Dipartimento di Matematica
Corso di Laurea Magistrale in Matematica
Tesi di Laurea Magistrale
The cubic Dirac equation with Hartree type
nonlinearity: global existence in H
12(R
2)
Candidato:
Relatore:
Boris Shakarov
Prof. Vladimir Georgiev
Contro - relatore:
Prof. Nicola Visciglia
Abstract
Local and global well - posedness for the Dirac equation with Hartree type nonlinearity with initial data in the critical space H12(R2) with finite L2(R2)
norm is established. The proof is based on Strichartz estimates, conservation laws, Coifman - Meyer theorem and Paley - Littlewood decomposition.
Contents
1 Introduction 3
2 The Dirac Equation 6
2.1 The Boson Star Equation . . . 9
2.2 Main Results . . . 10 3 Preliminaries 12 3.1 Harmonic Analysis . . . 12 3.2 Contraction Principle . . . 17 3.3 Strichartz Estimates . . . 19 3.4 Nonlinearity . . . 20 3.5 Functional Calculus . . . 24
3.6 Other Functional Spaces . . . 25
3.6.1 Hardy Spaces . . . 26
3.6.2 Besov and Triebel - Lizorkin Spaces . . . 27
4 Local Well - Posedness 29 4.1 Evaluating the Nonlinearity . . . 29
4.2 Local Existence . . . 32
4.3 Another Approach . . . 35
5 Global Existence 37 5.1 Conservation Laws . . . 37
5.2 Global Existence . . . 40
5.3 Proof of the Theorem 2.2.3 . . . 41
Chapter 1
Introduction
In this thesis we will speak about a particular dispersive partial differential equation. The term ”dispersion”, informally, refers to the fact that different frequencies of the solution to this equation travel at different velocities, thus dispersing the solution over time. The most know example of this type of equations is the wave equation ∂2
tu(t, x)−∆xu(t, x) = 0 which is only partially
dispersive, in the sense that in this case the frequencies determine directions of propagation and not the speed. Let’s look at a simple example:
Example 1.0.1. Given the Cauchy problem in a finite dimensional Hilbert space D
(
∂tu(t) − Lu(t) = 0,
u(0) = u0,
where u : R → D and L ∈ End(D), the unique global solution is the given by the linear propagator u(t) = etLu0 =
P∞
n=0 tn
n!L nu
0. Then, if we consider the
cases in which L is skew - adjoint, from the spectral theorem, we could infer that there exists an orthogonal Fourier basis (ej)j∈J of D of eigenvectors for
L with some imaginary eigenvalues ih(j). From the fact that Lej = ih(j)ej
we have that if u0 = ej the solution is u(t) = eih(j)tej. Now, for any f ∈ D,
we can define the Fourier coefficients ˆf (j) := hf, eji. A general solution to
this problem can be seen as du(t)(j) = eith(j)uˆ0(j). In particular, each Fourier
mode oscillates independently.
In general, all the dispersive equations, although having very different behaviours, present a similar structure. In fact, considering only equations with the first - order in time derivative, the general form of a dispersive equation is
where u : R × Rd→ V with values in a finite dimensional Hilbert space and h(D)u = F−1(h(ξ)F (u)), where F is the Fourier transform with respect to x.
One of the most renowned dispersive equation comes from physics, and it is the Dirac equation
(1.1) iγα∂αu =
mc }
u,
where (γ0, ..., γ3) ∈ End(V ) are Gamma matrices, u : R1+3→ V , V = C4
is the spinor space and } > 0 is the Planck’s constant. There are plenty of works referring to this homogeneous equation and to its variations with nonlinearities. For instance, in this work [10] of A. Tesfahun, global solution and scattering result are presented with small initial data in Hs , s > 0, and in this work [11] of Bournaveas and Candy, global well - posedness is found in the massless case with Lorentz nonlinearity. In general, a lot of results have been found in the case where u : R1+3→ C4 but little is explored about
the case in which u : R1+2→ C2 and in this work we will explore and proof
a new global existence result for this type of problem.
Our strategy is to use a special projection to change substantially the equation from Dirac to the Boson Star equation. This method is very useful, and it is used in a lot of articles such as [13] and [25]. From this point, a classical study of the existence and uniqueness will be done, proving firstly the local existence and uniqueness and then, thanks to conservation laws, the global existence. In 2013, Herr and Lenzmann [21] showed local well posedness in Hs(R3) for s > 1
4 and for s > 0 with radial initial data. In [22],
Enno Lenzmann and others proved the existence of travelling solitary waves with velocity |v| < c. A similar work to ours is [17] of Enno Lenzmann, in which the same result is proved for the R1+3 space. The difference with our
work is that he makes use of Sobolev Embedding theorem which depends on the space dimension. In our work, we use it, but with a different initial space, and so our proves are quite different.
The rest of the paper is organized as follows. In the second chapter we will present the Dirac equation with Hartree type nonlinearity and, after some manipulations, we will arrive to a more suitable Boson Star equation. Then, we will present the main bilinear estimate which will lead to the global existence of the solution.
In the third chapter we will present some tools of Harmonic Analysis, and then we make a general introduction to the dispersive equation giving exam-ples with our equation.
In the fourth chapter, using Strichartz estimates and a contraction principle, we will proof the local existence of the solution. In particular we will find that, for every initial datum u0 ∈ H
1
2(R2), there is a small time interval
[0, Tu0] in which the solution exists in the space C
0([0, T u0], H
1 2(R2)).
In the fifth chapter we will deal with the task of extending the local solution to be global. The technique used here will be a non - perturbative one, and in particular we will do this extension thanks to two conservation laws. Then, the last part will be dedicated to proving the main bilinear estimate. In particular it will be crucial to distinguish a non critical situation in which a classical result works, and a critical situation in which Gagliardo - Nirenberg inequality will allow us to exhibit a proof.
Chapter 2
The Dirac Equation
We presented the original Dirac equation in the introduction (1.1). In rel-ativistic quantum mechanics the state of a free electron is represented by a wave function u(t, x) with u(t, ·) ∈ L2(R3, C4) for any t. This function satisfies the free Dirac equation:
(−i∂t+ α · D + mβ)u = 0
where m > 0 is the electron’s mass and Dirac operator α · D is defined as α · D = −iP3
k=1αk∂xk. α1, α2, α3 and β are Pauli’s 4 × 4 matrices:
β =I2 0 0 −I2 αk = 0 σk σk 0 with (2.1) σ1 = 0 1 1 0 σ2 = 0 −i i 0 σ3 = 1 0 0 −1 .
Dirac found that these matrices must satisfy the following anti - commu-tating properties:
{αj, αk} = αjαk+ αkαj = 2δjkI4,
{αj, β} = 0, β2 = I4,
where δjk is the Kronecker’s symbol and I4 is the identity matrice of
dimen-sion 4 × 4. These matrices could be defined also in 2 × 2 dimendimen-sion, as it will be done below.
with the Hartree type nonlinearity. This type of nonlinearity was originally derived by uncoupling the Dirac - Klein - Gordon system
(
(−i∂t+ α · D + mβ)u = φψu,
(∂2
t − ∆ + M2)φ = hψ, βψiC4.
If we suppose that φ(t, x) = eiλtρ(x) is a standing wave, then from the second Klein - Gordon equation we have that
(−∆ + (M2+ λ2))φ = hψ, βψiC4.
Using Green functions, if we call b2 = M2+ λ2, the solution of this equation is φ = ( c14π|x|1 ∗ hψ, βψiC2, λ = ±M, c2e −b|x| |x| ∗ hψ, βψiC4, |λ| < M,
for c1 and c2 constants. To be precise, this is not the most general definition
of the Hartree type nonlinearity, which is the following:
Definition 2.0.1. For γ ∈ [0, 2] the Hartree type nonlinearity is of the form F (u(t, x)) = λ(V ∗ hu, βuiC4)βu with the potential V satisfying the growing
condition bV ∈ C4(R3\ {0}) and for k ∈ [0, 4]
|∇ bV (ξ)| . |ξ|−γ−k f or |ξ| ≤ 1 and
|∇ bV (ξ)| . |ξ|−2−k f or |ξ| > 1.
The examples given before are referred to the cases in which γ = 0 and the Coulomb potential V (x) = |x|−1 is corresponding to γ = 2. A lot of important results have been already discovered for this equation such as scattering results in [14] and articles in which γ = 0 such as [15].
In this work however, we do not consider the standard Dirac equation with solution u : R4 → C4, but instead we prove some new results for the case
u : R1+2 → C2. So, given a mass parameter m ≥ 0, we will study the
behaviour of solutions to the following
(2.2)
(
(−i∂t+ α · D + mβ)u = (V ∗ hβu, uiC2)βu,
u(0, x) = u0(x) ∈ H
1 2(R2),
with α = (α1, α2) and D = −i∇x = (1i∂x1,
1
i∂x2). We will use the Pauli’s
matrices for the space 2 × 2 which are
(2.3) β =1 0 0 −1 , α1 = 0 i −i 0 , α2 =0 1 1 0 .
These matrices, as before, satisfy:
(2.4) {β, αi} = 0 {αj, αi} = 0 i 6= j,
and
(2.5) {β, β} = 2I {αi, αi} = 2I.
