Local Cauchy theory for the nonlinear Schrödinger equation in spaces of infinite mass

Abstract

We consider the Cauchy problem for the nonlinear Schrödinger equation on $\mathbb {R}^d$, where the initial data is in $\dot{H}^1(\mathbb {R}^d)\cap L^p(\mathbb {R}^d)$. We prove local well-posedness for large ranges of p and discuss some global well-posedness results.

1 Introduction

In this work, we consider the classical nonlinear Schrödinger equation over $\mathbb {R}^d$:

$$\begin{aligned} iu_t + \Delta u + \lambda |u|^\sigma u = 0, \quad u=u(t,x),\quad (t,x)\in \mathbb {R}\times \mathbb {R}^d, \ \lambda \in \mathbb {R}, \ 0<\sigma <4/(d-2)^+ \end{aligned}$$

and focus on the corresponding Cauchy problem $u(0)=u_0\in E$, where E is a suitable function space. This model equation is the subject of more than fifty years of intensive research, which makes us unable to give a complete list of important references (we simply refer the monographs [3, 11, 12] and references therein). The usual framework one considers is $E=H^1(\mathbb {R}^d)$, the so-called energy space, or more generally, $E=H^s(\mathbb {R}^d)$. A common property of these spaces is that they are $L^2$-based. The reason for this constraint comes from the fact that the linear group is bounded in $L^2$, but not in any other $L^p$.

In the sense of lifting the $L^2$ constraint, various approaches have been considered: in [7], one considers local well-posedness on Zhidkov spaces

$$\begin{aligned} E=X^k(\mathbb {R}^d)=\{u\in L^\infty (\mathbb {R}^d): \nabla u\in H^{k-1}(\mathbb {R}^d) \}. \end{aligned}$$

In [8], one takes the Gross–Pitaevskii equation and looks for local well-posedness on

$$\begin{aligned} E=\{u\in H^1_{loc}(\mathbb {R}^d): \nabla u\in L^2(\mathbb {R}^d), |u|^2-1\in L^2(\mathbb {R}^d)\}. \end{aligned}$$

In [1], one takes E as a modulation space, which allows to measure in a more precise way the concentration of a tempered distribution in both space and frequency. This leads to some local well-posedness results which require less regularity on the initial condition. In [15], one deals with the case $E=W^{s,p}(\mathbb {R}^d)$, $1<p<2$ (where the local well-posedness is proven for a weaker notion of solution). In [13], one takes the one-dimensional cubic (NLS) and considers

$$\begin{aligned} E=\left\{ u\in \mathcal {S}'(\mathbb {R}^d): \Vert e^{it\partial _{xx}}u\Vert _{L^3([-1,1]; L^6(\mathbb {R}))}<\infty \right\} . \end{aligned}$$

Finally, in [5], one considers $E=H^1(\mathbb {R}^2)+X$, where X is either a particular space of bounded functions with no decay or a subspace of $L^4(\mathbb {R}^2)$ (and not of $L^2(\mathbb {R}^2)$).

The aim of this paper is to look for local well-posedness results over another class of spaces, namely

$$\begin{aligned} E=X_p(\mathbb {R}^d)=\dot{H}^1(\mathbb {R}^d) \cap L^p(\mathbb {R}^d), \quad 2<p<2d/(d-2)^+. \end{aligned}$$

In particular, we obtain local well-posedness in the most general energy space $X_{\sigma +2}(\mathbb {R}^d)$ and obtain global well-posedness over $X_p(\mathbb {R}^d)$ in the defocusing case $\lambda <0$ for all $p\leqslant \sigma +2$.

Remark 1.1

Our results can be extended to more general nonlinearities f(u) as in the $H^1$ framework. We present our results for $f(u)=|u|^{\sigma }u$ so not to complicate unnecessarily the proofs and deviate from the main ideas.

We briefly explain the structure of this work: in Sect. 2, we derive the required group estimates and show that the Schrödinger group is well-defined over $X_p(\mathbb {R}^d)$. In Sect. 3, we show local well-posedness for $p\leqslant 2\sigma +2$, where the use of Strichartz estimates is available. We also prove global well-posedness for small $\sigma$ (cf. Proposition 3.6). In Sect. 4, we deal with the complementary case $p>2\sigma +2$ in dimensions $d=1,2$.

Notation The norm over $L^p(\mathbb {R}^d)$ will be denoted as $\Vert \cdot \Vert _p$ or $\Vert \cdot \Vert _{L^p}$, whichever is more convenient. The spatial domain $\mathbb {R}^d$ will often be ommited. The free Schrödinger group in $H^1(\mathbb {R}^d)$ is written as $\{S(t)\}_{t\in \mathbb {R}}$. We write $p^*=dp/(d-p)^+$. To avoid repetition, we hereby set $2<p<2^*$ and $0<\sigma <4/(d-2)^+$.

2 Linear estimates

We recall the essential Strichartz estimates. We say that (q, r) is an admissible pair if

$$\begin{aligned} 2\leqslant r \leqslant 2^*,\quad \frac{2}{q}=d\left( \frac{1}{2}-\frac{1}{r}\right) , \quad r\ne \infty \text{ if } d=2. \end{aligned}$$

Lemma 2.1

(Strichartz estimates) Given two admissible pairs (q, r) and $(\gamma , \rho )$, we have, for all sufficiently regular $u_0$ and f and for any interval $I\subset \mathbb {R}$,

$$\begin{aligned} \Vert S(\cdot )u_0\Vert _{L^q(I; L^r(\mathbb {R}^N))}\lesssim \Vert u_0\Vert _2 \end{aligned}$$

(2.0)

and

$$\begin{aligned} \left\| \int _{0<s<t} S(t-s)f(s) ds\right\| _{L^q(I; L^r(\mathbb {R}^N))}\lesssim \Vert f\Vert _{L^{\gamma '}(I; L^{\rho '}(\mathbb {R}^N))}. \end{aligned}$$

(2.1)

Remark 2.1

The estimate (2.1) may be extended to other sets of admisssible pairs: see [6] and [14]. However, the linear estimate (2.0) is not valid for any other pairs and for $u_0\notin L^2(\mathbb {R}^d)$.

Proposition 2.2

(Group estimates with loss of derivative) Define k so that (k, p) is admissible. Then

• (Linear estimate) For $\phi \in \mathcal {S}(\mathbb {R}^d)$,

  $$\begin{aligned} \Vert S(t)\phi \Vert _p^2\lesssim \Vert \phi \Vert _p^2 + |t|^{1-\frac{2}{k}}\Vert \nabla \phi \Vert _2^2,\quad t\in \mathbb {R}. \end{aligned}$$

• (Non-homogeneous estimate) For $f\in C([0,T]; \mathcal {S}(\mathbb {R}^d))$ and any (q, r) admissible,

  $$\begin{aligned}&\left\| \int _0^\cdot S(\cdot -s) f(s) ds\right\| _{L^\infty ((0,T); L^p(\mathbb {R}^d))}\nonumber \\&\quad \lesssim C(T)\left( \Vert f\Vert _{L^2((0,T); L^p(\mathbb {R}^d))} + \Vert \nabla f\Vert _{L^{q'}((0,T); L^{r'}(\mathbb {R}^d))}\right) , \end{aligned}$$

  (2.2)

  where $C(\cdot )$ is a increasing bounded function over bounded intervals of $\mathbb {R}$.

