Regularity and global behavior of solutions to
nonlinear Schrodinger equations
著者
Sato Takuya
学位授与機関
Tohoku University
博 士 論 文
Regularity and global behavior of solutions to
nonlinear Schr¨
odinger equations
佐藤
拓也
Regularity and global behavior of solutions to
nonlinear Schr¨
odinger equations
A thesis presented
by
Takuya Sato
to
Mathematical Institute
for the degree of
Doctor of Science
Tohoku University
Sendai, Japan
Contents
1 Introduction 1
1.1 The nonlinear Schr¨odinger equation 1 1.2 Mass resonance and analytic smoothing effect for the nonlinear
Schr¨odinger system 2
1.3 The lower bound estimate for the dissipative nonlinear Schr¨odinger
equation 7
1.4 L2-decay for the dissipative nonlinear Schr¨odinger equation 10
2 Preliminaries 15
2.1 The free Schr¨odinger equation 15
2.2 Elementary inequalities 17
2.3 The Banach fixed point theorem 17
3 Analytic smoothing effect for the nonlinear Schr¨odinger system
with general mass resonance 19
3.1 An operator associated an analytic continuation 19
3.2 Proof of Theorem 1.2.2 20
4 L2-lower bound for the dissipative nonlinear Schr¨odinger equation 23
4.1 Proof of Theorem 1.3.1 23
4.2 The energy estimate for the dissipative nonlinear Schr¨odinger equation 24
4.3 Proof of Theorem 1.3.2 26
5 L2-decay for the dissipative nonlinear Schr¨odinger equation 28
5.1 A priori estimate in the Sobolev space 28
5.2 Proof of Theorem 1.4.1 33
6 L2-decay for the dissipative nonlinear Schr¨odinger equation
in the Gevrey space 36
6.1 A priori estimate in the Gevrey space 36
Summary
1.1
The nonlinear Schr¨
odinger equation
The Schr¨odinger equation is classified as a dispersive partial differential equation by its principal linear symbol and with adding nonlinear interaction term, the problem is called the nonlinear Schr¨odinger equation. The nonlinear Schr¨odinger equation arises in various fields of applications such as in the nonlinear optics, quantum mechanics, and a simplified model in fluid mechanics and they have been developed largely by mathematical analysis. It is well-known that the one dimensional cubic nonlinear Schr¨odinger equation is complete integrable, namely, the solution of the equation is explicitly written by given data and satisfies infinite conservation laws. However the stability of the problem such as the well-posedness of the Cauchy problem is not obtained by the algebraic method and such a problem is considered in a method of analysis. In general, a various type of nonlinear Schr¨odinger equations have been studied in many aspects and there appear a large amount of results along the theory of partial differential equations. In this thesis, we consider regularity and global dynamics of solutions to nonlinear Schr¨odinger equations with or without dissipative coefficient. Let I = (−T, T ) be a time interval with T > 0 and let
u :R×Rn → C be the unknown function that solves the Cauchy problem of the nonlinear
Schr¨odinger equation with a p-th power nonlinearity:
(1.1.1) i∂tu + 1 2∆u = λ|u| p−1u, t∈ I, x ∈ Rn, u(0, x) = u0(x), x∈ Rn,
where u0 :Rn→ C is a given initial data, p > 1 and λ ∈ C \ {0}.
When we consider the Cauchy problem for the nonlinear Schr¨odinger equation (1.1.1), it is natural to consider in a framework of a energy spaces L2(Rn) or H1(Rn), since the Cauchy problem (1.1.1) has two useful conservation laws, L2 (mass) of the solution and the energy functional of solution. For λ∈ C \ {0}, the local well-posedness of the Cauchy problem (1.1.1) was established by Ginibre-Velo [7], [8], Tsutsumi [51], Kato [29], and Yajima [53] (see also [2], [4], [5] and [31]). More precisely, for any u0 ∈ H1(Rn), there
exists T = T (ku0kH1) > 0 and a unique solution u ∈ C([0, T ); H1(Rn)) to (1.1.1) for
1 < p < 1 + 4/(n− 2). Moreover, the solution exists globally in time for 1 < p < 1 + 4/n by aid of the conservation laws. Such a result is considered along the theory of evolution equations using the semi-group theory and it is common idea to introduce the notion of the mild solution.
