Random data Cauchy problem for the nonlinear Schr\"{o}dinger equation with derivative nonlinearity

We consider the Cauchy problem for the nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ) u= \pm \partial (\overline{u}^m)$ on $\R ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\frac{\partial}{\partial x_1} , \, \dots , \, \frac{\partial}{\partial x_d}$ or $|\nabla |= \mathcal{F}^{-1}[|\xi | \mathcal{F}]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s(\R ^d)$ with $s>\max \left( \frac{d-1}{d} s_c , \frac{s_c}{2}, s_c - \frac{d}{2(d+1)} \right)$ for $d+m \ge 5$, where $s$ is below the scaling critical regularity $s_c := \frac{d}{2}-\frac{1}{m-1}$.


Introduction
We consider the Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity: Here, m is a positive integer, u : R × R d → C is an unknown function, φ : R d → C is a given function, ∂ is a first order derivative with respect to the spatial variable, for example a linear combination of which implies that s c := d 2 − 1 m−1 is the scaling critical regularity. We mention the previous and related results for (1.1). Grünrock [11] proved local in time wellposedness of (1.1) in L 2 (R) when m = 2 and in H s (R d ) for s > s c when d ≥ 1, d + m ≥ 4. The first author [16], [17] proved that (1.1) is small data global well-posedness and scattering for s ≥ s c if m+ d ≥ 4. Well-posedness of the Cauchy problem for (1.1) in d = 1 whose ∂(u m ) is replaced by ∂ x (|u| 2 u) is intensively studied by many authors (see, for example, [13], [14], [21], [22], [8], [3], [9], [15], [23], [20] and references therein). Presence of derivative causes some difficulties. In the present paper, we impose that the nonlinear part of (1.1) has special structure which cancels out the worst interaction. Owing to this property, we can recover one derivative.
The above results are deterministic results. We consider well-posedness of (1.1) with randomized initial data. Following the papers [1], [2], we define the randomization. Let ψ ∈ S(R d ) satisfy Let {g n } be a sequence of independent mean zero complex valued random variables on a probability space (Ω, F , P ), where the real and imaginary parts of g n are independent and endowed with probability distributions µ (1) n and µ (2) n . Throughout this paper, we assume that there exists c > 0 such that R e κx dµ (j) n (x) ≤ e cκ 2 for all κ ∈ R, n ∈ Z d , j = 1, 2. This condition is satisfied by the standard complex valued Gaussian random variables and the standard Bernoulli random variables. We then define the Wiener randomization of φ by The randomization has no smoothing in terms of differentiability ( [5,Appendix B]). However, it improves the integrability (see for example Lemma 2.3 in [1]). ¿From this point of view, the randomization makes the problem subcritical in some sense. In the present paper, we focus on the case where the regularity is less than s c = d 2 − 1 m−1 because well-posedness in H s (R d ) with s ≥ s c holds in the deterministic setting. Given φ ∈ H s (R d ), let φ ω be its randomization defined by (1.2). Then, for almost all ω ∈ Ω, there exist T ω > 0 and a unique solution u to (1.1) with u(0, x) = φ ω (x) in a space continuously embedded in More precisely, there exist C, c > 0, γ > 0 such that for each 0 < T < 1, there exists Ω T ⊂ Ω with We find a solution u which is a perturbation of e it∆ φ ω . The linear evolution for the randomized initial data has better integrability than that for the initial data (see Lemma 2.3 below), but it remains C((−T ω , T ω ); H s (R d )). On the other hand, from the smoothing effect of the linear evolution and absence of resonance interaction, the difference u − e it∆ φ ω belongs to C((−T ω , T ω ); H sc (R d )) even if φ ∈ H s (R) with s < s c . , if d ≥ 2 and m ≥ 4. Next, we focus on global existence of the solution with small initial data. Theorem 1.3. Assume d ≥ 1, m ≥ 2, and d + m ≥ 5. Given φ ∈ H s (R d ), let φ ω be its randomization defined by (1.2). Then, for almost all ω ∈ Ω, there exists ε(ω) > 0 such that for every ε ∈ (0, ε(ω)), there exists a global in time solution u to (1.1) with u(0, x) = εφ ω (x) in a space continuously embedded in C(R; H s (R d )). Moreover, the solution is scattering in the following sense: The uniqueness holds in the space Y s defined by Definition 3.9 below, which is a subspace continuously embedded in S(t)φ ω + C(R; H sc (R d )).
Remark 1.4. Theorem 1.3 is a consequence of the following: there exist C, c > 0 and Ω φ ⊂ Ω such that with the following properties: The nonlinear part of (1.1) excludes the resonance, which is the worst interaction. In other words, if an output of the nonlinear interaction is on the characteristic curve, then the at least one of the inputs is off its curve (see (4.10) below). Therefore, by using the modulation estimate and the Fourier restriction norm, we can recover one derivative. These are also useful in the randomized initial data setting.
The number α(d, m) : On the other hand, Bényi, Oh, and Pocovnicu [2] showed that the cubic nonlinear Schrödinger equation without derivative is almost sure well-posed in 2 is the scaling critical regularity. Here, we note that This difference comes from the fact that we rely on not only the bilinear refinement of the Strichartz estimates but also the modulation bound. We obtain the almost sure well-posedness in d ≥ 2 if m ≥ 3, although the result of Bényi et. al. is required d ≥ 3. One reason for this is that the scaling critical regularity of (1.1) is bigger than that of the cubic nonlinear Schrödinger equation without derivative. More precisely, the scaling critical regularity is zero if d = 1, m = 3 in our case, while the scaling critical regularity is zero if d = 2 in the cubic nonlinear Schrödinger equation without derivative. Indeed, since the randomization does not improve regularity, we can not expect that almost sure well-posedness holds in the Sobolev space with negative regularity. ¿From the same reason, we need the condition d + m ≥ 5 in Theorems 1.1 and 1.3.
Put N m (u) = ∂(u m ). Let z(t) = z ω (t) := S(t)φ ω and v(t) = u(t) − z(t) be the linear and nonlinear parts of u respectively. As in [2], we consider the following perturbed equation: In the previous results of Bényi et. al. [2] and the authors [18], the lower bound of s comes from a nonlinearity part which only consists of the linear evolution of the probabilistic initial data. However, the lower bound in Theorems 1.1 and 1.3 appears in different nonlinear parts when m ≥ 3. More precisely, d−1 d s c and s c − d 2(d+1) are need to estimate N m (z · · · zv) and N m (v · · · vz) respectively. Hence, v, which has more regularity than z, behaves like a bad part for d ≥ 2 and m ≥ 4.
We now give a brief outline of this article. In Section 2, we collect lemmas which are used in the proof of our main results. In Section 3, we define the function spaces and show these properties. In Section 4, we show that the nonlinear estimates, which play a crucial role in the proof of our main results. In Section 5, we give a proof of almost sure well-posedness results, Theorems 1.1 and 1.3.

