Higher order expansion for the probabilistic local well-posedness theory for a cubic nonlinear Schrödinger equation
arxiv(2024)
摘要
In this paper, we study the probabilistic local well-posedness of the cubic
Schrödinger equation (cubic NLS):
(i∂_t + Δ) u = ± |u|^2
u on [0,T) ×ℝ^d,
with initial data being a
unit-scale Wiener randomization of a given function f. We prove that a
solution exists almost-surely locally in time provided f∈
H^S_x(ℝ^d) with S>max(d-3/4,d-4/2)
for d≥ 3. In particular, we establish that the local well-posedness holds
for any S>0 when d=3. We also show that, under appropriate smallness
conditions for the initial data, solutions are global in time and scatter. The
solutions are constructed as a sum of an explicit multilinear expansion of the
flow in terms of the random initial data and of an additional smoother
remainder term with deterministically subcritical regularity.
We develop the framework of directional space-time norms to control the
(probabilistic) multilinear expansion and the (deterministic) remainder term
and to obtain improved bilinear probabilistic-deterministic Strichartz
estimates.
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