On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces

0.22 MB·

by  Van Duong Dinh

On the Cauchy problem for the nonlinear
semi-relativistic equation in Sobolev spaces
Van Duong Dinh
Abstract
Weprovedthelocalwell-posednessforthepower-typenonlinearsemi-relativisticorhalf-
7
1 waveequation(NLHW)inSobolevspaces. Ourproofsmainlybasesonthecontractionmap-
0 pingargumentusingStrichartzestimate. WealsoapplythetechniqueofChrist-Colliander-
2 Tao in [7] to provethe ill-posedness for (NLHW)in some cases of thesuper-critical range.

n
a 1 Introduction and main results
J

3
We consider the Cauchy semi-relativistic or half-wave equation posed on Rd,d 1, namely

≥
]
AP (cid:26) i∂tu(t,x)+Λuu((0t,,xx)) ==−u0µ(|xu)|,ν−1xu(t,Rxd),. (t,x)∈R×Rd, (NLHW)
∈
.
h where ν >1,µ 1 and Λ=√ ∆ is the Fourier multiplier by ξ . The number µ=1 (resp.
t ∈{± } − | |
a µ= 1)correspondstothedefocusingcase(resp. focusingcase). TheCauchyproblemproblem
−
m such as (NLHW) arises in various physical contexts, such as water waves (see e.g. [19]), and
[ gravitationalcollapse (see e.g. [11], [12]).
It is worth to noticing that if we set for λ>0,
1
v uλ(t,x)=λ−ν−11u(λ−1t,λ−1x),
2
5 then the (NLHW) is invariant under this scaling that is for T (0,+ ],
8 ∈ ∞
0 u solves (NLHW) on ( T,T) u solves (NLHW) on ( λT,λT).
λ
0 − ⇐⇒ −
. We also have
1
0 kuλ(0)kH˙γ =λd2−ν−11−γku0kH˙γ.
7
From this, we define the critical regularity exponent for (NLHW) by
1
:
v d 1
γ = . (1.1)
i c 2 − ν 1
X −
r One said that Hγ is sub-critical (critical, super-critical) if γ >γc (γ =γc, γ <γc) respectively.
a
Another important property of (NLHW) is that the following mass and energy are formally
conserved under the flow of the equation,
1 µ
M(u(t))= u(t,x)2dx, E(u(t))= Λ1/2u(t,x)2+ u(t,x)ν+1dx.
Z | | Z 2| | ν+1| |
The nonlinear half-wave equation (NLHW) has attracted a lot of works in a past decay (see
e.g. [12], [22], [14], [6], [13] andreferences therein). The main purpose of this note is to give the
well-posedandill-posedresultsfor(NLHW)inSobolevspaces. Theproofsofthewell-posedness
base on Strichartz estimate and the standard contraction argument. We thus only focus on the
cased 2whereStrichartzestimateappearsandjustrecalltheknownresultsinonedimensional
≥
case. Precisely, we prove the well-posedness in Hγ with
γ >1 1/max(ν 1,4) when d=2,
− − (1.2)
(cid:26) γ >d/2 1/max(ν 1,2) when d 3,
− − ≥
1
2 Van Duong Dinh
and of course with some regularity assumption on ν. This remains a gap between γ and
c
1 1/max(ν 1,4) when d = 2 and d/2 1/max(ν 1,2) when d 3. Next, we can apply
− − − − ≥
successfullytheargumentof[18](seealso[10])toprovethelocalwell-posednesswithsmalldata
scattering in the critical case provided ν >5 for d=2 and ν >3 for d 3. The cases ν (1,5]
≥ ∈
when d = 2 and ν (1,3] when d 3 still remain open. It requires another technique rather
∈ ≥
than just Strichartzestimate. Finally, using the technique of Christ-Colliander-Taogivenin [7],
we are able to prove the ill-posedness for (NLHW) in some cases of the super-critical range,
precisely in Hγ with γ (( , d/2] ( ,γ )) [0,γ ). We expect that the ill-posedness
c c
∈ −∞ − ∩ −∞ ∪
still holds in the range γ ( d/2,0) ( ,γ ) as for the nonlinear Schr¨odinger equation (see
c
∈ − ∩ −∞
[7]). But it is not clear to us how to prove it at the moment. Recently, Hong and Sire in [18]
usedthetechniqueof[7]withthepseudo-Galileantransformationtogettheill-posednessforthe
nonlinear fractional Schr¨odinger equation with negative exponent. Unfortunately, it seem to be
difficult to control the error of the pseudo-Galilean transformation in high Sobolev norms and
so far only restricted in one dimension. Note also that one has a sharp ill-posed result for the
cubic (NLHW) in 1D (see [6]). Specifically, one has the ill-posedness for γ <1/2which is larger
than γ . The proof of this result mainly bases on the relation with the cubic Szego¨ equation
c
which can not extend easily to general nonlinearity.
