(\Omega )\). Multiplying (3.38)\(_1\) with \(S_0\) and taking the inner product in \(L^2(\Omega )\) with W, we obtain that

$$\begin{aligned} \begin{aligned} \langle S_0W_t, W \rangle&+ \langle S_0\mathcal {E}_1(U_s+U_1) W_x, W \rangle + \langle S_0\mathcal {E}_2(U_s+U_1) W_y, W \rangle + \langle S_0\ell (W), W \rangle \\&\qquad =\langle ( L_{U_s + U_2} - L_{U_s + U_1}) (U_2 + U_s) , W \rangle . \end{aligned} \end{aligned}$$
(3.39)

We are now going to estimate the terms in (3.39). Direct calculation and integration by parts give

$$\begin{aligned} \begin{aligned}&\langle S_0\ell (W), W \rangle = 0, \\&\langle S_0W_t, W \rangle = \frac{1}{2}\frac{{\mathrm {d}}}{{\mathrm {d}t}}\langle S_0W, W \rangle - \frac{1}{2}\langle (S_0)_tW, W \rangle . \end{aligned} \end{aligned}$$
(3.40)

Note that \(U_1\) satisfies the mixed hyperbolic condition (1.5), hence, the matrix \(S_0\mathcal {E}_2(U_s+U_1)\) is positive definite. Using the boundary conditions at \(y=0\) and integrating by parts yield

$$\begin{aligned} \langle S_0\mathcal {E}_2(U_s+U_1) W_y, W \rangle= & {} \frac{1}{2}\langle S_0\mathcal {E}_2(U_s+U_1) W, W \rangle _{L^2((0,1)_x)}\big |_{y=1}\nonumber \\&- \frac{1}{2}\langle (S_0\mathcal {E}_2(U_s+U_1))_y W, W \rangle \nonumber \\&\ge -\frac{1}{2}\langle (S_0\mathcal {E}_2(U_s+U_1))_y W, W \rangle ; \end{aligned}$$
(3.41)

integrating by parts also yields

$$\begin{aligned} \langle S_0\mathcal {E}_1(U_s+U_1) W_x, W \rangle= & {} \frac{1}{2}\langle S_0\mathcal {E}_1(U_s+U_1) W, W \rangle _{L^2((0,1)_y)}\big |_{x=0}^{x=1}\nonumber \\&- \frac{1}{2}\langle (S_0\mathcal {E}_1(U_s+U_1))_x W, W \rangle . \end{aligned}$$
(3.42)

We now recall relation (7) in [10, Sect. 2.1], which will be useful for handling the boundary terms at \(x=0\) and \(x=1\) in (3.42). The relation (7) in [10, Sect. 2.1] can be restated as the following: there exists \(\epsilon _0>0\) and \(C_0>0\) such that

$$\begin{aligned} \big (S_0\mathcal {E}_1(U_s+U_1) W, W\big )\big |_{x=0}^{x=1} \ge \epsilon _0|W |^2 - C_0|{\mathrm {d}} b(U_s+U_1) W |^2,\,\forall \, W\in \mathbb {R}^3, \end{aligned}$$
(3.43)

where \((\cdot ,\cdot )\) (resp. \(|\cdot |\)) denotes the standard inner product (resp. norm) on \(\mathbb {R}^3\). Taking (3.43) into account, we deduce from (3.42) that

$$\begin{aligned} \begin{aligned} \langle S_0\mathcal {E}_1(U_s+U_1) W_x, W \rangle&\ge \epsilon _0\Vert W \Vert _{L^2( (0,1)_y) }^2 - C_0\Vert {\mathrm {d}} b(U_s+U_1)W \Vert _{L^2( (0,1)_y) }^2\\&\quad - \frac{1}{2}\langle (S_0\mathcal {E}_1(U_s+U_1))_x W, W \rangle , \\ \end{aligned} \end{aligned}$$
(3.44)

where \(\epsilon _0, C_0>0\) only depend on \(c_0, c_1, c_2,g\).