1.2
Mass resonance and analytic smoothing effect for
the nonlinear Schr¨
odinger system
It is well known that solutions to the Cauchy problem of parabolic equations gain a smoothness of solutions against a non-smooth initial data. We call such properties as a smoothing effect of a solution. For the nonlinear Schr¨odinger equation, it is well-understood that there is almost no smoothing effect under general settings. However, if we restrict the initial data satisfying spatially weighted condition, then one can find that solutions to the nonlinear Schr¨odinger equations exhibit a local smoothing effect (cf. Kato [29]). In the first part of this thesis, we consider the existence of analytic solutions to the nonlinear Schr¨odinger equations for the non-regular initial data. We call this as the analytic smoothing effect of solutions.
The smoothing effects for the nonlinear Schr¨odinger equations are often considered for the case of the gauge invariant nonlinearity. The equation in the problem (1.1.1) satisfies the gauge invariant property and several properties follows from such a nonlinear structure. A typical example of the gauge invariant nonlinearity is f (u) = |u|p−1u for p > 1. In this case, the corresponding solution to (1.1.1) preserves the energy function.
Hayashi-Nakamitsu-Tsutsumi [13] showed that the solution is infinitely differentiable with the rapidly decaying initial data. Hayashi-Saitoh [17] proved that if the initial data decays exponentially, then the solution is analytic when p = 3 (see also [16], [17], [26], [30], [36]). Since the analytic smoothing effect is possible to prove for polynomial nonlinearities, the power p of the nonlinearity f (u) =|u|p−1u is restricted to the positive odd integer.
On the other hand, the Cauchy problem for the two components system originally derived from the Raman effect in optics has been considered by several authors:
(1.2.1) i∂tu1+ 1 2m1 ∆u1 = u1u2, t ∈ I, x ∈ Rn, i∂tu2+ 1 2m2 ∆u2 = u21, t ∈ I, x ∈ R n , u1(0, x) = φ1(x), u2(0, x) = φ2(x), x∈ Rn.
Hayashi-Li-Naumkin [10] and Hayashi-Li-Ozawa [12] showed the small data scattering for the solutions to (1.2.1) (see also [41]) and the finite time blow-up of the solution is shown by Hayashi-Ozawa-Tanaka [15] and Ogawa-Uriya [40] (see also [27]). Hoshino-Ozawa [22] proved the analytic smoothing effect for the solutions to (1.2.1) (see also [18], [19], [22]-[25], [43]). Those results are mainly shown under the mass resonance condition. We denote the nonlinearities for the above system (1.2.1) as
{
f1(u1, u2) = u1u2,
f2(u1, u2) = u21,
then the gauge invariant property can be identified by (1.2.2)
{
f1(eiθ1u1, eiθ2u2) = ei(θ2−θ1)u1u2 = eiθ1f1(u1, u2),
f2(eiθ1u1, eiθ2u2) = ei2θ1u21 = e
iθ2f
2(u1, u2),
under the special relation 2θ1 = θ2 ∈ R. On the other hand, the linear part of the
nonlinear Schr¨odinger equation shows asymptotic behavior given by the factrization of the free Schr¨odinger evolution such as
(1.2.3) u1(t, x) = 1 (it)n/2e i(m1|x| 2 2t )φ 1 (x t ) , u2(t, x) = 1 (it)n/2e i(m2|x|2t 2)φ 2 (x t )
as t → ±∞. Hence when θ1 = m1θ, θ2 = m2θ with θ = |x|2/2t ∈ R, the linear
funda-mental frequency has a resonance situation for the gauge invariance parameter (θ1, θ2).
From the above observation through (1.2.2) and (1.2.3), the gauge invariance invites the frequency resonance when 2m1 = m2 and we call such a special case for the coefficients
as the mass resonance structure.
In the first part of this thesis, we prove the analytic smoothing effect for the multi-component system including the problem (1.2.1) with p-th powered nonlinearity. For
k ∈ N\{0} and mj > 0, j = 1,· · · , k, we consider the Cauchy problem of the k-component
system of the nonlinear Schr¨odinger equations:
(1.2.4) i∂tuj + 1 2mj
∆uj = fj(u, u), t∈ I, x ∈ Rn,
uj(0, x) = φj(x), x∈ Rn,
where u = {uj}j=1,··· ,k : I × Rn → Ck is the unknown functions and φ = {φj}j=1,··· ,k :
Rn → Ck is given initial data. The nonlinearity f ={f
j}j=1,··· ,k : C2k → Ck satisfying a
generalized gauge invariant condition given by the following.
Definition (The gauge invaeiance). The nonlinearities fj(u1,· · · uk) : Ck → Ck (j =
1, 2,· · · , k) have the gauge invariant property if fj(u1,· · · uk) satisfies
(1.2.5) fj(eiθ1u1,· · · eiθkuk) = eiθjfj(u1,· · · , uk)
for θj ∈ R.