Let us firstly recall some known result about the local existence of (NLHW) in 1D. It is
well-known that (NLHW) is locally well-posed in Hγ(R) with γ >1/2 and of course with some
regularity condition using the energy method and the contraction mapping argument. When
ν =3,i.e. cubic nonlinearity,the (NLHW) is locallywell-posedinHγ(R) with γ 1/2(see e.g.
[22], [24]). This result is optimal in the sense that the (NLHW) is ill-posed in H≥γ(R) provided
γ < 1/2 (see e.g. [6]). To our knowledge, the local well-posedness for the generalized (NLHW)
in Hγ(R) with γ 1/2 seems to be an open question.
≤
Before stating our results, let us introduce some notations (see [15], Appendix, [28] or [3]).
Let ϕ C∞(Rd) be such that ϕ (ξ) = 1 for ξ 1 and supp(ϕ ) ξ Rd, ξ 2 . Set
ϕ(ξ) :=0 ∈ϕ (ξ0) ϕ (2ξ). We see th0at ϕ C∞(R|d|)≤and supp(ϕ) 0ξ ⊂R{d,∈1/2 |ξ| ≤ 2}. We
denote the0 Lit−tlew0ood-Paley projections∈by0P := χ (D),P :=⊂χ({N−∈1D) with≤N|=| ≤2k,}k Z
0 0 N
∈
where χ (D),χ(N−1D) are the Fourier multipliers by χ (ξ) and χ(N−1ξ) respectively. Given
0 0
γ R and 1 q , the Sobolev and Besov spaces are defined by
∈ ≤ ≤∞
Hqγ :=nu∈S′ | kukHqγ :=khΛiγukLq <∞o, hΛi:=p1+Λ2,
1/2
Bqγ :=nu∈S′ | kukBqγ :=kP0ukLq +(cid:16)NX∈2NN2γkPNuk2Lq(cid:17) <∞o,
where S′ is the space of tempered distributions. Now let S be a subspace of the Schwartz
0
space S consisting of functions φ satisfying Dαφˆ(0) = 0 for all α Nd whereˆis the Fourier
transform on S and S′ its topology dual space. One can see S′ as∈S′/P whe·re P is the set
0 0
of all polynomials on Rd. The homogeneous Sobolev and Besov spaces are defined by
H˙γ := u S′ u := Λγu < ,
q n ∈ 0 | k kH˙qγ k kLq ∞o
1/2
B˙γ := u S′ u := N2γ P u 2 < .
q n ∈ 0 | k kB˙qγ (cid:16)NX∈2Z k N kLq(cid:17) ∞o
It is easy to see that the norms kukBqγ and kukB˙qγ do not depend on the choice of ϕ0, and S0 is
dense in H˙γ,B˙γ. Under these settings, Hγ,Bγ,H˙γ and B˙γ are Banach spaces with the norms
q q q q q q
kukHqγ,kukBqγ,kukH˙qγ andkukB˙qγ respectively(seee.g. [28]). Inthenote,weshalluseHγ :=H2γ,
H˙γ := H˙γ. We note (see [3], [15]) that if 2 q < , then B˙γ H˙γ. The reverse inclusion
2 ≤ ∞ q ⊂ q
holds for 1 < r 2. In particular, B˙γ = H˙γ and B˙0 = H˙0 = L2. Moreover, if γ > 0, then
Hγ =Lq H˙γ a≤nd Bγ =Lq B˙γ. 2 2 2
q ∩ q q ∩ q
Throughout this sequel, a pair (p,q) is said to be admissible if
2 d 1 d 1
(p,q) [2, ]2, (p,q,d)=(2, ,3), + − − .
∈ ∞ 6 ∞ p q ≤ 2
Cauchy Problem Semi-relativistic Equation 3
We also denote for (p,q) [1, ]2,
∈ ∞
d d 1
γ = . (1.3)
p,q
2 − q − p
Our first result concerns the local well-posedness of (NLHW) in sub-critical case.
Theorem 1.1. Let γ 0 and ν >1 be such that (1.2), and also, if ν is not an odd integer,
≥
γ ν, (1.4)
⌈ ⌉≤
where γ is the smallest positive integer greater than or equal to γ. Then for all u Hγ, there
0
⌈ ⌉ ∈
exist T∗ (0, ] and a unique solution to (NLHW) satisfying
∈ ∞
u C([0,T∗),Hγ) Lp ([0,T∗),L∞),
∈ ∩ loc
for some p>max(ν 1,4) when d=2 and some p>max(ν 1,2) when d 3. Moreover, the
− − ≥
following properties hold:
(i) If T∗ < , then u(t) as t T∗.
