On the global well-posedness of the quadratic NLS on $H^1({{\mathbb {T}}}) + L^2({{\mathbb {R}}})$

Abstract

We study the one dimensional nonlinear Schrödinger equation with power nonlinearity $\left| u \right| ^{\alpha - 1} u$ for $\alpha \in [1,5]$ and initial data $u_0 \in H^1({{\mathbb {T}}}) + L^2({{\mathbb {R}}})$. We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity ($\alpha = 2$) we obtain global well-posedness in the space $C({{\mathbb {R}}}, H^1({{\mathbb {T}}}) + L^2({{\mathbb {R}}}))$ via Gronwall’s inequality.

1 Introduction and main results

We are interested in the Cauchy problem for the nonlinear Schrödinger equation (NLS) with power nonlinearity on the space $H^1({{\mathbb {T}}}) + L^2({{\mathbb {R}}})$, i.e.

(1)

where $u_0 = w_0 + v_0 \in H^1({{\mathbb {T}}}) + L^2({{\mathbb {R}}})$ and $\alpha \in [1,5]$. By ${{\mathbb {T}}}$ we denote the one-dimensional torus, i.e. ${{\mathbb {T}}}= {{\mathbb {R}}}/ 2\pi {{\mathbb {Z}}}$, where we consider functions on ${{\mathbb {T}}}$ to be $2\pi$-periodic functions on ${{\mathbb {R}}}$. Before we state our main results, let us mention that the NLS (1) is globally well-posed in $L^2({{\mathbb {R}}})$ via Strichartz estimates and mass conservation (see [15]) and it is globally well-posed in $L^{2}({{\mathbb {T}}})$ via the Fourier restriction norm method and mass conservation (see [1]). Motivation for the investigation of hybrid initial values $u_0 \in H^1({{\mathbb {T}}}) + L^2({{\mathbb {R}}})$ comes from high-speed optical fiber communications, where in a certain approximation the behavior of pulses in glass-fiber cables is described by a NLS equation. The NLS (1) with initial data in $H^s({{\mathbb {T}}}) + H^s({{\mathbb {R}}})$ was referred to in [5] as the tooth problem. A tooth, is, for example, $w_0$ restricted to one period. We think of the addition of $v_0$ to $w_0$ as eliminating finitely many of these teeth in the underlying periodic signal. A periodic signal is the simplest type of a non-decaying signal, encoding, for example, an infinite string of ones if there is exactly one tooth per period. However, such a purely periodic signal carries no information. One would like to be able to change it, at least locally. This leads necessarily to a hybrid formulation of the NLS where the signal is the sum of a periodic and a localized part, the localized part being able to remove one or more of the teeth in the underlying periodic signal. This way one can model, for example, a signal consisting of two infinite blocks of ones which are separated by a single zero, or even far more complicated patterns. In the optics literature the phenomenon of ghost pulses (see [12, 16]) occurs which in our terminology corresponds to the regrowth of missing teeth of the solution to the NLS (1). Notice, that this is a question of the stability of periodic solutions with respect to non-periodic, decaying perturbations (see, e.g., [8] and references therein for the stability of periodic solutions with respect to periodic perturbations).

The case of the cubic nonlinearity ($\alpha =3$) and the initial data $u_0 \in H^s({{\mathbb {T}}}) + H^s({{\mathbb {R}}})$, where $s \ge 0$, was studied by the authors in [5], where the existence of weak solutions in the extended sense was established. Moreover, under some further assumptions, unconditional uniqueness was obtained (that is, the solution is unique among all continuous functions with values in the space from which the initial data is taken). In this paper, due to the non-algebraic structure of the nonlinearity in (1) (for $\alpha \ne 3$) we have to use different methods. For the relation between the solutions of [5] and the solutions of Theorem 2 we refer to Remark 4.

To state the main results of this paper we need some preparation. Let $u = w + v \in C([0, T], H^1({{\mathbb {T}}}) + L^2({{\mathbb {R}}}))$ where w satisfies the periodic NLS on the torus with initial data $w_0$. The following is known about w (the case $\alpha \ge 2$ has been treated in [10, Theorem 2.1] while the remaining case $\alpha \in [1, 2)$ is presented in Theorem 30).

Theorem 1

The Cauchy problem for the periodic NLS

(2)

is locally well-posed in $H^1({{\mathbb {T}}})$ for $\alpha \ge 1$. That means that for any $w_0 \in H^1({{\mathbb {T}}})$ there is a unique $w \in C([0,T], H^1({{\mathbb {T}}}))$ satisfying (2) in the mild sense. The guaranteed time of existence T depends only on $\left\| w_0 \right\| _{H^1({{\mathbb {T}}})}$.

A solution w to the periodic NLS at hand dictates that the local part v has to be a solution of the Cauchy problem for the modified NLS

(3)

where

$$\begin{aligned} G_\alpha (w, v) :=\left| v + w \right| ^{\alpha -1}(v + w) - \left| w \right| ^{\alpha - 1} w. \end{aligned}$$

(4)

The main results of the paper are the following two theorems on local and global well-posedness of the modified NLS (3) and consequently the NLS (1).

Theorem 2

(Local well-posedness of the NLS (1)) For $\alpha \in [1, 5]$ the Cauchy problem (3) is locally well-posed in $C([0,T], L^2({{\mathbb {R}}})) \cap L^{\frac{4(\alpha +1)}{\alpha -1}}([0,T], L^{\alpha + 1}({{\mathbb {R}}}))$ for any $v_0 \in L^{2}({{\mathbb {R}}})$.

Hence, the original Cauchy problem (1) is locally well-posed.

In the case $\alpha \in [1, 5)$, the guaranteed time of existence T depends only on $\left\| v_0 \right\| _2$ and $\left\| w_0 \right\| _{H^1({{\mathbb {T}}})}$, whereas, for $\alpha = 5$, T depends on the profile of $v_0$ and $\left\| w_0 \right\| _{H^1({{\mathbb {T}}})}$.

Remark 3

In the case $\alpha \in [1, 2]$, the intersection in Theorem 2 is not needed, i.e. one has unconditional well-posedness for the perturbation v. However, it is not clear whether the Cauchy problem (1) is unconditionally well-posed, since the well-posedness we obtain for the periodic part w is only conditional (see the proof of Theorem 26).

Remark 4

Notice that the weak solution in the extended sense ${\tilde{u}}$ constructed in [5] and the solution u from Theorem 2 coincide. This can be seen as follows: u is a weak solution in the extended sense, which follows by the definition, Plancherel’s theorem and the dominated convergence theorem. Moreover, in the aforementioned paper it was observed that ${\tilde{u}}$ is unique among those solutions, which can be approximated by smooth solutions. This is true for u and hence ${\tilde{u}} = u$ follows.

For $\alpha = 2$, we need the Cauchy problem for the periodic NLS (2) to be globally well-posed in $H^{1}({{\mathbb {T}}})$. Although this result is well-known in the community, we could not find a suitable reference. The standard reference for this seems to be [1]. However, in [1, Proposition 5.73] $\alpha \ge 3$ is required (in our notation). Moreover, in part (ii) of the remark on page 152 in [1], Bourgain mentions that one could obtain the existence of a solution for the quadratic nonlinearity using Schauder’s fixed point theorem, but one would lose uniqueness. Notice, that although [1, Theorem 4.45] covers the case of initial data $w_0 \in H^1({{\mathbb {T}}}) \subseteq L^2({{\mathbb {T}}})$ the solution w is not guaranteed to stay in $H^1({{\mathbb {T}}})$, i.e. to satisfy $w(t) \in H^1({{\mathbb {T}}})$ for all $t \in {{\mathbb {R}}}$. Since it seems that this result is not to be found in the literature we provide a proof in the Appendix (Theorem 30). This global existence and uniqueness result on the torus, together with a close inspection of the mass $\int \left| v \right| ^2 \mathrm {d}{x}$ are essential ingredients in our proof of global well-posedness of (1) on the “tooth space” $H^1({{\mathbb {T}}}) + L^2({{\mathbb {R}}})$.

Theorem 5

(Global well-posedness of the quadratic NLS) For $\alpha = 2$ and $v_{0}\in L^{2}({{\mathbb {R}}})$ the unique solution v of (3) from Theorem 2 extends globally and obeys the bound

$$\begin{aligned} \left\| v(\cdot , t) \right\| _2 \le \left\| v_0 \right\| _2 \exp \left[ \left\| w \right\| _{L^\infty _t L^\infty _x} t \right] \qquad \forall t \in [0, \infty ). \end{aligned}$$

(5)

Hence, the original Cauchy problem (1) for $\alpha = 2$ is globally well-posed.

Although the local well-posedness result in Theorem 2 covers the whole range $\alpha \in [1, 5]$, the methods of the proof of Theorem 5 only work for $\alpha = 2$. A more precise explanation is given in Remark 16.

Of course, one can consider hybrid problems for other dispersive equations. Here we confine ourselves to a remark on the KdV.

Remark 6

Observe that the tooth problem for the KdV reduces to a known setting. More precisely, consider real solutions of

(6)

Let $u = v + w \in C([0, T], H^{s_1}({{\mathbb {R}}}) + H^{s_2}({{\mathbb {T}}}))$, where $s_2 \in {{\mathbb {N}}}$ and w is a global solution of the periodic KdV for the initial data $w_0$ (see [2, Theorem 5]). Then v solves

$$\begin{aligned} v_t + v_{xxx} + v_x v + (w v)_x = 0 \end{aligned}$$

with the initial data $v_0$, which is the KdV with the potential w. This problem has been studied in e.g. [6, Section 3.1] using parabolic regularization. There it has been shown that v satisfies an exponential bound similar to (5). Combining both results we obtain:

For $s_1, s_2 \in {{\mathbb {N}}}$ satisfying $s_1 \ge 2$ and $s_2 \ge s_1 + 1$ the KdV tooth problem, i.e., the Cauchy problem (6), is globally well-posed in $H^{s_1}({{\mathbb {R}}}) + H^{s_2}({{\mathbb {T}}})$.

The paper is organized as follows: In Sect. 2 we state the required prerequisites for the proofs of the main theorems. In Sect. 3 we present the proof of Theorem 2 and in Sect. 4 we present the proof of Theorem 5. Finally, in the Appendix we justify that the subquadratic periodic NLS (2) is globally well-posed in $H^1({{\mathbb {T}}})$.

2 Prerequisites

Let us fix the notation and state some results necessary for the proof of our main theorems. For the purpose of smoothing we will use the heat kernel $(\phi _\varepsilon )_{\varepsilon \ge 0}$. Recall, that $\phi _\varepsilon = \delta _0$, if $\varepsilon = 0$, and

$$\begin{aligned} \phi _\varepsilon (x) = \frac{1}{ 2 \sqrt{\pi \varepsilon }} \, e^{-\frac{\left| x \right| ^2}{4 \varepsilon }} \qquad \forall x \in {{\mathbb {R}}}, \end{aligned}$$

if $\varepsilon > 0$. We shall denote the convolution (in the space variable x) by e.g. $u *\phi _\varepsilon$.

For $s \in {{\mathbb {R}}}$ and $\Omega \in \left\{ {{\mathbb {R}}}, {{\mathbb {T}}} \right\}$ we shall denote by $H^s(\Omega )$ the Sobolev spaces on $\Omega$. Also, we set $H^\infty (\Omega ) :=\cap _{s \in {{\mathbb {R}}}} H^s(\Omega )$. By $\mathcal {F}$ we will denote the Fourier transform on ${{\mathbb {R}}}$.

To shorten the notation we often omit positive constants from inequalities and write e.g. $A \lesssim B$ instead of $A \le C B$. To indicate that $C = C(d)$ we sometimes write $A \lesssim _d B$. To emphasize that C is a universal constant we write $A \lesssim _1 B$. By $A \approx B$ we mean $A \lesssim B$ and $B \lesssim A$. By writing $A \overset{!}{\lesssim } B$ we indicate that we want a constant C to exist such that $A \le B C$ holds.

We will use the following simple lemma, which can be found e.g. in [4, Lemma 3.9].

Lemma 7

(Size estimate) Let $\alpha \ge 1$. Then the following size estimate

$$\begin{aligned}&\left| \left| v_1 + w \right| ^{(\alpha - 1)}(v_1 + w) - \left| v_2 + w \right| ^{(\alpha - 1)}(v_2 + w) \right| \end{aligned}$$

(7)

$$\begin{aligned}\le & {} \alpha \max \left\{ 1, 2^{\alpha - 1} \right\} \left( \left| v_1 \right| ^{\alpha - 1} + \left| v_2 \right| ^{\alpha - 1} + \left| w \right| ^{\alpha - 1} \right) \left| v_1 - v_2 \right| \end{aligned}$$

(8)

holds for any $v_1, v_2, w \in {\mathbb {C}}$.

A pair of exponents $(r, q) \in [2, \infty ]^2$ is called admissible (in one dimension), if

$$\begin{aligned} \frac{2}{q} + \frac{1}{ r} = \frac{1}{2}. \end{aligned}$$

(9)

Let us denote by $q_{\text {a}}(r)$ the solution of (9) for any $r \in [2, \infty ]$. Another pair of exponents $(\rho , \gamma ) \in [1, 2]$ shall be called dually admissible, if $(\rho ', \gamma ') \in [2, \infty ]$ is admissible, i.e. if

$$\begin{aligned} \frac{2}{\gamma } + \frac{1}{ \rho } = \frac{5}{2}. \end{aligned}$$

(10)

For any $\rho \in [1, 2]$ we denote by $\gamma _{\text {a}}(\rho )$ the solution of (10).

Proposition 8

(Strichartz estimates) (Cf. [9, Theorem 1.2]) Let $(r, q_{\text {a}}(r))$ be admissible and $(\rho , \gamma _{\text {a}}(\rho ))$ be dually admissible. Then there is a constant $C = C(r, \rho ) > 0$ such that for any $T > 0$, any $v_0 \in L^2({{\mathbb {R}}})$ and any $F \in L^\gamma _{\text {a}}(\rho )([0, T], L^\rho ({{\mathbb {R}}}))$ the homogeneous and inhomogeneous Strichartz estimates

$$\begin{aligned} \left\| e^{\mathrm {i}t \partial _{x}^{2}} v_0 \right\| _{L^{q_{\text {a}}(r)}([0,T], L^r({{\mathbb {R}}}))}\le & {} C \left\| v_0 \right\| _{L^2({{\mathbb {R}}})}, \end{aligned}$$

(11)

$$\begin{aligned} \left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} F(\cdot , \tau ) \right\| _{L^{q_{\text {a}}(r)}([0,T], L^r({{\mathbb {R}}}))}\le & {} C \left\| F \right\| _{L^{\gamma _{\text {a}}(\rho )}([0,T], L^\rho ({{\mathbb {R}}}))}. \end{aligned}$$

(12)

hold.

