Advances in Differential Equations

Local and global well-posedness for a class of nonlinear dispersive equations

Radu C. Cascaval

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Abstract

We study the local and global well-posedness of the initial-value problem for the class of nonlinear dispersive PDEs of the form \[ u_t - M u_x + F(u)_x = 0, \;\;\; t\in \mathbb{R}, \] where $u=u(x,t)$, $x\in \mathbb{R} $ or $x\in \mathbb{T} $. Here $M$ is a linear operator, given in the Fourier space by the multiplication operator: $\hat{Mv} ( \xi ) = |\xi |^{2\beta } \hat{v} (\xi ),$ $ \beta \ge \frac12 $ and $F$ is a nonlinear, (sufficiently) smooth function. This equation is a generalization of the Korteweg--de Vries (KdV) equation ($\beta = 1 $), the Benjamin--Ono (BO) equation ($\beta = \frac12$) and the fifth-order KdV equation ($\beta = 2$). The nonlinearity can be very general, but a certain growth condition must be imposed in order to obtain global results. Roughly speaking, we impose that $(F'(r))_{+}$ grows at most like $\left| r\right| ^p$ as $r\to\infty $, for some $p < 4\beta $. Global existence of solutions is, therefore, intimately related to the balance between the strength of the nonlinearity and the dispersion relation. The semigroup methods developed by Goldstein--Oharu--Takahashi are being successfully applied here. Most of the results are presented in the periodic case (i.e., $x\in\mathbb{T} $), but they are also valid in the real-line case (when $x\in \mathbb{R} )$.

Article information

Source
Adv. Differential Equations, Volume 9, Number 1-2 (2004), 85-132.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867969

Mathematical Reviews number (MathSciNet)
MR2099607

Zentralblatt MATH identifier
1103.35089

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 34G20: Nonlinear equations [See also 47Hxx, 47Jxx] 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 47H20: Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07] 47N20: Applications to differential and integral equations

Citation

Cascaval, Radu C. Local and global well-posedness for a class of nonlinear dispersive equations. Adv. Differential Equations 9 (2004), no. 1-2, 85--132. https://projecteuclid.org/euclid.ade/1355867969


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