Hγ
∞ k k →∞ →
(ii) u depends continuously on u in the following sense. There exists 0 < T < T∗ such that
0
if u u in Hγ and if u denotes the solution of (NLHW) with initial data u , then
0,n 0 n 0,n
→
0 < T < T∗(u ) for all n sufficiently large and u is bounded in La([0,T],Hγ−γa,b) for
0,n n b
any admissible pair (a,b) with b< . Moreover, u u in La([0,T],H−γa,b) as n .
∞ n → b →∞
In particular, u u in C([0,T],Hγ−ǫ) for all ǫ>0.
n
→
(iii) Let β > γ be such that if ν is not an odd integer, β ν. If u Hβ, then u
0
⌈ ⌉ ≤ ∈ ∈
C([0,T∗),Hβ).
ThecontinuousdependencecanbeimprovedtoholdinC([0,T],Hγ)ifweassumethatν >1
is an odd integer or γ ν 1 otherwise (see Remark 2.8).
⌈ ⌉≤ −
Theorem 1.2. Let
ν >5 when d=2,
(1.5)
(cid:26) γ >3 when d 3,
≥
and also, if ν is not an odd integer
γ ν. (1.6)
c
⌈ ⌉≤
Then for all u0 Hγc, there exist T∗ (0, ] and a unique solution to (NLHW) satisfying
∈ ∈ ∞
u C([0,T∗),Hγc) Lp ([0,T∗),Bγc−γp,q),
∈ ∩ loc q
where p = 4,q = when d = 2; 2 < p < ν 1,q = p⋆ = 2p/(p 2) when d = 3; p = 2,q =
∞ − −
2⋆ =2(d 1)/(d 3) when d 4. Moreover, if u <ε for some ε>0 small enough, then
T∗ = a−nd the−solution is sc≥attering in Hγc, i.ke.0tkhHe˙rγec exists u+ Hγc such that
∞ 0 ∈
lim u(t) eitΛu+ =0.
t→+∞k − 0kHγc
Our final result is the following ill-posedness for the (NLHW).
Theorem 1.3. Let ν > 1 be such that if ν is not an odd integer, ν k+1 for some integer
≥
k >d/2. Then (NLHW) is ill-posed in Hγ for γ (( , d/2] ( ,γ )) [0,γ ). Precisely,
c c
if γ (( , d/2] ( ,γ )) (0,γ ), then for∈an−y∞t>−0 the∩sol−ut∞ion map∪S u(0) u(t)
c c
∈ −∞ − ∩ −∞ ∪ ∋ 7→
of (NLHW) fails to be continuous at 0 in the Hγ topology. Moreover, if γ > 0, the solution
c
map fails to be uniformly continuous on L2.
4 Van Duong Dinh
2 Well-posedness
In this section, we will give the proofs of Theorem 1.1 and Theorem 1.2. Our proof is based on
the standard contraction mapping argument using Strichartz estimate and nonlinear fractional
derivative.
2.1 Linear estimates
In this subsection, we recall Strichartz estimate for the half-wave equation.
Theorem 2.1 ([2], [21]). Let d 2,γ R and u be a (weak) solution to the linear half-wave
≥ ∈
equation, namely
t
u(t)=eitΛu + ei(t−s)ΛF(s)ds,
0
Z
0
for some data u ,F. Then for all (p,q) and (a,b) admissible pairs,
0
kukLp(R,B˙qγ) .ku0kH˙γ+γp,q +kFkLa′(R,B˙qγ′+γp,q−γa′,b′−1), (2.1)
where γp,q and γa′,b′ are as in (1.3). In particular,
kukLp(R,B˙qγ−γp,q) .ku0kH˙γ +kFkL1(R,H˙γ). (2.2)
Here (a,a′) and (b,b′) are conjugate pairs.
The proof of the above result is based on the scaling argument and the Fourier transform of
spherical measure.
Corollary 2.2. Let d 2 and γ R. If u is a (weak) solution to the linear half-wave equation
≥ ∈
for some data u ,F, then for all (p,q) admissible satisfying q < ,
0
∞
kukLp(R,Hqγ−γp,q) .ku0kHγ +kFkL1(R,Hγ). (2.3)
Proof. We firstly remark that (2.2) together with the Littlewood-Paley theorem yield for any
(p,q) admissible satisfying q < ,
∞
kukLp(R,H˙qγ−γp,q) .ku0kH˙γ +kFkL1(R,H˙γ). (2.4)
We next write kukLp(R,Hqγ−γp,q) =khΛiγ−γp,qukLp(R,Lq) and apply (2.4) with γ =γp,q to get
kukLp(R,Hqγ−γp,q) .khΛiγ−γp,qu0kH˙γp,q +khΛiγ−γp,qFkL1(R,H˙γp,q).
