Dispersive smoothing effects for KdV type equations

[摘要] 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 (1) partial derivative(i)u=a(x,t)u(3)+f(u(2),u(1)u,x,t), where x is an element of R, u(j) = partial derivative(x)(j) 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-infinity 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. if (2) integral(R)u(2)(x,0)(1+\x(-)\(m))dx

[发布日期] 1997-05-20