We restrict ourselves that the nonlinear interactions to the system (1.2.4) as the power type nonlinearity, namely, let {uj}j=1,··· ,k be a set of solutions to (1.2.4) and let p ∈ N,
we set for αj ={αj,`}`=1,··· ,k and βj ={βj,`}`=1,··· ,k ⊂ Zk≥0,
(1.2.6) fj(u, u)≡ λj k ∏ `=1 uαj,` ` u`βj,`, p =αj,1+· · · + αj,k+ βj,1+· · · + βj,k,
Proposition 1.2.1. For p, k∈ N, let the nonlinearity {fj}j=1,··· ,k given by (1.2.6) and A = α1,1− β1,1 · · · α1k− β1k .. . . .. ... αk,1− βk,1 · · · αkk− βkk .
Then {fj}j=1,··· ,k satisfies the gauge invariant structure (1.2.5) if and only if
det(A− I) = det α1,1− β1,1− 1 · · · α1k− β1k .. . . .. ... αk,1− βk,1 · · · αkk− βkk− 1 = 0,
where I is the identity matrix.
In view of the consideration along the asymptotic formula (1.2.3), one can naturally generalize the mass-resonance property according to the generalization of the gauge in-variance. We define the generalized mass resonance as follows:
Definition (Generalized mass resonance structure). Let dim Ker(A − I) = k0 < k,
m = {mj}j=1,··· ,k ⊂ R and let the nonlinearity f = {fj}j=1,··· ,k : C2k → Ck satisfy
the gauge invariant condition (1.2.5). Then the coefficient m ={mj}j=1,··· ,k satisfies the
mass resonance if there exists a∈ R \ {0} such that
(1.2.7) m = aθ,
where θ ={θj}j=1,··· ,k ⊂ R is the gauge invariant parameter appeared in (1.2.5).
We define the following functional spaces to state our result for the analyticity. Let
S(Rn) is the set of smooth and rapidly decreasing functions.
Definition (The L2-admissible pair). For n≥ 1, a pair of exponents (θ, q) is L2-admissible, if (θ, q) satisfies (1.2.8) n 2 = n q + 2 θ
and q ≥ 2 satisfies q ≤ 2n/(n − 2), if n ≥ 3, q < ∞, if n = 2 and q ≤ ∞, if n = 1.
Definition (Weighted L2-spaces). For m > 0, let
L2exp,m(Rn)≡ { f ∈ L2(Rn) ; kfkL2 exp,m ≡ sup a∈B1(0) kema·xfk L2 <∞ } .
For t > 0 and m > 0, let the generator of the Galilei transform given by Jm(t)f ≡ (x + it
m∇)f for any f ∈ S(R
n) and we denote J
1(t) as J (t). Then we introduce the exponential
operator associated with the operator Jm(t). For any a∈ B1(0) ={y ∈ Rn; |x − y| < 1},
t6= 0 and m > 0, let ema·Jm(t)f ≡ ∞ ∑ k=0 mk k! (a· Jm(t)) kf, f ∈ D(ema·Jm(t)) 4
where the domain D(ema·Jm(t)) is defined by D(ema·Jm(t)) = { f ∈ S(Rn) ; sup a∈B1(0) kema·Jm(t)fk Lq <∞ }
for some q≥ 2. We also introduce the analytic class of a solution as the following:
Am(I)≡ { f ∈ C ∩ L∞(I; L2)∩ Lθ(I; Lq) ; kfkAm(I) ≡ sup a∈B1(0) kema·Jmfk
Lθ(I;Lq) <∞ for all L2-admissible pairs (θ, q)
}
.
we define the mild solution to the system (1.2.4) as following.
Definition (The mild solution). Let k ∈ N, T > 0 and f = {fj}j=1,··· ,k given by (1.2.6).
A set of functions u ={uj}j=1,··· ,k is the solution to (1.2.4) if uj satisfies for j = 1,· · · , k,
uj(t) = Umj(t)φj− i ∫ t 0 Umj(t− s)fj(u, u) ds in C((−T, T ); L2(R)), where U mj(t) = e 1
2mjit∆is the unitary operator of the free Schr¨odinger
evolution in L2(Rn).
We state our main theorem for the analytic smoothing effect of the solution to (1.2.4).