Lemma 9

(Gronwall, integral form) (See [14, Theorem 1.10].) Let $A, T \ge 0$ and $u, B \in C([0, T], {{\mathbb {R}}}^+_0)$ be such that

$$\begin{aligned} u(t) \le A + \int _0^t B(s) u(s) \mathrm {d}{s} \qquad \forall t \in [0, T]. \end{aligned}$$

Then

$$\begin{aligned} u(t) \le A \exp \left( \int _0^t B(s) \mathrm {d}{s}\right) \qquad \forall t \in [0, T]. \end{aligned}$$

Lemma 10

(Gronwall, differential form) (See [14, Theorem 1.12].) Let $T > 0$, $u: [0, T] \rightarrow {{\mathbb {R}}}^+_0$ be absolutely continuous and $B \in C([0,T], {{\mathbb {R}}}_0^+)$ such that

$$\begin{aligned} u'(t) \le B(t) u(t) \qquad \text {for almost every } t \in [0, T]. \end{aligned}$$

Then

$$\begin{aligned} u(t) \le u(0) \exp \left( \int _0^t B(s) \mathrm {d}{s}\right) \qquad \forall t \in [0, T]. \end{aligned}$$

Lemma 11

(See [5, Equation (18)]). Let $s \ge 0$. Then there is a constant $C = C(s) > 0$ such that for any $v \in H^s({{\mathbb {R}}})$ and any $w \in H^{s + 1}({{\mathbb {T}}})$ one has $v \cdot w \in H^s({{\mathbb {R}}})$ and

$$\begin{aligned} \left\| vw \right\| _{H^s({{\mathbb {R}}})} \le C \left\| v \right\| _{H^s({{\mathbb {R}}})} \left\| w \right\| _{H^{s + 1}({{\mathbb {T}}})}. \end{aligned}$$

The above estimate is not optimal w.r.t. the assumed regularity of w. However, we do not need a stronger version and the proof is straight forward.

3 Proof of Theorem 2

Consider first the case $\alpha \in [2, 5)$. Let us define the space

$$\begin{aligned} X :=C([0, T], L^2({{\mathbb {R}}})) \cap L^{q_{\text {a}}(\alpha + 1)}([0,T], L^{\alpha + 1}({{\mathbb {R}}})) \end{aligned}$$

(13)

equipped with the norm

$$\begin{aligned} \left\| v \right\| _X :=\left\| v \right\| _{L^\infty _t L^2_x} + \left\| v \right\| _{L^{q_{\text {a}}(\alpha + 1)}_t L^{\alpha + 1}} \qquad \forall v \in X, \end{aligned}$$

where T will be fixed later in the proof. The integral formulation of (3) reads as

$$\begin{aligned} v = e^{\mathrm {i}t \partial _{x}^{2}} v_0 \pm \mathrm {i}\int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} G_\alpha (w, v) \mathrm {d}{\tau } =:\mathcal {T}(v). \end{aligned}$$

(14)

By Banach’s fixed-point theorem, it suffices to show that there are $R, T > 0$ such that $\mathcal {T}$ is a contractive self-mapping of

$$\begin{aligned} M(R, T) :=\left\{ v \in X \Big | \, \left\| v \right\| _X \le R \right\} . \end{aligned}$$

Consider first the self-mapping property. For $r \in \left\{ 2, \alpha + 1 \right\}$ we have

$$\begin{aligned} \left\| \mathcal {T}v \right\| _{L^{q_{\text {a}}(r)}_t L^r_x} \le \left\| e^{\mathrm {i}t \partial _{x}^{2}} v_0 \right\| _{L^{q_{\text {a}}(r)}_t L^r_x} + \left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} G_\alpha (w, v) \mathrm {d}{\tau } \right\| _{L^{q_{\text {a}}(r)}_t L^r_x}. \end{aligned}$$

By the homogeneous Strichartz estimate (11), we have

$$\begin{aligned} \left\| e^{\mathrm {i}t \partial _{x}^{2}} v_0 \right\| _{L^{q_{\text {a}}(r)}_t L^r_x} \lesssim \left\| v_0 \right\| _2 \end{aligned}$$

for the first summand. This suggests the choice $R \approx \left\| v_0 \right\| _2$. For the second summand, whose norm also needs to be comparable with R, we will split the integral term. We proceed with the estimates for the contraction property of $\mathcal {T}$, because the self-mapping property follows from them by setting $v = v_1$ and $v_2 = 0$. To that end, let us define $G_\alpha (w, v_1, v_2) :=G_\alpha (w, v_1) - G_\alpha (w, v_2)$, set

$$\begin{aligned} A :=\left\{ x \in {{\mathbb {R}}}\, | \, \left| w \right| \le (\left| v_1 \right| + \left| v_2 \right| ) \right\} , \end{aligned}$$

(15)

and introduce

$$\begin{aligned} G_{\alpha , 1}(w, v_1, v_2) :=\mathbb {1}_{A} \left( \left| w + v_1 \right| ^{\alpha - 1} (w + v_1) - \left| w + v_2 \right| ^{\alpha - 1} (w + v_2) \right) \end{aligned}$$

and

$$\begin{aligned} G_{\alpha , 2}(w, v_1, v_2) :=\mathbb {1}_{A^c} \left( \left| w + v_1 \right| ^{\alpha - 1} (w + v_1) - \left| w + v_2 \right| ^{\alpha - 1} (w + v_2) \right) . \end{aligned}$$

By the triangle inequality one obtains for $r \in \left\{ 2, \alpha + 1 \right\}$ the estimate

$$\begin{aligned}&\left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} G_\alpha (w, v_1, v_2) \mathrm {d}{\tau } \right\| _{L^{q_{\text {a}}(r)}_t L^r_x} \nonumber \\&\quad \le \left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} G_{\alpha , 1}(w, v_1, v_2) \mathrm {d}{\tau } \right\| _{L^{q_{\text {a}}(r)}_t L^r_x}\nonumber \\&\qquad + \left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} G_{\alpha , 2}(w, v_1, v_2) \mathrm {d}{\tau } \right\| _{L^{q_{\text {a}}(r)}_t L^r_x}. \end{aligned}$$

(16)

We use the inhomogeneous Strichartz inequality and the size estimate (78) to bound the first summand of (16) by

$$\begin{aligned}&\left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _x^2} G_{\alpha , 1}(w, v_1, v_2) \mathrm {d}{\tau } \right\| _{L^{q_{\text {a}}(r)}_t L^r_x} \nonumber \\&\quad \le \left\| G_{\alpha , 1}(w, v_1, v_2) \right\| _{L^{q_{\text {a}}((\alpha + 1)')}_t L^{(\alpha + 1)'}_x} \nonumber \\&\quad \lesssim \left\| \mathbb {1}_A \left( \left| v_1 \right| ^{\alpha - 1} + \left| v_2 \right| ^{\alpha - 1} + \left| w \right| ^{\alpha - 1} \right) \left| v_1 - v_2 \right| \right\| _{L^{q_{\text {a}}((\alpha + 1)')}_t L^{(\alpha + 1)'}_x}. \end{aligned}$$

(17)

Using the definition of the set A and Hölder’s inequality for the space and time norms we arrive at the upper bound

$$\begin{aligned} \left\| (\left| v_1 \right| ^{\alpha - 1} + \left| v_2 \right| ^{\alpha - 1}) \left| v_1 - v_2 \right| \right\| _{L^{q_{\text {a}}((\alpha + 1)')}L^{(\alpha + 1)'}} \lesssim T^{1 - \frac{\alpha - 1}{4}} R^{\alpha - 1} \left\| v_1 - v_2 \right\| _X. \end{aligned}$$

For the second summand of (16) we obtain by the same methods the bound

$$\begin{aligned}&\left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _x^2} G_{\alpha , 2}(w, v_1, v_2) \mathrm {d}{\tau } \right\| _{L^{q_{\text {a}}(r)}_t L^r_x} \\&\quad \lesssim \left\| \left| w \right| ^{\alpha - 1} \left| v_1 - v_2 \right| \right\| _{L^1(L^2)} \lesssim T \left\| v_1 - v_2 \right\| _X. \end{aligned}$$

Choosing T small enough shows the contraction property of $\mathcal {T}$ and the proof, in the case $\alpha \in (2, 5)$, concludes.

The case $\alpha \in [1, 2]$ is treated in the same way, but instead of setting $\rho = (\alpha + 1)'$ one chooses $\rho = \frac{2}{\alpha }$ for the Strichartz exponent in (17). Applying Hölder’s inequality subsequently leads to the $L^\infty _t L^2_x$-norm and hence no intersection in (13) is required, i.e. we indeed have unconditional uniqueness.

For the remaining critical case $\alpha = 5$, consider the complete metric space

$$\begin{aligned} M(R, T) :=\left\{ v \in X \, \Big | \, \left\| v - e^{\mathrm {i}t \partial _x^2}v_0 \right\| _{L^\infty _t L^2_x} + \left\| v \right\| _{L^6_t L^6_x} \le R \right\} . \end{aligned}$$

We again show that $\mathcal {T}$ is a contractive self-mapping of M(R, T) for some $R, T > 0$. Candidates for R and T are determined from the first term of (16), corresponding to the effective power $\left| v \right| ^5$, exactly as in the treatment of the usual mass critical NLS (see e.g. [11, Theorem 5.3]). Subsequently, the remaining terms corresponding to the effective power $\left| v \right| ^1$ are treated via the Strichartz estimates as in the case $\alpha \in (2, 5)$ enforcing a possibly smaller choice of T. We omit the details. $\square$

4 Proof of Theorem 5

The proof of Theorem 5 will be done by looking at the mass $\frac{1}{ 2} \left\| v(t) \right\| _2^2$ of the solution. In order to make this rigorous we have to work with solutions which are differentiable in time. We will get time regularity from regularity in space. Hence we replace $G_2$ in (3) by its smooth version $G^\varepsilon$. We obtain

(18)

where

$$\begin{aligned} G^\varepsilon (w, v) :=[\left| v + w \right| *\phi _\varepsilon ] (v + w) - [\left| w \right| *\phi _\varepsilon ]w. \end{aligned}$$

(19)

Theorem 12

(Local well-posedness of the smoothened modified NLS) Let $\varepsilon \ge 0$. Then there is a constant $C > 0$ such that for any $v_0 \in L^2$ and any $w \in C({{\mathbb {R}}}, L_x^\infty )$ the Cauchy problem (18) has a unique solution in $C([0,T], L^2({{\mathbb {R}}}))$, provided

$$\begin{aligned} T \le C \min \left\{ \left\| v_0 \right\| _2^{-\frac{4}{3}}, \left\| w \right\| _{L_t^\infty , L_x^\infty }^{-1} \right\} . \end{aligned}$$

(20)

Observe, that the time T above does not depend on $\varepsilon$.

Proof

Consider the integral formulation of (18), i.e.

$$\begin{aligned} v = e^{\mathrm {i}t \partial _{x}^{2}} v_0 \pm \mathrm {i}\int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} G^\varepsilon (w, v) \mathrm {d}{\tau } =:\mathcal {T}^\varepsilon (v) \end{aligned}$$

(21)

and notice that

$$\begin{aligned} G^\varepsilon (w, v) = \underbrace{ \left( [\left| v + w \right| - \left| w \right| ]*\phi _\varepsilon \right) v }_{=:G_1^\varepsilon (w, v)} + \underbrace{ \left( [\left| v + w \right| - \left| w \right| ] *\phi _\varepsilon \right) w + [\left| w \right| *\phi _\varepsilon ] v }_{=:G_2^\varepsilon (w, v)}. \end{aligned}$$

By Banach’s fixed-point theorem, it suffices to show that there are $R, T > 0$ such that $\mathcal {T}^{\varepsilon }$ is a contractive self-mapping of

$$\begin{aligned} M(R, T) :=\left\{ v \in C([0, T], L^2({{\mathbb {R}}})) \Big | \, \left\| v \right\| \le R \right\} . \end{aligned}$$

Consider first the self-mapping property. We have

$$\begin{aligned} \left\| \mathcal {T}^\varepsilon v \right\| _{L^\infty _t L^2_x} \le \left\| e^{\mathrm {i}t \partial _{x}^{2}} v_0 \right\| _{L^\infty _t L^2_x} + \left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} G^\varepsilon (w, v) \mathrm {d}{\tau } \right\| _{L^\infty _t L^2_x}. \end{aligned}$$

Since the operator $e^{it\partial _{x}^{2}}$ is an isometry on $L^{2}$ we have

$$\begin{aligned} \left\| e^{\mathrm {i}t \partial _{x}^{2}} v_0 \right\| _{L^\infty _t L^2_x} = \left\| v_0 \right\| _2 \end{aligned}$$

for the first summand. This suggests the choice $R \approx \left\| v_0 \right\| _2$. For the second summand, whose norm needs to also be comparable with R, we split the integral term and obtain

$$\begin{aligned}&\left\| \int _0^T e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} G^\varepsilon (w, v) \mathrm {d}{\tau } \right\| _{L^\infty _t L^2_x} \\&\quad \le \left\| \int _0^T e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} G_1^\varepsilon (w, v) \mathrm {d}{\tau } \right\| _{L^\infty _t L^2_x} + \left\| \int _0^T e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} G_2^\varepsilon (w, v) \mathrm {d}{\tau } \right\| _{L^\infty _t L^2_x}. \end{aligned}$$

Now, both summands are treated via the inhomogeneous Strichartz estimate as in the proof of Theorem 2. More precisely, one has

$$\begin{aligned} \left\| \int _0^T e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} G_1^\varepsilon (w, v) \mathrm {d}{\tau } \right\| _{L^\infty _t L^2_x}\lesssim & {} \left\| ([\left| v + w \right| - \left| w \right| ] *\phi _\epsilon ) v) \right\| _{L^\gamma (L^\rho )} \\\le & {} \left\| \left\| [\left| v + w \right| - \left| w \right| ] *\phi _\epsilon \right\| _{L_x^{2 \rho }} \left\| v \right\| _{L_x^{2 \rho }} \right\| _{L^\gamma _t} \\\le & {} \left\| \left\| v \right\| _{L_x^{2 \rho }}^2 \right\| _{L_t^{\gamma }} = \left\| v \right\| _{L^{2 \gamma } (L^{2 \rho })}^2. \end{aligned}$$

Above, we used the Cauchy–Schwarz inequality to arrive at the second line and Young’s inequality (if $\varepsilon \ne 0$) and a size estimate to pass to the last line (all in the space variable).

As we want to arrive at the norm in $C([0, T], L^2({{\mathbb {R}}}))$, we put $2 \rho = 2$, i.e. $\rho = 1$. Then, from the admissibility condition (9) for $(\rho ', \gamma ')$, one obtains $\gamma = \frac{4}{3}$. As $2 \gamma = \frac{8}{3} < \infty = q_{\text {a}}(2)$, one can raise the time exponent to $\infty$ by Hölder’s inequality for the time variable, i.e.

$$\begin{aligned} \left\| v \right\| _{L^{2 \gamma } (L^{2 \rho })}^2 \le T^{\frac{3}{4}} \left\| v \right\| _{L^\infty (L^2)}^2 \le T^{\frac{3}{4}} R^2 \overset{!}{\lesssim } R. \end{aligned}$$

(22)

This inequality holds under the condition

$$\begin{aligned} T \lesssim _1 \left\| v_0 \right\| _2^{-\frac{4}{3}}, \end{aligned}$$

which is satisfied by (20).