In the nonlinear part, we decided to take Hartree type nonlinearity
F (u) := (Vb∗ |u|2)u = ((b − ∆)−1|u|2)u
where Vb is a convolution kernel. We will derive a precise form of it in the
next chapter. Informally, regarding only infinitive and infinite behaviour, Vb
could be seen as
(2.6) Vb(x) = C(1 − log(|x|))e−b|x|,
where b > 0 and C > 0 is a constant. Note that in has a critical point in zero. Moreover, as we will do in the next chapter, observe that, without losing of generality, we could take b = 1 so that convolution with V := V1
is, up to a constant, equivalent to using the operator hDi−2 = (1 − ∆)−1. In fact, considering their symbols σ and σb, it is clear that the functions
σ(ξ) = (1 + |ξ|2)−1 and σ
b(ξ) = (b + |ξ|2)−1 has a similar behaviour and
same integrability and smoothness properties. At this point, to study this equation, one could take different ways. One of the standard ways is to study the integral form of the solution
u(t, x) = Um(t, D)u0(x) +
Z t
0
Um(t − τ, D)(V ∗ hu, βui)(τ )βu(τ )dτ,
where the linear propagator is defined by
Um(t, D) = I cos t(m2− ∆) 1 2 − (α + iβ)(m2− ∆)− 1 2 sin t(m2− ∆) 1 2.
One can check that the essential part of the linear propagator is e±ithDim.
This consideration is one of the reasons why it is convenient to take the second way, which will be taken by us. This way will be presented in the next chapter, and it was used in lot of other works on the argument such as [13], [6].
2.1
The Boson Star Equation
As written in (2.2), the Dirac equation has a linear part whose coefficients are matrices. There is another favourable way to see this system as two different Boson Star equations. The new setup is usually used to identify a null - structure in the nonlinearity as it was done for example in [13]. The idea is to diagonalize the operator Dm(ξ) :=
P2 i=1α iξ i+ mβ. We have that (2.7) Dm(ξ) = m ξ2+ iξ1 ξ2− iξ1 −m. .
Its spectrum is σ(Dm(ξ)) = {±p|ξ|2+ m2 = ± hξim}, and
Dm(ξ)2 = |ξ|2+ m2 0 0 |ξ|2+ m2 , and so |Dm(ξ)| = hξim 0 0 hξim .
Let’s define two projection operators Π±(ξ) = 12(I ± hξi1
m
Dm(ξ)), where
hξim = pm2+ |ξ|2. These operators are idempotent and they decompose
Dm(ξ). Moreover, all these properties are true:
Π±(ξ)Dm = 1 2(±|Dm| + Dm), Π + (ξ)Dm+ Π−(ξ)Dm = Dm, Π±(ξ)Π±(ξ) = Π±(ξ), Π±(ξ)Π∓(ξ) = 0, and βΠ±(D) = Π∓(D)β ± mhDi−1m . In particular, observe that
(α · D + mβ) = hDim(Π+(D) − Π−(D)).
By applying the operators Π±(D) to the equation (2.2), we obtain the fol-lowing system
(
(−i∂t+ hDim)u+ = Π+(D)(V ∗ hu, βuiC2)βu),
(−i∂t− hDim)u−= Π−(D)(V ∗ hu, βuiC2)βu),
where u± = Π±(D)u and initial data are u±(0, x) = u±0(x) ∈ H
1
2(R2). Note
that these equations are of the form of the another important dispersive equa-tion: the so called Boson Star equation (or Klein - Gordon semi - relativistic equation), which is
(2.8)
(
(−i∂t+
√
m2− ∆)u = (V ∗ |u|2)u,
u(0, x) = u0 ∈ H
1 2(R2),
This equation is used to describe gravitational collapse and dynamics of boson stars (see recent work [23] for a rigorous derivation of this equation). The constant m can be normalized. In literature, two different options are considered to be important: the massless case (m = 0) and the case in which m could be taken equal to 1. In this work we will focus on the case in which m = 1. The key observation in the case in which m = 0 is that, with x ∈ R3, the equation (2.9) is invariant under the scaling
u(t, x) → uλ(t, x) = λ
3
2u(λt, λx),
for fixed λ > 0. This rescaling leaves the L2(R3) norm invariant, and so the
equation (2.9) is L2−critical.
So, we consider m = 1 for the rest of the work. Defining hDi := √1 − ∆, and using a particular Hartree type nonlinearity (the Yukawa potential), we have that, up to a constant, (V ∗ |u|2)u = (hDi−2|u|2)u. Then, from now on,
we will work with this version of the equation:
(2.9)
(
(i∂t− hDi)u = λ(hDi −2
|u|2)u,
u(0, x) = u0(x) ∈ H
1 2(R2),
where λ ∈ R\{0} in the place of −1. As it will be seen in the next chapter, changing the sign in front of the nonlinearity could drastically change the behaviour of the equation. Note that λ could be taken equal to +1, which is the defocusing case, or −1 which is the focusing case. Other values (with the exception of λ = 0) could be obtained with a re - scaling.
2.2
Main Results
This thesis is divided into two parts. In the first part, we proof the local existence of the solution using a contraction method and Strichartz estimates. Theorem 2.2.1 (Local existence). There exists a function T : H12(R2) →
(0, ∞] such that for any u0 ∈ H
1
2(R2), there exists a u ∈ C([0, T (u0)); H 1 2(R2))
such that for all t ∈ (0, T (u0)), u is the unique local solution to the equation
(2.9) in the sense that, for all t < T (u0) it is true that
ku(t, x)k H 1 2 x(R2) < ∞.
In the second part we will proof that such a solution is actually global, and so the time existence for all initial data u0 is unbounded.
Theorem 2.2.2 (Global existence). The local solution given by the previous theorem is actually global in C([0, ∞); H12(R2)) for every λ ∈ R, in the sense
that, given an initial datum u0(x) ∈ H
1
2(R2), for any t ∈ [0, ∞), there is a
constant Cu0 such that
ku(t, x)k H 1 2 x(R2) ≤ Cu0.
For these two results, we will use tools coming from Harmonic Analysis and from general Partial Differential Equation theory. Some of them could be found in the next chapter. The key result, from which global existence will follow, is this bilinear estimate:
Theorem 2.2.3. There exists C > 0 such that for any f, g ∈ L2(R2) we have
(2.10) k(1 − ∆)−12hf, gi
C2kL2(R2) ≤ Ckf kL2(R2)kgkL2(R2).
This theorem is the novelty of this work and it will be proved in chapter 5. Note that this result depends on the dimension, and it is proven only for dimension two. We will use the following notation: A . B stays for A ≤ cB with c > 0 constant.
Chapter 3
Preliminaries
3.1
Harmonic Analysis
In this section we want to present definitions, theorems and instruments used in this thesis which come from Harmonic Analysis, such as Fourier transform, Sobolev spaces and Paley - Littlewood decomposition. Let’s start from the most useful and powerful instrument:
Definition 3.1.1. We define the spacial Fourier transform F : f → ˆf for a Schwartz function f ∈ S(Rd) by the formula
(3.1) f (ξ) :=ˆ Z
Rd
f (x)e−ix·ξdx.
It is well know that the Fourier transform is an automorphism on the space S(Rd) and so it is well defined the inverse formula
(3.2) f (x) = 1 (2π)d Z Rd ˆ f (ξ)eiξ·xdξ.
The spacial Fourier transform is used to focus on the oscillations of a func-tion in the space. With abuse of notafunc-tion, we may neglect the multiplying constant in front of the inverse formula when it is insignificant. Thanks to the Plancherel identity R
Rd|f (x)|
2dx = 1 (2π)d
R
Rd| ˆf (ξ)|
2dξ, the Fourier
trans-form could be extended to the L2(Rd) space.
Next, we will define inhomogeneous Sobolev spaces Ws,p(Rd) and the
homo-geneous Sobolev spaces ˙Ws,p(Rd):
Definition 3.1.2. The two spaces are defined as the closure of the Schwartz functions under these two norms
(3.3) kf kWs,p(Rd) := k(1 − ∆) s 2f k
(3.4) kf kW˙s,p(Rd) := k(−∆) s 2f k
Lp(Rd).
In case of p = 2, we call them Hs and ˙Hs and by Plancherel’s theorem
we have that (3.5) kf kHs(Rd) = 1 (2π)d/2khξi sˆ f kL2 ξ(Rd).
There are some other equivalent definition of Sobolev spaces. The classi-cal definition of the non - fractional Sobolev space is the following
Definition 3.1.3. For every k ∈ N and let p ∈ (1, ∞). The Sobolev space Lk,p(Rn) is defined as the space of functions f in Lp(Rn) all of whose
dis-tributional derivatives ∂αf are also in Lp(Rn), for all multi - indices α that satisfy |α| ≤ k. This space is normed by
(3.6) kf kLk,p(Rn) = X |α|≤k k∂αf k Lp(Rn), where ∂0f = f .
Actually, this two norms are equivalent in the case when s = k.
