Local Cauchy theory for the nonlinear Schr\"odinger equation in spaces of infinite mass

TL;DR

This paper establishes local well-posedness for the nonlinear Schrödinger equation with initial data in certain infinite mass spaces, expanding understanding of solution behavior in these function spaces.

Contribution

It introduces a local Cauchy theory for NLS in spaces combining homogeneous Sobolev and Lebesgue spaces, addressing infinite mass initial data.

Findings

Proves local well-posedness for a broad range of p in the initial data space.

Discusses conditions under which global well-posedness can be achieved.

Extends the theory of NLS to new function spaces with infinite mass.

Abstract

We consider the Cauchy problem for the nonlinear Schr\"odinger 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.

Equations

$$i{u}_t+\Delta u+\lambda |u{|}^{\sigma }u=0,u=u(t,x),(t,x)\in \Re \times {}^d,\lambda \in \Re ,0<\sigma <4/(d-2{)}^{+}$$

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

$$E=\{u\in H_{loc}^1({}^d):\nabla u\in L^2({}^d),|u{|}^2-1\in L^2({}^d)\}.$$

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

$$2\le r\le 2^{\ast },\frac{2}{q}=d\left(\frac{1}{2}-\frac{1}{r}\right),r\ne \infty \text{mbox}ifd=2.$$

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

$${\Vert {\int }_{0<s<t}S(t-s)f(s)ds\Vert }_{L^q(I;L^r({}^N))}\lesssim \Vert f{\Vert }_{L^{{\gamma }^{\prime }}(I;L^{{\rho }^{\prime }}({}^N))}.$$

$$\Vert S(t)\varphi {\Vert }_p^2\lesssim \Vert \varphi {\Vert }_p^2+|t{|}^{1-\frac{2}{k}}\Vert \nabla \varphi {\Vert }_2^2,t\in \Re .$$

$${\Vert {\int }_0^{\cdot }S(\cdot -s)f(s)ds\Vert }_{L^{\infty }((0,T);L^p({}^d))}\lesssim C(T)\left(\Vert f{\Vert }_{L^2((0,T);L^p({}^d))}+\Vert \nabla f{\Vert }_{L^{q^{\prime }}((0,T);L^{r^{\prime }}({}^d))}\right),$$

$$i{u}_t+\Delta u=0,u(0)=\varphi .$$

$$\frac{1}{p}\frac{d}{dt}\Vert u(t){\Vert }_p^p\le \left|\mathrm{Im}\int |u{|}^{p-2}\bar{u}\Delta u\right|\le \frac{p-2}{2}\int |u{|}^{p-2}|\nabla u{|}^2\le \frac{p-2}{2}\Vert u(t){\Vert }_p^{p-2}\Vert \nabla u(t){\Vert }_p^2$$

$$\frac{d}{dt}\Vert u(t){\Vert }_p^2\le (p-2)\Vert \nabla u(t){\Vert }_p^2.$$

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

$$i{v}_t+\Delta v=f,v(0)=0.$$

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

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

$${\Vert {\int }_0^{\cdot }S(\cdot -s)f(s)ds\Vert }_{L^{\infty }((0,T),L^p({}^d))}\lesssim C(T,q)\Vert f{\Vert }_{L^{q^{\prime }}((0,T),L^{p^{\prime }}({}^d))},\frac{1}{q}>d\left(\frac{1}{2}-\frac{1}{p}\right).$$

$${\Vert {\int }_0^{t}S(t-s)f(s)ds\Vert }_{L^p({}^d)}$$

$$\lesssim {\int }_0^T\frac{1}{|t-s{|}^{d\left(\frac{1}{2}-\frac{1}{p}\right)}}\Vert f(s){\Vert }_{L^{p^{\prime }}({}^d)}ds$$

$$\lesssim {\left({\int }_0^T\frac{1}{|t-s{|}^{qd\left(\frac{1}{2}-\frac{1}{p}\right)}}ds\right)}^{\frac{1}{q}}\Vert f{\Vert }_{L^{q^{\prime }}((0,T),L^{p^{\prime }}({}^d))}.$$

$$X_p({}^d)=L^p({}^d)\cap {\dot{H}}^1({}^d).$$

$$\Vert S(t)\varphi {\Vert }_{X_p}\lesssim (1+|t{|}^{1-\frac{2}{k}}{)}^{1/2}\Vert \varphi {\Vert }_{X_p},t\in \Re .$$

