Dispersive Smoothing Effects for KdV Type Equations

✍ Scribed by Hongsheng Cai

• Book ID: 102586969
• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 424 KB
• Volume: 136
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1996.3245

✦ Synopsis

In this paper we study the smoothness properties of solutions of some nonlinear equations of Korteweg de Vries (KdV) type, which are of the form

where x # R, u j = 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 solutions u(x, t) are obtained for t>0 if the initial data u(x, 0) decays faster than it does polynomially on R & 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.

for an integer m 0 and the solution obeys &u& Hs 0 < for an existence time 0<t<T, then u(x, t) # H m loc (R) for all 0<t T, and u(x, t) # 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).

1997 Academic Press t u=a(x, t) u 3 + f (u 2 , u 1 , u, x, t)

(1) u(x, 0)=.(x),