For $G_2^\varepsilon$ we similarly obtain

$$\begin{aligned}&\left\| \int _0^T e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} G_2^\varepsilon (w, v) \mathrm {d}{\tau } \right\| _{L^\infty _t L^2_x} \\&\quad \lesssim \left\| ([\left| v + w \right| - \left| w \right| ] *\phi _\varepsilon ])w \right\| _{L^{\tilde{\gamma }}(L^{\tilde{\rho }})} + \left\| [\left| w \right| *\phi _\varepsilon ] v \right\| _{L^{\tilde{\gamma }}(L^{\tilde{\rho }})} \\&\quad \le \left\| w \right\| _{L^\infty (L^\infty )} \left\| [\left| v + w \right| - \left| w \right| ] *\phi _\varepsilon ] \right\| _{L^{\tilde{\gamma }}(L^{\tilde{\rho }})} + \left\| [\left| w \right| *\phi _\varepsilon ] \right\| _{L^\infty (L^\infty )} \left\| v \right\| _{L^{\tilde{\gamma }}(L^{\tilde{\rho }})} \\&\quad \lesssim \left\| w \right\| _{L^\infty (L^\infty )} \left\| v \right\| _{L^{\tilde{\gamma }}(L^{\tilde{\rho }})}, \end{aligned}$$

where we employed Young’s inequality and a size estimate to obtain the last line. In contrast to the $G_1$-case, we choose $\tilde{\rho } = 2$ to arrive at the norm in $C([0, T], L^2({{\mathbb {R}}}))$. Then, by the admissibility condition (9), $\tilde{\gamma } = 1 < \infty = q_{\text {a}}(2)$. Hence, by exploiting again the Hölder’s inequality for the time variable, we get

$$\begin{aligned} \left\| w \right\| _{L^\infty (L^\infty )} \left\| v \right\| _{L^{\tilde{\gamma }}(L^{\tilde{\rho }})} =&\left\| w \right\| _{L^\infty (L^\infty )} \left\| v \right\| _{L^1(L^2)} \\ \le&\left\| w \right\| _{L^\infty (L^\infty )} T \left\| v \right\| _{L^\infty (L^2)} \\ \le&\left\| w \right\| _{L^\infty (L^\infty )} R T \\ \overset{!}{\lesssim _1}&R. \end{aligned}$$

From this we obtain the additional condition

$$\begin{aligned} T \lesssim \left\| w \right\| _{L^\infty (L^\infty )}^{-1}, \end{aligned}$$

which is also satisfied by (20).

For the contraction property, consider the splitting

$$\begin{aligned} G^\varepsilon (w, v_1, v_2):= & {} G^\varepsilon (w, v_1) - G^\varepsilon (w, v_2) \\= & {} [\left| v_1 + w \right| *\phi _\varepsilon ](v_1 + w) - [\left| v_2 + w \right| *\phi _\varepsilon ](v_2 + w) \\= & {} \underbrace{ ([\left| v_1 + w \right| - \left| w \right| ] *\phi _\varepsilon )(v_1 - v_2) + ([\left| v_1 + w \right| - \left| v_2 + w \right| ] *\phi _\varepsilon ) v_2 }_{=:G_1^\varepsilon (w, v_1, v_2)} \\&+ \underbrace{([\left| v_1 + w \right| - \left| v_2 + w \right| ] *\phi _\varepsilon ) w + [\left| w \right| *\phi _\varepsilon ] (v_1 - v_2) }_{=:G_2^\varepsilon (w, v_1, v_2)}. \end{aligned}$$

Arguments similar to those used in the proof of the self-mapping property shown above yield the contraction property of $\mathcal {T}^\varepsilon$, possibly requiring an even smaller implicit constant in (20). $\square$

Lemma 13

(Convergence of the solutions for vanishing smoothing) Fix $v_0 \in L^2$ and $w \in C({{\mathbb {R}}}, C({{\mathbb {T}}}))$, and for all $\varepsilon \ge 0$ denote by $v^\varepsilon \in C([0, T], L^2({{\mathbb {R}}}))$ the unique solution of the Cauchy problem (18) from Theorem 12. Then,

$$\begin{aligned} \left\| v^\varepsilon - v^0 \right\| _{L^\infty _t L^2_x} \xrightarrow {\varepsilon \rightarrow 0+} 0. \end{aligned}$$

Proof

Recall, that by construction $v^\varepsilon$ and $v^0$ are fixed points of $\mathcal {T}^\varepsilon$ and $\mathcal {T}^0$ respectively and hence

$$\begin{aligned} \left\| v^\varepsilon - v^0 \right\| _{L^\infty _t L^2_x}\le & {} \left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} \left( G^\varepsilon (w, v^\varepsilon ) - G^0(w, v^0) \right) \mathrm {d}{\tau } \right\| _{L^\infty _t L^2_x} \\\le & {} \left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} \left( G^\varepsilon (w, v^\varepsilon ) - G^\varepsilon (w, v^0) \right) \mathrm {d}{\tau } \right\| _{L^\infty _t L^2_x} \\&+ \left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} \left( G^\varepsilon (w, v^0) - G^0(w, v^0) \right) \mathrm {d}{\tau } \right\| _{L^\infty _t L^2_x}. \end{aligned}$$

Due to the fact that $\mathcal {T}^\varepsilon$ is contractive, the first summand is controlled by

$$\begin{aligned} \left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} \left( G^\varepsilon (w, v^\varepsilon ) - G^\varepsilon (w, v^0) \right) \mathrm {d}{\tau } \right\| _{L^\infty _t L^2_x} \le C \left\| v^0 - v^\varepsilon \right\| _{L^\infty _t L^2_x}, \end{aligned}$$

where $C < 1$ is the contraction constant. Thus, it suffices to show that the second summand converges to zero. To that end we first gather terms with the same effective powers of $v^0$ and w, i.e.

$$\begin{aligned}&\int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} \left( G^\varepsilon (w, v^0) - G^0(w, v^0) \right) \mathrm {d}{\tau }\nonumber \\&\quad = \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} \left( \left[ \left| w + v^0 \right| *\phi _\varepsilon \right] (v^0 + w) - \left[ \left| w \right| *\phi _\varepsilon \right] w \right. \nonumber \\&\qquad - \left. \left| w + v^0 \right| (v^0 + w) + \left| w \right| w \right) \mathrm {d}{\tau } \nonumber \\&\quad = \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} \left( \left[ \left( \left| w + v^0 \right| - \left| w \right| \right) *\phi _\varepsilon - \left( \left| w + v^0 \right| - \left| w \right| \right) \right] v^0 \right) \mathrm {d}{\tau } \nonumber \\&\qquad + \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} \left( \left[ \left( \left| w + v^0 \right| - \left| w \right| \right) *\phi _\varepsilon - \left( \left| w + v^0 \right| - \left| w \right| \right) \right] w \right. \nonumber \\&\qquad + \left. \left( \left| w \right| *\phi _\varepsilon - \left| w \right| \right) v^0 \right) \mathrm {d}{\tau }. \end{aligned}$$

(23)

The first summand corresponding to $\left| v^0 \right| ^2$ is treated in the same way as the $G_1^\varepsilon$-term in the proof of Theorem 12, i.e. via a Strichartz estimate and Hölder’s inequality. We arrive at

$$\begin{aligned}&\left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} \left( \left[ \left( \left| w + v^0 \right| - \left| w \right| \right) *\phi _\varepsilon - \left( \left| w + v^0 \right| - \left| w \right| \right) \right] v^0 \right) \mathrm {d}{\tau } \right\| _{L^\infty _t L^2_x} \\&\quad \le \left\| \left( \left| w + v^0 \right| - \left| w \right| \right) *\phi _\varepsilon - \left( \left| w + v^0 \right| - \left| w \right| \right) \right\| _{L_t^{\frac{4}{3}}L_x^2} \cdot \left\| v^0 \right\| _{L_t^\infty L_x^2}. \end{aligned}$$

It suffices to show that the first factor above tends to zero, as $\varepsilon$ tends to zero. For almost every $t \in [0, T]$ we have that $\left( \left| w + v^0 \right| - \left| w \right| \right) \in L^2$, which implies, due to the fact that $(\phi _\varepsilon )_{\varepsilon > 0}$ is an approximation to the identity, that

$$\begin{aligned} \left\| \left( \left| w + v^0 \right| - \left| w \right| \right) *\phi _\varepsilon - \left( \left| w + v^0 \right| - \left| w \right| \right) \right\| _{L_x^2} \xrightarrow {\varepsilon \rightarrow 0+} 0. \end{aligned}$$

Furthermore, by Young’s inequality,

$$\begin{aligned} \left\| \left( \left| w + v^0 \right| - \left| w \right| \right) *\phi _\varepsilon - \left( \left| w + v^0 \right| - \left| w \right| \right) \right\| _{L_x^2}^{\frac{4}{3}} \lesssim \left\| v^0 \right\| _{L_x^2}^{\frac{4}{3}} \end{aligned}$$

for every $\varepsilon > 0$ and almost every $t \in [0, T]$. Also,

$$\begin{aligned} \int _0^T \left\| v^0(\cdot , \tau ) \right\| _{L_x^2}^{\frac{4}{3}} \mathrm {d}{\tau } = \left\| v^0 \right\| _{L^{\frac{4}{3}}_t L^2_x}^{\frac{4}{3}} \lesssim _T \left\| v^0 \right\| _{L^\infty _t L^2_x}^{\frac{4}{3}} \end{aligned}$$

and hence the claim follows by the dominated convergence theorem.

The last two summands on the right-hand side of (23) corresponding to $\left| v^0w \right|$ are treated like the $G_2^\varepsilon$-term and we arrive at

$$\begin{aligned}&\left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} \left[ \left( \left( \left| w + v^0 \right| - \left| w \right| \right) *\phi _\varepsilon - \left( \left| w + v^0 \right| - \left| w \right| \right) \right) w \right] \mathrm {d}{\tau } \right\| _{L^\infty _t L^2_x} \\&\qquad + \left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} \left[ \left( \left| w \right| *\phi _\varepsilon - \left| w \right| \right) v^0 \right] \mathrm {d}{\tau } \right\| _{L^\infty _t L^2_x} \\&\quad \le \left\| \left( \left| w + v^0 \right| - \left| w \right| \right) *\phi _\varepsilon - \left( \left| w + v^0 \right| - \left| w \right| \right) \right\| _{L_t^1 L_x^2} \cdot \left\| w \right\| _{L_t^\infty L_x^\infty } \\&\qquad + \left\| v^0 \right\| _{L_t^\infty L_x^2} \left\| \left| w \right| *\phi _\varepsilon - \left| w \right| \right\| _{L_t^1 L_x^\infty }. \end{aligned}$$

Observe, that $\left| w \right|$ is uniformly continuous in the x-variable on the whole of ${{\mathbb {R}}}$. Hence, as for the term in (23), the fact that $(\phi _\varepsilon )_{\varepsilon > 0}$ is an approximation to the identity implies the convergence to zero of the last two summands of the right-hand side of (23). $\square$

Lemma 14

(Smooth solutions for smooth initial data) (Cf. [14, Proposition 3.11].) Let $\varepsilon > 0$, $w \in C([0, T], H^\infty ({{\mathbb {T}}}))$ and $v_0 \in \mathcal {S}$ and let v denote the unique solution of (18). Then $v \in C^1([0, T], H^\infty ({{\mathbb {R}}}))$ and for any $s > \frac{1}{ 2}$ one has

$$\begin{aligned} \left\| v \right\| _{L_t^\infty H_x^s} \le C \left\| v_0 \right\| _{H^s} \exp \left( \left\| v \right\| _{L_t^1 L_x^\infty } + T \left\| w \right\| _{C(H^{s + 1}({{\mathbb {T}}}))} \right) \end{aligned}$$

(24)

for some $C = C(\varepsilon , s) > 0$.

Proof

We begin by showing that $v \in C([0, T], H^s({{\mathbb {R}}}))$ for any $s \in {{\mathbb {N}}}$. It suffices to prove that the operator $\mathcal {T}^\varepsilon$ from Theorem 12 is a contractive self-mapping in $M(R, T') \subseteq H^s$, for a possibly smaller $T' \le T$. We only show the self-mapping property. To that end, observe that

$$\begin{aligned} \left\| \mathcal {T}^\varepsilon v \right\| _{H^s}\le & {} \left\| e^{\mathrm {i}t \partial _{x}^{2}} v_0 \right\| _{H^s} + \left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} G^\varepsilon (w, v) \mathrm {d}{\tau } \right\| _{H^s} \\\le & {} \left\| v_0 \right\| _{H^s} + \int _0^t \left\| G^\varepsilon (w, v) \right\| _{H^s} \mathrm {d}{\tau }. \end{aligned}$$

The first summand fixes $R \approx \left\| v_0 \right\| _{H^s}$. For the integrand in the second summand we have (the variable $\tau$ is omitted in the notation)

$$\begin{aligned}&\left\| G^\varepsilon (w, v) \right\| _{H^s} \nonumber \\&\quad \le \underbrace{\left\| \left( \left[ \left| w + v \right| - \left| w \right| \right] *\phi _\varepsilon \right) v \right\| _{H^s}}_{=:I} + \underbrace{\left\| \left( \left| w \right| *\phi _\varepsilon \right) v \right\| _{H^s} }_{=:II} + \underbrace{\left\| \left( \left[ \left| w + v \right| - \left| w \right| \right] *\phi _\varepsilon \right) w \right\| _{H^s}}_{=:III}.\nonumber \\ \end{aligned}$$

(25)

As $H^s({{\mathbb {R}}})$ is an algebra with respect to point-wise multiplication, the first summand is estimated against

$$\begin{aligned} \left\| \left( \left[ \left| w + v \right| - \left| w \right| \right] *\phi _\varepsilon \right) v \right\| _{H^s} \lesssim \left\| \left[ \left| w + v \right| - \left| w \right| \right] *\phi _\varepsilon \right\| _{H^s} \left\| v \right\| _{H^s}. \end{aligned}$$

The first product above is further estimated via the definition of the $H^s$-norm as

$$\begin{aligned} \left\| \left[ \left| w + v \right| - \left| w \right| \right] *\phi _\varepsilon \right\| _{H^s} \lesssim \left\| \langle \cdot \rangle ^s \mathcal {F}\phi _\varepsilon \right\| _{L^\infty } \left\| \left| w + v \right| - \left| w \right| \right\| _2. \end{aligned}$$

(26)

Further estimating $\left\| v \right\| _2 \le \left\| v \right\| _{H^s} \le R$ and recalling the integral concludes the discussion of this term. The second summand (II) is treated via Lemma 11:

$$\begin{aligned} \left\| \left( \left| w \right| *\phi _\varepsilon \right) v \right\| _{H^s} \lesssim _s \left\| \left| w \right| *\phi _\varepsilon \right\| _{H^{s + 1}({{\mathbb {T}}})} \left\| v \right\| _{H^s} \end{aligned}$$

