Self similar solutions and Besov spaces for semi linear Schrodinger and wave equations

pefav - Fabrice Planchon

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Self-similar solutions and Besov spaces for semi-linear Schrodinger and wave equations Fabrice Planchon ? Abstract We prove that the initial value problem for the semi-linear Schrodinger and wave equations is well-posed in the Besov space B˙ n 2? 2 p ,∞ 2 (R n), when the nonlinearity is of type up, for p ? N. This allows us to obtain self-similar solutions, as well as to recover previously known results for the solutions under weaker smallness assumptions on the data. Introduction In this introduction we focus on the Schrodinger equation; remarks relevant to the wave equation will be made in the last section. We are interested in the Cauchy problem { i∂u∂t + ∆u = ±u p, u(x, 0) = u0(x), x ? Rn , t ≥ 0, (1) where n ≥ 2. The exact form of the non-linearity is relevant only with respect to the methods which will be used. One can deal with more general non-linearities, but this requires a lot more technicalities which are irrelevant to the equation itself and have to do with the composition of Besov spaces. By restricting ourselves to non-linearities of type u¯p1up2 where p1 and p2 are integers, we don't have to worry about further regularity assumptions on the non-linearity, and having an estimate on the non-linearity up gives immediatly an estimate on up ? vp.

besov spaces

global solution

can construct initial

let u0 ?

sobolev spaces

analysis can

regularity assumptions

?j ?

Sujets

Planchon

Sobolev space

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Publié par	pefav
Nombre de lectures	18
Langue	English