Combining these estimates, we first obtain from (3.39) that

$$\begin{aligned} \begin{aligned} \frac{1}{2}\frac{{\mathrm {d}}}{{\mathrm {d}t}}&\langle S_0W, W \rangle + \epsilon _0\Vert W \Vert _{L^2( (0,1)_y) }^2 \\ \le \,&\frac{1}{2}\langle (S_0)_tW, W \rangle +\frac{1}{2}\langle (S_0\mathcal {E}_2(U_s+U_1))_y W, W \rangle \\&+\frac{1}{2}\langle (S_0\mathcal {E}_1(U_s+U_1))_x W, W \rangle +\langle ( L_{U_s + U_2} - L_{U_s + U_1}) (U_2 + U_s) , W \rangle \\&+C_0\Vert {\mathrm {d}} b(U_s+U_1)W \Vert _{L^2( (0,1)_y) }^2 . \end{aligned} \end{aligned}$$
(3.45)

As a preliminary, since \(\phi _1\) satisfies the mixed hyperbolic condition (1.5), we first have

$$\begin{aligned} \langle W, W \rangle = \langle S_0^{-1} S_0W, W \rangle \le \Vert S_0^{-1} \Vert _{L^\infty } \langle S_0W, W \rangle \le \max (1, g/c_0 )\langle S_0W, W \rangle . \end{aligned}$$

We now estimate the right-hand side of (3.45) term by term.

$$\begin{aligned} \begin{aligned} \langle (S_0)_tW, W \rangle&=\langle (S_0)_tW, W \rangle \le \Vert (S_0)_t \Vert _{L^\infty } \langle W, W \rangle \\&\le C( \Vert (U_s+U_1)_t \Vert _{L^\infty } ) \langle W, W \rangle \le C( \Vert U_s \Vert _{H^3}, \Vert U_1 \Vert _{H^3}) \langle S_0W, W \rangle , \end{aligned} \end{aligned}$$
(3.46)

where we have used the Sobolev embedding theorem and similarly, we have

$$\begin{aligned} \begin{aligned} \langle (S_0\mathcal {E}_2(U_s+U_1))_y W, W \rangle \le C( \Vert U_s \Vert _{H^3}, \Vert U_1 \Vert _{H^3} )\langle S_0W, W \rangle , \\ \langle (S_0\mathcal {E}_1(U_s+U_1))_x W, W \rangle \le C( \Vert U_s \Vert _{H^3}, \Vert U_1 \Vert _{H^3} )\langle S_0W, W \rangle , \end{aligned} \end{aligned}$$
(3.47)

where \(C( \cdots ) >0 \) only depends on the parameters in its parenthesis and may vary from line to line. By the mean-value theorem, we obtain \(c' > 0\), depending only on the \(L^\infty \)-norms of \(U_s+U_1\) and \(U_s+U_2\) such that

$$\begin{aligned} \begin{aligned}&\langle ( L_{U_s + U_2} - L_{U_s + U_1}) (U_2 + U_s) , W \rangle \\&\quad =\langle (\mathcal {E}_1(U_s+U_2) - \mathcal {E}_1(U_s+U_1)) (U_2+U_s)_x , W \rangle \\&\qquad +\langle (\mathcal {E}_2(U_s+U_2) - \mathcal {E}_2(U_s+U_1)) (U_2+U_s)_y , W \rangle \\&\quad \le c' ( \Vert (U_2+U_s)_x \Vert _{L^\infty } + \Vert (U_2+U_s)_y \Vert _{L^\infty } ) \langle W, W \rangle \\&\quad \le C( \Vert U_s \Vert _{H^3}, \Vert U_1 \Vert _{H^3}, \Vert U_2 \Vert _{H^3} ) \langle S_0W, W \rangle . \end{aligned} \end{aligned}$$
(3.48)