Notice that, due to the scaling invariance of the Schrödinger equation, the polynomial growth in time in the linear estimate is unavoidable.

Proof

For the linear estimate, write $u=S(t)\phi$. Then $u\in C^1(\mathbb {R}; H^2(\mathbb {R}^d))$ satisfies

$$\begin{aligned} iu_t + \Delta u=0,\quad u(0)=\phi . \end{aligned}$$

Multiplying the equation by $|u|^{p-2}\bar{u}$, integrating over $\mathbb {R}^d$ and taking the imaginary part, we obtain

$$\begin{aligned} \frac{1}{p}\frac{d}{dt} \Vert u(t)\Vert _p^p\leqslant & {} \left| {{\mathrm{Im}}}\int |u|^{p-2}\bar{u}\Delta u\right| \leqslant \frac{p-2}{2} \int |u|^{p-2}|\nabla u|^2 \\\leqslant & {} \frac{p-2}{2}\Vert u(t)\Vert _{p}^{p-2}\Vert \nabla u(t)\Vert _p^2 \end{aligned}$$

Thus we have

$$\begin{aligned} \frac{d}{dt} \Vert u(t)\Vert _p^2 \leqslant (p-2)\Vert \nabla u(t)\Vert _p^2. \end{aligned}$$

An integration between 0 and $t\in \mathbb {R}$ and the linear Strichartz estimate yield

$$\begin{aligned} \Vert u(t)\Vert _p^2\lesssim \Vert \phi \Vert _p^2 + \int _0^t \Vert \nabla u(s)\Vert _p^2 ds\lesssim & {} \Vert \phi \Vert _p^2 + |t|^{1-\frac{2}{k}}\left( \int _0^t \Vert \nabla u(s)\Vert _p^k ds\right) ^{\frac{2}{k}} \\\lesssim & {} \Vert \phi \Vert _p^2 + |t|^{1-\frac{2}{k}}\Vert \nabla \phi \Vert _2^2. \end{aligned}$$

For the non-homogeneous estimate, set $v(t)=-i\int _0^t S(t-s)f(s)ds$. Then $v\in C^1([0,T]; H^1(\mathbb {R}^d))$ satisfies

$$\begin{aligned} iv_t+\Delta v = f,\quad v(0)=0. \end{aligned}$$

As for the previous estimate, we have

$$\begin{aligned} \frac{1}{p}\frac{d}{dt} \Vert v(t)\Vert _p^p \lesssim \Vert v(t)\Vert _{p}^{p-2}\Vert \nabla v(t)\Vert _p^2 + \Vert v(t)\Vert _p^{p-1}\Vert f(t)\Vert _p \end{aligned}$$

and so

$$\begin{aligned} \frac{d}{dt} \Vert v(t)\Vert _p^2 \lesssim \Vert \nabla v(t)\Vert _p^2 + \Vert v(t)\Vert _p\Vert f(t)\Vert _p \lesssim \Vert \nabla v(t)\Vert _p^2 + \Vert v(t)\Vert _p^2 + \Vert f(t)\Vert _p^2. \end{aligned}$$

The required estimate now follows by direct integration in (0, t), $0<t<T$, and by the non-homogeneous Strichartz estimate. $\square$

Lemma 2.3

(Local Strichartz estimate without loss of derivatives) Given $f\in C([0,T], \mathcal {S}(\mathbb {R}^d))$,

$$\begin{aligned} \left\| \int _0^\cdot S(\cdot -s)f(s)ds\right\| _{L^\infty ((0,T), L^p(\mathbb {R}^d))}\lesssim & {} C(T,q)\Vert f\Vert _{L^{q'}((0,T), L^{p'}(\mathbb {R}^d))}, \quad \frac{1}{q}\nonumber \\> & {} d\left( \frac{1}{2}-\frac{1}{p}\right) . \end{aligned}$$

(2.3)

Proof

This estimate follows easily from the decay estimates of the Schrödinger group: indeed, given $0<t<T$,

$$\begin{aligned} \left\| \int _0^t S(t-s)f(s)ds\right\| _{L^p(\mathbb {R}^d)}&\lesssim \int _0^T \Vert S(t-s)f(s)\Vert _{L^p(\mathbb {R}^d)}ds \\&\lesssim \int _0^T \frac{1}{|t-s|^{d\left( \frac{1}{2}-\frac{1}{p}\right) }}\Vert f(s)\Vert _{L^{p'}(\mathbb {R}^d)}ds \\&\lesssim \left( \int _0^T \frac{1}{|t-s|^{qd\left( \frac{1}{2}-\frac{1}{p}\right) }} ds\right) ^{\frac{1}{q}}\Vert f\Vert _{L^{q'}((0,T), L^{p'}(\mathbb {R}^d))}. \end{aligned}$$

$\square$

We set

$$\begin{aligned} X_p(\mathbb {R}^d)=L^p(\mathbb {R}^d)\cap \dot{H}^1(\mathbb {R}^d). \end{aligned}$$

Remark 2.2

From the Gagliardo–Nirenberg inequality, we have $H^1(\mathbb {R}^d)\hookrightarrow X_p(\mathbb {R}^d)$.

Proposition 2.4

The Schrödinger group $\{S(t)\}_{t\in \mathbb {R}}$ over $H^1(\mathbb {R}^d)$ defines, by continuous extension, a one-parameter continuous group on $X_p(\mathbb {R}^d)$.

Proof

Given any $\phi \in \dot{H}^1(\mathbb {R}^d)$, we have $\Vert S(t)\nabla \phi \Vert _2=\Vert \nabla \phi \Vert _2$. Together with Proposition 2.2, this implies that

$$\begin{aligned} \Vert S(t)\phi \Vert _{X_p}\lesssim (1+|t|^{1-\frac{2}{k}})^{1/2}\Vert \phi \Vert _{X_p}, \ t\in \mathbb {R}. \end{aligned}$$

Therefore, for each fixed $t\in \mathbb {R}$, S(t) may be extended continuosly to $X_p$. By density, it follows easily that $S(t+s)=S(t)S(s)$, $t,s\in \mathbb {R}$, and $S(0)=I$ on $X_p$. Finally, we prove continuity at $t=0$: given $\phi \in X_p(\mathbb {R}^d)$ and $\epsilon >0$, take $\phi _\epsilon \in H^1(\mathbb {R}^d)$ such that

$$\begin{aligned} \Vert \phi _\epsilon -\phi \Vert _{X_p}<\epsilon . \end{aligned}$$