Theorem 1.2.2 ([39]). Let 1≤ n ≤ 4, 1 < p ≤ 1 + 4/n, (θ, q) be the L2-admissible given
by (1.2.8) and {mj}j=1,··· ,k ⊂ R \ {0}. Assume that the nonlinear term (1.2.6) satisfies
the gauge invariance condition (1.2.5) and the mass resonance structure (1.2.7). Then for any φ = {φj}j=1,··· ,k ∈
∏k j=1L
2
exp,mj(R
n), there exists T > 0 and a unique solution
u = {uj}j=1,··· ,k to (1.2.4) such that uj ∈ Amj(I), where I = (−T, T ). In particular, for
each t∈ I \ {0}, the solution uj(t, x) is real analytic in x∈ Rn.
Since the integrability of ema·Jmf implies the real analyticity to a-direction for each
fixed a ∈ B1(0), any function f ∈ Am(I) is analytic for all direction (cf. Stein-Weiss
[49]) hence the result for the anlyticity in Theorem 1.2.2 follows immediately and one can extend such a function into a holomorphic function over the complex region Rn+ iB
ta(0)
for each t∈ I \ {0}.
1.3
The lower bound estimate for the dissipative
non-linear Schr¨
odinger equation
In the second part of this thesis, we consider a global behavior of solutions to the Cauchy problem (1.1.1) under the dissipative coefficient conditions; Imλ < 0. According to such a condition, the L2-norm of solution is not preserved. Indeed, any solution to (1.1.1) in
such a case no longer conserves the total mass of the solution because of the dissipative L2-identity: (1.3.1) ku(t)k2L2 + |Im λ| ∫ t 0 ku(τ)kp+1 Lp+1dτ = ku0k2L2, t≥ 0,
namely, the L2-norm of a dissipative solution to (1.1.1) decreases monotonically in t≥ 0.
Thus the solution maintains not only a dispersive property but also a dissipative nature. Hence we call such a case, a dissipative solution to the nonlinear Schr¨odinger equations. For the nonlinear Schr¨odinger equation (1.1.1) with Im λ = 0, i.e., the pure dispersive case, the exponent p = 1 + 2/n is known to be the borderline for the scattering theory. Namely, when p > 1 + 2/n, the solution to (1.1.1) behaves as a free solution for the large time, while there is no scattering state for the case p ≤ 1 + 2/n (see for details [1], [8], [14], [50], [52]). The limiting case p = 1 + 2/n exhibits that the solution to (1.1.1) shows so-called nonlinear long range scattering discovered by Ozawa [42] (cf. [6], [9], [40]). For the dissipative case, Im λ < 0, Cazenave-Naumkin [3] and Hoshino [20] showed that there exists a scattering state u+ ∈ L2(Rn) for the dissipative solution u(t) to (1.1.1) as t→ ∞,
when p > 1 + 2/n, i.e.,
(1.3.2) ku(t) − U(t)u+k2 → 0 as t → ∞.
However by the dissipative structure observed by (1.3.1), it is not clear if the scattering state u+ ∈ L2(Rn) in (1.3.2) might not be trivial. Then we show that the mass of
dissipative solution to (1.1.1) has a lower bound as t → ∞ and it ensures that the existence of a non-zero scattering state u+ which is shown in [3] and [20].
To state the main theorem in chapter 3, we introduce a weighted Sobolev space. For
s > 0 and r > 0, let Hrs(Rn)≡ { f ∈ Hs(Rn); kfkHs r ≡ khxi rh∇isfk L2 <∞ } ,
where hxi ≡ (1 + |x|2)1/2 and h∇isf ≡ F−1[hξisf ] and we denote Hb r0(Rn) = L2r(Rn),
H0s(Rn) = Hs(Rn). Finally, we define the solution to (1.1.1) as the mild solution. Let
p(n) = ∞, if n = 1, 2 and p(n) = 1 + 4/(n − 2), if n ≥ 3. We state that any small
solutions to (1.1.1) with the dissipative nonlinearity have the positive L2-lower bound
when p > 1 + 2/n.
Theorem 1.3.1 ([45]). Let n ≥ 1, 1 + 2/n < p < p(n), n/2 < s < p, Re λ ∈ R and
Im λ < 0. Assume that v0 ∈ Hs(Rn)∩ L2s(Rn). Then, there exists small δ0 > 0 such that
for any 0 < δ < δ0, the Cauchy problem (1.1.1) with u0 = δv0 has a unique global solution
u∈ C([0, ∞); Hs(Rn)) with Js(t)u∈ C([0, ∞); L2(Rn)) and it satisfies
ku(t)kL2 ≥ Cδ, t ≥ 0,
where C > 0 is independent of t≥ 0.
