On the Fully Non-Linear Cauchy Problem with Small Data. II.

Abstract

This paper is one of a series devoted to the Cauchy problem for an equation of the form

$$ \square u = G(u,u',u'') $$

((1.1))

with small Cauchy data

$$ u = \varepsilon {u_0},\quad {\partial_t}u = \varepsilon {u_1},\quad {\text{when}}\,{\text{t = 0}} $$

((1.2))

Here □ = ² _t − Δ is the wave operator in R ¹⁺ⁿ, with variables denoted by t = x ₀ and x = (x1,..., x _n ), G is a C ^∞ function vanishing of second order at the origin, u′ and u″ denote all first and second order derivates of u, and u _j ∈ C ^∞₀ . General results have been obtained with simple proofs based on the idea of Klainerman [13] to use enrgy integral estimates for all equations obtained from (1.1) by multiplication with any product Ẑ ^Iof │I│ vector fields ∂/∂x _j , j = 0, …, n, the infinitesimal generators of the Lorents group

$$ \begin{gathered} {Z_{{jk}}} = {x_k}\partial /\partial {x_j} - {x_j}\partial /\partial {x_k},\quad j,\,k = 1,...,n, \hfill \\ {Z_{{0k}}} = {x_0}\partial /\partial {x_k} + {x_k}\partial /\partial {x_0} = - {Z_{{k0}}},\quad k = 1,...,n, \hfill \\ \end{gathered} $$

((1.3))

which commute with □, and the radial vector field

$$ {Z_0} = \sum\limits_0^n {{x_j}\partial /\partial {x_j}} $$

((1.4))

(We shall use the notation Z ^I for products of the vector fields (1.3), (1.4) only; note that these preserve homogeneity.) However, in low dimension the results are not always optimal when G depends on u itself.