Theestimate(2.3)thenfollowsbyusingthefactthatγ >0forall(p,q)isadmissiblesatisfying
p,q
q < .
∞
2.2 Nonlinear estimates
Inthissubsection,werecallsomenonlinearfractionalderivativeestimatesrelatedtoourpurpose.
Let us start with the following fractional Leibniz rule (or Kato-Ponce inequality).
Proposition 2.3. Let γ 0,1<r < and 1<p ,p ,q ,q satisfying
1 2 1 2
≥ ∞ ≤∞
1 1 1 1 1
= + = + .
r p q p q
1 1 2 2
Then there exists C =C(d,γ,r,p ,q ,p ,q )>0 such that for all u,v S,
1 1 2 2
∈
Λγ(uv) C Λγu v + u Λγv , (2.5)
k kLr ≤ (cid:16)k kLp1k kLq1 k kLp2k kLq2(cid:17)
Λ γ(uv) C Λ γu v + u Λ γv . (2.6)
kh i kLr ≤ (cid:16)kh i kLp1k kLq1 k kLp2kh i kLq2(cid:17)
Cauchy Problem Semi-relativistic Equation 5
We refer to [17] to the proof of the above inequality and more general result. We also have
the following fractional chain rule.
Proposition 2.4. Let F C1(C,C) and G C(C,R+) such that F(0)=0 and
∈ ∈
F′(θz+(1 θ)ζ) µ(θ)(G(z)+G(ζ)), z,ζ C, 0 θ 1,
| − |≤ ∈ ≤ ≤
where µ L1((0,1)). Then for γ (0,1) and 1<r,p< , 1<q satisfying
∈ ∈ ∞ ≤∞
1 1 1
= + ,
r p q
there exists C =C(d,µ,γ,r,p,q)>0 such that for all u S,
∈
ΛγF(u) C F′(u) Λγu , (2.7)
Lr Lq Lp
k k ≤ k k k k
Λ γF(u) C F′(u) Λ γu . (2.8)
Lr Lq Lp
kh i k ≤ k k kh i k
We refer the reader to [8] (see also [25]) for the proof of (2.7) and [27] for (2.8). Combining
the fractional Leibniz rule and the fractional chain rule, one has the following result (see [20]).
Lemma 2.5. Let F Ck(C,C),k N 0 . Assume that there is ν k such that
∈ ∈ \{ } ≥
DiF(z) C z ν−i, z C, i=1,2,....,k.
| |≤ | | ∈
Then for γ [0,k] and 1 < r,p < , 1 < q satisfying 1 = 1 + ν−1, there exists
∈ ∞ ≤ ∞ r p q
C =C(d,ν,γ,r,p,q)>0 such that for all u S,
∈
ΛγF(u) C u ν−1 Λγu , (2.9)
k kLr ≤ k kLq k kLp
Λ γF(u) C u ν−1 Λ γu . (2.10)
kh i kLr ≤ k kLq kh i kLp
Moreover, if F is a polynomial in u and u, then (2.9) and (2.10) hold true for any γ 0.
≥
Corollary 2.6. Let F(z)= z ν−1z with ν > 1, γ 0 and 1 <r,p < , 1< q satisfying
| | ≥ ∞ ≤∞
1 = 1 + ν−1.
r p q
(i) If ν is an odd integer or γ ν otherwise, then there exists C = C(d,ν,γ,r,p,q) > 0
such that for all u S, ⌈ ⌉ ≤
∈ F(u) C u ν−1 u .
k kH˙rγ ≤ k kLq k kH˙pγ
A similar estimate holds with H˙γ,H˙γ-norms are replaced by Hγ,Hγ-norms respectively.
r p r p
(ii) If ν is an odd integer or γ ν 1 otherwise, then there exists C =C(d,ν,γ,r,p,q)>0
such that for all u,v S⌈, ⌉≤ −
∈
F(u) F(v) C ( u ν−1+ v ν−1) u v
k − kH˙rγ ≤ (cid:16) k kLq k kLq k − kH˙pγ
+( u ν−2+ v ν−2)( u + v ) u v .
k kLq k kLq k kH˙pγ k kH˙pγ k − kLq(cid:17)
A similar estimate holds with H˙γ,H˙γ-norms are replaced by Hγ,Hγ-norms respectively.
r p r p
A next result will give a good control on the nonlinear term which allows us to use the
contraction mapping argument.
Lemma 2.7. Let ν be as in Theorem 1.2 and γ as in (1.1). Then
c
u 4 u ν−5 when d=2,
kukLν−ν−11(R,L∞) . kkukkLpL4p((RR,,BB˙˙∞pγγ⋆cc−−γγ4p,,∞p⋆k)kukkLLν∞−∞(1R(−R,B,p˙B2γ˙c2γ)c) where 2<p<ν−1 when d=3,
u 2 u ν−3 when d 4,
 k kL2(R,B˙2γ⋆c−γ2,2⋆)k kL∞(R,B˙2γc) ≥
where p⋆ =2p/(p 2) and 2⋆ =2(d 1)/(d 3).
