Use of a contraction mapping principle
Similarly to what is done for the study of diﬀerential equations, we begin the study of a dispersive equation by an integral formula for our equation: this formula is called a Duhamel formula.
If we integrate (2.1.2) between 0 and t, we get ˆ t f (t, x) = f (0, x) +e−isΔQ eisΔf (s, x), eisΔ f (s, x) ds, (2.2.1) 0 or, in terms of u, ˆ t u(t, x) = eitΔ f (0, x) + ei(t−s)Δ Q (u(s, x), u(s, x)) ds, (2.2.2) 0 Equation (2.2.1) is a Duhamel formula for (2.0.1): it allows us to rewrite (2.0.1) as a fixed point equation f = A(f ), (2.2.3) where A(f ) = f0 + A(f ), and ˆ t A (f )(t, x) := e−isΔQ eisΔf (s, x), eisΔf (s, x) ds.
We can similarly write it as u = A(u), with A(u) = eitΔ u0 + A(u). Solving the fixed point problem (2.2.3) can be done by proving that A (or A) is a contraction in a ball BS (0, ε) of a well-chosen space S i.e. ∀(f, g) ∈ BS (0, ε), A(f ) − A(g) S ≤ c f − g S , with c < 1.
A simple theorem: direct energy estimates
Here we give a first example of an existence theorem established with a contraction esti-mate, without any fine study of the structure of the equation.
Theorem 2.3.1. Let N > 1/2. There exists C > 0 such that for all ε > 0, for all u0 with u0 HN ≤ ε/2, there exists a solution u to (2.0.1) in the space L∞([0, T ), H N ) with T := 2C1ε, and u L∞ ([0,T ),HN ) ≤ ε. Moreover, u is unique in the ball of radius ε of H N . Remark 2.3.2. This result does not give a very long existence time, but it only needs the solution to be in H N .
Heuristics of dispersion
Physical observations show that in a large variety of frameworks (Schrödinger’s equation, Korteweg-de Vries’ equation, wave equation, etc.), a free wave will tend to spread out, leading to a L∞ decay in time. This phenomenon called dispersion occurs in so-called dispersive equations, in particular in equations of the form ∂tu + iL(D)u = 0, (2.4.1).
with D = i∂, L : R R and L(D) is the Fourier multiplier associated to L, i.e. for all function , F ˆ . To call this equation dispersive we moreover ask f (L(D)f )(ξ) = L(ξ)f (ξ) Hess(L) = 0.
Now we consider an initial data localized at a frequency ξ0, and we determine the velocity of the resulting solution. This solution is a wave packet, i.e. of the form, ˆ u(t, x) = A(ξ − ξ0)ei(x·ξ−ω(ξ)t) dξ, (2.4.2) R with A localizing around 0 and exponentially decreasing.
The space-time resonances method
The toy models studied in Sections 2.5.2.a and 2.5.2.b highlight the importance of the vanishing (or the non-vanishing) of the phase φ or its derivative ∂η φ. Let us develop mathematically what happens in those two diﬀerent cases. We are going to use the fol-lowing version of Duhamel’s formula (2.5.1): fˆ(t, ξ) = f0(ξ) + ˆ t ˆ e− f (η)f (ξ − η)dηds. ˆ isφ(ξ,η) ˆˆ 0 R Our aim is to be able to estimate this integral in a better way than in proofs of Theo-rems 2.3.1 and 2.4.4, by performing some transforms, depending on the behavior of φ and ∂η φ.
Table of contents :
1.1 Étude à données petites d’équations dispersives
1.2 Confinement d’ondes
1.3 Principaux résultats
1.4 Applications des résultats de la thèse – perspectives
1.5 Organisation de la thèse
2 Existence theorems for dispersive equations
2.1 The notion of profile
2.2 Use of a contraction mapping principle
2.3 A simple theorem: direct energy estimates
2.4 Use of the dispersive effect
2.5 The notion of space-time resonances
2.6 Other models of dispersive equations
2.7 Notations used throughout the manuscript
3 The quadratic Klein-Gordon equation R × T
3.2 Functional framework and main result
3.3 Duhamel formula
3.4 Study of the phase
3.5 Proof of the existence theorem
3.6 Conclusion and perspectives
4 The wave equation with a harmonic potential
4.3 High regularity results
4.4 General strategy for the HM x2H 3 2 x1(x1) norm
4.5 Estimates for high frequencies
4.6 Estimates for high Hermite modes
4.7 Estimates for low frequencies and low Hermite modes
5 Resonant system
5.2 Derivation of the resonant system
5.3 Long-time existence for the resonant system
5.4 Validity of the approximation
5.5 Perspective: understanding the dynamics of the resonant system
A Some harmonic analysis tools
A.1 Linear Fourier multiplier estimates
A.2 Behavior with dilation operators
A.3 Bilinear multiplier estimates
A.4 A L2 stationary phase lemma
A.5 Interaction between Hermite functions
B Study of the phase
B.1 Main results
B.2 Proof of the main results
C Paraproduct for the Hermite expansions
C.1 Statement of the theorem