$$\Vert {\varphi }_{ϵ}-\varphi {\Vert }_{X_p}<ϵ.$$

$$\underset{t\to 0}{\mathrm{limsup}}\Vert S(t)\varphi -\varphi {\Vert }_{X_p}$$

$$\lesssim \underset{t\to 0}{\mathrm{limsup}}\left((1+|t{|}^{1-\frac{2}{k}}{)}^{1/2}\Vert \varphi -{\varphi }_{ϵ}{\Vert }_{X_p}+\Vert S(t){\varphi }_{ϵ}-{\varphi }_{ϵ}{\Vert }_{H^1}\right)\lesssim ϵ.$$

$$X^2(\Re )=\{u\in L^{\infty }(\Re ):\nabla u\in H^1(\Re )\}.$$

$$\frac{d}{dt}\Vert u(t){\Vert }_p\le \Vert \Delta u(t){\Vert }_p.$$

$$\Vert u(t){\Vert }_p\le \Vert u_0{\Vert }_p+{\int }_0^t\Vert \Delta u(s){\Vert }_{p}ds\le \Vert u_0{\Vert }_p+C{t}^{1-\frac{1}{k}}\Vert \Delta u{\Vert }_{L^k((0,t),L^p)}\le \Vert u_0{\Vert }_p+C{t}^{1-\frac{1}{k}}\Vert \Delta u_0{\Vert }_2,$$

$$\Vert u(t){\Vert }_{\infty }\lesssim \Vert u_0{\Vert }_{\infty }+t^{\frac{3}{4}}\Vert \Delta u_0{\Vert }_2,t>0,u_0\in H^2(\Re ).$$

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

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

$$r=(\sigma +1){\rho }^{\prime }=\max \{\sigma +2,p\}.$$

Full text

Introduction

In this work, we consider the classical nonlinear Schrödinger equation over d:

[TABLE]

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 [2], [10], [11] and references therein). The usual framework one considers is $E=H^{1}({}^{d})$, the so-called energy space, or more generally, $E=H^{s}({}^{d})$. 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, we refer the papers [6], [7] and [4]. In the first paper, one considers local well-posedness on Zhidkov spaces

[TABLE]

In the second, one takes the Gross-Pitaevskii equation and looks for local well-posedness on

[TABLE]

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

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

[TABLE]

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

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

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

Linear estimates

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

[TABLE]

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 $I\subset\real$,

[TABLE]

and

[TABLE]

Remark 2.1.

The estimate (2.2) may be extended to other sets of admisssible pairs: see [5] and [12]. However, the linear estimate (2.1) is not valid for any other pairs and for $u_{0}\notin L^{2}({}^{d})$.

Proposition 2.2** (Group estimates with loss of derivative).**

Define $k$ so that $(k,p)$ is admissible. Then

• •

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

[TABLE]

• •

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

[TABLE]

*where $C(\cdot)$ is a increasing bounded function over bounded intervals of *.

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 $u\in C^{1}(\real;H^{2}({}^{d}))$ satisfies

[TABLE]

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

[TABLE]

Thus we have

[TABLE]

An integration between [math] and $t\in\real$ and the linear Strichartz estimate yield

[TABLE]

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

[TABLE]

As for the previous estimate, we have

[TABLE]

and so

[TABLE]

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

Lemma 2.3** (Local Strichartz estimate without loss of derivatives).**

Given $f\in C([0,T],\mathcal{S}({}^{d}))$,

[TABLE]

Proof.

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

[TABLE]

∎

We set

[TABLE]

Remark 2.2.

From the Gagliardo-Nirenberg inequality, we have $H^{1}({}^{d})\hookrightarrow X_{p}({}^{d})$.

Proposition 2.4**.**

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

Proof.

Given any $\phi\in\dot{H}^{1}({}^{d})$, we have $\|S(t)\nabla\phi\|_{2}=\|\nabla\phi\|_{2}$. Together with Proposition 2.2, this implies that

[TABLE]

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

[TABLE]

Then

[TABLE]

∎

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

[TABLE]

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

[TABLE]

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

[TABLE]

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 $L^{4}_{t}L^{\infty}_{x}$). Then, taking the limit $p\to\infty$, we obtain

[TABLE]

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 [6], one may see that

[TABLE]

Remark 2.4.