Theorem 3.1.1. If s = k is a nonnegative integer and p ∈ (1, ∞), then the norm of the space Lk,p(Rn) is comparable to the norm of the space Ws,p(Rn).
Proof. Firstly, suppose that f ∈ Ws,p(Rn). Then, for all |α| ≤ k we have
that the distributional derivatives ∂αf are equal to
∂αf = ∂
α
(1 − ∆)k2
(1 − ∆)k2f.
Theorem 6.2.7 in [3] gives that ∂α
(1−∆)k2 is a L
p multiplier. It follows that
X |α|≤k k∂αf k Lp(Rn)≤ Cn,p,kk(1 − ∆) k 2f kLp(Rn) < ∞.
Now suppose that f ∈ Lk,p(Rn). Then, using the fact that
(1 + |ξ|2)k2 = X |α|≤k k! α1!...αn!(k − |α|)! ξα ξ α (1 + |ξ|2)k2 ,
and, again, the fact that ξα
(1+|ξ|2)k2
are Lp multipliers whenever |α| ≤ k, we
have
k(1 − ∆)k2f kLp(Rn) = kF−1((1 + |ξ|2) k
= k X |α|≤k Cα,kF−1( ξα (1 + |ξ|2)k2 ξαF (f))kLp(Rn) = k X |α|≤k Cα,kF−1( ξα (1 + |ξ|2)k2 F (∂αf ))k Lp(Rn) . kf kLk,p(Rn) < ∞.
Next, let’s speak about propagators. From a general point of view, in Harmonic Analysis, propagators could be seen as Fourier multipliers in suit-able spaces. If u ∈ C1
t,locSx(R × Rd) is a classical solution to ∂tu(t, x) =
ih(D)u(t, x), then we have that ∂tu(t)(ξ) = ih(ξ) dd u(t)(ξ), with a unique solu-tion given by du(t)(ξ) = eith(ξ)uˆ0(ξ). In this thesis we consider only the case in
which h : Rd→ R. So h(ξ) is real and
b
u0is Schwartz, the function eith(ξ)ub0(ξ) is also Schwartz for any t ∈ R and then using the Fourier inversion formula we could define the propagator as
(3.7) u(t, x) = eith(D)u0(x) :=
Z
Rd
eith(ξ)+iξ·xub0(ξ)dξ.
Although we define this propagator in the Schwartz space, it can be easily extended by density to the spaces L2(Rd), Hs(Rd) and ˙Hs(Rd). Being an
unitary operator in this spaces, it is clear that (3.8) keith(D)f k
Hs(Rd)= kf kHs(Rd),
and the same is true for the other two norms. Moreover, being Fourier multipliers, propagators commute with other multipliers and in particular with fractional differential operators h∇is for any s ∈ R.
Another important tool coming from Harmonic Analysis is the so called Paley - Littlewood decomposition. Let ψ(ξ) be a real-valued radial and symmetric bump function with support supp(ψ(ξ)) = {ξ ∈ Rn : kξk ≤ 2}
which is equal to 1 in the ball B = {ξ ∈ Rn : kξk ≤ 1}. Now, for j ∈
Z, let φj(ξ) = ψ(2−jξ) − ψ(2−j+1ξ) be a bump function supported in the
annulus {(12)−j+1 ≤ kξk ≤ (2)j+1} whose derivatives satisfy the inequality
2j|α||∂αφ
j(ξ)| ≤ cα for some positive number cα and for all multi - indices
α ∈ Zn. By construction, the bump functions φj satisfy
(3.9) X
j∈Z
φj(ξ) = 1
for all ξ 6= 0, thus they provide a specific partition of unity which allows to decompose an arbitrary function u as
(3.10) u =X j∈Z Pju = X j∈Z uj,
where Pj is a projection operator defined by Pj(u) = (φju)ˆ ∨. A clear way
to see this decomposition is seeing these projections as Fourier multipliers calling (3.11) P[≤jf (ξ) := φ( ξ 2j) ˆf (ξ), (3.12) P[>jf (ξ) := (1 − φ( ξ 2j)) ˆf (ξ), (3.13) Pdjf (ξ) := (φ( ξ 2j) − φ( 2ξ 2j)) ˆf (ξ).
Paley - Littlewood decomposition is used to separate high - frequency com-ponents of a solution from the low - frequency comcom-ponents. It’s easy to see that, for example,
(3.14) P≤jf = X k≤j Pkf and (3.15) P≥jf = X k≥j Pkf.
Moreover, we will use the classical Sobolev Embedding theorem
Theorem 3.1.2 (Sobolev embedding theorem, first case). Let 0 < s < np and 1 < p < ∞. Then the Sobolev space Ws,p(Rn) embeds continuously in Lq(Rn) when the following homogeneity condition
(3.16) 1 p − 1 q = s n is satisfied. In particular H12(R2) ⊂ L4(R2).
Theorem 3.1.3 (Sobolev embedding theorem, second case). Let Ω be a domain in R2, eventually the whole space. If either sp > n or s = n and
p = 1, then
Ws,p(Rn) ⊂ L∞(Rn). In particular H2(R2) ⊂ L∞(R2).
Theorem 3.1.4 (Gagliardo - Nirenberg). Let u : Rn → R be a sufficiently regular function. Fix 1 ≤ q, r ≤ ∞ and a natural number m. Suppose also that a real number α ∈ [0, 1] satisfies
(3.17) 1 p = 1 r − m n α + 1 − α q Then there exists a constant Cm,n,p such that
kukLp(Rn)≤ Cm,n,pkDmukLαr(Rn)kuk1−αLq(Rn).
In particular we will use that
(3.18) kukL2(R2) ≤ Ck∇ukL1(R2).
We will also use the following two theorems: let’s define two operators Js
and Is as
(3.19) Jsf (x) := hDisf (x) = ((1 + |ξ|2)s/2f )ˆ∨,
(3.20) Isf (x) = (|ξ|sf )ˆ∨.
There is a special property which links this operators with the Paley - Lit-tlewood decomposition, and it is
Theorem 3.1.5. For p ∈ [1, ∞] and s ∈ R, we have the estimates 1. kJsukkLp ≤ C2skkukkLp (k ≥ 1),
2. kIsukkLp ≤ C2skkukkLp (∀k).
More over observe that the two operators commute. The proof can be found, e.g, in [2] lemma 6.2.1 Moreover
Theorem 3.1.6. The Bessel operator Js maps Lr(Rn) to itself with norm 1
for all s < 0 and all r ∈ [1, ∞].
Proof. For all s < 0 there exists a strictly positive function Vs(x) such that
hDisu = V s∗ u,
and kVskL1(Rn) = 1 (see e.g [4], proposition 1.2.5). We can conclude using
Young inequality that for all r ∈ [1, ∞]
In the final proof we will use the following inequality:
Theorem 3.1.7 (Bernstein inequality). For all k ∈ Z and all 1 ≤ p ≤ q ≤ ∞ it is true that
kPk(f )kLq(Rn) ≤ Cn,p,q2kskf kLp(Rn),
where Pk is the kth Paley - Littlewood projection and
1 p− 1 q = s n.
3.2
Contraction Principle
In this section we present a method which is often useful to proof the existence of a local solution to a dispersive PDE. This method is general, but we will present it using as example our equation (2.9) with λ = 1 (the case with λ = −1 has a similar development). This equation, by the Duhamel’s principle, have the integral form
(3.21) u(t, x) = e−ithDiu(0, x) + i Z t
0
ei(s−t)hDi(hDi−2|u(s, x)|2)u(s, x)ds,
which is equivalent to
(3.22) u(t) = e−ithDiu0+ (i∂t− hDi)−1(F (u)),
where F is the nonlinearity F (u(s, x)) := (hDi−2|u(s, x)|2)u(s, x). (i∂ t−
hDi)−1 is called the Duhamel’s operator. In this way we divided the solution
u(t) in two parts: the linear one which is given by the propagator and the perturbation given by the nonlinearity and we could view the solution as a fixed point of the map u → ulin+ (i∂t− hDi)−1(F (u)). From this
consider-ations, in the literature where were developed a surprisingly high number of iteration methods to find a fixed point is this PDE context. The idea is the as usual to proceed finding a sequence (uj) : I × Rd → Cm, for j ∈ N such
that u0 = 0 and then
(3.23) uj+1 = ulin+ (i∂t− hDi)−1(F (uj)).
The hope is that there exist a suitable Banach space in which this sequence converge to a limit u. Then, provided a certain continuity, the limit will be a fixed point.