The probabilistic lemmas
Firstly, we recall the probabilistic estimate. The randomization keeps differentiability of the function.
Proposition 2.2. Let (q, r) be admissible. Then, we have x . By the randomization, improved Strichartz type estimates hold.
, let φ ω be its randomization defined by (1.2). Let (q, r) be admissible with q, r < ∞ and r ≤r < ∞. Then, there exist C, c > 0 such that x for all λ > 0.

Function spaces and their properties
3.1. Definition of U p , V p spaces. In this section, we define the U p -and V p -type function spaces. We refer the reader to §2 in [12] for proofs of the basic properties.
Let Z be the set of finite partitions −∞ < t 0 < t 1 < · · · < t K ≤ ∞ of the real line and we put v(t K ) := 0 for all functions v : R → L 2 if t K = ∞.
a U p -atom. Furthermore, we define the atomic space Definition 3.2. (i) Let 1 ≤ p < ∞. We define V p as the space of all functions v : R → L 2 (R d ) such that the limits lim t→±∞ v(t) exist in L 2 (R d ) and the norm (ii) Let V p −,rc be the closed subspace of all v ∈ V p such that v is right continuous and lim t→−∞ v(t) = 0, endowed with the norm (3.1).
Combining the interpolation (see Proposition 2.20 in [12]) with it, we obtain the bilinear refinement of the Strichartz estimate in the V 2 space settings.
with N min ≪ N max , and sufficiently small δ > 0, we have the estimate where the implicit constants depending only on d.
Remark 3.8. By the Strichartz estimate, the same estimate holds in the case N min ∼ N max except for d = 1. Hence, we neglect the condition N min ≪ N max if d ≥ 2.
Definition 3.9. For s ∈ R, we define Y s and Z s as the closure of C(R; S(R d ))∩V 2 −,∆ and C(R; S(R d ))∩ U 2 ∆ with respect to the norm

respectively.
We also use the time restricted space.
The space E T is a Banach space. For any T ∈ (0, ∞], we have the embeddings for t ≥ 0 and Γ[f ](t) = 0 otherwise. For the integral operator, we have the following.
Proposition 3.11. Let d ≥ 1, s ∈ R, and T ∈ (0, ∞]. Then the estimate where the implicit constant is dependent only on d, s. This estimate follows from Proposition 2.10, Remark 2.11 in [12].