One may ask if the required regularity is optimal: can we define the Schrödinger group on $X_{p}^{s}({}^{d}):=\dot{H}^{s}({}^{d})\cap L^{p}({}^{d})$? 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 [6].

Local well-posedness for $p\leq 2\sigma+2$

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

Definition 3.1** (Solution over $X_{p}({}^{d})$).**

Given $u_{0}\in X_{p}({}^{d})$, we say that $u\in C([0,T],X_{p}({}^{d}))$ is a solution of (NLS) with initial data $u_{0}$ if the Duhamel formula is valid:

[TABLE]

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

[TABLE]

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

Proposition 3.2** (Uniqueness over $X_{p}({}^{d})$).**

Suppose that $p\leq 2\sigma+2$. Let $u_{1},u_{2}\in C([0,T],X_{p}({}^{d}))$ be two solutions of (NLS) with initial data $u_{0}\in X_{p}({}^{d})$. Then $u_{1}\equiv u_{2}$.

Proof.

Taking the difference between the Duhamel formula for $u_{1}$ and $u_{2}$,

[TABLE]

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

[TABLE]

The claimed result now follows from [2, Lemma 4.2.2]. ∎

Theorem 3.3** (Local well-posedness on $X_{p}({}^{d})$, $p\leq 2\sigma+2$).**

Given $u_{0}\in X_{p}({}^{d})$, there exists $T=T(\|u_{0}\|_{X_{p}})>0$ and an unique solution

[TABLE]

of (NLS) with initial data $u_{0}$. One has

[TABLE]

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

[TABLE]

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 [7]). This insight allows the use of Strichartz estimates at the zero derivatives level. Without this possibility, one would be restricted to the estimate (2.5) and the possible ranges of $\sigma$ and $p$ would be significantly smaller.

Proof.

Step 1. Define

[TABLE]

and

[TABLE]

Consider the space

[TABLE]

endowed with the distance

[TABLE]

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 $u\in\mathcal{D}^{\prime}([0,T]\times{}^{d})$ such that $u_{n}-S(\cdot)u_{0}\to u-S(\cdot)u_{0}$ in $\mathcal{S}_{0}$. By [2, Theorem 1.2.5], this convergence implies that

[TABLE]

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

[TABLE]

and so $u_{n}\to u$ in $L^{\infty}((0,T),L^{p})$.

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

[TABLE]

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

[TABLE]

It follows that, for $M\sim 2\|u_{0}\|_{X_{p}}$ and $T$ sufficiently small, we have $\Phi:\mathcal{E}\mapsto\mathcal{E}$.

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

[TABLE]

Therefore, for $T=T(\|u_{0}\|_{X_{p}})$ 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 $u\in C([0,T],X_{p}({}^{d}))$ 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 $\|u(t)\|_{X_{p}}\to\infty$ as $t\to T^{*}(u_{0})$. The continuous dependence on the initial data follows as in the $H^{1}$ case (see, for example, the proof of [2, Theorem 4.4.1]) ∎

Remark 3.2.

The condition $p\leq 2\sigma+2$ 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

[TABLE]

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 $p\leq\rho^{\prime}(\sigma+1)\leq 2\sigma+2$ (because $\rho\geq 2$).

Proposition 3.4** (Persistence of integrability).**

Suppose that $\tilde{p}<p\leq 2\sigma+2$. Given $u_{0}\in X_{\tilde{p}}({}^{d})$, consider the $X_{p}({}^{d})$-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 $X_{\tilde{p}}({}^{d})$ and by the uniqueness over $X_{p}({}^{d})$, there exists a time $T_{0}>0$ such that $u\in C([0,T_{0}],X_{\tilde{p}}({}^{d}))$. Thus the statement of the proposition is equivalent to saying that $u$ does not blow-up in $X_{\tilde{p}}({}^{d})$ 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

[TABLE]

Then, for any $0<t<T$,

[TABLE]

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

Proposition 3.5** (Conservation of energy).**

Suppose that $p\leq\sigma+2$. Given $u_{0}\in X_{p}({}^{d})$, the corresponding solution $u$ of (NLS) with initial data $u_{0}$ satisfies

[TABLE]

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

[TABLE]

Proof.

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

[TABLE]

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

Proposition 3.6**.**

Fix $\lambda<0$. If

[TABLE]

then, for any $u_{0}\in X_{\sigma+2}({}^{d})$, 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.

