| Abstract: |
| Considered here is a class of higher-order models for the unidirectional propagation of small amplitude long waves on the surface of an ideal fluid, namely,
$$ \begin{split}
\eta_t+\eta_x-\gamma_1\beta{\eta}_{xxt}+\gamma_2\beta\eta_{xxx}+\delta_1\beta^2{\eta}_{xxxxt}
+\delta_2\beta^2\eta_{xxxxx}
\
+ \frac34\alpha(\eta^2)_x+
\alpha\beta\Big(\gamma (\eta^2)_{xx}-\frac7{48}\eta_x^2\Big)_x-\frac18\alpha^2(\eta^3)_x=0.
\end{split}\quad \quad (1)$$
It was derived by Bona, Carvajal, Panthee and Scialom, where $\eta=\eta(x,t)$ is the deviation of the free surface from
its rest position at the point corresponding to $x$ at time $t$.
The two physical parameters
$\alpha, \beta >0$ are
small compared to one and the Stokes' number $S = \alpha/\beta$ is of order one.
The five parameters $\gamma_1, \gamma_2, \delta_1, \delta_2$
and $\gamma $ are not arbitrary.
The restrictions are spelled out in the details of the work by Bona and his collaborators just mentioned.
The pure initial-value problem is the problem (1) with initial data
$$
\eta(x, 0)=\eta_0(x), \quad x\in\mathbb R \qquad\qquad\qquad (2)
$$
viewed as known. Assuming that
$\delta_1$ appearing in front of the $\eta_{xxxxt}$-term is positive and $\gamma_1$ is not too negative, and
if the initial data
is selected from the $L_2$-based Sobolev space $H^s(\mathbb R)$ for $s\geq 1$, then the initial-value problem (1)-(2) is locally well-posed in $H^s(\mathbb R)$. If, in addition,
$\gamma=\frac{7}{48}$, the well-posedness is global. This is a published result.
In my talk, I will address the initial-value
problem when the initial data is not a localized function, say bore-like initial data. In such a case, conservation laws do not work directly. |
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