Probabilistic nonlinear estimates
First of all, we recall the notations which are introduced in §1.
be the linear and nonlinear parts of u respectively. We consider the following perturbed equation: To state probabilistic nonlinear estimates, we define the following sets: The set S 2 δ does not depend on δ. But for convenience, we use this notation. For an interval I ⊂ R and δ > 0, The followings are the main results in this section.
< s < s c and δ > 0 be sufficiently small depending only on d, m and s. Given φ ∈ H s (R d ), let φ ω be its randomization defined by (1.2). For R > 0, we put Here, the constants C 1 and C 2 are depending only on d and m.
Remark 4.2. In the quadratic case, the condition s > d−1 d s c comes from the estimate for zz. In the cubic case, the condition s > d−1 d s c needs to treat the zzv case (see §4.2 below), while the other cases are less restricted. On the other hand, the lower bound of the regularity in [2] appears in the zzz case.

Remark 4.3. Note that the pairs
are admissible. Accordingly, Lemmas 2.1 and 2.3 imply that E R in Lemma 4.1 satisfies the bound To show the local in time nonlinear estimates, we define the norm Since Hölder's inequality yields for any Banach space X, we obtain the following (see the proof of Lemma 4.1 and Remark 4.5 below).
< s < s c and δ > 0 be sufficiently small depending only on d, m and s.

Proof of Lemma 4.1
We only prove (4.2) because (4.3) follows from a similar manner. Thanks to Proposition 3.11, it suffices to show (4.9) We use the dyadic decomposition as follows.
Here, we divide the integration on the right hand side into 2 m+1 parts of the form Here, we note that at least one of the modulations is bounded below. More precisely, for (τ j , ξ j ) ∈ R 1+d (j = 0, 1, . . . , m) with m j=0 τ j = 0 and m j=0 ξ j = 0, by the triangle inequality, we have Thus, let us assume that one of Q j is Q >max 0≤j≤m N 2 j otherwise the integration becomes zero. Putting R j := Q j P Nj , we mainly focus on the estimate of which is bigger than the left hand side of (4.9) because of l 1 ֒→ l 2 . For a set N ⊂ (2 N0 ) m+1 (for example, N is defined by {N 2 , . . . , N m ≤ N 0 ∼ N 1 }), we use the notation I N as We separately treat the cases m = 2 and m ≥ 3.

4.1.
The case m = 2. In this subsection, we consider the case m = 2, where we have Although this case is essentially treated in [16], we give a proof for completeness.
Case 2: zz case. Without loss of generality, we may assume N 1 ≤ N 2 . Moreover, Q 0 = Q >N 2 0 holds in this case.
We consider only N 2 N 0 since the case N 0 N 2 is simpler. (In fact, if N 0 N 2 , then N for s ≥ 0.) In this case, Q 2 = Q >N 2 0 or Q 0 = Q >N 2 0 holds in this case. We deal with only Q 2 = Q >N 2 0 because the case Q 0 = Q >N 2 0 follows from the same manner. Subcase 3-1: By Hölder's inequality, Lemma 3.5, and Corollary 3.7, we have , v 0 V 2 ∆ = 1 and δ > 0 is sufficiently small in the last inequality.
By Hölder's inequality, Lemmas 3.4 and 3.5, we have By Hölder's inequality, Lemmas 3.4 and 3.5, we have Remark 4.5. ¿From (4.5), we get the factor T δ in the case 3-3 if ω ∈ E m,L R . In the other cases, from x , we get the factor T δ .