By the second-order Taylor expansion of b, we obtain that

$$\begin{aligned} \Vert {\mathrm {d}} b(U_s+U_1)W \Vert _{L^2( (0,1)_y) }^2= & {} \Vert b(U_s + U_1)-b(U_s+U_2) -{\mathrm {d}} b(U_s+U_1)\cdot W \Vert _{L^2( (0,1)_y) }^2\nonumber \\&\le C( \Vert U_s \Vert _{L^\infty }, \Vert U_1 \Vert _{L^\infty }, \Vert U_2 \Vert _{L^\infty } ) \Vert W^2 \Vert _{L^2( (0,1)_y) }^2\nonumber \\&\le C(\Vert U_s \Vert _{H^3}, \Vert U_1 \Vert _{H^3}, \Vert U_2 \Vert _{H^3} ) \Vert W \Vert _{L^\infty ((0,1)_y)}^2 \Vert W \Vert _{L^2( (0,1)_y) }^2;\nonumber \\ \end{aligned}$$
(3.49)

Using the Cauchy-Schwarz inequality and noting that \(W=0\) at \(t=0\), we find

$$\begin{aligned} \Vert W \Vert _{L^\infty ((0,1)_y)} \le \Vert W \Vert _{L^\infty }^2 \le T^2\Vert W_t \Vert ^2\le T^2 ( \Vert U_1 \Vert _{H^1} + \Vert U_2 \Vert _{H^1} )^2. \end{aligned}$$

Therefore, upon diminishing T again such that

$$\begin{aligned} T^2 C(\Vert U_s \Vert _{H^3}, \Vert U_1 \Vert _{H^3}, \Vert U_2 \Vert _{H^3} )( \Vert U_1 \Vert _{H^1} + \Vert U_2 \Vert _{H^1} )^2 \le \epsilon _0, \end{aligned}$$

the boundary term in the right-hand side of (3.45) is less than the left-hand side of (3.45), and hence we can derive from (3.45) the following differential equation

$$\begin{aligned} \frac{{\mathrm {d}}}{{\mathrm {d}t}}\langle S_0W, W \rangle \le C(\Vert U_s \Vert _{H^3}, \Vert U_1 \Vert _{H^3}, \Vert U_2 \Vert _{H^3} ) \langle S_0W, W \rangle . \end{aligned}$$

Since the solutions \(U_1, U_2\) and the stationary solution \(U_s\) belong to \(H^3\), and together with the initial condition \(W=0\) at \(t=0\), the Gronwall lemma implies that \(\langle S_0W, W \rangle \equiv 0\) and hence \(W\equiv 0\). We thus completed the proof of Theorem 3.1. \(\square \)

3.6 An Example of the Compatibility Conditions

Since the compatibility conditions stated in Sect. 3.2 are very dense and technical, we now aim to present those compatibility conditions explicitly in the (least) case when \(m=3\) Footnote 2.

Recall that \(H(U+U_s)\), \(\widetilde{H}(U+U_s)\), and \(\widetilde{\mathcal {E}_1}(U+U_s)\) are already defined in Sect. 3.2. The compatibility conditions at \(t=0\) are (3.15) and (3.16), which can be written explicitly as

(3.50)

where \(V_1\) and \(V_2\) are defined by

$$\begin{aligned} \begin{aligned} V_1&= H( U_0 + U_s ) - \mathcal {E}_1(U_0 + U_s)U_{0,x} - \mathcal {E}_2(U_0 + U_s)U_{0,y} - \ell (U_0),\\ V_2&={\mathrm {d}} H(U_0 + U_s)\cdot V_1 - ({\mathrm {d}} \mathcal {E}_1(U_0 + U_s)\cdot V_1)U_{0,x} - ({\mathrm {d}} \mathcal {E}_2(U_0 + U_s)\cdot V_1)U_{0,y}\\&\quad -\mathcal {E}_1(U_0 + U_s)V_{1,x} - \mathcal {E}_2(U_0 + U_s)V_{1,y} - \ell (V_1). \end{aligned} \end{aligned}$$