Then

$$\begin{aligned} \limsup _{t\rightarrow 0}\Vert S(t)\phi - \phi \Vert _{X_p}&\leqslant \limsup _{t\rightarrow 0}\left( \Vert S(t)(\phi -\phi _\epsilon )\Vert _{X_p} + \Vert S(t)\phi _\epsilon \right. \\&\quad \left. -\,\phi _\epsilon \Vert _{X_p} + \Vert \phi _\epsilon -\phi \Vert _{X_p}\right) \\&\lesssim \limsup _{t\rightarrow 0}\left( (1+|t|^{1-\frac{2}{k}})^{1/2}\Vert \phi -\phi _\epsilon \Vert _{X_p}\right. \\&\quad \left. +\, \Vert S(t)\phi _\epsilon -\phi _\epsilon \Vert _{H^1} \right) \lesssim \epsilon . \end{aligned}$$

$\square$

Remark 2.3

Fix $d=1$. Using the same ideas, one may easily observe that the Schrödinger group is well-defined on the Zhidkov space

$$\begin{aligned} X^2(\mathbb {R})=\{u\in L^\infty (\mathbb {R}): \nabla u\in H^1(\mathbb {R})\}. \end{aligned}$$

Indeed, for any $2\leqslant p\leqslant \infty$ a direct integration of the equation gives

$$\begin{aligned} \frac{d}{dt}\Vert u(t)\Vert _p \leqslant \Vert \Delta u(t)\Vert _p. \end{aligned}$$

Hence, choosing k so that (k, p) is an admissible pair,

$$\begin{aligned} \Vert u(t)\Vert _p\leqslant & {} \Vert u_0\Vert _p + \int _0^t \Vert \Delta u(s)\Vert _p ds \leqslant \Vert u_0\Vert _p + Ct^{1-\frac{1}{k}}\Vert \Delta u\Vert _{L^k((0,t), L^p)}\\\leqslant & {} \Vert u_0\Vert _p + Ct^{1-\frac{1}{k}}\Vert \Delta u_0\Vert _{2}, \end{aligned}$$

where C is a constant independent on p (this comes from the fact that such a constant may be obtained via the interpolation between $L^{\infty }_t L^2_x$ and $L^{4}_tL^\infty _x$). Then, taking the limit $p\rightarrow \infty$, we obtain

$$\begin{aligned} \Vert u(t)\Vert _\infty \lesssim \Vert u_0\Vert _\infty + t^{\frac{3}{4}}\Vert \Delta u_0\Vert _2, \quad t>0,\ u_0\in H^2(\mathbb {R}). \end{aligned}$$

For higher dimensions, a similar procedure may be applied, at the expense of some derivatives (one must use Sobolev injection to control $L^p$, with p large). As one might expect, this argument does not provide the best possible estimate: in [7], one may see that

$$\begin{aligned} \Vert u(t)\Vert _\infty \lesssim (1+t^{\frac{1}{4}})\left( \Vert u_0\Vert _\infty + \Vert \nabla u_0\Vert _2\right) ,\quad t>0, u_0\in H^1(\mathbb {R}). \end{aligned}$$

Remark 2.4

One may ask if the required regularity is optimal: can we define the Schrödinger group on $X_p^s(\mathbb {R}^d):=\dot{H}^s(\mathbb {R}^d)\cap L^p(\mathbb {R}^d)$? What is the optimal s? Taking into consideration the previous remark, we conjecture that it should be possible to lower the regularity assumption. This entails a deeper analysis of the Schrödinger group, as it was done in [7].

3 Local well-posedness for $p\leqslant 2\sigma +2$

In order to clarify what do we mean by a solution of (NLS), we give the following

Definition 3.1

(Solution over $X_p(\mathbb {R}^d)$) Given $u_0\in X_p(\mathbb {R}^d)$, we say that $u\in C([0,T], X_p(\mathbb {R}^d))$ is a solution of (NLS) with initial data $u_0$ if the Duhamel formula is valid:

$$\begin{aligned} u(t)=S(t)u_0 + i\lambda \int _0^t S(t-s)|u(s)|^\sigma u(s)ds,\quad t\in [0,T]. \end{aligned}$$

Throughout this section, let $(\gamma ,\rho )$ and (q, r) be admissible pairs such that

$$\begin{aligned} r=(\sigma +1)\rho '=\max \{\sigma +2, p\}. \end{aligned}$$

(3.1)

It is easy to check that such pairs are well-defined for $p\leqslant 2\sigma +2$.

Proposition 3.2

(Uniqueness over $X_p(\mathbb {R}^d)$) Suppose that $p\leqslant 2\sigma +2$. Let $u_1, u_2\in C([0,T], X_p(\mathbb {R}^d))$ be two solutions of (NLS) with initial data $u_0\in X_p(\mathbb {R}^d)$. Then $u_1\equiv u_2$.

Proof

Taking the difference between the Duhamel formula for $u_1$ and $u_2$,

$$\begin{aligned} u_1(t)-u_2(t)=i\lambda \int _0^t S(t-s)\left( |u_1(s)|^\sigma u_1(s) - |u_2(s)|^\sigma u_2(s)\right) ds \end{aligned}$$

Then, for any interval $J=[0,t]\subset [0,T]$, since $X_p(\mathbb {R}^d)\hookrightarrow L^r(\mathbb {R}^d)$,

$$\begin{aligned} \Vert u_1-u_2\Vert _{L^q(J, L^r)}&\lesssim \Vert |u_1|^\sigma u_1- |u_2|^\sigma u_2\Vert _{L^{\gamma '}(J, L^{\rho '})} \\&\lesssim \left\| (\Vert u_1\Vert ^\sigma _{r} + \Vert u_2\Vert _r^\sigma )\Vert u_1-u_2\Vert _r \right\| _{L^{\gamma '}(J)} \\&\lesssim \left( \Vert u_1\Vert _{L^\infty ([0,T], X_p(\mathbb {R}^d))} \right. \\&\quad \left. +\, \Vert u_2\Vert _{L^\infty ([0,T], X_p(\mathbb {R}^d))}\right) \Vert u_1-u_2\Vert _{L^{\gamma '}(J, L^r)}\\&\lesssim C(T)\Vert u_1-u_2\Vert _{L^{\gamma '}(J, L^r)} \end{aligned}$$

The claimed result now follows from [3, Lemma 4.2.2]. $\square$

Theorem 3.3

(Local well-posedness on $X_p(\mathbb {R}^d)$, $p\leqslant 2\sigma +2$) Given $u_0\in X_p(\mathbb {R}^d)$, there exists $T=T(\Vert u_0\Vert _{X_p})>0$ and an unique solution

$$\begin{aligned} u\in C([0,T), X_p(\mathbb {R}^d))\cap L^\gamma ((0,T),\dot{W}^{1,\rho }(\mathbb {R}^d))\cap L^q((0,T), \dot{W}^{1,r}(\mathbb {R}^d)) \end{aligned}$$

of (NLS) with initial data $u_0$. One has

$$\begin{aligned} u-S(\cdot )u_0 \in C([0,T]. L^2(\mathbb {R}^d))\cap L^q((0,T), L^{r}(\mathbb {R}^d)) \cap L^\gamma ((0,T), L^{\rho }(\mathbb {R}^d)) . \end{aligned}$$

(3.2)

Moreover, the solution depends continuously on the initial data and may be extended in an unique way to a maximal time interval $[0,T^*(u_0))$. If $T^*(u_0)<\infty$, then

$$\begin{aligned} \lim _{t\rightarrow T^*(u_0)} \Vert u(t)\Vert _{X_p} = +\infty . \end{aligned}$$

Remark 3.1

The property (3.2) is a type of nonlinear “smoothing” effect: the integral term in Duhamel’s formula turns out to have more integrability than the solution itself (a similar property was seen in [8, 13]). This insight allows the use of Strichartz estimates at the zero derivatives level. Without this possibility, one would be restricted to the estimate (2.3) and the possible ranges of $\sigma$ and p would be significantly smaller.

