From: Brenda Sue Cook (brenda@math.missouri.edu)
Date: Thu Feb 07 2008 - 16:12:03 CST
SPECIAL DIFFERENTIAL EQUATIONS SEMINAR
Department of Mathematics
Hongqiu Chen
(University of Memphis)
Sharp Results of Well-posedness
Abstract: Consider the initial-value problem
\begin{equation}\label{**} \left.\begin{split}
& u_t+u_x+g(u)_x+Lu_t=0, \qquad x\in\mathbb R,\quad t>0,
\\
&u(x,0)=u_0(x), \qquad x\in\mathbb R,
\end{split} \right\}\end{equation}
where $u=u(x,t)$ is a real-valued function, $L$ is a
Fourier multiplier operator with real symbol $\alpha(\xi),$
that is $\widehat{Lv}(\xi)=\alpha(\xi)\widehat{v}(\xi)$,
and $g$ is a smooth, real-valued function
of a real variable. Equations of this form arise as models of
wave propagation in a variety of physical contexts. Here,
fundamental issues of local and global well-posedness are
established for $L_p$, $H^s$ and bore-like or kink-like initial
data. In the special case where $\alpha(\xi)=|\xi|^{r}$
wherein $r>1$ and $g(u)=\frac12u^2,$
the initial value problem \eqref{**} is locally
well-posed in $H^s$ if $r$ and $s$ satisfy one of the following
three conditions:
(a) $r\geq 1$ %<r\leq \frac54$
and $s>\frac12;$
(b) $r>\frac54$ %<r\leq \frac32$
and $s>\frac14;$
(c) $r>\frac32 $ and $s\geq 0.$
In addition, if $r>1$ and $s\geq 1-\frac{r}2,$ then the well-
posedness
is global.
Tuesday, February 12
1:00 p.m.
208 Strickland (GCB)
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