The compatibility conditions at \(y=0\) are (3.22) and (3.23) and can also be written explicitly as

$$\begin{aligned} \Pi (0, t)&= b(U_s + \mathsf {G}),\quad U_0|_{y=0} = \mathsf {G}|_{t=0} ,\nonumber \\ \partial _y\Pi (0, t)&= {\mathrm {d}} b(U_s + \mathsf {G})\cdot (U_{s,y} + W_1), \quad \partial _y U_0|_{y=0} = W_1|_{t=0},\nonumber \\ \partial _{yy}\Pi (0,t)&= ({\mathrm {d}}^2 b(U_s + \mathsf {G})\cdot (U_{s,y} + W_1))(U_{s,y} + W_1) \nonumber \\&\quad + {\mathrm {d}} b(U_s + \mathsf {G})\cdot (U_{s,yy} + W_2), \quad \partial _{yy} U_0|_{y=0} = W_2|_{t=0}, \end{aligned}$$
(3.51)

where \(W_1\) and \(W_2\) are defined by

$$\begin{aligned} \begin{aligned} W_1&= \widetilde{H}(\mathsf {G}+U_s) - \widetilde{\mathcal {E}}_1(\mathsf {G}+U_s)\mathsf {G}_x - \mathcal {E}_2(\mathsf {G}+U_s)^{-1} (\mathsf {G}_t + \ell (\mathsf {G})),\\ W_2&={\mathrm {d}} \widetilde{H}(\mathsf {G}+U_s)\cdot ( W_1 + U_{s,y}) - ( {\mathrm {d}} \widetilde{\mathcal {E}}_1(\mathsf {G}+U_s) \cdot ( W_1 + U_{s,y} ))\mathsf {G}_x\\&\quad -({\mathrm {d}} ( \mathcal {E}_2(\mathsf {G}+U_s)^{-1})\cdot ( W_1 + U_{s,y} ))( \mathsf {G}_t +\ell (\mathsf {G}))\\&\quad -\widetilde{\mathcal {E}}_1(\mathsf {G}+U_s) W_{1,x} - \mathcal {E}_2(\mathsf {G}+U_s)^{-1} ( W_{1,t} + \ell ( W_1 ) ). \end{aligned} \end{aligned}$$

Note that written component by component, the first compatibility condition (3.50)\(_1\) is equivalent to (3.31).

4 An Invariance Property for the Shallow Water Equations and Application

In this section, the goal is to show that we are able to solve the IBVP for the 2d inviscid SWE under the fully hyperbolic condition (1.3) for a more general orientation of the rectangular domain \(\Omega \) as long as we choose the rectangular domain properly. In order to achieve this goal, we first prove an invariance property for the fluid equations (in particular for the 2d inviscid SWE) and then show how to choose the domain. The results of Sect. 4.1 are essentially well-known but necessary to classify the notations.

4.1 An Invariance Property for the Fluid Equations

The partial differential equations arising from geophysical fluid dynamics are generally derived from physical laws, in particular, the conservation of mass and conservation of momentum. A basic principle in physics is that the physical laws should be independent of the reference frame chosen. Hence, we expect that the fluid equations are also independent of the coordinate system chosen and we call that the invariance property for the fluid equations. We first prove the invariance property for the 2d inviscid SWE and then extend it to more general fluid equations. We recall the 2d inviscid SWE (1.1) as

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t+uu_x + vu_y + g\phi _x -fv= 0, \\ v_t+uv_x + vv_y + g\phi _y + fu= 0. \\ \phi _t+u\phi _x + v\phi _y + \phi (u_x+v_y) = 0.\\ \end{array}\right. } \end{aligned}$$

For the purpose of unifying notations in this section, we set \({\varvec{u}}=(u, v)^t\) and \(\varvec{x}=(x_1, x_2)=(x,y)\); then we can rewrite the 2d inviscid SWE as

$$\begin{aligned} {\left\{ \begin{array}{ll} \phi _t + ({\varvec{u}}\cdot \nabla )\phi + \phi \nabla \cdot {\varvec{u}}=0,\\ {\varvec{u}}_t + ({\varvec{u}}\cdot \nabla ){\varvec{u}} + g\nabla \phi + \mathcal {F} {\varvec{u}}=0,\\ \end{array}\right. } \end{aligned}$$
(4.1)

where

$$\begin{aligned} \mathcal {F} = \begin{pmatrix} 0&{}-f\\ f&{}0 \end{pmatrix}. \end{aligned}$$