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Self-similar solutions and Besov spaces for semi-linear
Schr¨odinger and wave equations
∗Fabrice Planchon
Abstract
We prove that the initial value problem for the semi-linear Schro¨dinger and wave
n 2
− ,∞
2 p n˙equations is well-posed in the Besov space B ( ), when the nonlinearity is of2
ptypeu , forp∈ . Thisallows ustoobtain self-similar solutions, aswell astorecover
previouslyknownresultsforthesolutions underweaker smallnessassumptionsonthe
data.
Introduction
In this introduction we focus on the Schr¨odinger equation; remarks relevant to the wave
equation will be made in the last section. We are interested in the Cauchy problem
(
∂u pi +Δu = ±u ,
(1) ∂t
nu(x,0) = u (x),x∈ , t≥ 0,0
where n ≥ 2. The exact form of the non-linearity is relevant only with respect to the
methods which will be used. One can deal with more general non-linearities, but this
requires a lot more technicalities which are irrelevant to the equation itself and have to
do with the composition of Besov spaces. By restricting ourselves to non-linearities of
p p21type u¯ u where p1 and p2 are integers, we don’t have to worry about further regularity
passumptions on the non-linearity, and having an estimate on the non-linearity u gives
p pimmediatly an estimate on u −v .
The following invariance by scaling of (1) will play an important role
(
2
p−1u (x) −→ u (x) = λ u (λx)0 0,λ 0(2) 2
2p−1u(x,t) −→ u (x,t) = λ u(λx,λ t).λ
n 2 sp˙Let s be such that s − =− . The homogeneous Sobolev space H is expected to bep p 2 p−1
the “critical” space for well-posedness as its norm is invariant by scaling (2). This result
∗Laboratoire d’Analyse Num´erique, URA CNRS 189, Universit´e Pierre et Marie Curie, 4 place Jussieu
BP 187, 75 252 Paris Cedex
1
R
R
Nsp˙is already known, see ([5]), there exists a (weak) solution of (1) which is C([0,T],H ),
s˙ punique under an additional assumption. Such a solution is global in time, if the H norm
of the initial data is small. There are of course other results on global well-posedness for
appropriate non-linearities, and we refer the reader (without any claim to be exhaustive)
to ([8]) or to ([3]) for recent developments.
Our present motivations are of a diﬀerent nature. They came out of understanding some
recent work on self-similar solutions for (1) in ([6, 7, 20, 12]). A self-similar solution is
by deﬁnition a solution which is invariant by scaling (2). Since this forces the initial data
to be homogeneous, such solutions cannot be obtained by the well-posedness results in
Sobolev spaces. In [6], under some restrictions on p, the authors introduce a functional
space, namely the space of functions u such that
β(3) supt ku(x,t)k <∞p+1
t
in which β is to be chosen to preserve the scaling invariance. The authors construct
solutions by a ﬁxed point argument in such a space, provided
β itΔ(4) supt ke u (x)k < ε .0 p+1 0
t
ε˜0By direct computations, one can prove that u (x) = satisﬁes (4), thus giving a self-0 2
p−1|x|
1 x√similar solution u(x,t) = U( ). More generally, ε˜ could be replaced by a small2√ 0tp−1t
n nC (S ) function ([20]).
Our goal will be to draw a connection between such a construction and the usual one in
s˙ αSobolevspaces. Having this inmind, a naturalextension toH isthe homogeneous Besov
s ,∞p˙space B , and unlike its Sobolev counterpart, it contains homogeneous functions. Let2
us recall
Z ZX
s 2s 2 2s 2α α α˙ ˆ ˆf(x)∈ H ⇔ |ξ| |f(ξ)| dξ 2 |f(ξ)| dξ < +∞,
j j+12 <|ξ|<2j
and one can weaken this requirement to
Z
s ,∞α 2s 2α ˆ˙(5) f(x)∈ B ⇔ sup2 |f(ξ)| dξ < +∞.2
j j+1j 2 <|ξ|<2
s ,∞1 α˙From this deﬁnition, we obtain immediately that ∈ B . Thus solving the Cauchy2 2
α|x|
problem (1) in such a space will, among other things, give self-similar solutions.
In the next section, we will treat the Schr¨odinger equation, and in the last one, the wave
equation for which an equivalent analysis can be carried. To end this section let us recall
the deﬁnition of Besov spaces, their characterizations via frequency localization, and some
useful results on Besov spaces.
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 inS ( ).j j j j j+1 j
2
R
R
tn 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 s,q˙• If s > , or s = and q = 1, let us denote m =E(s− ). Then B is the space ofpp p p
distributions f, modulo polynomials of degree less than m+1, such that
P∞
– We have f = Δ (f) for the quotient topology.j−∞
js q
p– The sequence ǫ = 2 kΔ (f)k belongs to ℓ .j j L
pNote that the choice of L as the “base” space is in no way an obligation. We will later
p p,ruse more general Besov spaces, with L replaced by the Lorentz space L . We will denote
s,q˙such a modiﬁedspace asB . Werefer thereader to [1,14]forthedeﬁnition anddetailed
(p,r)
properties of Lorentz spaces.
Another type of space will also be of help
Definition 2
ρ s,q˙Let u(x,t)∈S. We will say that u∈L (B ) iﬀt (p,r)
js q
ρ p,r(6) 2 kΔ uk = ε ∈ ℓ .j jL (L )xt
Lastly, we recall two lemmas, which allow for an easy characterization of Besov spaces,
depending on the sign of s.
Lemma 1
Let s> 0, E a Banach functional space, q∈ [0,+∞], and deﬁne
s,q js q˙f ∈ B ≡ 2 kΔ fk = ε ∈ ℓj E jE
P
s,qj js qˆ ˙Then, if f = f , where suppf ∈ B(0,2 ) and (2 kf k ) ∈ ℓ , we have f ∈ B .j j j E jj E
Its counterpart for s< 0 reads
Lemma 2
s,q s,q˙ ˙Let s< 0, B deﬁned as in lemma 1. Then, an equivalent characterization of f ∈ B isE E
js q(7) 2 kS uk = ε ∈ ℓ .j E j
We omit both proofs, which involve summation over large or small frequencies along with
Young inequality for discrete sequences.
31 The semi-linear Schr¨odinger equation
In the introduction we did not set any restriction (other than being an integer) on the
s ⋆pvalue of p. However, well-posedness in H holds only if p≥ p = 1+4/n, which amounts
⋆to dealing with a situation where s ≥ 0. For the critical value p , the equation (1) isp
2invariant by the pseudo-conformal transformation, and well-posed in L . We will have
⋆to restrict ourselves to strictly positive regularity, namely s > 0, and thus p > p , forp
technical reasons which will be clear from the proof. Such a restriction doesn’t appear in
[20], where one can go up to p > 1 +2/n. However the corresponding Cauchy problem
spin the appropriate Sobolev space H is not known to be well-posed, as s < 0, and ourp
approach fails for such cases. Let us state our main result
Theorem 1
s ,∞p⋆ ˙ s ,∞Let n≥ 2, p∈ , p > p , u ∈ B , such that ku k p < ǫ (p,n). Then there exists a0 0 ˙ 02 B2
global solution of (1) such that
s ,∞p∞ ˙(8) u(x,t)∈ L (B ),t 2
(9) u(x,t)−→ u (x) weakly.0
t→0
Moreover, this solution is unique under the condition
s ,∞(10) ku(x,t)k p <ǫ ,2 ˙ 1L (B )
t 2n( ,2)
n−2
for n≥ 3 and
(11) ku(x,t)k 2 < ǫ˜ ,p −1 1
,∞ p+1,∞2L (L )xt
for n = 2.
The uniqueness conditions (10) and (11), as customary in problems for which solutions are
obtained by a ﬁxed-point argument, are related to the auxiliary spaces needed for such an
argument. We refer to [15] for a more detailed discussion on this issue, as there are several
ways to chose such an auxiliary space. The condition (9) relates to the Besov spaces we
consider (see [4, 17] for discussions on such problems). Indeed strong continuity at t = 0 is
forbidden, and therefore we obtain a somewhat weaker result than what is usually meant
for “well-posedness”. However, if one has some additional regularity on the initial data,
then this regularity is preserved for the solution, namely we obtain
Theorem 2
s˙ pLet u ∈ H verify the hypothesis of Theorem 1. Then the global solution obtained by0
Theorem 1 is such that
sp˙(12) u(x,t)∈ C (H ).t
4
NThis result can be viewed as an extension of global well-posedness in Sobolev spaces for
small data, for one can construct initial data with an arbit