− − −
6 Van Duong Dinh
The above lemma follows the same spirit as Lemma 3.5 in [18] (see also [10]) using the
argument of Lemma 3.1 in [9].
Proof. We only give a sketch of the proof in the case d 4, the cases d = 2,3 are treated
similarly. By interpolation, we can assume that ν 1=m/≥n>2,m,n N with gcd(m,n)=1.
− ∈
We proceed as in [18] and set
cN(t)=Nγc−γ2,2⋆kPNu(t)kL2⋆(Rd), c′N(t)=NγckPNu(t)kL2(Rd).
By Bernstein’s inequality, we have
kPNu(t)kL∞(Rd) .N2d⋆−γc+γ2,2⋆cN(t)=Nmn−12cN(t), (2.11)
kPNu(t)kL∞(Rd) .Nd2−γcc′N(t)=Nmnc′N(t).
This implies that for θ (0,1) which will be chosen later,
∈
kPNu(t)kL∞(Rd) .Nmn−θ2(cN(t))θ(c′N(t))1−θ. (2.12)
We next use
m
m
A(t):=(cid:16)NX∈2ZkPNu(t)kL∞(Rd)(cid:17) .N1≥X···≥NmjY=1kPNju(t)kL∞(Rd).
Estimating the n highest frequencies by (2.11) and the rest by (2.12), we get
n m
A(t). Nmn−12c (t) Nmn−θ2(c (t))θ(c′ (t))1−θ .
N1≥X···≥Nm(cid:16)jY=1 j Nj (cid:17)(cid:16)j=Yn+1 j Nj Nj (cid:17)
For an arbitrary δ >0, we set
c˜N(t)= min(N/N′,N′/N)δcN′(t), c˜′N(t)= min(N/N′,N′/N)δc′N′(t).
NX′∈2Z NX′∈2Z
Using the factthat c (t) c˜ (t)andc˜ (t).(N /N )δc˜ (t) forj =2,...,mandsimilarlyfor
N ≤ N Nj 1 j N1
primes, we see that
n m
A(t). Nmn−12(N /N )δc˜ (t) Nmn−θ2(N /N )δ(c˜ (t))θ(c˜′ (t))1−θ .
N1≥X···≥Nm(cid:16)jY=1 j 1 j N1 (cid:17)(cid:16)j=Yn+1 j 1 j N1 N1 (cid:17)
We can rewrite the above quantity in the right hand side as
m n
Nmn−σ2θ−δ Nmn−12−δ Nmn−21+(m−1)δ(c˜ (t))n+(m−n)θ(c˜′ (t))(m−n)(1−θ).
N1≥X···≥Nm(cid:16)j=Yn+1 j (cid:17)(cid:16)jY=2 j (cid:17) 1 N1 N1
By choosing θ =1/(ν 2) (0,1) and δ >0 so that
− ∈
n θ n 1 m 2n
δ >0, +(m 1)δ <0 or δ < − .
m − 2 − m − 2 − 2m(m 1)
−
Here condition ν >3 ensures that m 2n>0. Summing in N , then in N ,..., then in N ,
m m−1 2
−
we have
A(t). (c˜ (t))2n(c˜′ (t))(ν−3)n.
N1 N1
NX1∈2Z
The Ho¨lder inequality with the fact that (ν 3)n 1 implies
− ≥
A(t).k(c˜(t))2nkℓ2(2Z)k(c˜′(t))(ν−3)nkℓ2(2Z)
= c˜(t) 2n c˜′(t) (ν−3)n c˜(t) 2n c˜′(t) (ν−3)n,
k kℓ4n(2Z)k kℓ2(ν−3)n(2Z) ≤k kℓ2(2Z)k kℓ2(2Z)
Cauchy Problem Semi-relativistic Equation 7
1/q
where kc˜(t)kℓq(2Z) := (cid:16) N∈2Z|c˜N(t)|q(cid:17) and similarly for prime. The Minkowski inequality
then implies P
A(t). c(t) 2n c′(t) (ν−3)n.
k kℓ2(2Z)k kℓ2(2Z)
This implies that A(t)<∞ for amost allwhere t, hence that NkPNu(t)kL∞(Rd) <∞. There-
fore NkPNu(t)kL∞(Rd) convergesin L∞(Rd). Since it convePrgesto u in the ditribution sense,
so thPe limit is u(t). Thus
u ν−1 = u(t) m/n dt. c(t) 2 c′(t) ν−3 dt
k kLν−1(R,L∞(Rd)) ZRk kL∞(Rd) ZRk kℓ2(2Z)k kℓ2(2Z)
. c 2 c′ ν−3 = u 2 u ν−3 .
k kLpRℓ2(2Z)k kL∞R ℓ2(2Z) k kLp(R,B˙2γ⋆c−γ2,2⋆(Rd))k kL∞(R,B˙2γc(Rd))
The proof is complete.