Proof

Step 1 Define

$$\begin{aligned} \mathcal {S}_0=L^\infty ((0,T), L^2) \cap L^q((0,T), L^r) \cap L^\gamma ((0,T), L^\rho ). \end{aligned}$$

and

$$\begin{aligned} \mathcal {S}_1=L^\infty ((0,T), H^1) \cap L^q((0,T), W^{1,r}) \cap L^\gamma ((0,T), W^{1,\rho }). \end{aligned}$$

Consider the space

$$\begin{aligned} \mathcal {E}=\Big \{&u\in L^\infty ((0,T), X_p)\cap L^\gamma ((0,T), \dot{W}^{1,\rho })\cap L^q((0,T), \dot{W}^{1,r}):\\&|||u|||:= \Vert u\Vert _{L^\infty ((0,T), L^p)} + \Vert u - S(\cdot )u_0\Vert _{\mathcal {S}_1} \leqslant M\Big \}. \end{aligned}$$

endowed with the distance

$$\begin{aligned} d(u,v)= \Vert u - v \Vert _{\mathcal {S}_0}. \end{aligned}$$

It is not hard to check that $(\mathcal {E}, d)$ is a complete metric space: indeed, if $\{u_n\}_{n\in \mathbb {N}}$ is a Cauchy sequence in $\mathcal {E}$, then $\{u_n-S(\cdot )u_0\}_{n\in \mathbb {N}}$ is a Cauchy sequence in $\mathcal {S}_0$. Then there exists $u\in \mathcal {D}'([0,T]\times \mathbb {R}^d)$ such that $u_n-S(\cdot )u_0\rightarrow u-S(\cdot )u_0$ in $\mathcal {S}_0$. By [3, Theorem 1.2.5], this convergence implies that

$$\begin{aligned} u-S(\cdot )u_0\in \mathcal {S}_1,\quad \Vert u-S(\cdot )u_0\Vert _{\mathcal {S}_1} \leqslant \liminf \Vert u_n-S(\cdot )u_0\Vert _{\mathcal {S}_1} \end{aligned}$$

Finally, it follows from the Gagliardo–Nirenberg inequality that, for some $0<\theta <1$,

$$\begin{aligned} \Vert u_n-u\Vert _{L^\infty ((0,T), L^p)}\lesssim \Vert u_n-u\Vert _{L^\infty ((0,T), L^2)}^{1-\theta }\Vert \nabla u_n-\nabla u\Vert _{L^\infty ((0,T), L^2)}^\theta \rightarrow 0 \end{aligned}$$

and so $u_n\rightarrow u$ in $L^\infty ((0,T), L^p)$.

Step 2 Define, for any $u\in \mathcal {E}$,

$$\begin{aligned} (\Phi u)(t)= S(t)u_0 +i\lambda \int _0^t S(t-s)|u(s)|^\sigma u(s) ds, \quad 0<t<T. \end{aligned}$$

It follows from the definition of r (see (3.1)) that $X_p(\mathbb {R}^d)\hookrightarrow L^r(\mathbb {R}^d)$. Then

$$\begin{aligned} \Vert \Phi u - S(\cdot )u_0\Vert _{\mathcal {S}_1}&\lesssim \Vert |u|^\sigma u\Vert _{L^{\gamma '}((0,T), W^{1,\rho '})} \\&\lesssim \left\| \Vert u\Vert _{r}^{\sigma }( \Vert u\Vert _{r} + \Vert \nabla u\Vert _{r})\right\| _{L^{\gamma '}(0,T)}\\&\lesssim \left\| \Vert u\Vert _{X_p}^{\sigma }( \Vert u\Vert _{X_p} + \Vert \nabla (u -S(\cdot )u_0)\Vert _{r} + \Vert S(\cdot )\nabla u_0\Vert _{r}\right\| _{L^{\gamma '}(0,T)}\\&\lesssim T^{\frac{1}{\gamma '}}\Vert u\Vert _{L^\infty ((0,T), X_p)}^{\sigma +1} \\&\quad + T^{\frac{1}{\gamma '}-\frac{1}{q}}\Vert u\Vert _{L^\infty ((0,T), X_p)}^{\sigma }\Vert \nabla (u -S(\cdot )u_0)\Vert _{L^{q}((0,T), L^{r})} \\&\quad + T^{\frac{1}{\gamma '}-\frac{1}{q}}\Vert u\Vert _{L^\infty ((0,T), X_p)}^{\sigma }\Vert S(\cdot )\nabla u_0\Vert _{L^{q}((0,T), L^{r})}\\&\lesssim \left( T^{\frac{1}{q'}} + T^{\frac{1}{q'}-\frac{1}{q}}\right) (M^{\sigma +1} + \Vert u_0\Vert _{X_p}^{\sigma +1}). \end{aligned}$$

It follows that, for $M \sim 2\Vert u_0\Vert _{X_p}$ and T sufficiently small, we have $\Phi :\mathcal {E}\mapsto \mathcal {E}$.

Step 3 Now we show a contraction estimate: given $u,v\in \mathcal {E}$,

$$\begin{aligned} d(\Phi (u), \Phi (v))&\lesssim \Vert |u|^\sigma u - |v|^\sigma v\Vert _{L^{\gamma '}(L^{\rho '})} \\&\lesssim \left\| (\Vert u\Vert _{r}^\sigma + \Vert v\Vert _{r}^\sigma )\Vert u-v\Vert _{r} \right\| _{L^{\gamma '}(0,T)} \\&\lesssim T^{\frac{1}{\gamma '}-\frac{1}{q}}\left( \Vert u\Vert _{L^\infty ((0,T), X_p(\mathbb {R}^d)}^\sigma \right. \\&\quad \left. +\, \Vert u\Vert _{L^\infty ((0,T), X_p(\mathbb {R}^d)}^\sigma \right) \Vert u-v\Vert _{L^q((0,T), L^{r})} \\&\lesssim T^{\frac{1}{q'}-\frac{1}{q}}\left( M^\sigma + \Vert u_0\Vert _{X_p}^\sigma \right) d(u,v). \end{aligned}$$