We again estimate $\left\| v \right\| _{H^s} \le R$ and observe for the other factor that

$$\begin{aligned} \left\| \left| w \right| *\phi _\varepsilon \right\| _{H^{s + 1}({{\mathbb {T}}})}&\approx \sum _{\left| \alpha \right| \le \lceil s + 1 \rceil } \left\| \left| w \right| *\left[ D^\alpha \phi _\varepsilon \right] \right\| _{L^2({{\mathbb {T}}})} \\&\le \left\| w \right\| _{\infty } \sum _{\left| \alpha \right| \le \lceil s + 1 \rceil } \left\| D^\alpha \phi _\varepsilon \right\| _{L^1({{\mathbb {R}}})} \\&\lesssim _{\varepsilon , s} \left\| w \right\| _{H^{s + 1}({{\mathbb {T}}})}. \end{aligned}$$

The last summand (III) is estimated via

$$\begin{aligned} \left\| \left( \left[ \left| w + v \right| - \left| w \right| \right] *\phi _\varepsilon \right) w \right\| _{H^s} \lesssim _{\varepsilon , s} \left\| v \right\| _{H^s} \left\| w \right\| _{H^{s + 1}({{\mathbb {T}}})}. \end{aligned}$$

The proof of the above requires no new techniques and is omitted. All in all this shows the local well-posedness of (18) in $C([0, T'], H^s)$, where the guaranteed time of existence is

$$\begin{aligned} T' \approx _{\varepsilon , s} \left\{ \left\| w \right\| _{H^{s + 1}({{\mathbb {T}}})}^{-1}, \left\| v_0 \right\| _{H^s({{\mathbb {R}}})}^{-1} \right\} . \end{aligned}$$

To prove the estimate (24), we will employ Lemma 9 (Gronwall’s inequality). To that end, let $T'$ be now the maximal time of existence of the solution $v \in C([0, T'), H^s)$. Observe that

$$\begin{aligned} \left\| v(\cdot , t) \right\| _{H^s} = \left\| (\mathcal {T}^\varepsilon v)(\cdot , t) \right\| _{H^s} \le \left\| v_0 \right\| _{H^s} + \int _0^t \left\| G^\varepsilon (w, v)(\cdot , \tau ) \right\| _{H^s} \mathrm {d}{\tau } \qquad \forall t \in [0, T'). \end{aligned}$$

The integrand above is estimated as in inequality (25). The first term (I), however, needs retreatment, as it is quadratic in $\left\| v \right\| _{H^s}$. The algebra property of $H^s({{\mathbb {R}}}) \cap L^\infty ({{\mathbb {R}}})$ implies

$$\begin{aligned} I \le \left\| \left( \left[ \left| w + v \right| - \left| w \right| \right] *\phi _\varepsilon \right) \right\| _{H^s} \left\| v \right\| _\infty + \left\| \left( \left[ \left| w + v \right| - \left| w \right| \right] *\phi _\varepsilon \right) \right\| _\infty \left\| v \right\| _{H^s}. \end{aligned}$$

We estimate the first factor in the first summand by (26). For the first factor of the second summand we have

$$\begin{aligned} \left\| \left( \left[ \left| w + v \right| - \left| w \right| \right] *\phi _\varepsilon \right) \right\| _\infty \le \left\| \left[ \left| w + v \right| - \left| w \right| \right] \right\| _\infty \left\| \phi _\varepsilon \right\| _1 \le \left\| v \right\| _\infty \end{aligned}$$

by Young’s inequality. Reinserting the estimates for the terms (II) and (III) yields

$$\begin{aligned} \left\| v(\cdot , t) \right\| _{H^s} \lesssim _{s, \varepsilon } \left\| v_0 \right\| _{H^s} + \int _0^t \left( \left\| v(\cdot , \tau ) \right\| _{\infty } + \left\| w(\cdot , \tau ) \right\| _{H^{s + 1}({{\mathbb {T}}})} \right) \left\| v(\cdot , \tau ) \right\| _{H^s} \mathrm {d}{\tau }. \end{aligned}$$

Gronwall’s inequality now implies

$$\begin{aligned} \left\| v(\cdot , t) \right\| _{H^s}&\lesssim _{\varepsilon , s} \left\| v_0 \right\| _{H^s} \exp \left( \int _0^t \left( \left\| v(\cdot , \tau ) \right\| _{\infty } + \left\| w(\cdot , \tau ) \right\| _{H^{s + 1}({{\mathbb {T}}})} \right) \mathrm {d}{\tau } \right) \\&\le \left\| v_0 \right\| _{H^s} \exp \left( \left\| v \right\| _{L_t^1 L_x^\infty } + T' \left\| w \right\| _{C(H^{s + 1}({{\mathbb {T}}}))} \right) \qquad \forall t \in [0, T'). \end{aligned}$$

Thus we see that a blowup cannot occur for any $T' < T$ and so $T' = T$.

This indeed shows that $v \in C([0, T], H^s)$. As $v_0 \in \mathcal {S}$ and $w \in C([0, T], H^\infty ({{\mathbb {T}}}))$ are smooth, a classical result from semi-group theory (see [13, Theorem 4.2.4]) implies that $v \in C^1([0, T], H^s)$. Since $s > \frac{1}{ 2}$ was arbitrary, the proof is complete. $\square$

Proposition 15

The unique solution v of (18) from Theorem 12 satisfies

$$\begin{aligned} \left\| v(\cdot , t) \right\| _2 \le \left\| v_0 \right\| _2 \exp \left[ \left\| w \right\| _{L^\infty _t L^\infty _x} t \right] \qquad \forall t \in [0, T]. \end{aligned}$$

(27)

Proof

Let $w^{n} \in C([0 ,T], H^\infty ({{\mathbb {T}}}))$ be functions with the property

$$\begin{aligned} \left\| w^n - w \right\| _{C([0,T],H^1({{\mathbb {T}}}))} \xrightarrow {n \rightarrow \infty } 0 \end{aligned}$$

and let $v_n \xrightarrow {n \rightarrow \infty } v_0$ in the $L^2$-norm where $v_n \in \mathcal {S}$ for all $n \in {{\mathbb {N}}}$. Moreover, let $v^{\varepsilon , n} \in C^1([0, T], L^2))$ be the solution of (18) with initial data $v_n$ and nonlinearity $G^{\varepsilon }(w^n, v^{\varepsilon , n})$ (the smoothness of $v^{\varepsilon , n}$ follows from Lemma 14). We have

$$\begin{aligned} \frac{1}{ 2} \frac{\mathrm {d}}{\mathrm {d}{t}} \left\| v^{\varepsilon , n}(\cdot , t) \right\| _2^2= & {} {\text {Re}}\left\langle {\dot{v}}^{\varepsilon , n}(\cdot , t), v^{\varepsilon , n}(\cdot , t) \right\rangle = {\text {Re}}\left\langle \mathrm {i}\partial _{x}^{2} v^{\varepsilon , n} \pm \mathrm {i}G^\varepsilon (w^n, v^{\varepsilon , n}), v^{\varepsilon , n} \right\rangle \nonumber \\= & {} \underbrace{-{\text {Re}}\mathrm {i}\left\langle \nabla v^{\varepsilon , n}, \nabla v^{\varepsilon , n} \right\rangle }_{= 0} \nonumber \\&\pm {\text {Re}}\mathrm {i}\left\langle (\left| v^{\varepsilon , n} + w^n \right| *\phi _\varepsilon ) (v^{\varepsilon , n} + w^n) - (\left| w^n \right| *\phi _\varepsilon ) w^n, v^{\varepsilon , n} \right\rangle \nonumber \\= & {} \pm \underbrace{{\text {Re}}\mathrm {i}\left\langle (\left| v^{\varepsilon , n} + w^n \right| *\phi _\varepsilon ) v^{\varepsilon , n}, v^{\varepsilon , n} \right\rangle }_{= 0} \nonumber \\&\pm {\text {Re}}\mathrm {i}\left\langle ([\left| v^{\varepsilon , n} + w^n \right| - \left| w^n \right| ] *\phi _\varepsilon ) w^n, v^{\varepsilon , n} \right\rangle \end{aligned}$$

(28)

and hence

$$\begin{aligned} \frac{1}{ 2} \frac{\mathrm {d}}{\mathrm {d}{t}} \left\| v^{\varepsilon , n}(\cdot , t) \right\| _2^2\le & {} \left| \left\langle [\left| v^{\varepsilon , n} + w^n \right| - \left| w^n \right| ] *\phi _\varepsilon ) w^n, v^{\varepsilon , n} \right\rangle \right| \nonumber \\\le & {} \left\| [\left| v^{\varepsilon , n} + w^n \right| - \left| w^n \right| ] *\phi _\varepsilon ) w^n \right\| _{L^2_x} \left\| v^{\varepsilon , n} \right\| _{L^2_x} \nonumber \\\le & {} \left\| w^n \right\| _{L^\infty _t L^\infty _x} \left\| v^{\varepsilon , n} \right\| _{L^2_x}^2 \end{aligned}$$

(29)

for all $t \in [0, T]$. Above, we obtained the first estimate by the Cauchy–Schwarz inequality and the second one by Hölder’s inequality, Young’s inequality and the size estimate. By the differential form of the Gronwall’s inequality from Lemma 10, we obtain

$$\begin{aligned} \left\| v^{\varepsilon , n}(\cdot , t) \right\| _2 \le \left\| v_n \right\| _2 \exp \left[ \left\| w^n \right\| _{L^\infty _t L^\infty _x} t \right] \qquad \forall t \in [0, T]. \end{aligned}$$

In the limit $n \rightarrow \infty$, the right-hand side above converges to the right-hand side of (27). It remains to show

$$\begin{aligned} \left\| v^{\varepsilon , n} - v^{\varepsilon } \right\| _{L^\infty L^2} \xrightarrow {n \rightarrow \infty } 0, \end{aligned}$$

(30)

because then the left-hand side converges to $\left\| v^\varepsilon \right\| _{L^\infty _t L^2_x}$ in the limit $n \rightarrow \infty$. Finally, Lemma 13 yields

$$\begin{aligned} \left\| v^\varepsilon \right\| _{L^\infty _t L^2_x} \xrightarrow {\varepsilon \rightarrow 0} \left\| v^0 \right\| _{L^\infty _t L^2_x}. \end{aligned}$$

To prove (30), observe that the linear evolution poses no problems and hence it suffices to control the integral term

$$\begin{aligned} \left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} \left[ G^\varepsilon (w, v^{\varepsilon }) - G^\varepsilon (w^n, v^{\varepsilon , n}) \right] \mathrm {d}{\tau } \right\| _{L^\infty L^2}. \end{aligned}$$

To that end, we will split the difference of the nonlinear terms according to their effective power up to one exception. We begin by observing that

$$\begin{aligned}&G^\varepsilon (w, v^{\varepsilon }) - G^\varepsilon (w^n, v^{\varepsilon , n}) \\&\quad = (\left| w + v^\varepsilon \right| *\phi _\varepsilon ) v^\varepsilon - (\left| w^n + v^{\varepsilon , n} \right| *\phi _\varepsilon ) v^{\varepsilon , n} \\&\qquad +\, ([\left| w + v^\varepsilon \right| - \left| w \right| ] *\phi _\varepsilon ) w - ([\left| w^n + v^{\varepsilon ,n} \right| - \left| w^n \right| ] *\phi _\varepsilon ) w^n \end{aligned}$$

and gather the first and the second summand, as well as the third and the last summand. In the first sum we have

$$\begin{aligned}&(\left| w + v^{\varepsilon } \right| *\phi _\varepsilon ) v^\varepsilon - (\left| w^n + v^{\varepsilon , n} \right| *\phi _\varepsilon ) v^{\varepsilon , n} \\&\quad = \underbrace{ (\left| w + v^\varepsilon \right| *\phi _\varepsilon ) v^\varepsilon - (\left| w + v^\varepsilon \right| *\phi _\varepsilon ) v^{\varepsilon , n}}_{=:I} \\&\qquad + \underbrace{ (\left| w + v^\varepsilon \right| *\phi _\varepsilon ) v^{\varepsilon , n} - (\left| w^n + v^{\varepsilon , n} \right| *\phi _\varepsilon ) v^{\varepsilon , n}}_{=:II}, \end{aligned}$$

whereas for the second sum

$$\begin{aligned}&([\left| w + v^\varepsilon \right| - \left| w \right| ] *\phi _\varepsilon )w - ([\left| w^n + v^{\varepsilon , n} \right| - \left| w^n \right| ] *\phi _\varepsilon ) w^n \\&\quad = \underbrace{ ([\left| w^n + v^\varepsilon \right| - \left| w^n \right| ] *\phi _\varepsilon ) w^n - ([\left| w^n + v^{\varepsilon , n} \right| - \left| w^n \right| ] *\phi _\varepsilon ) w^n }_{=:III} \\&\qquad + \underbrace{ ([\left| w + v^\varepsilon \right| - \left| w \right| ] *\phi _\varepsilon ) w - ([\left| w^n + v^\varepsilon \right| - \left| w^n \right| ] *\phi _\varepsilon ) w^n}_{=:IV} \end{aligned}$$

holds. We now complete the splitting of $G^\varepsilon (w, v^{\varepsilon , n}) - G^\varepsilon (w^n, v^{\varepsilon , n})$ into terms of the same effective powers. We have

$$\begin{aligned} I= & {} (\left| w + v^\varepsilon \right| *\phi _\varepsilon ) (v^\varepsilon - v^{\varepsilon , n}) \\= & {} ([\left| w + v^\varepsilon \right| - \left| w \right| ] *\phi _\varepsilon ) (v^\varepsilon - v^{\varepsilon , n}) + (\left| w \right| *\phi _\varepsilon )(v^\varepsilon - v^{\varepsilon , n}), \\ II= & {} ([\left| w + v^\varepsilon \right| - \left| w^n + v^{\varepsilon , n} \right| ] *\phi _\varepsilon ) v^{\varepsilon , n} \\= & {} ([\left| w + v^\varepsilon \right| - \left| w + v^{\varepsilon , n} \right| ] *\phi _\varepsilon ) v^{\varepsilon ,n} + ([\left| w + v^{\varepsilon , n} \right| - \left| w^n + v^{\varepsilon , n} \right| ] *\phi _\varepsilon ) v^{\varepsilon , n}, \\ III= & {} ([\left| w^n + v^\varepsilon \right| - \left| w^n + v^{\varepsilon , n} \right| ] *\phi _\varepsilon ) w^n \text { and} \\ IV= & {} ([\left| w + v^\varepsilon \right| - \left| w \right| ] *\phi _\varepsilon ) w - ([\left| w + v^\varepsilon \right| - \left| w \right| ] *\phi _\varepsilon ) w^n \\&-\, ([\left| w^n + v^\varepsilon \right| - \left| w^n \right| ] *\phi _\varepsilon ) w^n + ([\left| w + v^\varepsilon \right| - \left| w \right| ] *\phi _\varepsilon ) w^n \\= & {} ([\left| w + v^\varepsilon \right| - \left| w \right| ] *\phi _\varepsilon )(w - w^n) \\&+\, ([\left| w + v^\varepsilon \right| - \left| w \right| - \left| w^n + v^\varepsilon \right| + \left| w^n \right| ] *\phi _\varepsilon ) w^n, \end{aligned}$$

from which the effective powers are obvious, and put

$$\begin{aligned} {\tilde{G}}_1^\varepsilon (w, w^n, v^\varepsilon , v^{\varepsilon , n}):= & {} ([\left| w + v^\varepsilon \right| - \left| w \right| ] *\phi _\varepsilon ) (v^\varepsilon - v^{\varepsilon , n}) \\&+\, ([\left| w + v^\varepsilon \right| - \left| w + v^{\varepsilon , n} \right| ] *\phi _\varepsilon ) v^{\varepsilon , n}, \\ {\tilde{G}}_2^\varepsilon (w, w^n, v^\varepsilon , v^{\varepsilon , n}):= & {} (\left| w \right| *\phi _\varepsilon )(v^\varepsilon - v^{\varepsilon , n}) + ([\left| w^n + v^\varepsilon \right| - \left| w^n + v^{\varepsilon , n} \right| ] *\phi _\varepsilon ) w^n \\&+\, ([\left| w + v^{\varepsilon , n} \right| - \left| w^n + v^{\varepsilon , n} \right| ] *\phi _\varepsilon ) v^{\varepsilon , n} \\&+\, ([\left| w + v^\varepsilon \right| - \left| w \right| ] *\phi _\varepsilon )(w - w^n) \\&+\, ([\left| w + v^\varepsilon \right| - \left| w \right| - \left| w^n + v^\varepsilon \right| + \left| w^n \right| ] *\phi _\varepsilon ) w^n. \end{aligned}$$