Maybe the most standard way to approach this question is proving that there exists a suitable Banach space B in which the map u → ulin+ (i∂t−
hDi)−1(F (u)) is not only continuous, but also a Lipschitz map (typically
from a closed ball into itself). Then the existence of the (unique) fixed point is guaranteed by the classical contraction mapping theorem. We general idea to achieve this is to find two Banach spaces E and S on a timely local slab [0, T0) × Rd such that two estimates are true: the linear one
k(i∂t− hDi)−1f kE ≤ c0kf kS
and the non linear one
kF (u)kS ≤ c1kukE
whenever kukE ≤ R or more generally
kF (u) − F (v)kS ≤ c1ku − vkE
whenever kukE, kvkE ≤ R. Then, if c0c1 ≤ 12 and kulinkE ≤ R2, that map is
a contraction in the ball {u ∈ E : kukE ≤ R} with constant at most 12. The
precise argument is:
Theorem 3.2.1 (Abstract iteration argument). Let E and S be two Banach space. Given the equation u = ulin + DF (u), suppose that D is a linear
operator D : E → S such that kDf kS ≤ c0kf kE, for all f ∈ N and c0 >
0. Suppose that F : S → E is a nonlinear operator with F (0) = 0 and kF (u) − F (v)kE ≤ 2c10ku − vkS, for all u, v ∈ BR:= {u ∈ S : kukS ≤ R}, for
some R > 0. Then, for all ulin ∈ BR
2, there exists a unique solution u ∈ BR
to the equation and the map ulin → u is Lipschitz with constant 2 and we
have kukS ≤ 2kulinkS.
The proof is a simply observation that, with the given hypothesis, ku − vkS ≤ kulin−vlinkS+12ku−vkS. In this work we will use a simple modification
of this method.
This method has 3 difficulties: first of all, it’s not always clear which Banach spaces to use in the places of E and S. Moreover we need to have a certain smallness of the constants c0 and c1, for instance c0c1 ≤ 12. This difficulty will
be overcome in our work taking a small enough time interval. Another way used in literature is to take small initial data u0 in some norm or combining
both methods. The last difficulty is to assure in some they that the linear part of the solution ulin stays in a chosen ball BR
2 in such a way that the solution
is in BR. The idea here is to restrict the given map u = ulin+ DF (u) to be
BR→ BR in order to use the classical fixed point argument. Fortunately in
the case of Dirac equation, we have Strichartz estimates which bound in a suitable norm the linear part of the equation with constants depending on the initial data. This will be the argument of the next section.
3.3
Strichartz Estimates
As stated in the previous section, to have a complete local control of the solution to a dispersive equation, we need a way to control the smallness of the linear solution in terms of the size of the initial datum. The most direct way to do this is using the fundamental solution. This is a general method but for our purpose we will apply it directly to our case. The fundamental solution to the free Dirac equation of the form
( (i∂t− hDi)u = 0, u(0, x) = u0(x), is given by (3.24) u(t, x) = e−ithDiu0(x) = Ct Z R2 e−ithξieiξ·xub0(ξ)dξ.
Firstly we are concerned about finding fixed time estimates. Avoiding den-sity arguments, let’s suppose that the solution is Schwartz. Then, using Plancherel equality, it is easy to see that
ke−ithDiu0(x)kL2
x(R2)≤ ku0kL2x(R2),
and, since the propagator commutes with other Fourier multipliers
ke−ithDiu0(x)kHs
x(R2)≤ ku0kHxs(R2).
Moreover, it’s straightforward to obtain the dispersive inequality
ke−ithDiu0(x)kL∞ x (R2).
1
tku0kL1x(R2).
Then, we can interpolate this two estimates using the Marcinkeiwicz inter-polation theorem to obtain that
ke−ithDiu0(x)kLp0
x(R2)≤ Ctku0kL p x(Rd)
for p ∈ [1, 2]. Inserting fractional differential operators we conclude that
ke−ithDiu
0(x)kWs,p0
x (R2)≤ Ctku0kW s,p(R2).
From here, in some cases, it is possible to gain Strichartz estimates which involve also some norm in the time space. For our purposes it is completely sufficient this theorem proved in [1] and in [26].
Theorem 3.3.1. Given the dispersive equation i∂tu − h(|∇|)u = 0 in R1+n
with u(0, x) = u0(x), suppose that h ∈ C∞(0, ∞), and given these conditions:
1. h0(ρ) > 0, and either h00(ρ) > 0 or h00(ρ) < 0;
2. |h(k)(ρ1)| ∼ |h(k)(ρ2)| for 0 < ρ1 < ρ2 < 2ρ1;
3. ρ|h(k+1)| . |h(k)|
suppose that h satisfies conditions 1 and 2 for k = 1, 2 and 3 for k ∈ [1, [n/2]+ 1]. Then defining (3.25) F (Ds1,s2 h f )(ξ) = h0(|ξ|) |ξ| s1 |h00(|ξ|)|s2f (ξ),ˆ
the solution to the equation before satisfies
kukLq tL p x ≤ kD s1,s2 h f kH˙s,
where 2 ≤ p, q ≤ ∞, 2/q + n/p ≤ n/2 and (n, p, q) 6= (2, ∞, 2) with
s1 = 1 4 − 1 2p − 1 q, s2 = −1 4+ 1 2p , s = n 1 2− 1 p − 2 q.
3.4
Nonlinearity
In this section we want to turn our attention briefly to the effects of adding a nonlinearity to a dispersive equation. After this, we will focus on our Hartree type nonlinearity and we will proof some useful facts about it. This argument could be treated in the general case, but for our purpose, we will apply the consideration only to the equation
(3.26)
(
i∂tu(t, x) − hDiu(t, x) = µF (u(t, x)),
u(0, x) = u0(x) ∈ Hs(Rd),
in which µ ∈ {−1, +1} and F (u) is the non linear term, which generally has a power p greater than one.
Adding a nonlinearity makes the problem generally more complicated be-cause the behaviour depends also on F , p and µ. The sign µ decides if the nonlinearity is defocusing or focusing, which could be very important as in this example
Example 3.4.1. Let’s explore some special solutions of the equation
(3.27) i∂tu(t, x) − hDiu(t, x) = µ|u(t, x)|p−1u(t, x),
given by the classical method of separation of variables. As it will be proven in chapter 5, the operator −∆ has a strictly positive spectre σ(∆) = [0, ∞) in the space R2. Let’s define the function f ∈ C(σ(−∆), C) given by f (x) = (1 + x)12. By the functional calculus presented in the next section,
we can see that, if u ∈ H12(R2) is such that −∆u = λu, then it is true that
f (−∆)u = f (λ)u. For any ξ ∈ R2, the plane wave eix·ξ is an eigenfunction of −∆, and also has the amplitude one, which leads to consider
(3.28) u(t, x) = eix·ξv(t).
Inserting this function in the equation (3.27), with some calculation, we get that
(3.29) ∂tv(t) = −i(f (λ) + µ|v(t)|p−1)v(t).
Considering α to be the amplitude, we have the solution
(3.30) u(t, x) = αeix·ξe−if (λ)te−µ|α|p−1t.
Note that, since f (x) is always positive in R+, f (λ) is positive, so the time
oscillation of e−if (λ)t arising from the linear evolution is clockwise. Then, in the defocusing case µ = +1, the nonlinearity will amplify the dispersive effect of the linear equation. In the focusing case µ = −1, the nonlinearity will try to cancel the dispersive effect. In this case, if the amplitude α is small compared to the eigenvalue f (λ), then the dispersive effect is stronger. If the amplitude is large then the focusing effect takes over. This is a general heuristic: the focusing and defocusing behave are similar when the initial data are small or then f (λ) = (1 + |ξ|2)12 (λ = |ξ|2) is high (generally at high
frequencies).
In this work we will study the case of low regularity, in particular u0 ∈
H12(R2). The reason for this is that low regularity theory gives more control
on the nature of the singularities of the solution. Moreover, this choice will allow to gain some conserved quantities.
We will consider the so called Hartree type nonlinearity. Dirac equation with simpler Coulomb potential V was firstly derived by Chadam and Glassey in their work [5] by uncoupling the Maxwell - Dirac equations under assumption of vanishing magnetic field. In the same work, they conjured that uncoupling
Dirac - Klein - Gordon equations, it could be possible to obtain the Yukawa potential which is considered in this work. In chapter 2 we have already showed the idea behind this nonlinearity.
As we said before, we consider a particular Hartree type nonlinearity: that is, we take the nonlinearity to be (not considering the sign) F (u) = ((1 − ∆)−1|u|2)u. A lot of authors consider this nonlinearity to be equal to the
convolution with a precise kernel, that is
(3.31) F (u) = (V (x) ∗ |u|2)u,
where
(3.32) V (x) = C(1 + log(|x|−1))e−|x|
. Although this is a cheap way to represent asymptotic behaviour of this kernel, ( the behaviour near zero and at infinity), the most precise way to represent this operator is more complicated and will be shown below. So we want to search for a function V (x) such that, for every Schwartz function u, we have that
(3.33) (1 − ∆)−1u =F−1( bV ˆu) = u ∗ V, where V (x) = CF−1(1 + |ξ|2)−1)(x).
Theorem 3.4.1. The function V (x) is smooth on Rn\ {0}, V (x) is strictly positive and kV kL1(Rn) = 1. Moreover, there exist two constants Cn, Mn such
that (3.34) V (x) ≤ Cne− |x| 2 , when |x| ≥ 2, and (3.35) Mn−1h(x) ≤ V (x) ≤ Mnh(x),
when |x| ≤ 2, where h(x) = 1 − log(|x|2 ) + O(|x2|) and |O(|x|2
)| . |x|2. Proof. We start with the gamma function identity
(3.36) A−1 = 1
Γ(1) Z ∞
0
e−tAdt,
which we use to obtain
(3.37) (1 + |ξ|2)−1 = 1 Γ(1) Z ∞ 0 e−te−| √ tξ|2 dt.