Therefore, for $T=T(\Vert u_0\Vert _{X_p})$ small enough, the mapping $\Phi :\mathcal {E}\mapsto \mathcal {E}$ is a strict contraction and so, by Banach’s fixed point theorem, $\Phi$ has a unique fixed point over $\mathcal {E}$. This gives the local existence of a solution $u\in C([0,T], X_p(\mathbb {R}^d))$ of (NLS) with initial data $u_0$. From the uniqueness result, such a solution can then be extended to a maximal interval of existence $(0,T^*(u_0))$. If such an interval is bounded, then necessarily one has $\Vert u(t)\Vert _{X_p}\rightarrow \infty$ as $t\rightarrow T^*(u_0)$. The continuous dependence on the initial data follows as in the $H^1$ case (see, for example, the proof of [3, Theorem 4.4.1]). $\square$

Remark 3.2

The condition $p\leqslant 2\sigma +2$ is necessary for one to use Strichartz estimates with no derivatives. Indeed, when one applies Strichartz to the integral term of the Duhamel formula, one has

$$\begin{aligned} \left\| \int _0^\cdot S(\cdot - s)|u(s)|^{\sigma }u(s)ds\right\| _{L^{q}((0,T), L^r)}\lesssim \Vert u\Vert _{L^{\gamma '(\sigma +1)}((0,T), L^{\rho '(\sigma +1)})}^{\sigma +1} \end{aligned}$$

for any admissible pairs (q, r) and $(\gamma , \rho )$. Since the solution u only lies on spaces with spatial integrability larger or equal than p, one must have $p\leqslant \rho '(\sigma +1) \leqslant 2\sigma +2$ (because $\rho \geqslant 2$).

Proposition 3.4

(Persistence of integrability) Suppose that $\tilde{p}<p\leqslant 2\sigma +2$. Given $u_0\in X_{\tilde{p}}(\mathbb {R}^d)$, consider the $X_p(\mathbb {R}^d)$-solution $u\in C([0,T^*(u_0)),X_{p})$ of (NLS) with initial data $u_0$. Then $u\in C([0,T^*(u_0)),X_{\tilde{p}})$.

Proof

By the local well-posedness result over $X_{\tilde{p}}(\mathbb {R}^d)$ and by the uniqueness over $X_p(\mathbb {R}^d)$, there exists a time $T_0>0$ such that $u\in C([0,T_0], X_{\tilde{p}}(\mathbb {R}^d))$. Thus the statement of the proposition is equivalent to saying that u does not blow-up in $X_{\tilde{p}}(\mathbb {R}^d)$ at a time $T_0<T<T^*(u_0)$. Since u is bounded in $X_p$ over [0, T], it follows from the local existence theorem that

$$\begin{aligned} \Vert u-S(\cdot )u_0\Vert _{L^\infty ((0,T), H^1)} <\infty . \end{aligned}$$

Then, by Proposition 2.2, for any $0<t<T$,

$$\begin{aligned} \Vert u\Vert _{L^\infty ((0,t), X_{\tilde{p}})}&\lesssim \Vert S(\cdot )u_0\Vert _{L^\infty ((0,t), X_{\tilde{p}})} + \Vert u-S(\cdot )u_0\Vert _{L^\infty ((0,t), X_{\tilde{p}})} \\&\lesssim \Vert u_0\Vert _{X_{\tilde{p}}} + \Vert u-S(\cdot )u_0\Vert _{L^\infty ((0,t), H^1)} <\infty , \end{aligned}$$

which implies that u does not blow-up at time $t=T$. $\square$

Proposition 3.5

(Conservation of energy) Suppose that $p\leqslant \sigma +2$. Given $u_0\in X_p(\mathbb {R}^d)$, the corresponding solution u of (NLS) with initial data $u_0$ satisfies

$$\begin{aligned} E(u(t))=E(u_0):=\frac{1}{2}\Vert \nabla u_0\Vert _2^2 - \frac{\lambda }{\sigma +2}\Vert u_0\Vert _{\sigma +2}^{\sigma +2}, \ 0<t<T(u_0). \end{aligned}$$

Consequently, if $\lambda <0$, then $T^*(u_0)=\infty$. Moreover, if $\lambda >0$ and $T^*(u_0)<\infty$, then

$$\begin{aligned} \lim _{t\rightarrow T^*(u_0)} \Vert \nabla u(t)\Vert _2 = \lim _{t\rightarrow T^*(u_0)} \Vert u(t)\Vert _{\sigma +2} = \infty . \end{aligned}$$

(3.3)

Proof

Since the conservation law is valid for $u_0\in H^1(\mathbb {R}^d)$, through a regularization argument, the same is true for any $u_0\in X_p(\mathbb {R}^d)$. If $\lambda <0$, one has

$$\begin{aligned} \Vert u(t)\Vert _{X_{\sigma +2}(\mathbb {R}^d)} \lesssim E(u_0), \quad 0<t<T^*(u_0). \end{aligned}$$

By the blow-up alternative, this implies that u, as a $X_{\sigma +2}(\mathbb {R}^d)$ solution, is globally defined. By persistence of integrability, this implies that u is global in $X_p(\mathbb {R}^d)$. If $\lambda >0$, suppose by contradiction that (3.3) is not true. Then, by conservation of energy, u is bounded in $X_{\sigma +2}(\mathbb {R}^d)$ and therefore it is globally defined (as an $X_{\sigma +2}(\mathbb {R}^d)$ solution, but also as an $X_p(\mathbb {R}^d)$ solution, by persistence of integrability). $\square$

Proposition 3.6

Fix $\lambda >0$. If

$$\begin{aligned} 2\sigma ^2 + (d+2)\sigma \leqslant 4, \end{aligned}$$

(3.4)

then, for any $u_0\in X_{\sigma +2}(\mathbb {R}^d)$, the corresponding solution u of (NLS) is globally defined.

Remark 3.3

Notice that the condition on $\sigma$ implies that $\sigma<\min \{\sqrt{2}, 4/(d+2)\}<4/d$.

Proof

Using a suitable scaling, one may reduce the case $\lambda >0$ to $\lambda =1$. By contradiction, assume that u blows-up at time $t=T$. The previous proposition then implies that

$$\begin{aligned} \lim _{t\rightarrow T} \Vert \nabla u(t)\Vert _2 =\infty . \end{aligned}$$

The first step is to obtain a corrected mass conservation estimate: indeed, by direct integration of the equation,

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \Vert u(t)-S(t)u_0\Vert _2^2&= - {{\mathrm{Im}}}\int |u(t)|^\sigma u(t) \overline{(u(t)-S(t)u_0)} \\&= {{\mathrm{Im}}}\int |u(t)|^\sigma u(t) \overline{S(t)u_0} \\&\lesssim \Vert u(t)\Vert _{\sigma +2}^{\sigma +1}\Vert S(t)u_0\Vert _{\sigma +2}. \end{aligned}$$

