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Information Center for Mathematical Science

논문검색

Information Center for Mathematical Science

논문검색

Journal of Differential Equations
( Vol. 136 NO.2 / (1997))
Dispersive Smooting Effects for KdV Type Equations
Hongsheng Cai,
Pages. 191-221
Abstract In this paper we study the smoothness properties of solutions some
nonlinear equations of Korteweg-de Vries (KdV) type, which are of the form
$$partial_t u = a(x,t)u_3 + f(u_2, u_1, u, x, t), hspace{10mm} (1)$$
where $x in R, u_j = partial^j_x u$, and $k$ and $j$ are nonnegative
integers. Our principal condition is that $a(x,t)$ be positive and bounded, so
that the dispersion is dominant. It is shown under certain additional
conditions on $a$ and $f$ that $C^infty$ solutions $u(x,t)$ are obtained for
$t > 0$ if the initial data $u(x,0)$ decays faster than it does polynomially on
$R^{-1}$ and has certain initial Sobolev regularity.
A quantitative relationship between the rate of decay and the amount of gain of
smoothness is given. Let $s_0$ be the Sobolev index. If
$$int_R u^2(x,0)(1 + |x_{-}|^m)dx < infty, hspace{10mm}(2)$$
for an integer $m geqslant 0$ and the solution obeys $|u|_{H_{s_0}} < infty$
for an existence time $0 < t < T$, then $u(x,t) in H^m_{log}(R)$ for all $0 <
t leqslant T$, and $u(x,t) in L^1([0,T]) : H^{(m+1)}_[loc}(R))$. Our method
can also be extended to address the fully nonlinear dispersive equations
related to (1).
Contents 1. Introduction
2. The main identity
3. An important a priori estimate
4. Uniqueness theorem
5. Existence theorem
6. Presistence theorem
7. Gain of regularity
Key words
Mathmatical Subject Classification