The dissipative solutions corresponding to the large data to (1.1.1) also have the positive L2-lower bound by adding a special oscillation to the initial data.
Theorem 1.3.2 ([45]). Let n ≥ 1, 1 + 2/n < p < p(n), Re λ > 0 and Im λ < 0 andlet v0 ∈ H1(Rn)∩ L21(Rn). Then, there exists large b0 > 0 such that for any b > b0,
the Cauchy problem (1.1.1) with u0 = (exp i|x|
2
4 b0) v0 has a unique global solution u ∈
C([0,∞); H1(Rn)). Moreover, the solution satisfies that for any t≥ 0,
ku(t)kL2 ≥ C,
where C > 0 is independent of t≥ 0.
1.4
L
2-decay for the dissipative nonlinear Schr¨
odinger
equation
As is shown in Theorem 1.3.1 and 1.3.2 in the previous section, the dissipative solution to (1.1.1) has the L2-lower bound for p > 1 + 2/n in spite of the dissipative condition
Im λ < 0. On the other hand, if p ≤ 1 + 2/n, Kita-Shimomura [33], [34] proved the
L2-decay for the dissipative solution to (1.1.1) (cf. Shimomura [47] and Sunagawa [46], see also [28], [32] and [37]). Hayashi-Li-Naumkin [11] even obtained the L2-decay rate of the dissipative solution with H1(Rn)-regularity. Hoshino [21] showed the L2-decay order
of the dissipative solution to (1.1.1) in the Sobolev space Hr(Rn) for 0 < r < 1. Due
to the above decay results and Theorem 1.3.1 and 1.3.2, the power p = 1 + 2/n of the nonlinearity is a critical exponent for the L2-decay of the dissipative solution to (1.1.1).
In Chapter 4, we show that higher regularity of the solution has a strong influence to the
L2-decay of the solution under the case p ≤ 1 + 2/n. In order to classify the quantity estimate, we restrict our case to n = 1 and p = 3:
(1.4.1) i∂tu + 1 2∂ 2 xu = λ|u| 2 u, t∈ R, x ∈ R, u(0, x) = u0(x), x∈ R, where λ∈ C, Im λ < 0.
Our results are the followings.
Theorem 1.4.1 ([38]). Let k ∈ N. There exists a small constant ε0 > 0 such that for
any u0 ∈ Hk+1(R) ∩ H1k(R) with ku0kHk+1+ku0kHk
1 ≤ ε0, the Cauchy problem (1.4.1) has
a unique global solution u satisfying
sup
t≥0 k∂ k
xu(t)kL2 <∞
and for some C > 0,
(1.4.2) ku(t)kL2 ≤ C(log t)− 1 2+
1
We should like to emphasize that the estimate (1.4.2) is a generalization of the previous results [11], [21] and the smoothness of the solution indeed gives the faster decay of solution for the critical dissipative case. Since the regularity exponent can be arbitrary large, one can expect that the decay rate of the solution to (1.4.1) should be (log t)−12+ε as m → ∞.
In particular the threshold case p = 1 + 2/n, the decay order of mass of the solution is explicitly determined by the higher order regularity of the solution such as the Gevrey class: For s≥ 1, Gsv(R) ≡ { f ∈ L2(R); kfkGs v = ∞ ∑ k=0 1 (k!)sk∂ k xfkL2 <∞ } ,
which is the class between C∞-class and an analytic regular class. The Gevrey-regular so-lution to (1.4.1) shows a slightly rapid L2-decay than the L2-decay of dissipative solutions with the Hm-regularity.
Theorem 1.4.2 ([38], [44]). Let s ≥ 1. Then, there exists a small constant ε0 > 0 such
that for any u0 ∈ Gsv(R) with k∂xu0kGs v + ∞ ∑ k=0 1 (k!)skx∂ k
xu0kL2 ≤ ε0, the Cauchy problem
(1.4.1) has a unique global solution u in C([0,∞); Gsv(R)) such that sup
t≥0 ku(t)k Gs
v <∞
and there exists C > 0 such that for any t≥ ee,
ku(t)kL2 ≤ C(log log t) s
2(log t)− 1 2.
In the analytic case, s = 1, Ogawa-Sato [38] proved the L2-decay of dissipative
solu-tions by using an operator eiy∂x which is used by Hoshino [19] to obtain the analyticity
of solutions to the nonlinear Schr¨odinger system. Li-Sunagawa [35] showed an analogous dissipative L2-decay estimate for the derivative nonlinear Schr¨odinger equation.
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