We are going to show that the equations in (4.1) are invariant under a coordinate transformation and a variable change of the velocity (adapted to the coordinate transformation). We adopt the convention that the vectors in \(\mathbb {R}^2\) are viewed as column vectors and the dot product on \(\mathbb {R}^2\) are defined as

$$\begin{aligned} \varvec{y}\cdot \varvec{z}=\varvec{y}^t\varvec{z},\qquad \forall \,\varvec{y},\varvec{z}\in \mathbb {R}^2. \end{aligned}$$

Let T be a \(2\times 2\) orthogonal matrix, that is \(T^tT=I_2\), where \(I_2\) is the \(2\times 2\) identity matrix. Since T is orthogonal, we have

$$\begin{aligned} T\varvec{y}\cdot T\varvec{z} = T^t\varvec{y}\cdot T^t\varvec{z}= \varvec{y}\cdot \varvec{z},\qquad \forall \,\varvec{y},\varvec{z}\in \mathbb {R}^2. \end{aligned}$$
(4.2)

We now introduce the new coordinate system \(\varvec{x}'\) by setting

$$\begin{aligned} \varvec{x}' = T\varvec{x},\qquad \varvec{x}=T^t\varvec{x}', \end{aligned}$$

and the new variables \({\varvec{u}}'\) (adapted to the coordinate transformation) by setting

$$\begin{aligned} {\varvec{u}}' = T{\varvec{u}},\qquad {\varvec{u}}=T^t{\varvec{u}}'. \end{aligned}$$
(4.3)

Writing the gradient \(\nabla \) as a column vector \(\nabla =(\partial _{x_1}, \partial _{x_2})^t\), direct computations show that

$$\begin{aligned} \begin{aligned} \nabla '&= T\nabla ,\qquad \nabla =T^t\nabla ', \\ \Delta '&=\nabla '\cdot \nabla '=T\nabla \cdot T\nabla =\nabla \cdot \nabla =\Delta , \end{aligned} \end{aligned}$$
(4.4)

where \(\nabla '\) and \(\Delta '\) are the gradient and laplacian in the new coordinate system \(\varvec{x}'\).

The invariance property for the 2d inviscid SWE (4.1) reads

Proposition 4.1

In the new coordinate system \(\varvec{x}'\), the variables \(({\varvec{u}}', \phi ')\) defined by \({\varvec{u}}'={\varvec{u}}'(x') = T{\varvec{u}}(x')\) and \(\phi '=\phi (x')\) satisfy the same set of equations (4.1) as \(({\varvec{u}}, \phi )\), that is

$$\begin{aligned} {\left\{ \begin{array}{ll} \phi _t' + ({\varvec{u}}'\cdot \nabla ')\phi ' + \phi '\nabla '\cdot {\varvec{u}}'=0,\\ {\varvec{u}}_t' + ({\varvec{u}}'\cdot \nabla '){\varvec{u}}' + g\nabla '\phi '+ \mathcal {F} {\varvec{u}}'=0.\\ \end{array}\right. } \end{aligned}$$
(4.5)

Proof

We first show that \(({\varvec{u}}', \phi ')\) satisfies the first equation (4.5)\(_1\). Using (4.2)–(4.4), we compute

$$\begin{aligned} ({\varvec{u}}'\cdot \nabla ' )\phi ' = (T{\varvec{u}}\cdot T\nabla ) \phi ' = ({\varvec{u}}\cdot \nabla )\phi , \end{aligned}$$

and

$$\begin{aligned} \nabla '\cdot {\varvec{u}}' = T\nabla \cdot T{\varvec{u}} = \nabla \cdot {\varvec{u}}, \end{aligned}$$

which, together with (4.1)\(_1\), implies that in the new coordinate system \(\varvec{x}'\), the new variables \(({\varvec{u}}', \phi ')\) satisfy (4.5)\(_1\).

