On the Geophysical Green-Naghdi System

Abstract

In this paper, a modified Green-Naghdi system with the effect of the Coriolis force is derived, which is a model in the equatorial oceanography to describe the propagation of large amplitude surface waves. The effects of the Coriolis force caused by the Earth’s rotation and nonlinearities on local well-posedness and traveling wave solutions are then investigated. Employing Kato’s theory, the local well-posedness in Sobolev space $H^s$ with $s>\frac{5}{2}$ is established. Based on the qualitative method combined with the bifurcation method of dynamical systems, the classification of all traveling wave solutions, all possible phase portraits of bifurcations and exact traveling wave solutions to this system are obtained under various conditions about the parameters depending on the value of the rotation $\Omega$

Appendix

For the sake of completeness, we present Kato’s theorem in this section. We begin by fixing some notations. Let A be an operator. We denote D(A) the domain of the operator A, and $\Vert \cdot \Vert _{X}$ the norm of the Banach space X.

Consider the abstract quasilinear equation,

$$\begin{aligned} \left\{ \begin{array}{l} \frac{dv}{dt}+A(t,v)v=f(t,v),\; t\ge 0,\\ v(0)=v_0. \end{array}\right. \end{aligned}$$

(5.1)

Let X and Y be Hilbert spaces, such that Y is continuously and densely embedded in X, and let $Q:Y\rightarrow X$ be a topological isomorphism. Let L(Y, X) denote the space of all bounded linear operators from Y to X (L(X), if $X=Y$). Assume that

(i) For each $t \ge 0 ,A(t,y)\in L(Y,X)$ for $y \in X$ with

$$\begin{aligned} \Vert (A(t,y)-A(t,z))w\Vert _X \le \mu _1\Vert y-z\Vert _X\Vert w\Vert _Y,\;t\ge 0,y,z,w\in Y, \end{aligned}$$

and $A(t,y)\in G(X,1,\beta )$ (i.e., A(t, y) is quasi-m-accretive), uniformly on bounded sets in Y.

(ii) $QA(t,y)Q^{-1}=A(t,y)+B(t,y)$, where $B(t,y)\in L(X)$ is bounded for each $t\ge 0$, uniformly on bounded sets in Y. Moreover,

$$\begin{aligned} \Vert (B(t,y)-B(t,z))w\Vert _X \le \mu _2\Vert y-z\Vert _Y\Vert w\Vert _X,\;t\ge 0,y,z\;\in Y,w\;\in X. \end{aligned}$$

(iii) For each $y\in Y$, $t\mapsto f(t,y)$ is continuous on $[0,+\infty )$. For each $t\ge 0$, $f(t,y):Y \rightarrow Y$ and extends also to a map from X into X. f is uniformly bounded on bounded sets in Y, and

$$\begin{aligned} \Vert (f(t,y)-f(t,z))\Vert _Y \le \mu _3\Vert y-z\Vert _Y,\;t\ge 0,y,z\;\in Y, \end{aligned}$$

$$\begin{aligned} \displaystyle \Vert (f(t,y)-f(t,z))\Vert _X \le \mu _4\Vert y-z\Vert _X,\;t\ge 0,y,z,\;\in X. \end{aligned}$$

Here, $\mu _1$, $\mu _2$, $\mu _3$, and $\mu _4$ are constants depending only on max$\{\Vert y\Vert _Y,\Vert z\Vert _Y\}$.

Theorem 5.1

(Kato’s Theorem) (Kato 1975) If assumptions (i)–(iii) hold, given $v_0\in Y$, there is a maximal $T>0$ depending only on $\Vert v_0\Vert _Y$ and a unique solution v to Eq. (5.1), such that

$$\begin{aligned} v=v(.;v_0)\;\in \;C([0,T);Y)\cap C^1([0,T);X). \end{aligned}$$

Moreover, the map $v_0\mapsto v(.;v_0)$ is continuous from Y to $C([0,T);Y)\cap C^1([0,T);X)$.