2.3 Proof of Theorem 1.1
We now give the proof of Theorem1.1 by using the standardfixed point argument in a suitable
Banach space. Thanks to (1.2), we are able to choose p > max(ν 1,4) when d = 2 and
−
p>max(ν 1,2) when d 3 such that γ >d/2 1/p and then choose q [2, ) such that
− ≥ − ∈ ∞
2 d 1 d 1
+ − − .
p q ≤ 2
Step 1. Existence. Let us consider
X :=nu∈L∞(I,Hγ)∩Lp(I,Hqγ−γp,q) | kukL∞(I,Hγ)+kukLp(I,Hqγ−γp,q) ≤Mo,
equipped with the distance
d(u,v):=ku−vkL∞(I,L2)+ku−vkLp(I,Hq−γp,q),
where I =[0,T] and M,T >0 to be chosen later. By the Duhamel formula, it suffices to prove
that the functional
t
Φ(u)(t)=eitΛu +iµ ei(t−s)Λ u(s)ν−1u(s)ds (2.13)
0
Z | |
0
is a contraction on (X,d). The Strichartz estimate (2.3) yields
kΦ(u)kL∞(I,Hγ)+kΦ(u)kLp(I,Hqγ−γp,q) .ku0kHγ +kF(u)kL1(I,Hγ),
kΦ(u)−Φ(v)kL∞(I,L2)+kΦ(u)−Φ(v)kLp(I,Hq−γp,q) .kF(u)−F(v)kL1(I,L2),
where F(u)= uν−1u and similarly for F(v). By our assumptions on ν, Corollary 2.6 gives
| |
kF(u)kL1(I,Hγ) .kukLν−ν−11(I,L∞)kukL∞(I,Hγ) .T1−ν−p1kukLν−p(1I,L∞)kukL∞(I,Hγ), (2.14)
kF(u)−F(v)kL1(I,L2) .(cid:16)kukLν−ν−11(I,L∞)+kvkLν−ν−11(I,L∞)(cid:17)ku−vkL∞(I,L2)
.T1−ν−p1(cid:16)kukLν−p(1I,L∞)+kvkLν−p(1I,L∞)(cid:17)ku−vkL∞(I,L2). (2.15)
The Sobolev embedding with the fact that γ γ > d/q implies Lp(I,Hγ−γp,q) Lp(I,L∞).
p,q q
− ⊂
Thus, we get
kΦ(u)kL∞(I,Hγ)+kΦ(u)kLp(I,Hqγ−γp,q) .ku0kHγ +T1−ν−p1kukLν−p(1I,Hqγ−γp,q)kukL∞(I,Hγ),
and
d(Φ(u),Φ(v)).T1−ν−p1(cid:16)kukLν−p(1I,Hqγ−γp,q)+kvkLν−p(1I,Hqγ−γp,q)(cid:17)ku−vkL∞(I,L2).
8 Van Duong Dinh
This shows that for all u,v X, there exists C >0 independent of u Hγ and T such that
0
∈ ∈
kΦ(u)kL∞(I,Hγ)+kΦ(u)kLp(I,Hqγ−γp,q) ≤Cku0kHγ +CT1−ν−p1Mν,
d(Φ(u),Φ(v)) CT1−ν−p1Mν−1d(u,v).
≤
Therefore,ifwesetM =2Cku0kHγ andchooseT >0smallenoughsothatCT1−ν−p1Mν−1 ≤ 21,
then X is stable by Φ and Φ is a contraction on X. By the fixed point theorem, there exists a
unique u X so that Φ(u)=u.
∈
Step 2. Uniqueness. Consideru,v C(I,Hγ) Lp(I,L∞)two solutionsof(NLHW). Since the
∈ ∩
uniqueness is a local property (see [5]), it suffices to show u = v for T is small. We have from
(2.15) that
d(u,v)≤CT1−ν−p1(cid:16)kukLν−p(1I,L∞)+kvkLν−p(1I,L∞)(cid:17)d(u,v).
Since u Lp(I,L∞) is small if T is small and similarly for v, we see that if T >0 small enough,
k k
1
d(u,v) d(u,v) or u=v.
≤ 2
Step 3. Item (i). Since the time of existence constructed in Step 1 only depends on Hγ-norm
of the initial data. The blowup alternative follows by standard argument (see e.g. [5]).