Now, by the triangle inequality and the inhomogeneous Strichartz estimate, one has

$$\begin{aligned}&\left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} \left[ G^\varepsilon (w, v^{\varepsilon , n}) - G^\varepsilon (w^n, v^{\varepsilon , n}) \right] \mathrm {d}{\tau } \right\| _{L^\infty L^2} \\&\quad \le \left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} {\tilde{G}}_1^\varepsilon (w, w^n, v^\varepsilon , v^{\varepsilon , n}) \mathrm {d}{\tau } \right\| _{L^\infty L^2} \\&\qquad + \left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} {\tilde{G}}_2^\varepsilon (w, w^n, v^\varepsilon , v^{\varepsilon , n}) \mathrm {d}{\tau } \right\| _{L^\infty L^2} \\&\quad \lesssim \left\| {\tilde{G}}_1^\varepsilon (w, w^n, v^\varepsilon , v^{\varepsilon , n}) \right\| _{L^\frac{4}{3}_t L^1_x} + \left\| {\tilde{G}}_2^\varepsilon (w, w^n, v^\varepsilon , v^{\varepsilon , n}) \right\| _{L^1_t L^2_x}. \end{aligned}$$

We begin by estimating the first summand above. In fact, we have

$$\begin{aligned}&\left\| ([\left| w + v^\varepsilon \right| - \left| w \right| ] *\phi _\varepsilon ) (v^\varepsilon - v^{\varepsilon , n}) \right\| _{L^\frac{4}{3}_t L^1_x} \\&\quad \le \left\| t \mapsto \left\| [\left| w + v^\varepsilon \right| - \left| w \right| ] *\phi _\varepsilon \right\| _{L^2_x} \left\| v^\varepsilon - v^{\varepsilon , n} \right\| _{L^2_x} \right\| _{\frac{4}{3}} \\&\quad \le \left\| t \mapsto \left\| v^\varepsilon \right\| _{L^2_x} \left\| v^\varepsilon - v^{\varepsilon , n} \right\| _{L^2_x} \right\| _{\frac{4}{3}} \\&\quad \le T^{\frac{3}{4}} \left\| v^\varepsilon \right\| _{L^\infty _t L^2_x} \left\| v^\varepsilon - v^{\varepsilon , n} \right\| _{L^\infty _t L^2_x}. \end{aligned}$$

by the Cauchy–Schwarz, Young’s and the inverse triangle inequalities for the space variable and Hölder’s inequality for the time variable. Choosing T sufficiently small shows that

$$\begin{aligned} \left\| ([\left| w + v^\varepsilon \right| - \left| w \right| ] *\phi _\varepsilon ) (v^\varepsilon - v^{\varepsilon , n}) \right\| _{L^\frac{4}{3}_t L^1_x} \le \frac{1}{ 5} \left\| v^\varepsilon - v^{\varepsilon , n} \right\| _{L^\infty _t L^2_x}. \end{aligned}$$

For the second term in the definition of ${\tilde{G}}_1^\varepsilon$ the same techniques are applied which yields the bound

$$\begin{aligned} \left\| ([\left| w + v^\varepsilon \right| - \left| w + v^{\varepsilon , n} \right| ] *\phi _\varepsilon ) v^{\varepsilon ,n} \right\| _{L^\frac{4}{3}_t L^1_x} \le T^{\frac{3}{4}} \left\| v^{\varepsilon , n} \right\| _{L^\infty _t L^2_x} \left\| v^\varepsilon - v^{\varepsilon , n} \right\| _{L^\infty _t L^2_x}. \end{aligned}$$

By the proof of Theorem 12, one has

$$\begin{aligned} \left\| v^{\varepsilon , n} \right\| _{L^\infty _t L^2_x} \lesssim \left\| v_n \right\| _2 \approx \left\| v_0 \right\| _2 \end{aligned}$$

(31)

and thus choosing T sufficiently small again yields

$$\begin{aligned} \left\| ([\left| w + v^\varepsilon \right| - \left| w + v^{\varepsilon , n} \right| ] *\phi _\varepsilon ) v^{\varepsilon ,n} \right\| _{L^\frac{4}{3}_t L^1_x} \le \frac{1}{ 5} \left\| v^\varepsilon - v^{\varepsilon , n} \right\| _{L^\infty _t L^2_x}. \end{aligned}$$

The first term in the definition of ${\tilde{G}}_2^\varepsilon$ is treated similarly to the above. The same is true for the second term, where we additionally observe that

$$\begin{aligned} \sup _{n \in {{\mathbb {N}}}} \left\| w^n \right\| _{C([0, T], H^1({{\mathbb {T}}}))} < \infty . \end{aligned}$$

(32)

For the third term, we have

$$\begin{aligned}&\left\| ([\left| w + v^{\varepsilon , n} \right| - \left| w^n + v^{\varepsilon , n} \right| ] *\phi _\varepsilon ) v^{\varepsilon , n} \right\| _{L^1_t L^2_x} \\&\quad \le \left\| [\left| w + v^{\varepsilon , n} \right| - \left| w^n + v^{\varepsilon , n} \right| ] *\phi _\varepsilon \right\| _{L^\infty _t L^\infty _x} \left\| v^{\varepsilon , n} \right\| _{L^\infty _t L^2_x} \\&\quad \le \left\| w - w^n \right\| _{L^\infty _t L^\infty _x} \left\| v^{\varepsilon , n} \right\| _{L^\infty _t L^2_x} \\&\quad \lesssim \left\| w - w^n \right\| _{L^\infty _t H^1_x({{\mathbb {T}}})} \xrightarrow {n \rightarrow \infty } 0, \end{aligned}$$

where the Cauchy–Schwarz inequality was used for the first estimate, the embedding $L^\infty _t \hookrightarrow L^1_t$, Young’s inequality and the inverse triangle inequality for the second estimate and the embedding $C([0,T], H^1({{\mathbb {T}}})) \hookrightarrow L^\infty _t L^\infty _x$ together with (31) for the last estimate. By the same techniques, one obtains the convergence of the fourth term to zero.

Finally, for the last term in the definition of ${\tilde{G}}_2^\varepsilon$, one has

$$\begin{aligned}&\left\| ([\left| w + v^\varepsilon \right| - \left| w \right| - \left| w^n + v^\varepsilon \right| + \left| w^n \right| ] *\phi _\varepsilon ) w^n \right\| _{L^1_t L^2_x} \\&\quad \le \left\| \left| w + v^\varepsilon \right| - \left| w \right| - \left| w^n + v^\varepsilon \right| + \left| w^n \right| \right\| _{L^1_t L^2_x} \left\| w^n \right\| _{L^\infty H^1_x({{\mathbb {T}}})} \\&\quad \lesssim \left\| \left| w + v^\varepsilon \right| - \left| w \right| - \left| w^n + v^\varepsilon \right| + \left| w^n \right| \right\| _{L^1_t L^2_x}, \end{aligned}$$

where Hölder’s inequality, the embedding $C([0,T], H^1({{\mathbb {T}}})) \hookrightarrow L^\infty _t L^\infty _x$ and Young’s inequality were used for the first estimate and (32) for the second estimate. Observe that by the inverse triangle inequality, the bound

$$\begin{aligned} \left| \left| w + v^\varepsilon \right| - \left| w \right| - \left| w^n + v^\varepsilon \right| + \left| w^n \right| \right| \le 2 \min \left\{ \left| w - w^n \right| , \left| v^\varepsilon \right| \right\} \le 2 \left| v^\varepsilon \right| \end{aligned}$$

holds pointwise (in t and x). This implies that

$$\begin{aligned} \left| w + v^\varepsilon \right| - \left| w \right| - \left| w^n + v^\varepsilon \right| + \left| w^n \right| \xrightarrow {n \rightarrow \infty } 0 \end{aligned}$$

and hence, by the theorem of dominated convergence for the space variable,

$$\begin{aligned} g_n(t) :=\left\| \left| w + v^\varepsilon \right| - \left| w \right| - \left| w^n + v^\varepsilon \right| + \left| w^n \right| \right\| _{L^2_x} \xrightarrow {n \rightarrow \infty } 0 \qquad \forall t \in [0, T]. \end{aligned}$$

Moreover, for all $t \in [0, T]$, we have $g_n(t) \le 2 \left\| v^\varepsilon (\cdot , t) \right\| _2$ and $\left\| v^\varepsilon \right\| _{L^1_t L^2_x} \lesssim \left\| v^\varepsilon \right\| _{L^\infty _t L^2_x} < \infty$. Hence, reapplying the theorem of dominated convergence for the time variable yields

$$\begin{aligned} \left\| ([\left| w + v^\varepsilon \right| - \left| w \right| - \left| w^n + v^\varepsilon \right| + \left| w^n \right| ] *\phi _\varepsilon ) w^n \right\| _{L^1_t L^2_x} \xrightarrow {n \rightarrow \infty } 0 \end{aligned}$$

as claimed. $\square$

Notice that (27) together with the local well-posedness of the modNLS (3) from Theorem 12 imply that the modNLS (3) is globally well-posed, i.e. Theorem 5 is proved.

Remark 16

Observe, that in the case $\alpha \ne 2$, the proof would proceed roughly unchanged up to Eq. (28). However, the differential inequality (29) would then become

$$\begin{aligned} \frac{1}{ 2} \frac{\mathrm {d}}{\mathrm {d}{t}} \left\| v^{\varepsilon , n}(\cdot , t) \right\| _2^2 \lesssim \left\| w^n \right\| _{L^\infty _t L^\infty _x}^{\alpha - 1} \left\| v^{\varepsilon , n} \right\| _{L^2_x}^2 + \left\| w^n \right\| _{L^\infty _t L^\infty _x} \left\| v^{\varepsilon , n} \right\| _{L^{\alpha }_x}^\alpha \end{aligned}$$

and this bound is not sufficient to exclude a blow-up of the $L^2$-norm (for $\alpha > 2$ the exponent on the right-hand side is too big and for $\alpha < 2$ it seems that the $L^\alpha$-norm cannot be controlled within the given setting).

Appendix A. Quadratic and subquadratic NLS on the torus

To prove global existence of solutions to the Cauchy problem of the (sub-)quadratic nonlinear Schrödinger equation on ${{\mathbb {T}}}$ (that is (2) with $\alpha \in [1, 2]$), we will employ the mass and energy conservation laws. The justification of conservation laws requires solutions which are differentiable in time. Again, time regularity will be obtained from the regularity in space. To that end we will smoothen out the rough (sub-)quadratic nonlinearity, in such a way that the solutions of the resulting equation still admit suitable conservation laws. The regularization is slightly different from the one used in the proof of Theorem 5. Let us mention that the ideas presented here are borrowed from [7] where the same problem was studied on ${{\mathbb {R}}}^d$, using a contraction argument and conservation laws. Since our setting is based on the torus, we have to work with Bourgain spaces. For the convenience of the reader, we present some of the arguments in detail.

Observe that if w is a sufficiently nice $2 \pi$-periodic function and $\varepsilon > 0$, then

$$\begin{aligned} (w *\phi _\varepsilon )(x)= & {} \int _{-\infty }^\infty w(y) \phi _\varepsilon (x - y) \mathrm {d}{y} = \sum _{n \in {{\mathbb {Z}}}} \int _{(2n - 1) \pi }^{(2n + 1)\pi } w(y) \phi _\varepsilon (x - y) \mathrm {d}{y} \\= & {} \int _{-\pi }^{\pi } w(y) \sum _{n \in {{\mathbb {Z}}}} \phi _\varepsilon (x - y - 2n \pi ) \mathrm {d}{y}. \end{aligned}$$

Hence, the convolution of w with $\phi _\varepsilon$ on ${{\mathbb {R}}}$ corresponds to the convolution of w with the periodization of $\phi _\varepsilon$ on ${{\mathbb {T}}}$. For the rest of the paper we will slightly abuse the notation and denote this periodization also by $\phi _\varepsilon$. In the same spirit we will use from now on $*$ to denote the convolution on ${{\mathbb {T}}}$.

The smooth version of (2) for $\alpha \in [1, 2]$ reads as

(33)

and the corresponding Duhamel’s formula is (cf. [7, Equations (2.14), (2.13), (2.11) and (1.15)])

$$\begin{aligned} w(\cdot , t) = e^{\mathrm {i}t \partial _{x}^{2}} (w_0 *\phi _\varepsilon ) \pm \mathrm {i}\int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} \left[ \left( \left| w *\phi _\varepsilon \right| ^{\alpha - 1} (w *\phi _\varepsilon )\right) *\phi _\varepsilon (\cdot , \tau )\right] \mathrm {d}{\tau }. \end{aligned}$$

(34)

From now on, we denote by $\mathcal {F}$ and $\mathcal {F}^{(-1)}$ the Fourier transform and the inverse Fourier transform, on the torus, respectively. We use the symmetric choice of constants and write also

$$\begin{aligned} {\hat{f}}(\xi ):= & {} (\mathcal {F}f)(\xi ) = \frac{1}{ \sqrt{2 \pi }} \int _{-\pi }^\pi e^{-\mathrm {i}\xi \cdot x} f(x) \mathrm {d}{x}, \\ {\check{g}}(x):= & {} \left( \mathcal {F}^{(-1)} g\right) (x) = \frac{1}{ \sqrt{2 \pi }} \sum _{\xi \in {{\mathbb {Z}}}} e^{\mathrm {i}\xi \cdot x} g(\xi ). \end{aligned}$$

One has $\mathcal {F}(f *g) = \sqrt{2 \pi } {\hat{f}} {\hat{g}}$. Furthermore, let $\langle \xi \rangle :=\sqrt{1 + \left| \xi \right| ^2}$ for any $\xi \in {{\mathbb {R}}}$ and $J^s w :=\mathcal {F}^{(-1)} \langle \cdot \rangle ^s \mathcal {F}w$ for any $w \in (C^\infty ({{\mathbb {T}}}))'$.

