Smoothing properties and retarded estimates for some dispersive evolution equations. (English) Zbl 0762.35008

Ginibre, J.; Velo, G.

Commun. Math. Phys. 144, No. 1, 163-188 (1992).

Dispersive partial differential equations of the type \[ \partial_ tu- Lu=f \] with skew adjoint operator \(L\) acting in some Hilbert space, for instance, \(L^ 2(\mathbb{R}^ n)\), are considered. The Cauchy problem with initial data \(u(t=0)=u_ 0\) is formally solved by \[ u(t)=U(t)u_ 0+\int^ t_ 0d\tau U(t-\tau)f(\tau), \] therefore an essential role in the treatment of the problem is played by the properties of the operators \[ u_ 0\to U(t)u_ 0,\quad f\to\int^ t_ 0d\tau U(t-\tau)f(\tau). \] “Smoothing” means that less derivatives occur on \(u_ 0\) and \(f\) than on their images. “Retardation” refers to the specific form of the second operator.
In this paper, the mentioned operators are treated in terms of abstract Banach space theory. The obtained estimates are applied to the Cauchy problem for the generalized Benjamin-Ono equation that corresponds to \(L=i\Phi(-i\partial/\partial x)\), \(f=(\partial/\partial x)v'(u)\), \(\Phi(\xi)=\xi|\xi|^{2\mu}\), \(v(0)=v'(0)=0\).

Reviewer: I.J.Dorfman (Moskva)

MSC:

35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35G10	Initial value problems for linear higher-order PDEs
35K25	Higher-order parabolic equations
35Q53	KdV equations (Korteweg-de Vries equations)

Keywords:

estimates for solution; space-time integrability properties; Cauchy problem; Benjamin-Ono equation

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