The integral above converges in both ends. Now take the inverse Fourier transform and use the fact that the function e−|ξ|2 is an eigenfunction of the Fourier transform to obtain that
(3.38) V (x) = (2 √ π)−n Γ(1) Z ∞ 0 e−te−|x|24t t −n 2 dt.
This identity shows that V (x) is smooth on Rn\ {0} and that V (x) > 0 for
all x ∈ Rn. Moreover, kV kL1 =Fx(V )(0) = 1.
Now suppose that |x| ≥ 2. Then t + |x|4t2 ≥ t + 1
t and t + |x|2 4t ≥ |x|. This implies that (3.39) −t − |x| 2 4t ≤ − 1 2(t + 1 t) − 1 2|x|, from which follows that, for n = 2,
(3.40) V (x) ≤ C0 Z ∞ 0 e−t2e− 1 2tt−1dt e−|x|2 = C2e− |x| 2 .
Now suppose that |x| ≤ 2. Write V (x) = V1(x) + V2(x) + V3(x) where
(3.41) V1(x) = (2 √ π)−n Γ(1) Z |x|2 0 e−te−|x|24t t −n 2 dt, (3.42) V2(x) = (2 √ π)−n Γ(1) Z 4 |x|2 e−te−|x|24t t −n 2 dt, (3.43) V3(x) = (2 √ π)−n Γ(1) Z ∞ 4 e−te−|x|24t t −n 2 dt.
Since t|x|2 ≤ 4 in V1(x), we have that e−t|x|2
= 1 + O(t|x|2) and thus, for
n = 2, (3.44) V1(x) = (2 √ π)−n Γ(1) Z 1 0 e−4t1t−1dt + O(|x|2)(2 √ π)−n Γ(1) Z 1 0 e−4t1t−1dt. In V2(x) we have that 0 ≤ |x|2 4t ≤ 1 4 and t ∈ [0, 4]. So e −17 4 ≤ e−t− |x|2 4t ≤ 1; thus (3.45) V2(x) ∼ Z 4 |x|2 t−1dt = −2 log(|x| 2 ).
Finally, in V3(x) we have that e−14 ≤ e− |x|2
4t ≤ 1 and so V3(x) is bounded
3.5
Functional Calculus
Since a very important part of this work depends on a non standard oper-ator such as (1 − ∆)12, we decided to insert this very short presentation of
the vast theory of functional analysis. There are several apparently distinct formulations of this theory such as e.g a part of theory of operator algebras, through spectral theorem, or as pseudo differential operators. In some sense they are all equivalent. The purpose of this theory is to answer the question: Given an operator A on a Hilbert space, for which f can we define f (A)? Let’s begin with an example.
Example 3.5.1. Let A be a bounded operator on an Hilbert space H, and f = P∞
n=1anxn, with a radius of convergence R. If kAk < R, it is natural
to set f (A) = P∞
n=0anAn. In this case f is an analytic function in a domain
including all σ(A).
We want to extend this result on polynomials to all f ∈ C0(σ(A), C).
Lemma 3.5.1. Let T : V → B be a bounded operator between a normed space and a Banach vector space. Then T could be uniquely extended to the completion of V keeping the same norm.
Theorem 3.5.1 (Continuous Function Calculus). Let A be a bounded, self - adjoint operator on a Hilbert space H. Then there exists a unique φA= φ :
C(σ(A), C) → L(H) = {linear, bounded operators on H} such that 1. φ is a ∗-homomorphism of algebras.
2. φ is continuous, that is, kφ(f )kL(H) . kf kC0.
3. φ(id) = A.
Moreover, φ has the additional properties:
• If Au = λu then φ(f )u = f (λ)u. • σ(φ(f )) = {f (λ); λ ∈ σ(A)}. • If f ≥ 0 then φ(f ) ≥ 0. • kφ(f )kL(H)= kf kC0.
Proof. The idea behind the proof is simple: the first and the third properties determine φ(p) for any polynomial p(x). By the Weierstrass theorem, the
set of polynomials is dense in C(σ(A), C), so to use the lemma 3.5.1, it is sufficient to prove that
(3.46) kp(A)kL(H) = kp(x)kC(σ(A) = sup λ∈σ(A)
|p(λ)|. For a complete proof and further extensions see for example [20].
It is convenient to write f (A) instead of φ(f ) as it underlines the depen-dence of the homomorphism from A. In this work A will be always −∆ and fs(x) = (1 + x)
s
2, and the Hilbert space will be H 1 2(R2).
For the differential operators there is a simpler way to justify the existence of functional calculus: it is done using the Fourier transform and the operators resulting from this process are called pseudo differential operators.
The Fourier transform is an important tool in the theory of PDE because of its very convenient property of replacing differentiation by multiplication by a polynomial
(3.47) F (−i∂xiu) = ξiu.ˆ
For this reason it is used to define pseudo differential operators in a very simple manner. Fixed a continuous function σ we can define the operator with symbol σ(ξ) by the formula
(3.48) Op(σ(ξ))u(x) = Fξ−1(σ(ξ)Fy(u(y))).
This is a simple and operative way to treat PDE with pseudo differential operators.
Example 3.5.2. Given a function u ∈ H12(R2) the operator
√
1 − ∆ is defined using it’s symbol (1 + |ξ|2)12, and we have that
(3.49) (1 − ∆)12u(x) =F−1
ξ ((1 + |ξ| 2
)12F
y(u(y))).
Observe that this result could be obtained using the Stone theorem in the theory of strongly continuous semigroups.
3.6
Other Functional Spaces
In this section we introduce other functional space which appear in this work. We define them and we show some of their useful properties. We put together theory taken from [3], [4] and [24]. We use the Paley - Littlewood decomposition introduced before, with the same notations.
3.6.1
Hardy Spaces
Definition 3.6.1. A tempered distribution u belongs to the Hardy space hp(Rn) if and only if (3.50) kukhp(Rn)= k X j∈Z |Pj(u)|2 !12 kLp(Rn)< ∞.
This is the most economical definition, but there are other equivalent definitions useful to prove interesting properties of this space. For example, Hardy space could be defined also in this way:
Definition 3.6.2. Let (3.51) P (x) = Γ n+1 2 πn+12 1 (1 + |x|2)n+12
be the Poisson Kernel and Pt(x) = t−nP (xt). Let M (u, P ) := supt>0|Pt∗ u|
be the maximal operator associated with P . Then
(3.52) kukhp(Rn) = kM (u, P )kLp(Rn).
This second definition is useful to prove the equivalence of this space with a more known Lebesgue space Lp(Rn).
Theorem 3.6.1. 1. If p ∈ (1, ∞) then the space hp(Rn) is equivalent to
Lp(Rn) in the sense that
(3.53) kukLp(Rn) ≤ kukhp(Rn) ≤ Cn,pkukLp(Rn).
2. If p = 1 then h1(Rn) ⊂ Lp(Rn) and so
(3.54) kukL1(Rn) ≤ kukh1(Rn).
Where is another useful characterization, which uses Riesz transforms.
Definition 3.6.3. For j ∈ [1, n]∩N and f a Schwartz function, the jth Riesz transform of f is given by (3.55) Rj(f )(x) = Γ n+12 πn+12 lim ε→0 Z |y|≥ε xj − yj |x − y|n+1f (y)dy,
Lemma 3.6.1. The jth Riesz transform Rj has the symbol −iξj |ξ| , that is, (3.56) Rj(f )(x) =Fξ−1 −iξj |ξ|Fy(f (y)) .
Combining two results, we could we could define h1(Rn) Hardy spaces in terms of the Riesz transforms.
Theorem 3.6.2. With n ≥ 2,
1. If u ∈ h1(Rn), then R
i(u) ∈ L1(Rn) for every i ∈ [1, n] ∪ N.
2. Conversely, there exists a constant Cn such that, for u ∈ L1(Rn), we
have (3.57) Cnkukh1(Rn) ≤ kukL1(Rn)+ n X k=1 kRk(u)kL1(Rn).
Actually, point 2 of the theorem could be used to give an alternative definition of the h1(Rn) norm.
Corollary 3.6.1. For n ≥ 2, an integrable function on Rn lies in the Hardy
space h1(Rn) if and only if its Riesz transforms are also in L1(Rn).
Example 3.6.1. h1(Rn) 6= L1(Rn) because if u ∈ h1(Rn) then R
1(u) ∈
L1(Rn) and thusF
x(R1(u(x))) is uniformly continuous. But since
Fx(R1(u(x)) = −i
ξ1
|ξ|F (u)(ξ),
it follows thatFx(R1(u(x))) is continuous at zero if and only ifF (u)(0) = 0.
But this happens exactly when u has integral zero. So every function in h1(Rn) has zero integral.
3.6.2
Besov and Triebel - Lizorkin Spaces
With the notations of Paley - Littlewood decomposition above, let S0 be such
that S0+
P
j≥1Pj = I.