1.1 A.1. Prerequisites for the periodic case

In this section, we present some technical results from the literature, needed for the treatment of the (sub-)quadratic nonlinearity on the torus.

Lemma 17

Let $p \in [1, \infty ]$ and $\varepsilon \ge 0$. Then for any $w \in L^p({{\mathbb {T}}})$ one has

$$\begin{aligned} \left\| w *\phi _\varepsilon \right\| _{L^p({{\mathbb {T}}})} \le \left\| w \right\| _{L^p({{\mathbb {T}}})}. \end{aligned}$$

Lemma 18

Let $s \in {{\mathbb {R}}}$ and $w \in H^s({{\mathbb {T}}})$. Then

$$\begin{aligned} \left\| w *\phi _\varepsilon \right\| _{H^s({{\mathbb {T}}})} \le \left\| w \right\| _{H^s({{\mathbb {T}}})} \qquad \text {and} \qquad \left\| w *\phi _\varepsilon \right\| _{{\dot{H}}^s({{\mathbb {T}}})} \le \left\| w \right\| _{{\dot{H}}^s({{\mathbb {T}}})} \qquad \forall \varepsilon \ge 0, \end{aligned}$$

where $\left\| \cdot \right\| _{{\dot{H}}^s({{\mathbb {T}}})}$ is the homogeneous Sobolev norm on the torus. Furthermore, if $\varepsilon > 0$, then

$$\begin{aligned} \left\| w *\phi _\varepsilon \right\| _{H^s({{\mathbb {T}}})} \lesssim _{\varepsilon , s} \left\| w \right\| _{L^2({{\mathbb {T}}})}. \end{aligned}$$

Lemma 19

(à la Banach-Alaoglu) (Cf. [3, Theorem 3.16].) Let $w^n \xrightarrow {n \rightarrow \infty } w$ in $L^2({{\mathbb {T}}})$ and $\sup _{n \in {{\mathbb {N}}}} \left\| w^n \right\| _{H^1({{\mathbb {T}}})} < \infty$. Then $w \in H^1({{\mathbb {T}}})$,

$$\begin{aligned} \left\| w \right\| _{H^1({{\mathbb {T}}})} \le \liminf _{n \rightarrow \infty } \left\| w^n \right\| _{H^1({{\mathbb {T}}})}, \qquad \left\| w \right\| _{{\dot{H}}^1({{\mathbb {T}}})} \le \liminf _{n \rightarrow \infty } \left\| w^n \right\| _{{\dot{H}}^1({{\mathbb {T}}})}, \end{aligned}$$

(35)

and $w^n \rightharpoonup w$ in $H^1({{\mathbb {T}}})$, i.e. for any $u \in H^1({{\mathbb {T}}})$ one has

$$\begin{aligned} \lim _{n \rightarrow \infty } \left\langle w^n, u \right\rangle _{H^1({{\mathbb {T}}})} = \lim _{n \rightarrow \infty } \sum _{k \in {{\mathbb {Z}}}} \langle k \rangle ^2 \overline{{\hat{w}}^n_k} {\hat{u}}_k = \left\langle w, u \right\rangle _{H^1({{\mathbb {T}}})}. \end{aligned}$$

(36)

If additionally $\left\| w^n \right\| _{H^1} \xrightarrow {n \rightarrow \infty } \left\| w \right\| _{H^1}$, then $w^n \xrightarrow {n \rightarrow \infty } w$ in $H^1({{\mathbb {T}}})$.

In the following we are going to use the $X^{s,b}$ spaces on the torus where $s, b\in {{\mathbb {R}}}$. They are defined via the norm (see equation (3.49) in [6])

$$\begin{aligned} \Vert w\Vert _{X^{s,b}}=\Vert \langle k\rangle ^{s}\langle \tau +k^{2}\rangle ^{b}{\hat{w}}(\tau ,k)\Vert _{L^{2}_{\tau }l^{2}_{k}}. \end{aligned}$$

(37)

Lemma 20

$(X^{0, \frac{3}{8}} \hookrightarrow L^4({{\mathbb {T}}}\times {{\mathbb {R}}}))$ (See [14, Proposition 2.13].) We have

$$\begin{aligned} \left\| w \right\| _{L^4({{\mathbb {T}}}\times {{\mathbb {R}}})} \lesssim \left\| w \right\| _{X^{0, \frac{3}{8}}} \end{aligned}$$

for any $w \in \mathcal {S}({{\mathbb {R}}}, C^\infty ({{\mathbb {T}}}))$.

Lemma 21

$(X_\delta ^{s, b} \hookrightarrow C(H^s))$ (Cf. [6, Lemma 3.9].) Let $b > \frac{1}{ 2}$ and $s \in {{\mathbb {R}}}$. Then

$$\begin{aligned} \left\| w \right\| _{C([0, \delta ], H^s({{\mathbb {T}}}))} \lesssim \left\| w \right\| _{X_\delta ^{s, b}}. \end{aligned}$$

Lemma 22

(Linear Schrödinger evolution in $X_\delta ^{s, b}$) (Cf. [6, Lemma 3.10].) Let $b, s \in {{\mathbb {R}}}$, $\delta \in (0, 1]$ and $\eta$ a smooth cut-off in time. Then

$$\begin{aligned} \left\| \eta (t) e^{\mathrm {i}t \partial _{x}^{2}} w_0 \right\| _{X_{\delta }^{s, b}} \lesssim \left\| w_0 \right\| _{H^s({{\mathbb {T}}})} \qquad \forall w_0 \in H^s({{\mathbb {T}}}). \end{aligned}$$

Lemma 23

(Treating the integral term in $X_\delta ^{s, b})$ (Cf. [6, Lemma 3.12].) Let $b \in \left( \frac{1}{ 2}, 1 \right]$, $s \in {{\mathbb {R}}}$ and $\delta \le 1$. Set $b' :=b - 1$. Then

$$\begin{aligned} \left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} F(\tau ) \mathrm {d}{\tau } \right\| _{X_\delta ^{s, b}} \lesssim _b \left\| F \right\| _{X_\delta ^{s, b'}} \qquad \forall F \in X_\delta ^{s, b'}. \end{aligned}$$

Lemma 24

(Changing b in $X_\delta ^{s, b}$) (Cf. [6, Lemma 3.11].) Let $b, b' \in \left( -\frac{1}{ 2}, \frac{1}{ 2} \right)$ with $b' < b$, $s \in {{\mathbb {R}}}$ and $\delta \in (0, 1]$. Then

$$\begin{aligned} \left\| w \right\| _{X_\delta ^{s, b'}} \lesssim \delta ^{b - b'} \left\| w \right\| _{X_\delta ^{s, b}} \qquad \forall w. \end{aligned}$$

The next proposition appears in [6] for the case of the cubic nonlinearity and $\varepsilon = 0$. Since we need the corresponding result for the (sub-)quadratic nonlinearity, which is more complicated than the cubic nonlinearity because it is not algebraic, we present the proof, too.

Proposition 25

(Control of the nonlinearity in $X_\delta ^{s, b}$) (Cf. [6, Proposition 3.26].) Let $s \ge 0$ and $\varepsilon > 0$ or $\varepsilon = s = 0$. Then, for all $w_1, w_2$ we have

$$\begin{aligned}&\left\| \left( \left| w_1 *\phi _\varepsilon \right| ^{\alpha - 1} (w_1 *\phi _\varepsilon )\right) *\phi _\varepsilon - \left( \left| w_2 *\phi _\varepsilon \right| ^{\alpha - 1} (w_2 *\phi _\varepsilon )\right) *\phi _\varepsilon \right\| _{X_\delta ^{s, -\frac{3}{8}}}\\&\quad \lesssim _{\varepsilon , s} \left( \left\| w_1 \right\| ^{\alpha - 1}_{X_\delta ^{0, \frac{3}{8}}} + \left\| w_2 \right\| ^{\alpha - 1}_{X_\delta ^{0, \frac{3}{8}}}\right) \left( \left\| w_1 - w_2 \right\| _{X_\delta ^{0, \frac{3}{8}}} \right) . \end{aligned}$$

Proof

Fix $w_1, w_2$. Then, by Plancherel theorem and duality in $L^2({{\mathbb {R}}}\times {{\mathbb {T}}})$, one has

$$\begin{aligned}&\left\| \left( \left| w_1 *\phi _\varepsilon \right| ^{\alpha - 1} (w_1 *\phi _\varepsilon )\right) *\phi _\varepsilon - \left( \left| w_2 *\phi _\varepsilon \right| ^{\alpha - 1} (w_2 *\phi _\varepsilon )\right) *\phi _\varepsilon \right\| _{X_\delta ^{s, -\frac{3}{8}}} \\&\quad = \sup _{\left\| w \right\| _{X_\delta ^{-s, \frac{3}{8}}} = 1} \left| \left\langle \left( \left| w_1 *\phi _\varepsilon \right| ^{\alpha - 1} (w_1 *\phi _\varepsilon ) - \left| w_2 *\phi _\varepsilon \right| ^{\alpha - 1} (w_2 *\phi _\varepsilon )\right) *\phi _\varepsilon , w \right\rangle _{L^2({{\mathbb {R}}}\times {{\mathbb {T}}})} \right| . \end{aligned}$$

Fix any $w \in X_\delta ^{-s, \frac{3}{8}}$ with $\left\| w \right\| _{X_\delta ^{-s, \frac{3}{8}}} = 1$. Then

$$\begin{aligned}&\left| \left\langle \left( \left| w_1 *\phi _\varepsilon \right| ^{\alpha - 1} (w_1 *\phi _\varepsilon ) - \left| w_2 *\phi _\varepsilon \right| ^{\alpha - 1} (w_2 *\phi _\varepsilon )\right) *\phi _\varepsilon , w \right\rangle _{L^2({{\mathbb {R}}}\times {{\mathbb {T}}})} \right| \\&\quad = \left| \left\langle J^s \left[ \left( \left| w_1 *\phi _\varepsilon \right| ^{\alpha - 1} (w_1 *\phi _\varepsilon ) - \left| w_2 *\phi _\varepsilon \right| ^{\alpha - 1} (w_2 *\phi _\varepsilon )\right) *\phi _\varepsilon \right] , J^{-s} w \right\rangle _{L^2({{\mathbb {R}}}\times {{\mathbb {T}}})} \right| \\&\quad \le \left\| \left( \left| w_1 *\phi _\varepsilon \right| ^{\alpha - 1} (w_1 *\phi _\varepsilon ) - \left| w_2 *\phi _\varepsilon \right| ^{\alpha - 1} (w_2 *\phi _\varepsilon )\right) *(J^s \phi _\varepsilon ) \right\| _{L^\frac{4}{3}({{\mathbb {R}}}\times {{\mathbb {T}}})} \\&\qquad \cdot \left\| J^{-s} w \right\| _{L^4({{\mathbb {R}}}\times {{\mathbb {T}}})} \\&\quad \lesssim _{\varepsilon , s} \left\| \left| w_1 *\phi _\varepsilon \right| ^{\alpha - 1} (w_1 *\phi _\varepsilon ) - \left| w_2 *\phi _\varepsilon \right| ^{\alpha - 1} (w_2 *\phi _\varepsilon ) \right\| _{L^\frac{4}{3}({{\mathbb {R}}}\times {{\mathbb {T}}})} \underbrace{\left\| J^{-s} w \right\| _{X_\delta ^{0, \frac{3}{8}}} }_{= \left\| w \right\| _{X_\delta ^{-s, \frac{3}{8}}} = 1} \\&\quad \le \left\| \left| w_1 *\phi _\varepsilon \right| ^{\alpha - 1} (w_1 *\phi _\varepsilon ) - \left| w_2 *\phi _\varepsilon \right| ^{\alpha - 1} (w_2 *\phi _\varepsilon ) \right\| _{L^\frac{4}{3}({{\mathbb {R}}}\times {{\mathbb {T}}})} \\&\quad \le \left\| \left( \left| w_1 *\phi _\varepsilon \right| ^{\alpha - 1} + \left| w_1 *\phi _\varepsilon \right| ^{\alpha - 1} \right) ((w_1 - w_2) *\phi _\varepsilon ) \right\| _{L^\frac{4}{3}({{\mathbb {R}}}\times {{\mathbb {T}}})}, \end{aligned}$$

where, for the first estimate, we used Hölder’s inequality and Young’s inequality, Lemma 20 for the second and the size estimate (78) for the last inequality. Applying Hölder’s inequality again yields the upper bound

$$\begin{aligned} \left( \left\| \left| w_1 *\phi _\varepsilon \right| ^{\alpha - 1} \right\| _{L^4({{\mathbb {R}}}\times {{\mathbb {T}}})} + \left\| \left| w_1 *\phi _\varepsilon \right| ^{\alpha - 1} \right\| _{L^4({{\mathbb {R}}}\times {{\mathbb {T}}})} \right) \left\| (w_1 - w_2) *\phi _\varepsilon \right\| _{L^2({{\mathbb {R}}}\times {{\mathbb {T}}})}. \end{aligned}$$

For the first factor, we apply Hölder’s and Young’s inequalities as well as the embedding from Lemma 20 and arrive at the upper bound of

$$\begin{aligned} \left\| w_1 \right\| _{L^4({{\mathbb {R}}}\times {{\mathbb {T}}})}^{\alpha - 1} + \left\| w_2 \right\| _{L^4({{\mathbb {R}}}\times {{\mathbb {T}}})}^{\alpha - 1} \lesssim \left\| w_1 \right\| _{X^{0, \frac{3}{8}}}^{\alpha - 1} + \left\| w_2 \right\| _{X^{0, \frac{3}{8}}}^{\alpha - 1}. \end{aligned}$$

For the second factor, Young’s inequality and the definition of the norm in $X^{0, \frac{3}{8}}$ yield the upper bound of

$$\begin{aligned} \left\| w_1 - w_2 \right\| _{X^{0, \frac{3}{8}}}. \end{aligned}$$

This completes the proof. $\square$

1.2 A.2. Results in the periodic case

First, we consider local well-posedness:

Theorem 26

(Cf. [6, Theorem 3.27] for the cubic NLS.) Let $\varepsilon > 0$ and $s \ge 0$ or $\varepsilon = s = 0$. Then the (smoothened) (sub-)quadratic NLS (33) is locally well-posed in $H^s({{\mathbb {T}}})$.