Step 4. Item (ii). Let u u in Hγ and C,T =T(u ) be as in Step 1. Set M =4C u .
0,n 0 0 0 Hγ
→ k k
It follows that 2C u M for sufficiently large n. Thus the solution u constructed in
0,n Hγ n
k k ≤
Step1 belongsto X withT =T(u )for nlargeenough. We havefromStrichartzestimate (2.3)
0
and (2.14) that
kukLa(I,Hγ−γa,b) .ku0kHγ +T1−ν−p1kukLν−p(1I,L∞)kukL∞(I,Hγ),
b
provided (a,b) is admissible and b < . This shows the boundedness of u in La(I,Hγ−γa,b).
∞ n b
We also have from (2.15) and the choice of T that
1
d(u ,u) C u u + d(u ,u) or d(u ,u) 2C u u .
n ≤ k 0,n− 0kL2 2 n n ≤ k 0,n− 0kL2
This yields that u u in L∞(I,L2) Lp(I,H−γp,q). Strichartz estimate (2.3) again implies
n q
that u u in La(→I,H−γa,b) for any∩admissible pair (a,b) with b < . The convergence in
n → b ∞
C(I,Hγ−ǫ) follows from the boundedness in L∞(I,Hγ), the convergencein L∞(I,L2) and that
1−ǫ ǫ
kukHγ−ǫ ≤kukHγγkukLγ2.
Step 5. Item (iii). If u Hβ for some β > γ satisfying β ν if ν > 1 is not an odd
0
∈ ⌈ ⌉ ≤
integer,then Step1 showsthe existence ofHβ solutiondefined onsomemaximalinterval[0,T).
Since Hβ solution is also a Hγ solution, thus T T∗. Suppose that T <T∗. Then the unitary
≤
property of eitΛ and Lemma imply that
t
u(t) u +C u(s) ν−1 u(s) ds,
k kHβ ≤k 0kHβ Z k kL∞k kHβ
0
for all 0 t<T. The Gronwall’s inequality then gives
≤
t
u(t) u exp C u(s) ν−1ds ,
k kHβ ≤k 0kHβ (cid:16) Z0 k kL∞ (cid:17)
for all 0 t<T. Using the fact that u Lν−1([0,T∗),L∞), we see that limsup u(t) <
≤ ∈ loc k kHβ ∞
as t T which is a contradiction to the blowup alternative in Hβ. (cid:3)
→
Remark 2.8. If we assume that ν >1 is an odd integer or
γ ν 1
⌈ ⌉≤ −
Cauchy Problem Semi-relativistic Equation 9
otherwise, then the continuous dependence holds in C(I,Hγ). To see this, we consider X as
above equipped with the following metric
d(u,v):=ku−vkL∞(I,Hγ)+ku−vkLp(I,Hqγ−γp,q).
Using Item (ii) of Corollary 2.6, we have
kF(u)−F(v)kL1(I,Hγ) .(kukLν−ν−11(I,L∞)+kvkLν−ν−11(I,L∞))ku−vkL∞(I,Hγ)
+(kukLν−ν−21(I,L∞)+kvkLν−ν−21(I,L∞))(kukL∞(I,Hγ)+kvkL∞(I,Hγ))ku−vkLν−1(I,L∞).
The Sobolev embedding then implies for all u,v X,
∈
d(Φ(u),Φ(v)).T1−ν−p1Mν−1d(u,v).
Therefore, the continuity in C(I,Hγ) follows as in Step 4.
2.4 Proof of Theorem 1.2
We now turn to the proof of the local well-posedness and small data scattering in critical case
by following the same argument as in [10].
Step 1. Existence. We only treat for d 4, the ones for d = 2,d = 3 are completely similar.
≥
Let us consider
X :=nu∈L∞(I,Hγc)∩L2(I,B2γ⋆c−γ2,2⋆) | kukL∞(I,H˙γc) ≤M,kukL2(I,B˙2γ⋆c−γ2,2⋆) ≤No,
equipped with the distance
d(u,v):=ku−vkL∞(I,L2)+ku−vkL2(I,B˙2−⋆γ2,2⋆),
where I = [0,T] and T,M,N > 0 will be chosen later. One can check (see e.g. [4] or [5]) that
(X,d) is a complete metric space. Using the Duhamel formula
t
Φ(u)(t)=eitΛu +iµ ei(t−s)Λ u(s)ν−1u(s)ds=:u (t)+u (t), (2.16)
0 hom inh
Z | |
0
the Strichartz estimate (2.2) yields
kuhomkL2(I,B˙2γ⋆c−γ2,2⋆) .ku0kH˙γc.