Definition 3.6.4. For every s ∈ R and every 0 < p, q ≤ ∞ we define the norms (3.58) kukBs p,q(Rn) = kS0(u)kLp(Rn)+ ∞ X j=1 (2jskPj(u)kLp(Rn))q !1q ,
(3.59) kukFs p,q(Rn) = kS0(u)kLp(Rn)+ k ∞ X j=1 (2js|Pj(u)|)q !1q kLp(Rn).
The space of all tempered distributions u for which the quantity kukBs
p,q(Rn) is finite is called the (inhomogeneous) Besov space, and it’s
de-noted by Bs
p,q(Rn). The space of all tempered distributions u for which the
quantity kukFs
p,q(Rn) is finite is called the (inhomogeneous) Triebel - Lizorkin
space, and it’s denoted by Fs
p,q(Rn). The homogeneous versions of these
spaces are defined in a similar way but using the norm
(3.60) kukB˙s p,q(Rn)= X j∈Z (2jskPj(u)kLp(Rn))q !1q , (3.61) kukF˙s p,q(Rn) = k X j∈Z (2jsPj(u))q !1q kLp(Rn).
It is not difficult to observe that all the four quantities are quasi-norms. These spaces are important because they gives an alternative way to see a more familiar spaces as Lebesgue and Sobolev one. In fact
Theorem 3.6.3 (Table of Coincidences). In the following list, the equalities mean the norms equivalence of the respective spaces:
1. ˙Fp,20 (Rn) = Fp,20 (Rn) = Lp(Rn) = hp(Rn) for p ∈ (1, ∞). 2. Bs p,p(Rn) = Fp,ps (Rn) 3. Fp,2s (Rn) = Ws,p(Rn) for p ∈ (1, ∞). 4. ˙Fs p,2(Rn) = ˙Ws,p(Rn) for p ∈ (1, ∞). 5. F0 p,2(Rn) = hp(Rn) for p ∈ (0, 1].
Chapter 4
Local Well - Posedness
The main purpose of this chapter is to prove the local well - posedness of the solution to the equation (2.9). This result will be proven using a classical contraction method and the proof will be divided into two steps.
4.1
Evaluating the Nonlinearity
Let’s call the nonlinearity F (u) := λ(V ∗ |u|2)u. We show that the
nonlinear-ity is locally Lipschitz continuous from H12(R2) into itself. This is the main
point of the contraction principle exposed in the previous chapter.
Lemma 4.1.1. For all λ ∈ R, the map F (u) is locally Lipschitz continuous from H12(R2) into itself with
(4.1) kF (u) − F (v)k
H12(R2) ≤ LMku − vkH12(R2),
for all u, v ∈ H12(R2) where LM is a constant depending only on the H 1 2(R2)
norm of u and v, with M = max{kuk
H 1 2 x(R2) , kvk H 1 2 x(R2) }.
The proof relies on two theorems: Kato-Ponce inequality (or Fractional Liebniz rule):
Theorem 4.1.1 (Kato - Ponce inequality). For the operator hDis we have
the following inequality
(4.2) khDis(f g)k Lr(Rn)≤ C1kf kLp1khDisgkLp2 + C2kgkLq1khDisf kLq2, where s > 0 and 1r = p1 1 + 1 p2 = 1 q1 + 1 q2 for 1 < r < ∞ and 1 < p1, p2, q1, q2 ≤ ∞.
It relies also on theorem 3.1.6 (see for example [4]) and it uses Sobolev Embedding theorem, so it depends on the dimension of the space. The same result for the space R3 was proven in [17].
Proof. Fix u(x), v(x) ∈ H12(R2). Observe that
kF (u) − F (v)k
H12 = khDi 1
2((hDi−2|u|2)u) − hDi 1
2((hDi−2|v|2)v)kL2
= k1 2hDi
1
2(hDi−2(|u|2− |v|2)(u + v)) + 1
2hDi
1
2(hDi−2(|u|2+ |v|2)(u − v))kL2
. khDi12(hDi−2(|u|2− |v|2)(u + v))k
L2+ khDi 1
2(hDi−2(|u|2+ |v|2)(u − v))k
L2.
Now we want to bound the two parts separately. Starting from
(4.3) B := khDi12(hDi−2(|u|2 + |v|2)(u − v))k
L2,
and using the Fractional rule we have that
B . khDi−32(|u|2+|v|2)kL4ku−vkL4+khDi−2(|u|2+|v|2)kL∞khDi 1
2(u−v)kL2.
Using the Sobolev Embedding theorem (see theorem 3.1.2) in R2 we have
that ku − vkL4 ≤ ku − vk
H12. For the same reason
khDi−32(|u|2+ |v|2)k L4 ≤ khDi− 3 2(|u|2+ |v|2)k H12 = khDi −1 (|u|2+ |v|2)kL2.
Using the fact that Bessel operators are Lp Fourier multipliers (see theorem 3.1.6)
khDi−1(|u|2+ |v|2)kL2 ≤ k(|u|2 + |v|2)kL2 ≤ k|u|2kL2 + k|v|2kL2
= kuk2L4 + kvk2L4 ≤ kuk2
H12 + kvk
2 H12.
The last one is bounded using Sobolev Embedding theorem, second form (see theorem 3.1.3)
khDi−2(|u|2+ |v|2)kL∞ . k|u|2+ |v|2kL2 . kuk2
H12 + kvk
2 H12.
Putting all together, we have that
(4.4) B . (kuk2 H12 + kvk 2 H12)ku − vkH 1 2.
Now we will bound
(4.5) A := khDi12(hDi−2(|u|2− |v|2)(u + v))k
The idea is again to use Kato - Ponce inequality
A . khDi−32(|u|2−|v|2)kL4ku+vkL4+khDi−2(|u|2−|v|2)kL∞khDi 1
2(u+v)kL2.
Let’s start from khDi−32(|u|2− |v|2)kL4. Exactly as it was done before, using
the first form of Sobolev Embedding theorem and theorem 3.1.6, we have that:
khDi−32(|u|2− |v|2)kL4 . k|u|2− |v|2kL2,
and using H¨older inequality and first form of Sobolev Embedding theorem we can arrive to
(4.6)
k|u|2− |v|2k
L2 ≤ ku − vkL4(kukL4 + kvkL4) . ku − vk
H12(kukH12 + kvkH12).
Moreover, as before, using the second form of Sobolev Embedding theo-rem
khDi−2(|u|2− |v|2)k
L∞ ≤ k|u|2− |v|2kL2 . ku − vk
H12(kukH12 + kvkH12).
To bound the last two terms we use again the first form of Sobolev Embedding
ku + vkL4 . ku + vk H12 ≤ kukH12 + kvkH12, and khDi12(u + v)k L2 = ku + vk H12 ≤ kukH12 + kvkH12.
Putting all together, we have
(4.7) A . (kuk
H12 + kvkH12)
2ku − vk H12.
Then the final estimate is
(4.8) kF (u) − F (v)k H12 . (kuk 2 H12 + kvk 2 H12 + kukH 1 2kvkH12)ku − vkH12.
Finally, let’s prove (4.6). If we define the following function f (s) := |v + s(u − v)|2, then it is clear that f (1) = |u|2 , f (0) = |v|2 and
|u|2− |v|2 = f (1) − f (0) =
Z 1
0
f0(s)ds.
With some calculations,
f0(s) = (u − v)(¯v + s(¯u − ¯v)) + (¯u − ¯v)(v + s(u − v)).
Then, we have that
|u|2− |v|2 = (u − v)¯v + (u − v)(¯u − ¯v) + (¯u − ¯v)v = (u − v)¯u + (¯u − ¯v)v.
4.2
Local Existence
Using the Duhamel’s principle, we can conclude that a strong solution to the problem (2.9) is given by
(4.9) u(t, x) = e−ithDiu(0, x) + i Z t
0
ei(s−t)hDiλ(hDi−2|u(s, x)|2)u(s, x)ds.
In fact, calling the nonlinearity F (u(s, x)) := λ(hDi−2|u(s, x)|2)u(s, x),
if we make the ansatz u(t, x) = e−ithDiv(t, x) for some suitable v(t, x), the equation (2.9) is (formally) equivalent to
i(−ihDie−ithDiv(t, x) + ∂t(v(t, x))e−ithDi) − hDie−ithDiv(t, x) = F (u(t, x))
and thus it is equivalent to ∂tv(t, x) = −ieithDiF (u(t, x)), which by the
fun-damental theorem of calculus is equivalent to
(4.10) v(t, x) = v(0, x) + i Z t
0
eishDiF (u(s, x))ds.
The representation above follows by multiplying both sides by e−ithDi. We want to prove that our problem is locally well-posed with a fixed u0 ∈
H12(R2). Let’s call YT := C0([0, T ); H 1
2(R2)) with the norm
kukYT := supt∈[0,T )ku(t, x)k H
1 2 x
. We want to find a suitable time T in such a way that we can use the contraction principle presented in the previous chapter. To have an a prior bound to the linear part of the solution, we use Strichartz estimates.
Theorem 4.2.1. Given the equation (4.9) with the linear part of the solution ulin(t, x) = e−ithDiu0(x), the following Strichartz estimate is true:
(4.11) kulin(t, x)kL∞t (R),L2x(R2). ku0(x)kL2x(R2).
