Higher-order dispersive systems: Difference between revisions
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One can study more general dispersive equations of the general form | One can study more general dispersive equations of the general form | ||
u_t + P(nabla) u = F(u, nabla u, | :<math>u_t + P(\nabla) u = F(u, \nabla u, \dots)</math> | ||
where P(nabla) is an anti-selfadjoint constant coefficient operator, and F involves fewer derivatives on u than P and contains only quadratic and higher terms, and may possibly contain non-local operations such as Hilbert transforms, Hartree-type potentials, or Riesz transforms. Such equations arise as various approximations to wave equations, see e.g. [Dy1979], [Hog1985]. Smoothing effects for the linear part of the equation were established in [BenKocSau2003], [Hs-p]. Nonlinear local existence in the analytic category was established in [Bd1993]. For smooth but not analytic data some local existence results have been established in [Tar1995], [Tar1997], [Ci-p]. | where P(nabla) is an anti-selfadjoint constant coefficient operator, and F involves fewer derivatives on u than P and contains only quadratic and higher terms, and may possibly contain non-local operations such as Hilbert transforms, Hartree-type potentials, or Riesz transforms. Such equations arise as various approximations to wave equations, see e.g. [Dy1979], [Hog1985]. Smoothing effects for the linear part of the equation were established in [BenKocSau2003], [Hs-p]. Nonlinear local existence in the analytic category was established in [Bd1993]. For smooth but not analytic data some local existence results have been established in [Tar1995], [Tar1997], [Ci-p]. | ||
[[Category:Equations]] | |||
Revision as of 06:02, 27 July 2006
One can study more general dispersive equations of the general form
where P(nabla) is an anti-selfadjoint constant coefficient operator, and F involves fewer derivatives on u than P and contains only quadratic and higher terms, and may possibly contain non-local operations such as Hilbert transforms, Hartree-type potentials, or Riesz transforms. Such equations arise as various approximations to wave equations, see e.g. [Dy1979], [Hog1985]. Smoothing effects for the linear part of the equation were established in [BenKocSau2003], [Hs-p]. Nonlinear local existence in the analytic category was established in [Bd1993]. For smooth but not analytic data some local existence results have been established in [Tar1995], [Tar1997], [Ci-p].