Integrating on (0, t),

$$\begin{aligned} \Vert u(t)-S(t)u_0\Vert _2^2&\lesssim \int _0^t \Vert u(s)\Vert _{\sigma +2}^{\sigma +1}\Vert S(s)u_0\Vert _{\sigma +2} ds \\&\lesssim \Vert S(\cdot )u_0\Vert _{L^\infty ((0,T), L^{\sigma +2})} \int _0^t \Vert u(s)\Vert _{\sigma +2}^{\sigma +1} ds \\&\lesssim \int _0^t \Vert u(s)\Vert _{\sigma +2}^{\sigma +1} ds. \end{aligned}$$

All of these formal computations can be justified by a suitable regularization and approximation argument. The next step is to use the conservation of energy and the Gagliardo–Nirenberg inequality to obtain a bound on $\Vert \nabla u(t)\Vert _2$.

$$\begin{aligned} \frac{1}{2}\Vert \nabla u(t)\Vert _2^2&= E(u_0) + \frac{1}{\sigma +2}\Vert u(t)\Vert _{\sigma +2}^{\sigma +2} \\&\lesssim 1 + \Vert u(t)-S(t)u_0\Vert _{\sigma +2}^{\sigma +2} + \Vert S(t)u_0\Vert _{\sigma +2}^{\sigma +2} \\&\lesssim 1+ \Vert \nabla (u(t)- S(t)u_0)\Vert _2^{\frac{d\sigma }{2}} \Vert u(t)-S(t)u_0\Vert _2^{\frac{4-(d-2)\sigma }{2}}\\&\lesssim 1+ \Vert \nabla (u(t)- S(t)u_0)\Vert _2^{\frac{d\sigma }{2}} \left( \int _0^t \Vert u(s)\Vert _{\sigma +2}^{\sigma +1}ds\right) ^{\frac{4-(d-2)\sigma }{4}} \end{aligned}$$

For t close to T, $\Vert \nabla (u(t)- S(t)u_0)\Vert _2 \sim \Vert \nabla u(t)\Vert _2$ and, by conservation of energy,

$$\begin{aligned} \Vert u(t)\Vert _{\sigma +2}^{\sigma +1}\lesssim \Vert \nabla u(t)\Vert _2^{\frac{2\sigma +2}{\sigma +2}} \end{aligned}$$

Thus

$$\begin{aligned} \Vert \nabla u(t)\Vert _2^{\frac{4-d\sigma }{2}} \lesssim 1 + \left( \int _0^t \Vert \nabla u(s)\Vert _2^{\frac{2\sigma +2}{\sigma +2}}ds\right) ^{\frac{4-(d-2)\sigma }{4}} \end{aligned}$$

which, together with the condition on $\sigma$, implies that

$$\begin{aligned} g(t):=\Vert \nabla u(t)\Vert _2^{\frac{2\sigma +2}{\sigma +2}}\leqslant \Vert \nabla u(t)\Vert _2^{\frac{2(4-d\sigma )}{4-(d-2)\sigma }} \lesssim 1 + \int _0^t \Vert \nabla u(s)\Vert _2^{\frac{2\sigma +2}{\sigma +2}}ds \lesssim 1 + \int _0^t g(s) ds. \end{aligned}$$

The desired contradiction now follows from a standard application of Gronwall’s lemma. $\square$

Remark 3.4

(Global existence in the focusing $L^2$ -subcritical regime) As it is well-known, the global existence in $H^1(\mathbb {R}^d)$ for $\sigma <4/d$ follows easily from the conservation of mass and energy and from the Gagliardo–Nirenberg inequality. In Proposition 3.6, we managed to perform a similar argument by using the corrected mass

$$\begin{aligned} M(t)=\Vert u(t)-S(t)u_0\Vert _2^2. \end{aligned}$$

However, the range of exponents for which the result is valid still leaves much to be desired. We are left with some questions: Is there another choice for “corrected mass” that allows a larger range of exponents? Is it possible that the large tails of the initial data contribute to blow-up behaviour?

Remark 3.5

(Blow-up in the $L^2$ -(super)critical regime) One may ask whether the known blow-up results for $\sigma \geqslant 4/d$ can be extended to initial data in $X_p(\mathbb {R}^d)$ which do not lie in $L^2(\mathbb {R}^d)$. First of all, notice that

$$\begin{aligned} X_p(\mathbb {R}^d)\cap L^2(|x|^2dx) \hookrightarrow L^2(\mathbb {R}^d). \end{aligned}$$

Thus, in order to obtain blow-up outside $L^2$, one must first show blow-up in $H^1$ without the finite variance assumption. This is an open problem, which has been solved in [10] under radial hypothesis and relying heavily on the conservation of mass (which is unavailable on $X_p(\mathbb {R}^d)$). For the nonradial case, recent works (see, for example, [9]) only manage to prove unboundedness of solutions of negative energy. The problem of blow-up solutions strictly in $X_p(\mathbb {R}^d)$ is an even harder problem, requiring a better control on the tails of the solution.

Remark 3.6

(Scaling invariance) It is useful to understand how scalings affect the $X_p(\mathbb {R}^d)$ norm: recalling that the (NLS) is invariant under the scaling $u_\lambda (t,x)=\lambda ^{2/\sigma }u(\lambda ^2 t, \lambda x)$, we have

$$\begin{aligned} \Vert u_\lambda (t)\Vert _{p} = \lambda ^{\frac{2}{\sigma }-\frac{d}{p}} \Vert u(\lambda ^2 t)\Vert _p, \quad \Vert \nabla u_\lambda (t)\Vert _2 = \lambda ^{\frac{2}{\sigma }+1-\frac{d}{2}}\Vert \nabla u(\lambda ^2 t)\Vert _2. \end{aligned}$$

Thus the (NLS) is $X_p(\mathbb {R}^d)$-subcritical for $\sigma <2p/d$. In this situation, global existence for small data is equivalent to global existence for any data. Recall, however, that, for $\sigma \geqslant 4/d$, existence of blow-up phenomena is known for special initial data in $H^1(\mathbb {R}^d)\hookrightarrow X_p(\mathbb {R}^d)$. Therefore, it is impossible to obtain a global existence result for small data for $4/d\leqslant \sigma < 2p/d$. Notice that in the energy case $p=\sigma +2$, one has $\sigma < 2p/d$ for any $\sigma +2<2^*$.

Remark 3.7

(Global existence for small data) The main obstacle in proving global existence for small data turns out to be the linear part of the Duhamel formula $S(t)u_0$, since there isn’t, to our knowledge, a way to bound uniformly this term over $X_p(\mathbb {R}^d)$. The other possibility is to leave the linear term with a space-time norm: indeed, for some powers $\sigma >2/d$, it is well-known that, if $u_0\in H^1(\mathbb {R}^d)$ is such that

$$\begin{aligned} \Vert S(\cdot )u_0\Vert _{L^a((0,\infty ), L^{\sigma +2}(\mathbb {R}^d))} \text{ is } \text{ small }, \quad a=\frac{2\sigma (\sigma +2)}{4-\sigma (d-2)}, \end{aligned}$$

then the corresponding solution of (NLS) is globally defined (see [4]). It is not hard to check that the result can be extended to $u_0\in X_{\sigma +2}(\mathbb {R}^d)$.

