On the Cauchy problem in Besov spaces for a non linear Schrodinger equation

Fabrice Planchon

Description

On the Cauchy problem in Besov spaces for a non-linear Schrodinger equation Fabrice Planchon ? Abstract We prove that the initial value problem for a non-linear Schrodinger equation is well-posed in the Besov space B˙ n 2? 2 ? ,∞ 2 (R n), where the nonlinearity is of type |u|?u. This allows to obtain self-similar solutions, and to recover previous results under weaker smallness assumptions on the data. Introduction We are interested in the following equation: { i∂u∂t + ∆u = |u| ?u, u(x, 0) = u0(x), x ? Rn , t ≥ 0, (1) where is either 1 or ?1, and n ≥ 2. One important property of (1) is its invariance by scaling: { u0(x) ?? u0,?(x) = ? 2 ?u0(?x) u(x, t) ?? u?(x, t) = ? 2 ?u(?x, ?2t).(2) . Let s? be such that s?? n2 = ? 2? . One therefore expects the homogeneous Sobolev space H˙s? to be the “critical” space for well-posedness as its norm is invariant by rescaling. Theory ([5]) indeed asserts that (1) is locally well-posed in that Sobolev space provided s? ≥ 0.

• similar solution


• lorentz space


• let u0 ?


• point argument


• h˙s?


