Some kind of Besov spaces related to non-linear PDEs
by
Tosinobu Muramatu
Chuo University

Besov-type spaces related to linear parts of non-linear PDEs give good existence rusults for Cauchy problems.

The Besov spaces reported here are defined as follows: For an infinitely differentialble real-valued function P(\xi), a weight \rho on R₊, a real number b, 1 <= p <= \infty, and 1 <= q <= \infty the space B_{p, q, P}^{(\rho, b)}(R^d+1) is the space of tempered distributions f such that the norm

||f||B(\rho, b)p, q, P : = ||{\rho(2j)2bk||fjk, P(x, t)||Lp(Rd+1)}||lq

(\theequation)

is finite. We write the space by B_{p, q, P}^{(s, b)}(R^d+1) when \rho(t)=t^s. Here, [^f] denotes the Fourier transform of f,

^ f   jk, P  (\xi, \tau) = jj(|\xi|)jk(\tau-P(\xi)) ^ f   (\xi, \tau),

(\theequation)

and j_j(z), j=0, 1, ... , are infinitely differentiable functions of a real variable z having the following properties: j_j(z)=j_j(-z), suppj₀ subset {z;|z| < 2}, supp j₁ subset {z;1 < |z| < 4}, j_k(z)=j₁(2^-k+1z) ( for , k >= 1),   \sum_j=0^\inftyj_j(z)=1.

This norm is a Besov analogue of that given by

||f||Xs, b= æ è ó õ ó õ R2  (1+|\xi|)2s (1+|\tau-P(\xi)|)2b| ^ f   (\xi, \tau)|2d\xid\tau ö ø 1/2   ,

(\theequation)

which has been employed by Kenig-Ponce-Vega (J. A. M. S. 9, 1996, Trans. A. M. S. 348, 1996). They proved the existence of time-local solutions to the KdV equation

\labelkdvequation\partialtu+\partialx3u+\partialx(u2)=0,

(\theequation)

with u(x, 0)=u₀(x) in H^s(R), s > -3/4, by making use of the estimate

\labelestderfgcapx||\partialx(fg)||Xs, b-1 <= c||f||Xs, b||g||Xs, b with P=\xi3,

(\theequation)

which holds for any s > -3/4 and some b > 1/2, and fails for any s < -3/4 and any b in R. Nakanishi-Takaoka-Tsutsumi showed that () fails also when s=-3/4.

The corresponding estimates by means of our norm are the following inequalities:

||\partialx(fg)||B(\rho, -1/2)2, 1, \xi3 <= c||f||B(\rho, 1/2)2, 1, \xi3||g||B(\rho, 1/2)2, 1, \xi3,   if -3/4 <= s < 0,   ||\partialx(fg)||B(\rho, -1/2)2, 1, \xi3 <= c||f||B(s, b)2, 1, \xi3||g||B(s, 1/2)2, 1, \xi3, if -3/4 <= s < 0,  b > 1/2,

where \rho(t)=log(2+t)ts, which make it possible to improve Kenig-Ponce-Vega's result, i. e., it can be proved that there exists T=T(||u0||B2, 1-3/4(R)) and u(x, t) in B2, 1, \xi3(-3/4, 1/2)(R2) which satisfies the equation () when |t| <= T and the initial condition u(x, 0)=u0(x) in B2, 1-3/4(R).

Our method is very general. It gives not only an improvement of Kenig-Ponce-Vega's result for one-dimensional nonlinear Schrödinger equation with nonlinear terms of the form c₁u²+c₂u[`u]+c<sub>3</sub>[`u]², but it also is applicable to many kind of nonlinear PDEs.

Department of Mathematics, Chuo University

Date received: August 10, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahz-98.