Analytic smoothing effects for a class of dispersive equations

Hideki TAKUWA (Kyoto University) takuwa@amp.i.kyoto-u.ac.jp

1 Introduction and main result

Let m be an integer greater than or equal to 2. Let P (y, D_{y}) be an linear differential operator in ℝ,

P (y, D_{y}) = D_{y}^{m} + ∑_{0≤l≤m−1} c_{l}(y)D_{y}^{l}, (1)

where D_{y} = \frac{1}{i} \frac{∂}{∂y} and c_{l}(y) are analytic functions in ℝ.

We shall make the following assumption.

One can find positive constants C_{0}, R_{0}, K_{0} and σ_{0} ∈ (0, 1) such that for y ∈ ℝ, |y| > R_{0}, and k ∈ ℕ ∪ {0},

∑_{0≤l≤m−1} |D_{y}^{k}c_{l}(y)| ≤ C_{0} \frac{K_{0}^{k}k!}{|y|^{1+σ_{0}+k}} . (2)

In our case the principal symbol for P (y, D_{y}) is p(y, η) = p(η) = η^{m}. Let ρ = (y, η) ∈ T^{∗}ℝ\0, and let (Y (s; y, η), Θ(s; y, η)) be the solution to the equation,

\begin{cases}\frac{d}{ds}Y (s) & = \frac{∂p}{∂η}(Y (s), Θ(s)), Y (0) = y, \\ \frac{d}{ds}Θ(s) & = −\frac{∂p}{∂y}(Y (s), Θ(s)), Θ(0) = η. \end{cases} (3)

Therefore

\begin{cases}Y (s) & = y + msη^{m−1}, \\ Θ(s) & = Θ(0) = η. \end{cases} (4)

Let us introduce a space of the initial data,

Γ_{ρ_{0}} = {Y (s; y_{0}, η_{0}) ∈ ℝ, s ≥ 0}. (5)

X_{ρ_{0}} = \{v ∈ L^{2}(ℝ); ∃δ_{0} > 0, e^{δ_{0}|y|^{\frac{1}{m−1}}} v(y) ∈ L^{2}(Γ_{ρ_{0}})\}. (6)

For u_{0} ∈ L^{2}(ℝ) let u(t, ·) ∈ C(ℝ; L^{2}(ℝ)) be the solution of the initial value problem,

\{ \begin{matrix}D_{t}u + P (y, D_{y})u = 0, \\ u|_{t=0} = u_{0}(y). \end{matrix} (7)

Theorem 1 Let P (y, D_{y}) be defined in (1) satisfying (2)and ρ_{0} = (y_{0}, η_{0}) ∈ T^{∗}ℝ\0.

Let u_{0} ∈ L^{2}(ℝ) be in X_{ρ_{0}}. Then for all t < 0 ρ_{0} does not belong to the analytic wave front set W F_{A}[u(t, ·)] of the solution u(t, ·) for (7).

In fact we can extend this result in more general cases. The speaker will show the extension of this result in the talk.

Our approach is based on FBI transform. The speaker will explain the way to apply the theory of FBI transform into the study of the smoothing effects.

2 Analytic wave front set and FBI transform

Let ρ_{0} = (y_{0}, η_{0}) ∈ T^{∗}ℝ\0. Let φ(x, y) be a holomorphic function in a neighborhood U_{0} × V_{y_{0}} of (0, y_{0}) in ℂ × ℂ which satisfies

\frac{∂φ}{∂y}(0, y) = −η_{0}, (8)

Im \frac{∂^{2}φ}{∂y^{2}}(0, y) > 0, (9)

\frac{∂^{2}φ}{∂x∂y} (0, y) ≠ 0. (10)

For above φ(x, y) we can define,

Φ(x) = max_{y∈V_{y_{0}}}(− Im φ(x, y)), (11)

for x ∈ U_{0}.

Let a(x, y, λ) = ∑_{k≥0} a_{k}(x, y)λ^{−k} be a analytic symbol of order zero, elliptic in a neighborhood of (0, y_{0}). Let χ ∈ C_{0}^{∞} be a cutoff function with support in a neighberhood of y_{0}, 0 ≤ χ ≤ 1, and χ ≡ 1 near y_{0}.

The FBI transform of a distribution u ∈ D′(ℝ) is defined by

T u(x, λ) = ⟨χ(·)u, e^{iλφ(x,·)}a(x, ·, λ)⟩, λ > 1. (12)

Assume that u(t, ·) is a element of a family of distribution on ℝ depending of a real parameter t. Let t_{0} ∈ ℝ. We shall say that a point ρ_{0} ∈ T^{∗}ℝ\0 does not belong to the locally uniform analytic wave front set { \atop W F_{A}}[u(t_{0}, ·)] if there exist an FBI transform T, positive constants C, μ, λ_{0}, ε, and a neighberhood U_{0} of 0 such that

e^{−λΦ(x)}|T u(t, x, λ)| ≤ Ce^{−μλ}, for ∀x ∈ U_{0}, ∀λ ≥ λ_{0}, ∀t ∈ (t_{0} − ε, t_{0} + ε). (13)

References

[1] W. Craig, T. Kappeler and W. Strauss, Miclolocal dispersive smoothing for the Schrödinger equation, Comm. Pure. Appl. Math. 48, (1995), 769-860.

[2] L. Robbiano, and C. Zuily, Miclolocal analytic smoothing effect for the Schrödinger equation, Duke Math. Journal, 100, no.1, (1999), 93-129.

[3] J. Sjöstrand, Singularités analytiques microlocales, Astérisques, 95 (1982).