For the second equation (4.5)\(_2\), in the new variables \(({\varvec{u}}', \phi ')\), we infer from (4.1)\(_2\) that

$$\begin{aligned} T^t\partial _t{\varvec{u}}' +(T^t{\varvec{u}}'\cdot \nabla )T^t{\varvec{u}}'+ g \nabla \phi '+ \mathcal {F} T^t{\varvec{u}}'=0, \end{aligned}$$

which, together with (4.4), reads in the new coordinate system \(\varvec{x}'\):

$$\begin{aligned} T^t\partial _t{\varvec{u}}' + (T^t{\varvec{u}}'\cdot T^t\nabla ') T^t{\varvec{u}}' + g T^t\nabla '\phi '+ \mathcal {F} T^t{\varvec{u}}'=0. \end{aligned}$$
(4.6)

Observe that any \(2\times 2\) orthogonal matrix is of the form

$$\begin{aligned} T=\begin{pmatrix} \beta &{}\quad -\alpha \\ \alpha &{}\quad \beta \\ \end{pmatrix},\quad \text { for some }\alpha ,\quad \beta \in \mathbb {R},\;\alpha ^2+\beta ^2=1, \end{aligned}$$

and direct calculations show the commutation relation \(\mathcal {F} T^t = T^t\mathcal {F}\). We then can simplify (4.6) as

$$\begin{aligned} T^t\partial _t{\varvec{u}}' + T^t ({\varvec{u}}'\cdot \nabla ') {\varvec{u}}' + g T^t\nabla '\phi ' + T^t \mathcal {F} {\varvec{u}}'=0. \end{aligned}$$

which, multiplying by T on both sides, is (4.5)\(_2\). We thus completed the proof. \(\square \)

We now extend Proposition 4.1 to more general fluid equations, which read

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \rho +{\varvec{u}}\cdot \nabla \rho + \rho \nabla \cdot {\varvec{u}}=0,\\ \partial _t{\varvec{u}} -\mu \Delta {\varvec{u}}+({\varvec{u}}\cdot \nabla ){\varvec{u}}+\Phi (\rho )\nabla p=0, \end{array}\right. } \end{aligned}$$
(4.7)

where the gradient \(\nabla \) and Laplacian \(\Delta \) are with respect to \(\varvec{x}=(x_1,\cdots ,x_d)^t\in \mathbb {R}^d\), and \(\rho \in \mathbb {R}\) is the mass-like quantity (e.g. density), \({\varvec{u}}=(u_1,\cdots ,u_d)\in \mathbb {R}^d\) the velocity, \(p\in \mathbb {R}\) the pressure, \(\mu \) the viscosity, and \(\Phi (\rho )\in \mathbb {R}\) a scalar function of \(\rho \). The first equation in (4.7) generally comes from the conservation of mass and the second equation in (4.7) from the conservation of momentum. Again, the vectors in \(\mathbb {R}^d\) are viewed as column vectors and the dot product on \(\mathbb {R}^d\) are defined as

$$\begin{aligned} \varvec{y}\cdot \varvec{z}=\varvec{y}^t\varvec{z},\qquad \forall \,\varvec{y},\varvec{z}\in \mathbb {R}^d. \end{aligned}$$

Let T be a \(d\times d\) orthogonal matrix, that is \(T^tT=I_d\), where \(I_d\) is the \(d\times d\) identity matrix. Since T is orthogonal, we have

$$\begin{aligned} T\varvec{y}\cdot T\varvec{z} = T^t\varvec{y}\cdot T^t\varvec{z}= \varvec{y}\cdot \varvec{z},\qquad \forall \,\varvec{y},\varvec{z}\in \mathbb {R}^d. \end{aligned}$$
(4.8)

We now introduce the new coordinate system \(\varvec{x}'\) by setting

$$\begin{aligned} \varvec{x}' = T\varvec{x},\qquad \varvec{x}=T^t\varvec{x}', \end{aligned}$$

and the new variables \({\varvec{u}}'\) (adapted to the coordinate transformation) by setting

$$\begin{aligned} {\varvec{u}}' = T{\varvec{u}},\qquad {\varvec{u}}=T^t{\varvec{u}}'. \end{aligned}$$
(4.9)