4 Local well-posedness for $p>2\sigma +2$

As it was observed in Remark 3.2, the condition $p\leqslant 2\sigma +2$ was necessary in order to use Strichartz estimates with no loss in regularity. For $p>2\sigma +2$, in order to estimate $L^\infty _tL^p_x$, one must turn to estimate (2.2), which has a loss of one derivative. Therefore the distance one defines for the fixed-point argument must include norms with derivatives. This implies the need of a local Lipschitz condition

$$\begin{aligned} ||u|^\sigma \nabla u - |v|^\sigma \nabla v|\lesssim C(|u|, |v|, |\nabla u|, |\nabla v|)\left( |u-v| + |\nabla (u-v)|\right) , \end{aligned}$$

which we can only accomplish for $\sigma \geqslant 1$.

Because of the restriction $\sigma \geqslant 1$, one must have $4<p<2^*$, which excludes any dimension greater than three. For $d=3$, it turns out that no range of $p>2\sigma +2$ can be considered. Indeed, if one uses (2.2) with $f=|u|^\sigma u$,

$$\begin{aligned}&\left\| \int _0^\cdot S(\cdot -s)|u(s)|^\sigma u(s)ds\right\| _{L^\infty ((0,T), L^p)}\\&\quad \lesssim \Vert u\Vert _{L^{2\sigma +2}((0,T), L^{p(\sigma +1)})}^{\sigma +1} +\, \Vert \nabla (|u|^\sigma u)\Vert _{L^{q'}((0,T), L^{r'})}. \end{aligned}$$

We focus on the first norm on the right hand side. To control such a term, either $X_p\hookrightarrow L^{p(\sigma +1)}_x$ and

$$\begin{aligned} \Vert u\Vert _{L^{2\sigma +2}((0,T), L^{p(\sigma +1)}} \lesssim T^{\frac{1}{2\sigma +2}}\Vert u\Vert _{L^\infty ((0,T), X_p)}, \end{aligned}$$

or, setting $r\geqslant 2$ so that

$$\begin{aligned} 1-\frac{3}{r} = -\frac{3}{p(\sigma +1)}, \end{aligned}$$

one estimates

$$\begin{aligned} \Vert u\Vert _{L^{2\sigma +2}((0,T), L^{p(\sigma +1)})} \lesssim \Vert \nabla u\Vert _{L^{2\sigma +2}((0,T), L^r)} \lesssim T^{\frac{1}{2\sigma +2}-\frac{1}{q}}\Vert \nabla u\Vert _{L^{q}((0,T), L^r)}. \end{aligned}$$

In the first case, one needs $8<p(\sigma +1)<2^*=6$. In the second, one must impose $2\sigma +2<q$. A simple computation yields $p(3\sigma +1)<6$, which is again impossible, since $p(3\sigma +1)>16$.

Theorem 4.1

(Local well-posedness on $X_p(\mathbb {R}^d)$ for $d=1,2$) Given $u_0\in X_p(\mathbb {R}^d)$, there exists $T=T(\Vert u_0\Vert _{X_p})>0$ and an unique solution

$$\begin{aligned} u\in C([0,T], X_p(\mathbb {R}^d)) \end{aligned}$$

of (NLS) with initial data $u_0$. The solution depends continuously on the initial data and may be extended uniquely to a maximal interval $[0,T^*(u_0))$. If $T^*(u_0)<\infty$, then

$$\begin{aligned} \lim _{t\rightarrow T^*(u_0)} \Vert u(t)\Vert _{X_p} = +\infty . \end{aligned}$$

Proof

Consider the space

$$\begin{aligned} \mathcal {E}=\Big \{ u\in L^\infty ((0,T), X_p): |||u|||:= \Vert u\Vert _{L^\infty ((0,T), X_p)} \leqslant M \Big \}. \end{aligned}$$

endowed with the natural distance

$$\begin{aligned} d(u,v)= |||u-v|||. \end{aligned}$$

The space $(\mathcal {E}, d)$ is clearly a complete metric space. If $u,v\in \mathcal {E}$, then

$$\begin{aligned} \Vert |u|^\sigma u - |v|^\sigma v\Vert _{L^2((0,t), L^p)}^2\lesssim \int _0^t \left( \Vert u\Vert _{p(\sigma +1)}^{2\sigma } + \Vert v\Vert _{p(\sigma +1)}^{2\sigma }\right) \Vert u-v\Vert _{p(\sigma +1)}^2 ds \end{aligned}$$

Since $X_p(\mathbb {R}^d)\hookrightarrow L^{p(\sigma +1)}(\mathbb {R}^d)$,

$$\begin{aligned}&\Vert |u|^\sigma u - |v|^\sigma v\Vert _{L^2((0,t), L^p)}^2\nonumber \\&\quad \lesssim T\left( \Vert u\Vert _{L^\infty ((0,t). X_p)}^{2\sigma } +\, \Vert v\Vert _{L^\infty ((0,t). L^p)}^{2\sigma }\right) \Vert u-v\Vert _{L^\infty ((0,t). X_p)}^{2}. \end{aligned}$$

(4.1)

Choose an admissible pair $(\gamma ,\rho )$ with $\rho$ sufficiently close to 2. We have

$$\begin{aligned} \Vert \nabla \left( |u|^\sigma u - |v|^\sigma v\right) \Vert _{L^{\gamma '}((0,T), L^{\rho '})}&\lesssim \left\| \left( |u|^{\sigma -1} + |v|^{\sigma -1}\right) (|u-v||\nabla v|\right. \\&\left. +\, |v||\nabla (u-v)|) \right\| _{L^{\gamma '}((0,T), L^{\rho '})}. \end{aligned}$$

As an example, we treat the term $|u|^{\sigma -1}|u-v||\nabla v|$:

$$\begin{aligned} \Vert |u|^{\sigma -1}|u-v||\nabla v|\Vert _{\rho '}&\lesssim \Vert u\Vert _{\frac{2\sigma \rho '}{2-\rho '}}^{\sigma -1} \Vert u-v\Vert _{\frac{2\sigma \rho '}{2-\rho '}}\Vert \nabla v\Vert _{2}\\&\lesssim \Vert u\Vert _{X_p}^\sigma \Vert u-v\Vert _{X_p}, \end{aligned}$$

Therefore

$$\begin{aligned}&\Vert \nabla \left( |u|^\sigma u - |v|^\sigma v\right) \Vert _{L^{\gamma '}((0,T), L^{\rho '})}\nonumber \\&\quad \lesssim T^{\frac{1}{\gamma '}} \left( \Vert u\Vert _{L^\infty ((0,T), X_p)}^\sigma +\, \Vert v\Vert _{L^\infty ((0,T), X_p)}^\sigma \right) \Vert u-v\Vert _{L^\infty ((0,T), X_p)} \nonumber \\&\quad \lesssim T^{\frac{1}{\gamma '}} M^\sigma d(u,v). \end{aligned}$$

(4.2)