Proof

It suffices to show that the right-hand side of (34) defines a contractive self-mapping $\mathcal {T}: M(R, \delta ) \rightarrow M(R, \delta )$ for some $R, \delta > 0$, where

$$\begin{aligned} M(R, \delta ) :=\left\{ w \in Y \big | \, \left\| w \right\| _Y \le R \right\} \end{aligned}$$

and Y is a suitable subspace of $C([0, \delta ], H^s({{\mathbb {T}}}))$.

We consider the case $s \ge 1$ first. Put $Y = C([0, \delta ], H^s({{\mathbb {T}}}))$. Due to $e^{\mathrm {i}t \partial _{x}^{2}}$ being an isometry on $H^s({{\mathbb {T}}})$ for any $t \in {{\mathbb {R}}}$ and Lemma 18, we have

$$\begin{aligned}&\left\| \mathcal {T}w \right\| _Y \\&\quad \le \left\| e^{\mathrm {i}t \partial _{x}^{2}} (w_0 *\phi _\varepsilon ) \right\| _{H^s({{\mathbb {T}}})} + \left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _{x}^{2}} \left[ \left( \left| w *\phi _\varepsilon \right| ^{\alpha - 1} (w *\phi _\varepsilon )\right) *\phi _\varepsilon \right] \mathrm {d}{\tau } \right\| _Y \\&\quad \le \left\| w_0 \right\| _{H^s({{\mathbb {T}}})} + \delta \left\| (\left| w *\phi _\varepsilon \right| ^{\alpha - 1} (w *\phi _\varepsilon )) *\phi _\varepsilon \right\| _Y. \end{aligned}$$

This suggests the choice $R \approx \left\| w_0 \right\| _{H^s}$. Fix any $\tau \in [0, \delta ]$. Then, due to Lemma 18 and the embedding $H^s({{\mathbb {T}}}) \hookrightarrow L^\infty ({{\mathbb {T}}})$, we have that

$$\begin{aligned}&\left\| ((\left| w *\phi _\varepsilon \right| ^{\alpha - 1} (w *\phi _\varepsilon )) *\phi _\varepsilon )(\cdot , \tau ) \right\| _{H^s({{\mathbb {T}}})} \\&\quad \lesssim _{\varepsilon , s} \left\| (\left| w *\phi _\varepsilon \right| ^{\alpha - 1} (w *\phi _\varepsilon ))(\cdot , \tau ) \right\| _{L^2({{\mathbb {T}}})} \\&\quad \le \left\| (w *\phi _\varepsilon )(\cdot , \tau ) \right\| _{L^\infty ({{\mathbb {T}}})}^{\alpha - 1} \left\| (w *\phi _\varepsilon )(\cdot , \tau ) \right\| _{L^2({{\mathbb {T}}})} \\&\quad \lesssim R^\alpha . \end{aligned}$$

By the above, the condition $\left\| \mathcal {T}w \right\| _Y \le R$ is satisfied, if $\delta \lesssim _{\varepsilon , s} R^{1 - \alpha }$. The contraction property of $\mathcal {T}$ is shown in the same way, possibly requiring a smaller implicit constant in the last inequality.

In the case $s \in [0, 1)$ and $\varepsilon > 0$, consider any $b \in \left( \frac{1}{ 2}, \frac{5}{8} \right)$ and put $Y = X_{\delta }^{s, b}$ (by Lemma 21 one indeed has $Y \hookrightarrow C([0, \delta ], H^s({{\mathbb {T}}}))$). Then, by the triangle inequality and Lemmata 22 and 23 we have

$$\begin{aligned}&\left\| \mathcal {T}w \right\| _{X_{\delta }^{s, b}} \\&\quad \le \left\| e^{\mathrm {i}t \partial _{x}^{2}} (w_0 *\phi _\varepsilon ) \right\| _{X_{\delta }^{s, b}} + \left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _x^2} \left[ \left( \left| w *\phi _\varepsilon \right| ^{\alpha - 1} (w *\phi _\varepsilon ) \right) *\phi _\varepsilon \right] \mathrm {d}{\tau } \right\| _{X_{\delta }^{s, b}} \\&\quad \lesssim \left\| w_0 \right\| _{H^s({{\mathbb {T}}})} + \left\| \left( \left| w *\phi _\varepsilon \right| ^{\alpha - 1} (w *\phi _\varepsilon )\right) *\phi _\varepsilon \right\| _{X_\delta ^{s, b - 1}}. \end{aligned}$$

This estimate suggests $R \approx \left\| w_0 \right\| _{H^s({{\mathbb {T}}})}$. For the second summand, apply Lemma 24 and Proposition 25 (with $w = 0$) to obtain the upper bound

$$\begin{aligned} \left\| \left( \left| w *\phi _\varepsilon \right| ^{\alpha - 1} (w *\phi _\varepsilon )\right) *\phi _\varepsilon \right\| _{X_\delta ^{s, b - 1}}\lesssim & {} \delta ^{1 - b - \frac{3}{8}} \left\| \left( \left| w *\phi _\varepsilon \right| ^{\alpha - 1} (w *\phi _\varepsilon )\right) *\phi _\varepsilon \right\| _{X_\delta ^{s, -\frac{3}{8}}} \nonumber \\\lesssim & {} \delta ^{1 - b - \frac{3}{8}} \left\| w \right\| _{X_\delta ^{0, \frac{3}{8}}}^{\alpha - 1} \left\| w \right\| _{X_\delta ^{s, \frac{3}{8}}} \nonumber \\\le & {} \delta ^{1 - b - \frac{3}{8}} \left\| w \right\| _{X_\delta ^{s, \frac{3}{8}}}^\alpha \le \delta ^{1 - b - \frac{3}{8}} R^\alpha . \end{aligned}$$

(38)

As the exponent of $\delta$ is positive, we can choose $\delta$ small enough to make $\mathcal {T}$ a self-mapping of $M(R, \delta )$. The fact that $\mathcal {T}$ is contractive is proven similarly, possibly requiring a smaller $\delta$.

The remaining case $\varepsilon = s = 0$ is treated exactly as the last case. $\square$

In order to prove the conservation laws, we need to be able to approximate by smooth solutions.

Lemma 27

(Smooth solutions for smooth initial data) (Cf. [14, Proposition 3.11].) Let $\varepsilon > 0$, and $w_0 \in L^2({{\mathbb {T}}})$ and let w denote the unique solution of (34). Then $w \in C([0, \delta ], H^\infty ({{\mathbb {R}}}))$ and for any $s > \frac{1}{ 2}$ one has

$$\begin{aligned} \left\| w \right\| _{L_t^\infty H_x^s} \le C \left\| w_0 \right\| _{L^2} e^{Ct \left\| w \right\| ^{\alpha - 1}_{L_t^\infty H^1}} \end{aligned}$$

(39)

for some $C = C(\varepsilon , s) > 0$.

Proof

As w is the solution to (34), one immediately has

$$\begin{aligned} \left\| w(\cdot , t) \right\| _{H^s}&\le \left\| w_0 *\phi _\varepsilon \right\| _{H^s} + \int _0^t \left\| (\left| w *\phi _\varepsilon \right| ^{\alpha - 1} (w *\phi _\varepsilon )) *\phi _\varepsilon \right\| _{H^s} \mathrm {d}{\tau } \\&\lesssim _{\varepsilon , s} \left\| w_0 \right\| _{L^2} + \int _0^t \left\| \left| w *\phi _\varepsilon \right| ^{\alpha - 1} (w *\phi _\varepsilon ) \right\| _{L^2} \mathrm {d}{\tau } \\&\le \left\| w_0 \right\| _{L^2} + \left\| w *\phi _\varepsilon \right\| _{L^\infty _t L^\infty _x}^{\alpha - 1} \int _0^t \left\| w *\phi _\varepsilon \right\| _{L^2} \mathrm {d}{\tau } \\&\lesssim \left\| w_0 \right\| _{L^2} + \left\| w \right\| _{L^\infty _t H^1}^{\alpha - 1} \int _0^t \left\| w \right\| _{H^s} \mathrm {d}{\tau }. \end{aligned}$$

Now (39) follows from Lemma 9. $\square$

Theorem 28

Let $\varepsilon > 0$ and $s \in [1, \infty )$. Then the smoothened NLS (33) is globally well-posed in $H^s({{\mathbb {T}}})$.

Proof

Local well-posedness has already been shown in Theorem 26 and it remains to show that the solution w exists globally. By the blow-up alternative, it suffices to see that $\left\| w(\cdot , t) \right\| _{H^s({{\mathbb {T}}})}$ cannot explode. Moreover, by Lemma 27 it suffices to consider $s = 1$. By the same lemma, one has that $w \in C([0, \delta ], H^\infty ({{\mathbb {T}}}))$ and in particular, $w \in C^1([0, \delta ], H^1({{\mathbb {T}}}))$. Hence, the energy conservation (cf. [7, Equations (3.14) and (1.18)])

$$\begin{aligned} E_\varepsilon (w(\cdot , t)) :=\int _{{{\mathbb {T}}}} \frac{1}{ 2} \left| \nabla w(x, t) \right| ^2 \mp \frac{1}{ \alpha + 1} \left| (w *\phi _\varepsilon ) (x, t) \right| ^{\alpha + 1} \mathrm {d}{x} = E_\varepsilon (w_0 *\phi _\varepsilon ) \end{aligned}$$

(40)

is applicable to w. But

$$\begin{aligned} \left\| w(\cdot , t) \right\| _{H^1({{\mathbb {T}}})}^2 = \left\| w_0 \right\| _2^2 + 2 E_\varepsilon (w_0 *\phi _\varepsilon ) \pm \frac{2}{\alpha + 1} \left\| w(\cdot , t) \right\| _{\alpha + 1}^{\alpha + 1} \end{aligned}$$

(41)

and so $\left\| w(\cdot , t) \right\| _{{\dot{H}}^1({{\mathbb {T}}})}$ is controlled by $E_\varepsilon (w_0 *\phi _\varepsilon )$ in the defocusing case. In the focusing case we can assume w.l.o.g. that $\left\| w(\cdot , t) \right\| _{{\dot{H}}^1({{\mathbb {T}}})}^2$ is an unbounded function of t, (otherwise, there is nothing to show) and say that $\left\| w(\cdot , t) \right\| _{{\dot{H}}^1({{\mathbb {T}}})}^2$ is large. Then, by the Gagliardo-Nirenberg inequality from [3, Chapter 8, Eqn. (42)], we have

$$\begin{aligned} \left\| w(\cdot , t) \right\| _{\alpha + 1}^{\alpha + 1} \lesssim \left\| w(\cdot , t) \right\| _2^{\frac{\alpha + 3}{2}} \left\| w(\cdot , t) \right\| _{H^1({{\mathbb {T}}})}^{\frac{\alpha - 1}{2}} \le \frac{1}{ 2} \left\| w(\cdot , t) \right\| _{H^1({{\mathbb {T}}})}^2, \end{aligned}$$

(42)

where above we additionally used the mass conservation

$$\begin{aligned} \left\| w(\cdot , t) \right\| _{L^2({{\mathbb {T}}})} = \left\| w(\cdot , 0) \right\| _{L^2({{\mathbb {T}}})}. \end{aligned}$$

Hence, inserting (42) into (41) and rearranging the inequality shows that the quantity $\left\| w(\cdot , t) \right\| _{H^1({{\mathbb {T}}})}^2$ is bounded, in contradiction to the assumption. This completes the proof. $\square$

Theorem 29

(Cf. [6, Theorem 3.28] for the cubic NLS.) The Cauchy problem for the (sub-)quadratic periodic NLS ((2) with $\alpha \in [1, 2]$) is globally well-posed in $L^2({{\mathbb {T}}})$ and the solution w enjoys mass conservation $\left\| w(\cdot , t) \right\| _{L^2({{\mathbb {T}}})} = \left\| w_0 \right\| _{L^2({{\mathbb {T}}})}$.

Proof

Local well-posedness has already been shown in Theorem 26. Let w denote this local solution. By the blow-up alternative, it suffices to show mass conservation. To that end, let us denote by $w^\varepsilon$ the global solution of (33) for $\varepsilon > 0$ from Theorem 28. We will show that for any $b \in \left( \frac{1}{ 2}, \frac{5}{8}\right)$ one has $\left\| w^\varepsilon - w \right\| _{X_\delta ^{0, b}} \rightarrow 0$ as $\varepsilon \rightarrow 0+$. To that end, notice that

$$\begin{aligned}&\left\| w^\varepsilon - w \right\| _{X_\delta ^{0, b}} \end{aligned}$$

(43)

$$\begin{aligned}&\quad \le \left\| e^{\mathrm {i}t \partial _{x}^{2}} (w_0 *\phi _\varepsilon - w_0) \right\| _{X_\delta ^{0, b}} \nonumber \\&\qquad + \left\| \int _0^t e^{\mathrm {i}(t - \tau ) \partial _x^2} \left[ (\left| w^\varepsilon *\phi _\varepsilon \right| ^{\alpha - 1} (w^\varepsilon *\phi _\varepsilon )) *\phi _\varepsilon - \left| w \right| ^{\alpha - 1}w \right] \mathrm {d}{\tau } \right\| _{X_\delta ^{0, b}} \nonumber \\&\quad \lesssim \left\| w_0 *\phi _\varepsilon - w_0 \right\| _{L^2({{\mathbb {T}}})} + \left\| (\left| w^\varepsilon *\phi _\varepsilon \right| ^{\alpha - 1} (w^\varepsilon *\phi _\varepsilon )) *\phi _\varepsilon - \left| w \right| ^{\alpha - 1}w \right\| _{X_\delta ^{0, b - 1}} \nonumber \\&\quad \lesssim \left\| w_0 *\phi _\varepsilon - w_0 \right\| _{L^2({{\mathbb {T}}})} + \delta ^{1 - b} \left\| (\left| w^\varepsilon *\phi _\varepsilon \right| ^{\alpha - 1} (w^\varepsilon *\phi _\varepsilon )) *\phi _\varepsilon - \left| w \right| ^{\alpha - 1}w \right\| _{X_\delta ^{0, 0}},\nonumber \\ \end{aligned}$$

(44)

where we used the fact that w and $w^\varepsilon$ solve the corresponding fixed-point equations and Lemmata 22, 23 and 24.

For the first summand, observe that

$$\begin{aligned} \left\| w_0 *\phi _\varepsilon - w_0 \right\| _{L^2({{\mathbb {T}}})} = \left\| \left( \langle k \rangle ^s{\hat{w}}_0(k) (\sqrt{2 \pi } {\hat{\phi }}_\varepsilon (k) - 1) \right) _k \right\| _{l^2({{\mathbb {Z}}})} \end{aligned}$$

and the right-hand side above converges to 0 as $\varepsilon \rightarrow 0+$ by the dominated convergence theorem and the definition of $\phi _\varepsilon$.