Then the invariance property for the fluid equations (4.7) reads

Proposition 4.2

In the new coordinate system \(\varvec{x}'\), the variables \(({\varvec{u}}', \rho ')\) defined by \({\varvec{u}}'={\varvec{u}}'(x') = T{\varvec{u}}(x')\) and \(\rho '=\rho (x')\) satisfy the same set of equations (4.7) as \(({\varvec{u}}, \rho )\), that is

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \rho '+{\varvec{u}}'\cdot \nabla ' \rho ' + \rho '\nabla '\cdot {\varvec{u}}'=0,\\ \partial _t{\varvec{u}}' -\mu \Delta '{\varvec{u}}'+({\varvec{u}}'\cdot \nabla '){\varvec{u}}'+\Phi (\rho ')\nabla ' p=0. \end{array}\right. } \end{aligned}$$
(4.10)

The proof of Proposition 4.2 is similar to that of Proposition 4.1, we thus omit the details here.

We now consider some specific fluid equations, where the form of these equations is slightly different from (4.7).

Example 1—Navier–Stokes equations The famous (incompressible) Navier-Stokes equations read

$$\begin{aligned} {\left\{ \begin{array}{ll} {\varvec{u}}_t -\mu \Delta {\varvec{u}} + ({\varvec{u}}\cdot \nabla ){\varvec{u}}+\nabla p=0,\\ \nabla \cdot {\varvec{u}}=0, \end{array}\right. } \end{aligned}$$
(NSE)

where \({\varvec{u}}\) is the velocity, \(\mu \) the viscosity, and p is the pressure. We can infer from the proofs of Propositions 4.1–4.2, that the incompressible Navier-Stokes equations have the invariance property.

Example 2—Euler equations The motion of a compressible, inviscid fluid in the absence of heat convection is governed by the Euler equations:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \rho + {\varvec{u}}\cdot \nabla \rho + \rho \nabla \cdot {\varvec{u}} = 0,\\ \partial _t {\varvec{u}} + ({\varvec{u}}\cdot \nabla ){\varvec{u}} + \rho ^{-1}\nabla p=0,\\ \partial _t e + {\varvec{u}}\cdot \nabla e + \rho ^{-1}p\nabla \cdot {\varvec{u}} = 0. \end{array}\right. } \end{aligned}$$
(EE)

where \(\rho \) is the density, \({\varvec{u}}\) the velocity, e the internal energy, and p the pressure. The equation of state (pressure law) reads

$$\begin{aligned} p=p(\rho ,e). \end{aligned}$$

The last equation in (EE) representing energy conservation law is similar to the first equation in (EE) representing the mass conservation law so that from the proofs of Propositions 4.1–4.2, we can deduce that the Euler equations have the invariance property.

4.2 The Choice of the Domain

We return to the inviscid SWE and first introduce a notion to express the intrinsic structure of the 2d inviscid SWE and then show that the proper domain, which will lead to well-posedness result, is related to the intrinsic structure of the 2d inviscid SWE.

Definition 4.1

The 2d nonlinear inviscid SWE are said to be supercritical (resp. subcritical) in the direction \(\vec l=(\alpha , \beta )\) with \(\alpha ^2+\beta ^2=1\) (\(\alpha , \beta \) are constants) if the following holds

$$\begin{aligned} (u\alpha + v\beta )^2\; > (resp. <)\; g\phi . \end{aligned}$$
(4.11)