A similar estimate holds for kuhomkL∞(I,H˙γc). We see that kuhomkL2(I,B˙2γ⋆c−γ2,2⋆) ≤ ε for some
ε > 0 small enough which will be chosen later, provided that either u is small or it is
k 0kH˙γc
satisfied some T > 0 small enough by the dominated convergence theorem. Therefore, we can
take T = in the first case and T be this finite time in the second. On the other hand, using
∞
again (2.2), we have
kuinhkL2(I,B˙2γ⋆c−γ2,2⋆) .kF(u)kL1(I,H˙γc).
A same estimate holds for u . Corollary 2.6 and Lemma 2.7 give
k inhkL∞(I,H˙γc)
F(u) . u ν−1 u . u 2 u ν−2 . (2.17)
k kL1(I,H˙γc) k kLν−1(I,L∞)k kL∞(I,H˙γc) k kL2(I,B˙2γ⋆c−γ2,2⋆)k kL∞(I,H˙γc)
Similarly, we have
kF(u)−F(v)kL1(I,L2) .(cid:16)kukLν−ν−11(I,L∞)+kvkLν−ν−11(I,L∞)(cid:17)ku−vkL∞(I,L2) (2.18)
.(cid:16)kuk2L2(I,B˙2γ⋆c−γ2,2⋆)kukLν−∞3(I,H˙γc)+kvk2L2(I,B˙2γ⋆c−γ2,2⋆)kvkLν−∞3(I,H˙γc)(cid:17)ku−vkL∞(I,L2).
10 Van Duong Dinh
This implies for all u,v X, there exists C >0 independent of u0 Hγc such that
∈ ∈
kΦ(u)kL2(I,B˙2γ⋆c−γ2,2⋆) ≤ε+CN2Mν−2,
Φ(u) C u +CN2Mν−2,
k kL∞(I,H˙γc) ≤ k 0kH˙γc
d(Φ(u),Φ(v)) CN2Mν−3d(u,v).
≤
Now by setting N = 2ε and M = 2C u and choosing ε > 0 small enough such that
k 0kH˙γc
CN2Mν−3 min 1/2,ε/M ,we see thatX is stable byΦ andΦ is acontractiononX. Bythe
≤ { }
fixed point theorem, there exists a unique solutionu X to (NLHW). Note that when u
∈ k 0kH˙γc
is small enough, we can take T = .
Step 2. Uniqueness. The unique∞ness in C∞(I,Hγc) L2(I,Bγc−γ2,2⋆) follows as in Step 2 of
∩ 2⋆
the proof of Theorem 1.1 using (2.18). Here kukL2(I,B˙2γ⋆c−γ2,2⋆) can be small as T is small.
Step 3. Scattering. The global existence when u is small is given in Step 1. It remains
k 0kH˙γc
to show the scattering property. Thanks to (2.17), we see that
t2
ke−it2Λu(t2)−e−it1Λu(t1)kH˙γc =(cid:13)(cid:13)iµZt1 e−isΛ(|u|ν−1u)(s)ds(cid:13)(cid:13)H˙γc
F(u) (cid:13). u 2 u(cid:13)ν−2 0 (2.19)
≤k kL1([t1,t2],H˙γc) k kL2([t1,t2],B˙2γ⋆c−γ2,2⋆)k kL∞([t1,t2],H˙γc) →
as t ,t + . We have from (2.18) that
1 2
→ ∞
ke−it2Λu(t2)−e−it1Λu(t1)kL2 .kuk2L2([t1,t2],B˙2γ⋆c−γ2,2⋆)kukLν−∞3([t1,t2],H˙γc)kukL∞([t1,t2],L2), (2.20)
which also tends to zero as t ,t + . This implies that the limit
1 2
→ ∞
u+ := lim e−itΛu(t)
0
t→+∞
exists in Hγc. Moreover,we have
+∞
u(t) eitΛu+ = iµ ei(t−s)ΛF(u(s))ds.
− 0 − Z
t
The unitary property of eitΛ in L2, (2.19) and (2.20) imply that u(t) eitΛu+ 0 when
k − 0kHγc →
t + . This completes the proof of Theorem 1.2. (cid:3)
→ ∞
3 Ill-posedness
In this section, we will give the proof of Theorem 1.3. We follow closely the argument of [7]
using small dispersion analysis and decoherence arguments.
3.1 Small dispersion analysis
Now let us consider for 0<δ 1 the following equation
≪
i∂ φ(t,x)+δΛφ(t,x) = µφν−1φ(t,x), (t,x) R Rd,
(cid:26) t φ(0,x) =−φ0(|x)|, x Rd. ∈ × (3.1)
∈
Note that (3.1) can be transformed back to (NLHW) by using
u(t,x):=φ(t,δx).
Lemma 3.1. Let k > d/2 be an integer. If ν is not an odd integer, then we assume also
the additional regularity condition ν k + 1. Let φ be a Schwartz function. Then there
0
≥