By contradiction, assume that $u$ blows-up at time $t=T$. The previous proposition then implies that

[TABLE]

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

[TABLE]

Integrating on $(0,t)$,

[TABLE]

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 $\|\nabla u(t)\|_{2}$.

[TABLE]

For $t$ close to $T$, $\|\nabla(u(t)-S(t)u_{0})\|_{2}\sim\|\nabla u(t)\|_{2}$ and, by conservation of energy,

[TABLE]

Thus

[TABLE]

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

[TABLE]

The desired contradiction now follows from a standard application of Gronwall’s lemma. ∎

Remark 3.4 (Global existence in the focusing $L^{2}$-subcritical regime).

As it is well-known, the global existence in $H^{1}({}^{d})$ 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

[TABLE]

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 $\sigma\geq 4/d$ can be extended to initial data in $X_{p}({}^{d})$ which do not lie in $L^{2}({}^{d})$. First of all, notice that

[TABLE]

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 [9] under radial hypothesis and relying heavily on the conservation of mass (which is unavailable on $X_{p}({}^{d})$). For the nonradial case, recent works (see, for example, [8]) only manage to prove unboundedness of solutions of negative energy. The problem of blow-up solutions strictly in $X_{p}({}^{d})$ 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 $X_{p}({}^{d})$ 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

[TABLE]

Thus the (NLS) is $X_{p}({}^{d})$-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 $\sigma\geq 4/d$, existence of blow-up phenomena is known for special initial data in $H^{1}({}^{d})\hookrightarrow X_{p}({}^{d})$. Therefore, it is impossible to obtain a global existence result for small data for $4/d\leq\sigma<2p/d$. 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 $X_{p}({}^{d})$. 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 $u_{0}\in H^{1}({}^{d})$ is such that

[TABLE]

then the corresponding solution of (NLS) is globally defined (see [3]). It is not hard to check that the result can be extended to $u_{0}\in X_{\sigma+2}({}^{d})$.

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

As it was observed in Remark 3.2, the condition $p\leq 2\sigma+2$ was necessary in order to use Strichartz estimates with no loss in regularity. For $p>2\sigma+2$, in order to estimate $L^{\infty}_{t}L^{p}_{x}$, one must turn to estimate (2.4), 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

[TABLE]

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

Because of the restriction $\sigma\geq 1$, 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 (2.4) with $f=|u|^{\sigma}u$,

[TABLE]

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

[TABLE]

or, setting $r\geq 2$ so that

[TABLE]

one estimates

[TABLE]

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 $X_{p}({}^{d})$ for $d=1,2$).**

Given $u_{0}\in X_{p}({}^{d})$, there exists $T=T(\|u_{0}\|_{X_{p}})>0$ and an unique solution

[TABLE]

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

[TABLE]

Proof.

Consider the space

[TABLE]

endowed with the natural distance

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

Therefore

[TABLE]

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

[TABLE]

The estimates (4.1) and (4.2), together with (2.4) and Strichartz’s estimates then imply that

[TABLE]

and

[TABLE]

Choosing $M\sim 2\|u_{0}\|_{X_{p}}$, for $T=T(\|u_{0}\|_{X_{p}})$ 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 $u_{0},v_{0}\in X_{p}({}^{d})$, as in (4.3), one has

[TABLE]

Thus, for $T_{0}=T_{0}(\|u_{0}\|_{X_{p}},\|v_{0}\|_{X_{p}})$ small,

[TABLE]

and continuous dependence follows. ∎

Proposition 4.2** (Persistence of integrability).**

Fix $d=1,2$ and $p>\tilde{p}$. Given $u_{0}\in X_{\tilde{p}}({}^{d})$, consider the $X_{p}({}^{d})$-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 (2.4) to the Duhamel formula of $u$,

[TABLE]

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

[TABLE]

Choose $\rho$ sufficiently close to 2 so that $X_{p}({}^{d})\hookrightarrow L^{\frac{2\sigma\rho^{\prime}}{2-\rho^{\prime}}}({}^{d})$. Then

[TABLE]

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

Further comments

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

Local well-posedness: In dimensions $d\geq 3$, the local well-posedness in the case $p>2\sigma+2$ remains open. Is this optimal? As we have arqued in Remark 3.2, this case requires new estimates for the Schrödinger group. 2. 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. 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. 4.

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

Acknowledgements

The author was partially suported by Fundação para a Ciência e Tecnologia, through the grants UID/MAT/04561/2013 and SFRH/BD/96399/2013.