Proof. Observe that our equation is of the right form for applying theorem 3.3.1. In our case, we have that h(ρ) =p1 + |ρ|2, and
1. h0(ρ) = √ ρ 1+|ρ|2 > 0. 2. h00(ρ) = √ 1 1+|ρ|23 > 0. 3. h(3)(ρ) = √−3ρ 1+|ρ|25.
In particular the hypothesis are verified and choosing p = 2 , q = ∞ s1 =
s2 = s = 0, we have the estimate above.
Moreover, for the nonlinearity F (u(t, x)) we have that
(4.12) k Z t
0
ei(s−t)hDiF (u(s, x))dskL∞t (R),L2
x(R2) . kF (u(t, x))kL1t(R),L2x(R2),
because ei(s−t)hDi is a unitary operator.
Now we are ready to use the standard contraction principle, following for example [16]. The uniqueness is gained by this theorem
Theorem 4.2.2 (Gronwall’s lemma). Let T > 0, B(s) ∈ L1(0, T ), B(s) ≥ 0
a.e. and c1, c2 ≥ 0. Let w(t) ∈ L1(0, T ), w ≥ 0 a.e. be such that Bw ∈
L1(0, T ) and
(4.13) w(t) ≤ c1+ c2
Z t
0
B(s)w(s)ds,
for almost every t ∈ (0, T ). Then we have
(4.14) w(t) ≤ c1ec2 Rt
0B(s)ds,
for almost every t ∈ (0, T ).
In particular, if c1 = 0, we have that w = 0 a.e. Combining this
theo-rem with lemma 4.1.1 in which we proved that our nonlinearity F is locally Lipschitz continuous in the space H12(R2) with fixed t, that is,
(4.15) kF (u(t, x)) − F (v(t, x))k H 1 2 x(R2) ≤ LMku(t, x) − v(t, x)k H 1 2 x(R2)
for any u(t, x), v(t, x) ∈ B
H
1 2 x
(0, M ) and LM . M2, we can proof the following
local existence theorems:
Theorem 4.2.3. Let T > 0, u0 ∈ H
1
2(R2) and let u, v ∈ C([0, T ], H 1 2(R2))
be two solutions to integral form (4.9). Then u = v.
Proof. We set M = supt∈[0,T ]max{ku(t, x)k
H 1 2 x , kv(t, x)k H 1 2 x }. Then w(t) := ku(t, x) − v(t, x)k H 1 2 x ≤ Z t 0 kF (u(s, x)) − F (v(s, x))k H 1 2 x ds ≤ LM Z t 0 ku(s, x) − v(s, x)k H 1 2 x ds = LM Z t 0 w(s)ds.
Theorem 4.2.4 (Local Existence). Let M > 0 and fix u0 ∈ H
1
2(R2) be such that ku0k
H12 ≤ M . Then there exists
a unique solution u ∈ C([0, TM]; H
1
2(R2)) of the equation (4.9) with
(4.16) TM =
1 2L2M
> 0.
Proof. For this prove we use the notation H12 := H 1 2(R2).
The uniqueness is proven in the previous lemma. We fix u0 ∈ H
1 2 and we define E = {u ∈ C([0, TM]; H 1 2); ku(t, x)k H 1 2 x ≤ 2M, ∀t ∈ [0, TM]}.
We equip E with the distance generated by the norm of C([0, TM]; H
1 2) and we have d(u, v) = max t∈[0,TM] ku(t, x) − v(t, x)k H 1 2 x ,
which makes E a complete metric space since C([0, TM]; H
1
2) is a Banach
space. For all u ∈ E, we define φ(u) ∈ C([0, TM]; H
1 2) by (4.17) φ(u(t, x)) = e−ithDiu0(x) + i Z t 0 ei(s−t)hDiF (u(s, x))ds.
We have that F (0) = 0 and so kF (u(s, x))k
H 1 2 x ≤ 2M L2M = TM M. It follows
that, for all t ≤ TM,
kφ(u(t, x))k H 1 2 x ≤ ku0kH1 2 + Z t 0 kF (u(s, x))k H 1 2 x ds ≤ M + tM TM ≤ 2M. Consequently φ : E → E and for all u, v in E,
kφ(v(t, x)) − φ(u(t, x))k H 1 2 x ≤ L2M Z t 0 kv(s, x) − u(s, x)k H 1 2 x ≤ TML2Md(u, v) ≤ 1 2d(u, v).
Therefore, φ is a contraction in E and so φ has a fixed point u ∈ E, which solves the integral solution (4.9).
Corollary 4.2.1. There exists a function T : H12(R2) → (0, ∞] such that for
any u0 ∈ H
1
2(R2), T : u0 → Tu
0 and there exists a u ∈ C([0, Tu0); H 1 2(R2))
such that for all T ∈ (0, Tu0), u is the unique solution to the equation (4.9)
in C([0, T ]; H12(R2)). In addition, (4.18) 2L2ku(t,x)k H 1 2 x ≥ (Tu0 − t) −1 ,
1. Tu0 = ∞;
2. Tu0 < ∞ and limt→Tu0− ku(t, x)k H 1 2 x(R2) = ∞. In particular, if ku0kH1
2 = M , the time existence guaranteed by theorem 4.2.4
is Tu0 ∼
1 M2.
The proof is a standard argue by contradiction. In particular, there can be some initial data for which the solution is only local and other for which the solution is global (Tu0 = ∞).
4.3
Another Approach
There is another approach to prove that φu is a contraction, and that is using
a generalized Gronwall’s lemma:
Theorem 4.3.1. Given u0 ∈ H
1
2(R2), there exists a local solution to the
equation (2.9), that is there exists a finite time T0 such that the solution
u(t, x) belongs to C([0, T0); H
1 2(R2)).
Proof. We will proof that the operator φu is a contraction in the space
C([0, T0); H
1
2(R2)). We will use the generalized Gronwall’s lemma for which
we need to prove that
(4.19) kφu(u)k ≤ c1ku0k H 1 2 x(R2) + c2T0δkuk 3,
where δ is positive. Unless specified otherwise, the norms are taken in x, and we indicate with k.k the norm of the space C([0, T0); H
1
2(R2)). Note
that hDi commutes with eithDi and so
(4.20) k hDi12 eithDiu 0kL2 x(R2) ≤ c1k hDi 1 2 u 0kL2 x(R2)= c1ku0kH12 x(R2) ,
which is the first part of the right side of 4.19. Moreover
k hDi12 Z t 0 ei(t−s)hDiF (u(s, x))dskL∞ t ,L2x = k Z t 0 ei(t−s)hDiF (u(s, x))dsk L∞t ,H 1 2 x
≤ k hDi12 ((hDi−2|u|2)u)k
L1 t,L2x.
Now we will use the Kato - Ponce inequality [8]:
k hDi12 (hDi−2|u|2)u)k
ck hDi−32 |u|2k
L4kukL4 + ck hDi−2|u|2kL∞k hDi 1 2 uk
L2.
Note that, using the Sobolev Embedding theorem first form,
kukL4 . kuk
H12.
More over, using Bessel inequality 3.1.6, we have that
k hDi−1|u|2k
L2 . k|u|2kL2.
So, putting all together we have that
k hDi−32 |u|2k L4 . k hDi− 3 2 |u|2k H12 = k hDi 1 2 hDi− 3 2 |u|2k L2
. k hDi−1|u|2kL2 . k|u|2kL2 . kuk2L4 . kuk2
H12.
Finally we can get the following estimates
k hDi−2|u|2k
L∞ . k|u|2kL2 . kuk2
H12,
and thus
k hDi12 (hDi−2|u|2)u)k
L1 t[0,T0),L2x(R2) . kkuk 3 H 1 2 x(R2) kL1 t[0,T0) ≤ c2T0kuk 3.
So we have proven that
(4.21) kφu(u)k ≤ c1ku0kH1
2 + c2T0kuk
3,
and now we can conclude with a classical fixed point argument applied to φu.
Chapter 5
Global Existence
In the previous chapter, we proved the local existence and uniqueness of a solution to the equation (4.9) in the space C([0, T ); H12(R2)), showing also the
minimum guaranteed time of existence for all initial data u0 ∈ H
1
2(R2) and
showing a persistence of regularity for small time. The next step is proving that the time of existence is Tu0 = ∞. In other words, we want to show that,
given u0 ∈ H
1
2(R2), the solution will have a finite norm ku(t, x)k
H 1 2 x(R2) for any t > 0.
5.1
Conservation Laws
The first step for proving the global existence is obtaining some conservation laws. In particular we will proof the conservation of energy and L2 - mass that are respectively given by
(5.1) E[u] := 1 2 Z R2 ¯ uhDiudx +1 4λ Z R2
(hDi−2|u|2)|u|2dx,
(5.2) N [u] := Z R2 |u|2dx. Lemma 5.1.1. Given u0 ∈ H 1
2(R2), the local solution given by theorem 4.2.4
obeys to the conservation of energy and L2 - mass, i.e.,
E[u(t, x)] = E[u0(x)] and N [u(t, x)] = N [u0(x)],
Proof. This proof follows the traces in [17]. Fixed a initial datum u0, firstly,
we multiply the equation (2.9) for ¯u(t, x) and then we integrate it in x. Taking the imaginary part we have that
(5.3) ∂tkuk2L2(R2) = 0
for all t ∈ [0, Tu0). Consequently N [u(t, x)] = N [u0(x)].
