Abstract
We prove that the KP-I initial-value problem, { ∂ t u + ∂ x 3 u − ∂ x − 1 ∂ y 2 u + ∂ x ( u 2 / 2 ) = 0 on R x , y 2 × R t , u ( x , y , 0 ) = ϕ ( x , y ) , is locally well-posed in the space, H 1 , 0 ( R 2 ) = { ϕ ∈ L 2 ( R 2 ) : ‖ ϕ ‖ H 1 , 0 ( R 2 ) ≈ ‖ ϕ ‖ L 2 + ‖ ∂ x ϕ ‖ L 2 < ∞ } .
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