• lorentz spaces


• end-point strichartz


• can relax


• such

Sujets

• Planchon
• Lorentz space

Informations

• Publié par: pefav
• Nombre de lectures: 11
• Langue: English

Extrait

On the Cauchy problem in Besov spaces for a
non-linear Schr¨odinger equation
∗Fabrice Planchon
Abstract
We prove that the initial value problem for a non-linear Schro¨dinger equation is
n 2− ,∞ n α2 α˙well-posed in the Besov space B ( ), where the nonlinearity is of type |u| u.
2
This allows to obtain self-similar solutions, and to recover previous results under
weaker smallness assumptions on the data.
Introduction
We are interested in the following equation:
(
∂u
αi +Δu = ǫ|u| u,
(1) ∂t
nu(x,0) = u (x),x∈ , t≥ 0,0
where ǫ is either 1 or −1, and n ≥ 2. One important property of (1) is its invariance by
scaling:
2
αu (x) −→ u (x) =λ u (λx)0 0,λ 0(2) 2
2
αu(x,t) −→ u (x,t) =λ u(λx,λ t).λ
n 2. Let s be such that s − =− . One therefore expects the homogeneous Sobolev spaceα α 2 α
s˙ αH to be the “critical” space for well-posedness as its norm is invariant by rescaling.
Theory ([5]) indeed asserts that (1) is locally well-posed in that Sobolev space provided
s˙ αs ≥ 0. That is, there exists a (weak) solution of (1) which is C([0,T],H ), unique underα
s˙ αan additional assumption. Moreover, this solution is global in time if the H norm of the
initial data is small. For the sake of completeness we also recall ([9]) that (1) (with ε = 1)
1isgloballywell-posedintheenergyspaceH thankstotheconservationoftheHamiltonian
1 α+2 42H(u) = k∇uk +kuk , provided α < . Recent results by Bourgain ([3]) extend this2 α+22 n−2
1−ηglobal well-posedness to H for an appropriate (small) value of η.
Althoughmostofthepreviouslyresultsapplyfornonintegervaluesofα,underappropriate
∗Laboratoire d’Analyse Num´erique, URA CNRS 189, Universit´e Pierre et Marie Curie, 4 place Jussieu
BP 187, 75 252 Paris Cedex, [e-mail]
1
R
Rconditions involving the dimension n, we will restrict ourselves to α ∈ 2 \{0}. In fact
one could replace the nonlinearity by any homogeneous polynomial of u,u¯, with degree
α + 1. Such restrictions are mostly technical, and getting around them requires some
lengthy computations which are not directly related to the equation and will be presented
elsewhere.
Our motivations in the present work are slightly different from what has previously been
done. Firstly, we aimata better understanding ofthe recent construction ([6, 7, 17, 12])of
self-similar solutions for (1). A self-similar solution is by definition invariant by the scaling
(2), and therefore cannot be obtained by these aforementioned results in Sobolev spaces.
However, [6] shows how to obtain solutions such that
β(3) supt ku(x,t)k <∞α+2
t
where β is chosen to preserve the scaling invariance, provided
β itΔ(4) supt ke u (x)k < ε .0 α+2 0
t
ε˜0 satisfies (4), thus giving a self-similar solutionDirect calculations ([6]) prove that 2
α|x|
1 x n n√u(x,t) = U( ). More generally, ε˜ could be replaced by a small C (S ) function2 0√ tαt
([17]). It should be noted that the profile U does not a priori conserve the same regularity
sα˙displayed by such initial values. But a natural extension to H is the homogeneous Besov
s ,∞α˙space B , and we aim at considering initial data in such a space and solutions bounded2
in time with values in that space. Let us recall how these spaces can be defined:
Z Z j+12X
s 2s j 2 2s j 2α α α˙ ˆ ˆf(x)∈ H ⇔ |ξ| |f(ξ)| dξ 2 |f(ξ)| dξ < +∞,
j2j
and one can relax this definition to set
Z j+12
s ,∞α 2s j 2α˙ ˆ(5) f(x)∈ B ⇔ sup2 |f(ξ)| dξ < +∞.2
jj 2
1 sα,∞˙From this definition, one can easily check that ∈ B , and thus solving the Cauchy2 2
|x|α
problem in that space will allow self-similar solutions.
Secondly, Bourgain noticed ([3]) how the small data theory in Sobolev spaces can be im-
proved in the following way. Consider M > 0; then, for the equation (1) in dimension
s ,∞2, if ku k < M, there exists an ε(M) such that ku k α < ε(M) will be enough to0 ˙ s 0 ˙αH B2
get a global solution. Our aim is to extend such results in a unified framework with the
self-similar solutions. This will allow us to gain insight for such solutions as well as for
more regular solutions.
2
N
t1 Definitions and theorems
For sake of completeness, we first recall one (of the many) definitions of homogeneous
Besov spaces ([13],[20]). In order to consider more general indices, one has to replace the
rough Fourier cut-off from (5) by a smoother one.
Definition 1
n cb bLet φ ∈ S( ) such that φ ≡ 1 in B(0,1) and φ ≡ 0 in B(0,2) ,
nj j ′ nφ (x) = 2 φ(2 x), S = φ ∗·, Δ =S −S . Let f be in S ( ).j j j j j+1 j
n n s,q˙• If s < , or if s = and q = 1, f belongs to B if and only if the following twopp p
conditions are satisfied
Pm
– The partial sum Δ (f) converge to f as a tempered distribution.j−m
js q
p– The sequence ǫ = 2 kΔ (f)k belongs to l .j j L
n n n• If s > , or s = and q = 1, let us denote m the greatest integer less than s− .
p p p
s,q˙Then B is the space of distributions f, modulo polynomials of degree less thanp
m+1, such that
P∞– We have f = Δ (f) for the quotient topology.j−∞
js q– The sequence ǫ = 2 kΔ (f)k p belongs to l .j j L
pLater on, we actually work with an easy extension of these spaces where the space L is
p,rreplaced by the (more general) Lorentz space L . We will denote such a modified space
s,q˙as B . We refer to [1, 13] for definition and detailed properties of Lorentz spaces. For
(p,r)
our purposes, it suffices to know they behave like Lebesgue spaces, while providing better
accuracy.
In the proofs, we will deal with dimension n = 2 and then briefly indicate the (easier) case
n≥ 3 (note that n = 1 could be dealt the same way as n= 2 as well). The main difference
between these two cases is the the end-point Strichartz estimates proved in [10], which
allow us to carry a shorter and somewhat simpler proof in dimensions where it holds. In
4either case, a restriction on α will appear, namely α > . The meaning of the restriction
n
on α will be clear from the proof of our main result.
Theorem 1
s ,∞4 α˙ s ,∞Let α > , α∈ 2 \{0}, u ∈ B , such that ku k α <C (α,n). Then there exists a˙0 0 02 Bn 2
global solution of (1) such that
∞ s ,∞α˙(6) u(x,t)∈ L (B ),t 2
′(7) u(x,t)−→ u (x) weakly (in the sense ofσ(S,S )).0
t→0
Moreover, this solution is unique under an additional assumption, to be explained later.
3
R
N
R4The uniqueness condition can be formulated easily in dimension 2 (or when α < ),
n−2
via weak Lebesgue spaces (which are a particular case of Lorentz spaces). We then have
uniqueness in a ball, defined as follows:
α(α+2)
(8) ku(x,t)k β,∞ α+2,∞ < C , β = .1L (L )xt 2
Remark that uniqueness holds without specifying condition (6), which comes as an addi-
tional property of the solutions. While it should be possible to obtain a similar uniqueness
4assumption in bigger dimensions (when α≥ ), the proof given below gives only unique-
n−2
ness in (a ball of) a space defined as the intersection of (6) and an additional space used
β,∞1to carry a fixed-point argument. Note that (8) is a relaxed version of (3), as ∈ L .1 t
βt
The restriction on a ball for uniqueness and condition (7) have more to do with the na-
ture of the spaces under consideration than with the equation itself (see [4, 14] for other
instances of such situations). Indeed we cannot have strong continuity at t = 0, and there-
fore we obtain a somewhat weaker result than what is usually meant for “well-posedness”.
The solutions obtained in theorem 1 verify various additional space-time estimates, which
translate nicely in term of estimates on the profile for a self-similar solution:
Theorem 2
2Under the assumptions of theorem 1, if moreover u is homogeneous of degree − , the0 α
solution is self-similar,
1 x
(9) u(x,t) = U(√ ),
2√
α tt
and its profile is such that
s ,qα˙(10) U(x)∈ B ,p
2 n n+ = , q≥ 2, (q,p) = (2,∞).with
p q 2
We will see how to recover and extend the known results for Sobolev spaces, to obtain
Theorem 3
s˙ αLet u ∈ H verify the hypothesis of Theorem 1. Then the global solution obtained by0
Theorem 1 is such that
sα˙(11) u(x,t)W