The conservation of energy is more delicate. Formally, it is sufficient to multiply the equation (2.9) by ∂tu(t, x) ∈ H¯ −
1
2(R2) and then integrate over
R2. Taking the real part we have that
(5.4) 0 = ∂t 1 2 Z R2 ¯ uhDiudx + λ1 4 Z R2
(hDi−2|u|2)|u|2dx
.
In particular E[u(t, x)] = E[u0(x)]. The problem is pairing two elements of
the space H−12(R2) and then integrating them is generally not well defined. In
this case, we need to introduce a regularization procedure (see [17] and other regularization methods in [18], [19]). The idea is that we can approximate the operator hDi with the family of operators
(5.5) Mε:= (εhDi + 1)−1, f or ε > 0.
When using the fact that for all u ∈ Hs and s ∈ R, M
εu → u strongly, we
can approximate the difference
E[u(t2, x)] − E[u(t1, x)] = lim
ε→0+(E[Mεu(t2, x)] − E[Mεu(t1, x)]) .
Now it can be seen that, whenever ε > 0, there are not two H−12 elements
paired, in contrast to the case ε = 0. Then, using the dominated convergence theorem, it can be proven that limε→0+(E[Mεu(t2, x)] − E[Mεu(t1, x)]) =
0.
Now we want to proof that the operator hDi−swith s ≥ 0 is a symmetrical operator with respect to the norm of L2(R2). Firstly, we need the following
lemma
Lemma 5.1.2. The spectrum in the whole space R2 of the operator
−∆ : L2(R2) → L2(R2) is σ(−∆) = [0, ∞).
Proof. The domain of the operator −∆ is given by D(−∆) = {u ∈ L2 :
4π2|ξ|2u(ξ) ∈ Lˆ 2} , due to the fact thatF (−∆u)(ξ) = 4π4|ξ|2u(ξ). We defineˆ
the resolvent set ρ(−∆) = {λ ∈ C : (λI + ∆) : D(−∆) → L2 is a bijection}. The spectrum is σ(−∆) = C \ ρ(−∆).
We start proving that σ(−∆) ⊂ [0, ∞) : if we take a λ in C \ [0, ∞), then λ ∈ ρ(−∆) since
(λI + ∆)u = f ⇔ (λ − 4π2|ξ|2)ˆu(ξ) = ˆf (ξ) ⇔ ˆu = (λ − 4π2|ξ|2)−1f .ˆ
So (λI + ∆)−1u = F−1((λI + 4π2|ξ|2)−1f ) which is well defined if for allˆ
ξ ∈ R2, λ − 4π2|ξ|2 6= 0. So σ(−∆) ⊂ [0, ∞).
To prove the opposite inclusion, let’s fix λ ∈ [0, ∞) and let’s proof that (λI − ∆) is not bijective. We show that (λI + ∆)−1 is not bounded because there exists a sequence (uk) such that
kukL2
k(λI − ∆)ukkL2
→ ∞,
as k → ∞. Let’s take a vector x0 ∈ R2 such that |x0|2 = λ, and let’s define
u(x) := eix0·x. The trick is to define u
k := φku there φk ∈ C0∞ are a cut-off
functions such that φk(x) = 1 for x ∈ Bk(0), φk(x) = 0 for x ∈ Bk+1(0)c and
φk∈ [0, 1]. We choose them in such a way that their first two derivatives are
uniformly bounded in k. For this reason ((λI + ∆)uk) are uniformly bounded
in the sup norm, and having the support in the annulus Bk+1(0) − Bk(0), it
is simple to see that
k(λI + ∆)ukk2L2 . kn−1.
On the other hand, it is clear that kukk2L2 ≥ vol(Bk(0)) = ωnkn, and the last
two inequalities give the result.
Corollary 5.1.1. For all s ∈ R, the operator hDis is self-adjoint with respect
of the L2 scalar product.
Proof. This proof relays on functional calculus properties. σ(−∆) = [0, ∞) and the function f (x) = (1 + x)s is positive for all x ≥ 0 and all s ∈ R. Then
the operator f (−∆) = (1 − ∆)s is symmetric and positive. More particulars and basic functional calculus could be found here [20].
Due to this corollary, now we can simply distribute the two operators in the energy, gaining that the conservation law (5.1) could be written as
E[u0] = E[u] = 1 2 Z R2 ¯ uhDiudx + λ1 4 Z R2
(hDi−2|u|2)|u|2dx
= 1 2ku(t, x)k 2 H 1 2 x(R2) + λ1 4 Z R2 hDi−1|u|22 dx.
In particular, it is important to underline that
(5.6) kuk2 H 1 2 x(R2) = −1 2λk hDi −1 |u|2k2 L2 x(R2)+ 2E[u0],
where E[u0(x)] is the finite initial energy.
The idea is to search for a suitable estimate of k hDi−1|u|2k2
L2 in such a
way that, given a initial datum which satisfies some bounds, that quantity is limited. As it will be clear soon, to bound this element, we will use the key bilinear estimate (theorem 2.2.3), whose proof will be done in the final section of this chapter.
5.2
Global Existence
Definition 5.2.1. A solution u(t, x) to the equation (2.9) exists globally in H
1 2
x(R2) if and only if for any finite time t > 0 the norm ku(t, x)k H 1 2 x(R2) is finite.
The idea to gain the global solution is to proof a a priori bound of the H12 norm of the solution is such a way that
(5.7) ku(t, x)k H 1 2 x(R2) ≤ Cu0,
for all t > 0 and for all initial data u0(x) ∈ H
1
2, where Cu
0 is a constant,
depending on u0. In the case of a defocusing nonlinearity we have that λ = 1
and so (5.8) E[u] := 1 2 Z R2x ¯ uhDiudx +1 4 Z R2x
(hDi−2|u|2)|u|2dx.
This leads to the simple bound (5.9) ku(t, x)k2 H 1 2 x(R2) ≤ 2E[u] = 2E[u0].
In the case of the focusing nonlinearity (λ = −1), another term appears on the right and using theorem 2.2.3, we can obtain the following estimate
ku(t, x)k2 H 1 2 x(R2) = 1 2k hDi −1 |u(t, x)|2k2 L2 x(R2)+ 2E[u0(x)] . 2E[u0(x)] + ku(t, x)k4L2 x(R2).
It remains to prove that the right side of this two inequalities are finite. To do so, we combine this conservation law with the other one (5.1). Then (5.10) ku(t, x)k4L2
x(R2)= ku0(x)k
4 L2
x(R2)
which is finite, and
(5.11) E[u0(x)] = 1 2ku0(x)k 2 H 1 2 x(R2) −1 4k hDi −1 |u0(x)|2k2L2 x(R2)
5.3
Proof of the Theorem 2.2.3
We want to apply the theorem 2.2.3 in the case in which u = f and ¯u = ¯g. We start from the energy conservation
(5.12) kuk2 H 1 2 x(R2) = 1 2k hDi −1 |u(t, x)|2k2 L2 x(R2)+ E[u0(x)].
The main purpose is to bound the element k hDi−1|u(t, x)|2k2 L2
x(R2). Using
Gagliardo - Nirenberg inequality (see theorem 3.1.4), we get that
(5.13) k(1 − ∆)−12|u|2k2 L2 x(R2) . k∇(1 − ∆) −1 2|u|2k2 L1 x(R2).
One of the fundamental theorem to arrive to the desired estimate is the Coifman - Meyer theorem, (see e.g. [4]).
Theorem 5.3.1 (Coifman - Meyer). Suppose that a bounded function σ on (Rn)2\ {(0, 0)} satisfies
(5.14) |∂α1∂α2σ(ξ
1, ξ2)| . (|ξ1| + |ξ2|)−(|α1|+|α2|)
for all (ξ1, ξ2) 6= (0, 0) and all multi - indices α1, α2 with |α1| + |α2| ≤ 2n.
Then the bilinear operator
(5.15) Tσ(f1, f2)(x) =
Z
Rn
σ(ξ1, ξ2) ˆf1(ξ1) ˆf2(ξ2)eix(ξ1+ξ2)dξ1dξ2
is bounded from Lp× Lq to Lr where
p, q ∈ (1, ∞] r ∈ (1
2, ∞) 1/r = 1/p + 1/q
As it was showed in the section about pseudo differential operators, the operator ∇(1 − ∆)−12 applied to |u(t, x)|2 is defined in this way
(5.16) ∇(1 − ∆)−12|u(t, x)|2 =F−1
ξ
1 (1 + |ξ|2)12
ξFy(u(t, y)¯u(t, y))
! .
In particular, if we neglect a multiplicative constant in front of the inverse Fourier transform, we have that
(5.17) ∇(1 − ∆)−12|u(t, x)|2 = Z R2ξ eiξ·x " ξ 1 (1 + |ξ|2)12 Z R2y
e−iy·ξu(t, y)¯u(t, y)dy #