4.2.
The case m ≥ 3. In this subsection, we consider the case m ≥ 3.
Case 1: w j = v (j = 1, . . . , m) case. This is the deterministic case and the estimate is the same as in [17]. But, we repeat it for completeness. ¿From the symmetry, we may assume that N 1 ≤ · · · ≤ N m .
, the L 2 -orthogonality, and Lemma 3.5 yield that Similarly, we get Accordingly, from Hölder's inequality, Lemmas 3.4, 3.5, and above estimates, we have Since the case Q j = Q >N 2 0 (j = 2, . . . , m − 1) follows from a similar argument as above, we omit the details.
Secondly, we consider the case where Q m = Q >N 2 0 . By Hölder's inequality, Bernstein's inequality, Lemma 3.5, and Corollary 3.7, we have Hence, we obtain We skip the proof of the case Q 0 = Q >N 2 0 because it is the same as above.
Secondly, we consider the case where Q m = Q >N 2 m . ¿From Hölder's inequality, Corollary 3.7, and Lemma 3.5, we have Similarly, the case Q m−1 = Q >N 2 m follows from the same manner. Case 2: w j = z (j = 1, . . . , m) case.
Without loss of generality, we may assume N 1 ≤ · · · ≤ N m . Moreover, By Hölder's inequality, the Sobolev embedding H dδ 2(2+δ) (R d ) ֒→ L 2+δ (R d ), Lemmas 3.5 and 3.6, we get for ω ∈ E m R . Here, we have used the fact that s > sc 2 , and δ > 0 is sufficiently small in the last inequality. Case 3: The case where there exists l ∈ {1, . . . , m − 1} such that w j = v for 1 ≤ j ≤ l and w k = z for l + 1 ≤ k ≤ m.
Without loss of generality, we assume N 1 ≤ · · · ≤ N l and N l+1 ≤ · · · ≤ N m . We further split the proof into five subcases.
Firstly, we assume Q 0 = Q >N 2 0 and l = 1. By Hölder's inequality, the Sobolev embedding H dδ 2(2+δ) (R d ) ֒→ L 2+δ (R d ), Lemma 3.5, and Corollary 3.7, we get Here, we have used the fact that s > d 2 − 1 and δ > 0 is sufficiently small in the last inequality. We note that Since the case Q 1 = Q >N 2 0 and l = 1 is similarly handled, we omit the details.
for ω ∈ E m R . When l = 1, the part x , we get the same bound as above. We note that , we get the same bound. We consider the case l ≥ 2 and N l−1 N 1 d+1 0 . ¿From Hölder's inequality, the embeddings W Lemmas 3.4, and 3.5, we have for ω ∈ E m R . Here, we have used the fact that s > s c − d 2(d+1) and δ > 0 is sufficiently small in the last inequality.
Subsubcase 3-2-2: N 0 ∼ N m N l and Q 0 = Q ≪N 2 0 . We only consider the case Q 1 = Q >N 2 0 because the remaining cases are similarly handled. Firstly, we consider the case l ≥ 3. ¿From Hölder's inequality, the embeddings Lemma 3.5, and Corollary 3.7 we have for ω ∈ E m R . Here, we have used the fact that s > s c − 1 2 and δ > 0 is sufficiently small in the last inequality.
Thirdly, we assume l = 1 and N m−1 N  Lemmas 3.4, and 3.5, we have Here, we have used the fact that s > d−1 d s c and δ > 0 is sufficiently small in the last inequality.
disappears. Here, we have used the fact that s > max s c − 1 2 , d−1 d s c and δ > 0 is sufficiently small in the last inequality. Subcase 3-3: N l−1 ∼ N l N 0 , N m We assume l ≥ 2 because this is reduced the subcase 3-1 when l = 1.
Secondly, we consider the case Q l = Q >N 2 l . By Hölder's inequality, the Sobolev embedding H dδ 2(2+δ) (R d ) ֒→ L 2+δ (R d ), Lemma 3.5, and Corollary 3.7, we get Here, we have used the fact that 1 ≤ l ≤ m − 1 and δ > 0 is sufficiently small in the last inequality.
Secondly, we consider the case Q 1 = Q >N 2 l . ¿From Hölder's inequality, the embeddings Lemmas 3.4, and 3.5, we have for ω ∈ E m R . The case where Q j = Q >N 2 l (j = 2, . . . , l − 1) is similarly handled. Thirdly, we consider the case Q l = Q >N 2 l . ¿From Hölder's inequality, the embeddings H sc (R d ) ֒→ Lemmas 3.4, and 3.5, we have for ω ∈ E m R . Here, we have used the fact that s > max( d 2 − 1, − 1 m−1 ) and δ > 0 is sufficiently small in the last inequality.

Proof of Main results
By a standard contraction argument, we deduce Theorems 1.1 and 1.3 from Lemmas 4.4 and 4.1 respectively. We give a rough outline (see [2] and [18]).
Proof of Theorem 1.1. Let η be small enough such that where C ′ 1 and C ′ 2 are the constants as in (4.6) and (4.7). For any R > 0, we choose T = T (R) such that Then, by Lemma 4.4 and Z sc T ֒→ Y sc T , the mapping v → Γ[N m (v + z)] is a contraction on the ball B η defined by B η := {u ∈ Z sc T : u Z sc T ≤ η} outside a set of probability ≤ C exp(−c 1 T γ φ H s ) for some γ > 0, which leads the almost sure local in time well-posedness.
Proof of Theorem 1.3. The same argument as in Corollary 3.4 in [12] or Appendix C in [19] yields that (4.2), (4.3) with T = ∞ hold. Let η > 0 be sufficiently small such that where C 1 and C 2 are the constants as in (4.2) and (4.3). Then, by Lemma 4.1 with T = ∞ and Z sc ∞ ֒→ Y sc ∞ , the mapping v → Γ[N m (v + z)] is a contraction on the ball B R defined by B η := {u ∈ Z sc ∞ : u Z sc ∞ ≤ η} outside a set of probability ≤ C exp(−c η 2 φ H s ). We thus obtain the almost sure global in time wellposedness.
The scattering follows from (4.2) and the property of the U 2 -space.