For $u\in \mathcal {E}$, define

$$\begin{aligned} \Phi (u)(t)=S(t)u_0 +i\lambda \int _0^t S(t-s)|u(s)|^\sigma u(s)ds, \quad 0\leqslant t\leqslant T. \end{aligned}$$

The estimates (4.1) and (4.2), together with (2.2) and Strichartz’s estimates then imply that

$$\begin{aligned} |||\Phi (u)|||\lesssim&\Vert u_0\Vert _{X_p} + \left\| \int _0^\cdot S(\cdot -s)|u(s)|^\sigma u(s)ds\right\| _{L^\infty ((0,T), L^p)} \\&\quad + \left\| \int _0^\cdot S(\cdot -s)|u(s)|^\sigma u(s)ds\right\| _{L^\infty ((0,T), \dot{H}^1)}\\&\lesssim \Vert u_0\Vert _{X_p} + \left( \Vert |u|^\sigma u\Vert _{L^2((0,T); L^p)} + \Vert \nabla (|u|^\sigma u)\Vert _{L^{\gamma '}((0,T); L^{\rho '})}\right) \\&\lesssim \Vert u_0\Vert _{X_p} + \left( T^{\frac{1}{2}} + T^{\frac{1}{\gamma '}}\right) M^{\sigma +1} \end{aligned}$$

and

$$\begin{aligned} d(\Phi (u),\Phi (v))&\lesssim \left( \Vert |u|^\sigma u - |v|^\sigma v\Vert _{L^2((0,T); L^p)} + \Vert \nabla (|u|^\sigma u)\right. \nonumber \\&\quad \left. -\, \nabla ( |v|^\sigma v)\Vert _{L^{\gamma '}((0,T); L^{\rho '})}\right) \nonumber \\&\lesssim \left( T^{\frac{1}{2}} + T^{\frac{1}{\gamma '}}\right) M^{\sigma }d(u,v). \end{aligned}$$

(4.3)

Choosing $M\sim 2\Vert u_0\Vert _{X_p}$, for $T=T(\Vert u_0\Vert _{X_p})$ small enough, it follows that $\Phi :\mathcal {E}\mapsto \mathcal {E}$ is a strict contraction. Banach’s fixed point theorem now implies that $\Phi$ has a unique fixed point over $\mathcal {E}$, which is the unique solution u of (NLS) with initial data $u_0$ on the interval (0, T). This solution may then be extended uniquely to a maximal interval of existence $(0,T(u_0))$. The blow-up alternative follows by a standard continuation argument. Finally, if u, v are two solutions with initial data $u_0, v_0\in X_p(\mathbb {R}^d)$, as in (4.3), one has

$$\begin{aligned} d(u, v) = d(\Phi (u), \Phi (v))&\lesssim \Vert u_0-v_0\Vert _{X_p} + \left( T^{\frac{1}{2}} + T^{\frac{1}{\gamma '}}\right) M^{\sigma }d(u,v) \\&\lesssim \Vert u_0-v_0\Vert _{X_p} \\&\quad + \left( T^{\frac{1}{2}} + T^{\frac{1}{\gamma '}}\right) \left( \max \{\Vert u_0\Vert _{X_p}, \Vert v_0\Vert _{X_p}\}\right) ^{\sigma }d(u,v) \end{aligned}$$

Thus, for $T_0=T_0(\Vert u_0\Vert _{X_p}, \Vert v_0\Vert _{X_p})$ small,

$$\begin{aligned} d(u,v)\lesssim \Vert u_0-v_0\Vert _{X_p}, \end{aligned}$$

and continuous dependence follows. $\square$

Proposition 4.2

(Persistence of integrability) Fix $d=1,2$ and $p>\tilde{p}$. Given $u_0\in X_{\tilde{p}}(\mathbb {R}^d)$, consider the $X_p(\mathbb {R}^d)$-solution $u\in C([0,T^*(u_0)),X_{p})$ of (NLS) with initial data $u_0$. Then $u\in C([0,T^*(u_0)),X_{\tilde{p}})$.

Proof

As in the proof of Proposition 3.4, given $T<T^*(u_0)$, one must prove that the $L^{\tilde{p}}$ norm of u is bounded over (0, T). Applying (2.2) to the Duhamel formula of u,

$$\begin{aligned} \Vert u\Vert _{L^\infty ((0,T), L^{\tilde{p}})} \lesssim \Vert u_0\Vert _{X_{\tilde{p}}} + \Vert |u|^\sigma u\Vert _{L^2((0,T), L^{\tilde{p}})} + \Vert |u|^\sigma |\nabla u|\Vert _{L^{\gamma '}((0,T), L^{\rho '})}, \end{aligned}$$

for any admissible pair $(\gamma ,\rho )$. The penultimate term is treated using the injection $X_p(\mathbb {R}^d)\hookrightarrow L^{\tilde{p}(\sigma +1)}$:

$$\begin{aligned} \Vert |u|^\sigma u\Vert _{L^2((0,T), L^{\tilde{p}})} = \Vert u\Vert _{L^{2\sigma +2}((0,T), L^{\tilde{p}(\sigma +1)})}^{\sigma +1} \lesssim T^{\frac{1}{2}}\Vert u\Vert _{L^\infty ((0,T), X_p(\mathbb {R}^d)}^{\sigma +1}<\infty . \end{aligned}$$

Choose $\rho$ sufficiently close to 2 so that $X_p(\mathbb {R}^d)\hookrightarrow L^{\frac{2\sigma \rho '}{2-\rho '}}(\mathbb {R}^d)$. Then

$$\begin{aligned} \Vert |u|^\sigma |\nabla u|\Vert _{L^{\gamma '}((0,T), L^{\rho '})} \lesssim \left\| \Vert u\Vert _{\frac{2\sigma \rho '}{2-\rho '}}^\sigma \Vert \nabla u\Vert _2 \right\| _{L^{\gamma '}(0,T)}\lesssim T^{\frac{1}{\gamma '}} \Vert u\Vert _{L^\infty ((0,T), X_p)}^{\sigma +1}<\infty . \end{aligned}$$

Therefore $\Vert u\Vert _{L^\infty ((0,T), L^{\tilde{p}})}$ is finite and the proof is finished. $\square$

5 Further comments

In light of the results we have proven, we highlight some new questions that have risen:

• 1. Local well-posedness: In dimensions $d\geqslant 3$, the local well-posedness in the case $p>2\sigma +2$ remains open. Is this optimal? As we have argued in Remark 3.2, this case requires new estimates for the Schrödinger group.

• 2. Global well-posedness: this problem is completely open for $p>\sigma +2$. Even if the energy is well-defined, there are still several cases where global well-posedness (even for small data) remains unanswered.

• 3. New blow-up behaviour: in the opposite perspective, is it possible to exhibit new blow-up phenomena? This would be especially interesting either for the defocusing case or for the $L^2$-subcritical case, where blow-up behaviour in $H^1$ is impossible.

• 4. Stability of ground-states: in the $H^1$ framework, the work of [2] has shown that the ground-states are orbitally stable under $H^1$ perturbations. Does the result still hold if we consider $X_p$ perturbations?