For the second summand, note that $X_\delta ^{0, 0} = L^2([0, \delta ] \times {{\mathbb {T}}})$ and hence

$$\begin{aligned}&\left\| (\left| w^\varepsilon *\phi _\varepsilon \right| ^{\alpha - 1} (w^\varepsilon *\phi _\varepsilon )) *\phi _\varepsilon - \left| w \right| ^{\alpha - 1}w \right\| _{L^2([0, \delta ] \times {{\mathbb {T}}})} \\&\quad \le \left\| (\left| w \right| ^{\alpha - 1}w) *\phi _\varepsilon - \left| w \right| ^{\alpha - 1}w \right\| _{L^2([0, \delta ] \times {{\mathbb {T}}})} \\&\qquad + \left\| (\left| w^\varepsilon *\phi _\varepsilon \right| ^{\alpha - 1} (w^\varepsilon *\phi _\varepsilon ) - \left| w \right| ^{\alpha - 1}w) *\phi _\varepsilon \right\| _{L^2([0, \delta ] \times {{\mathbb {T}}})}. \end{aligned}$$

The first summand above goes to zero due to $(\phi _\varepsilon )_\varepsilon$ being an approximation to the identity on $L^{2 \alpha }({{\mathbb {T}}})$. The other summand is further estimated by

$$\begin{aligned}&\left\| (\left| w^\varepsilon *\phi _\varepsilon \right| ^{\alpha - 1} (w^\varepsilon *\phi _\varepsilon ) - \left| w \right| ^{\alpha - 1}w) *\phi _\varepsilon \right\| _{L^2([0, \delta ] \times {{\mathbb {T}}})} \nonumber \\&\quad \le \left\| \left| w^\varepsilon *\phi _\varepsilon \right| ^{\alpha - 1} (w^\varepsilon *\phi _\varepsilon ) - \left| w \right| ^{\alpha - 1}w \right\| _{L^2([0, \delta ] \times {{\mathbb {T}}})} \nonumber \\&\quad \lesssim \left\| \left( \left| w^\varepsilon *\phi _\varepsilon \right| ^{\alpha - 1} + \left| w \right| ^{\alpha - 1} \right) (w^\varepsilon *\phi _\varepsilon - w) \right\| _{L^2([0, \delta ] \times {{\mathbb {T}}})} \nonumber \\&\quad \le \left\| \left( \left| w^\varepsilon *\phi _\varepsilon \right| ^{\alpha - 1} + \left| w \right| ^{\alpha - 1} \right) [(w^\varepsilon - w) *\phi _\varepsilon ] \right\| _{L^2([0, \delta ] \times {{\mathbb {T}}})} \nonumber \\&\qquad + \left\| \left( \left| w^\varepsilon *\phi _\varepsilon \right| ^{\alpha - 1} + \left| w \right| ^{\alpha - 1} \right) (w *\phi _\varepsilon - w) \right\| _{L^2([0, \delta ] \times {{\mathbb {T}}})}. \end{aligned}$$

(45)

Let us introduce the set $A^\varepsilon = \left\{ \left| w^\varepsilon \right| *\phi _\varepsilon \le 1 \right\}$. Then the first summand above is further estimated by

$$\begin{aligned}&\left\| \left( \left| w^\varepsilon *\phi _\varepsilon \right| ^{\alpha - 1} + \left| w \right| ^{\alpha - 1} \right) [(w^\varepsilon - w) *\phi _\varepsilon ] \right\| _{L^2([0, \delta ] \times {{\mathbb {T}}})} \\&\quad \le \left\| \left( \mathbb {1}_{A^\varepsilon } \left| w^\varepsilon *\phi _\varepsilon \right| ^{\alpha - 1} + \mathbb {1}_{A^0} \left| w \right| ^{\alpha - 1} \right) [(w^\varepsilon - w) *\phi _\varepsilon ] \right\| _{L^2([0, \delta ] \times {{\mathbb {T}}})} \\&\qquad + \left\| \left( \mathbb {1}_{(A^\varepsilon )^c} \left| w^\varepsilon *\phi _\varepsilon \right| ^{\alpha - 1} + \mathbb {1}_{(A^0)^c} \left| w \right| ^{\alpha - 1} \right) [(w^\varepsilon - w) *\phi _\varepsilon ] \right\| _{L^2([0, \delta ] \times {{\mathbb {T}}})} \\&\quad \le \left\| w^\varepsilon - w \right\| _{L^2([0, \delta ] \times {{\mathbb {T}}})} + \left\| w \right\| _{L^4([0, \delta ] \times {{\mathbb {T}}})} \left\| w^\varepsilon - w \right\| _{L^4([0, \delta ] \times {{\mathbb {T}}})} \\&\quad \lesssim \left( 1 + \left\| w \right\| _{X_\delta ^{0, b}} \right) \left\| w^\varepsilon - w \right\| _{X_\delta ^{0, b}}, \end{aligned}$$

where we used Hölder’s and Young’s inequalities for the penultimate estimate and Lemma 20 for the last step. Recall that in front of this term is $\delta ^{1 - b}$ and, w.l.o.g., $\delta \ll 1$. Hence we can just move it to the left-hand side of (4344). The treatment of the last remaining term (45) does not require any new techniques.

By the above, $\left\| w^\varepsilon - w \right\| _{X_\delta ^{0, b}} \rightarrow 0$ as $\varepsilon \rightarrow 0+$. Applying Lemma 21, we see that

$$\begin{aligned} \left\| w \right\| _{C([0, T], L^2({{\mathbb {T}}}))}\le & {} \limsup _{\varepsilon \rightarrow 0+} \left[ \left\| w^\varepsilon - w \right\| _{X_\delta ^{0, b}} + \left\| w^\varepsilon \right\| _{C([0, T], L^2({{\mathbb {T}}}))} \right] \\\le & {} \limsup _{\varepsilon \rightarrow 0+} \left[ \left\| w_0 *\phi _\varepsilon \right\| _{L^2({{\mathbb {T}}})} \right] = \left\| w_0 \right\| _{L^2({{\mathbb {T}}})} \end{aligned}$$

and hence the solution w indeed enjoys mass conservation. This finishes the proof. $\square$

In addition to mass conservation, we also have conservation of the energy.

Theorem 30

(Cf. [7, Theorem 3.1] and [10, Theorem 2.1].) The Cauchy problem for the subquadratic periodic NLS ((2) with $\alpha \in [1, 2]$) is globally well-posed in $H^1({{\mathbb {T}}})$ and the solution w enjoys energy conservation $E(w(\cdot , t)) = E(w_0)$.

Remark 31

In [10] it is claimed that the quadratic NLS is globally well-posed on the torus. They refer to [7], where it is done on the real line. While our proof of Theorem 30 borrows some ideas from [7], we believe that in order to be able to do the torus case, one needs the result of Bourgain [1], in particular, the Bourgain spaces, which appeared 5 years after [10].

Proof of Theorem 30

Let $w_0 \in H^1({{\mathbb {T}}}) \subseteq L^2({{\mathbb {T}}})$. By Theorem 29, the subquadratic periodic NLS has the unique global solution $w \in C_{\text {b}}({{\mathbb {R}}}, L^2({{\mathbb {T}}}))$. It remains to show that $w \in C_{\text {b}}({{\mathbb {R}}}, H^1({{\mathbb {T}}}))$. To show that for any $t \in {{\mathbb {R}}}$ one has $w(\cdot , t) \in H^1({{\mathbb {T}}})$ we first prove that

$$\begin{aligned} \sup _{\varepsilon > 0} \left\| w^\varepsilon \right\| _{C({{\mathbb {R}}}, H^1({{\mathbb {T}}}))} < \infty . \end{aligned}$$

(46)

By the calculations similar to those in the proof of Theorem 29, it suffices to prove the corresponding bound for the energy $E_\varepsilon (w^\varepsilon (\cdot , t))$.

To that end let $w^\varepsilon$ be the unique global solution of the modified NLS (33) for $\varepsilon > 0$ from Theorem 28. The energy conservation from Equation (40) implies

$$\begin{aligned} E_\varepsilon (w^\varepsilon (\cdot , t)) = E_\varepsilon (w_0 *\phi _\varepsilon ) = \frac{1}{ 2} \left\| w_0 *\phi _\varepsilon \right\| _{{\dot{H}}^1({{\mathbb {T}}})}^2 \mp \frac{1}{ \alpha + 1} \left\| w_0 *\phi _\varepsilon \right\| _{L^3({{\mathbb {T}}})}^{\alpha + 1}. \end{aligned}$$

Observe that by Lemma 18 the first summand above satisfies

$$\begin{aligned} \left\| w_0 *\phi _\varepsilon \right\| _{{\dot{H}}^1({{\mathbb {T}}})}^2 \le \left\| w_0 \right\| _{{\dot{H}}^1({{\mathbb {T}}})}^2. \end{aligned}$$

If the sign of the second summand is negative (focusing case), there is nothing left to do. If the sign is positive (defocusing case), one has

$$\begin{aligned} \left\| w_0 *\phi _\varepsilon \right\| _{\alpha + 1}^{\alpha + 1} \le \left\| w_0 \right\| _{\alpha + 1}^{\alpha + 1} \le \left\| w_0 \right\| _{L^\infty ({{\mathbb {T}}})}^{\alpha - 1} \left\| w_0 \right\| _{L^2({{\mathbb {T}}})}^2 \le \left\| w_0 \right\| _{H^1({{\mathbb {T}}})}^{\alpha + 1} \end{aligned}$$

by Lemma 17. Therefore, the bound (46) holds.

Assume for now that $t \in [0, \delta ]$, where $\delta$ is the guaranteed time of existence of w in $L^2({{\mathbb {T}}})$. From the proof of Theorem 29, one has that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0+} \left\| w^\varepsilon - w \right\| _{C([0, T], L^2({{\mathbb {T}}}))} = 0. \end{aligned}$$

(47)

Hence, from Eqs. (46) and (47) and Lemma 19 it follows that

$$\begin{aligned} \left\| w(\cdot , t) \right\| _{H^1({{\mathbb {T}}})} \le \liminf _{\varepsilon \rightarrow 0+} \left\| w^\varepsilon (\cdot , t) \right\| _{H^1({{\mathbb {T}}})} < \infty . \end{aligned}$$

Observe, that by the above we have

$$\begin{aligned}&\left\| w^\varepsilon (\cdot , t) *\phi _\varepsilon - w \right\| _{L^{\alpha + 1}({{\mathbb {T}}})}^{\alpha + 1} \\&\quad \lesssim \left\| (w^\varepsilon (\cdot , t) - w(\cdot , t)) *\phi _\varepsilon \right\| _{L^{\alpha + 1}({{\mathbb {T}}})}^{\alpha + 1} + \left\| (w(\cdot , t) *\phi _\varepsilon - w(\cdot , t) \right\| _{L^{\alpha + 1}({{\mathbb {T}}})}^{\alpha + 1} \\&\quad \le \left( \left\| w^\varepsilon (\cdot , t) \right\| _{L^\infty }^{\alpha - 1} + \left\| w(\cdot , t) \right\| _{L^\infty }^{\alpha - 1}\right) \left\| w^\varepsilon (\cdot , t) - w(\cdot , t) \right\| _{L^2({{\mathbb {T}}})}^2 \\&\qquad + \left\| (w(\cdot , t) *\phi _\varepsilon - w(\cdot , t) \right\| _{L^{\alpha + 1}({{\mathbb {T}}})}^{\alpha + 1} \xrightarrow {\varepsilon \rightarrow 0+} 0 \end{aligned}$$

and hence

$$\begin{aligned} E_0(w(\cdot , t)) \le \liminf _{\varepsilon \rightarrow 0+} E_\varepsilon (w^\varepsilon (\cdot , t)) \le E_0(w_0). \end{aligned}$$

Interchanging 0 and t shows the reverse inequality and proves the energy conservation $E_0(w_0) = E_0(w(\cdot , t))$.

Reiterating the argument proves that $w \in L^\infty ({{\mathbb {R}}}, H^1({{\mathbb {T}}}))$. It remains to show that $w \in C({{\mathbb {R}}}, H^1({{\mathbb {T}}}))$. To that end, observe that $t \mapsto w(\cdot , t)$ is weakly continuous in $L^2({{\mathbb {T}}})$. But, by the above, $\sup _{t \in {{\mathbb {R}}}} \left\| w(\cdot , t) \right\| _{H^1({{\mathbb {T}}})} < \infty$ and hence $t \mapsto w(\cdot , t)$ is weakly continuous in $H^1({{\mathbb {T}}})$. By the observation

$$\begin{aligned} \left\| w(\cdot , t) - w(\cdot , s) \right\| _{H^1({{\mathbb {T}}})}^2 = \left\| w(\cdot , t) \right\| _{H^1({{\mathbb {T}}})}^2 + \left\| w(\cdot , s) \right\| _{H^1({{\mathbb {T}}})}^2 - 2 {\text {Re}}\left\langle w(\cdot , t), w(\cdot , s) \right\rangle _{H^1({{\mathbb {T}}})}, \end{aligned}$$

it is enough to show that $t \mapsto \left\| w(\cdot , t) \right\| _{H^1({{\mathbb {T}}})}$ is continuous. (See [3, Proposition 3.32] for this result in a more general setting.)

To that end, observe that by the mass and energy conservation we have

$$\begin{aligned} \left\| w(\cdot , t) \right\| _{H^1({{\mathbb {T}}})}^2= & {} 2 E(w(\cdot , t)) \pm \frac{2}{\alpha + 1} \left\| w(\cdot , t) \right\| _{L^{\alpha + 1}({{\mathbb {T}}})}^{\alpha + 1} + \left\| w(\cdot , t) \right\| _{L^2({{\mathbb {T}}})}^2 \\= & {} 2 E_0(w_0) \pm \frac{2}{\alpha + 1} \left\| w(\cdot , t) \right\| _{L^{\alpha + 1}({{\mathbb {T}}})}^{\alpha + 1} + \left\| w_0 \right\| _{L^2({{\mathbb {T}}})}^2. \end{aligned}$$

Moreover, for any $t, s \in {{\mathbb {R}}}$ we have

$$\begin{aligned}&\left| \left\| w(\cdot , t) \right\| _{L^{\alpha + 1}({{\mathbb {T}}})}^{\alpha + 1} - \left\| w(\cdot , s) \right\| _{L^{\alpha + 1}({{\mathbb {T}}})}^{\alpha + 1} \right| \\&\quad \lesssim \int _{{{\mathbb {T}}}} \left| w(x, t) - w(x, s) \right| \left( \left| w(x, t) \right| ^\alpha + \left| w(x, s) \right| ^\alpha \right) \mathrm {d}{x} \\&\quad \lesssim \left\| w \right\| _{L^\infty ({{\mathbb {R}}}, H^1({{\mathbb {T}}}))}^\alpha \left\| w(\cdot , t) - w(\cdot , s) \right\| _{L^2({{\mathbb {T}}})}. \end{aligned}$$

The fact that $w \in C_{\text {b}}({{\mathbb {R}}}, L^2({{\mathbb {T}}}))$ concludes the argument. $\square$