We observe that in the fully hyperbolic case, we are able to construct a supercritical direction for the 2d nonlinear inviscid SWE. We first rewrite the fully hyperbolic condition (1.3) as:

$$\begin{aligned} \Big ( u \cdot \frac{u}{ \sqrt{u^2+v^2} } + v \cdot \frac{ v }{ \sqrt{u^2+v^2} } \Big )^2 > g\phi , \end{aligned}$$

and since in this article we consider local smooth solutions, hence we could choose two constants \(\bar{u}, \bar{v}\) such that in a short time interval the differences \(| u - \bar{u} |\) and \(| v - \bar{v} |\) are sufficiently enough so that in the fully hyperbolic case, the 2d nonlinear inviscid SWE are supercritical in the direction \(\frac{(\bar{u}, \bar{v})}{\sqrt{\bar{u}^2+\bar{v}^2}}\). Therefore, without loss of generality, we can assume that

$$\begin{aligned} \text {The 2d nonlinear inviscid SWE is } supercritical \text { in the direction } \vec {l}, \end{aligned}$$
(4.12)

where the direction \(\vec l=(\alpha , \beta )\) with \(\alpha ^2+\beta ^2=1\). We then choose the domain \(\Omega \) to be a rectangle with one side parallel to the direction \(\vec l=(\alpha , \beta )\). As we already saw in Sect. 3, the assumption (4.12) enables us to extend the rectangular domain to a channel (smooth) domain, which allows us to apply the general results from [3] for the IBVP of the first-order hyperbolic equations in smooth domains. First, by the invariance property of the nonlinear SWE, we see that if we introduce the coordinate transformation

$$\begin{aligned} \begin{pmatrix} x' \\ y' \end{pmatrix} =T\begin{pmatrix} x \\ y \end{pmatrix}: =\begin{pmatrix} \beta &{}\quad -\alpha \\ \alpha &{}\quad \beta \\ \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}, \end{aligned}$$

and the corresponding variables change

$$\begin{aligned} \begin{pmatrix} u' \\ v' \end{pmatrix} =T\begin{pmatrix} u \\ v \end{pmatrix}: =\begin{pmatrix} \beta &{}\quad -\alpha \\ \alpha &{}\quad \beta \\ \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix}, \end{aligned}$$

then we know that in the new coordinate \((x',y')\) system, the variables \((u', v', \phi ')\) defined by \(u'=u'(x',y')\), \(v'=v'(x',y')\), and \(\phi '=\phi (x',y')\) satisfy the same set of equations (1.1) as \((u,v,\phi )\) and the new domain \(\Omega '\) denoting the image of \(\Omega \) under the coordinate transformation is a rectangular domain with one side parallel to the direction (0, 1). We now observe that the original assumption (4.12) becomes that the 2d inviscid SWE satisfied by \((u', v', \phi ')\) are supercritical in the direction (0, 1) in the new coordinate \((x', y')\) system, that is

$$\begin{aligned} v'^2 > g\phi '. \end{aligned}$$

Clearly, we have

$$\begin{aligned} u'^2 + v'^2 = (u', v')\cdot (u',v')^t = (u,v)T^tT(u,v)^t = (u,v)\cdot (u,v)^t = u^2 + v^2 > g\phi = g\phi '. \end{aligned}$$

Therefore, in the new coordinate system \((x',y')\), we are going back to the assumptions made in (1.3)–(1.4) for the 2d inviscid SWE satisfied by \((u',v',\phi ')\) and hence the local well-posedness result could be achieved.

Fig. 1
figure 1

The curvilinear polygonal domain

Full size image

Remark 4.1

In order to solve the IBVP associated to the SWE system (1.1), we need to properly choose the domain \(\Omega \) according to the intrinsic structure of the SWE system. Specifically, in the fully hyperbolic case, we know that the SWE system (1.1) must be supercritical in some direction (e.g. the direction \(\vec l=(0,1)\)) and we choose the domain \(\Omega \) to be a rectangle with one side parallel to the direction \(\vec l\). We remark that we could also choose a curvilinear polygonal domain as long as such a domain could be extended to a curvilinear channel (smooth) domain in the direction \(\vec l\) with periodicity. For example, Fig. 1 below provides a curvilinear polygonal domain that could be extended to a curvilinear channel (smooth) domain in the direction \(\vec l=(0,1)\). For the sake of simplicity, we consider the rectangular domain in this article in order to simplify the presentation.