On Well-Posedness for General Hierarchy Equations of Gross–Pitaevskii and Hartree Type

Abstract

Gross–Pitaevskii and Hartree hierarchies are infinite systems of coupled PDEs emerging naturally from the mean field theory of Bose gases. Their solutions are known to be related to initial value problems, in particular the Gross–Pitaevskii and Hartree equations. Due to their physical and mathematical relevance, the issues of well-posedness and uniqueness for these hierarchies have recently been studied thoroughly using specific nonlinear and combinatorial techniques. In this article, we introduce a new approach for the study of such hierarchy equations by firstly establishing a duality between them and certain Liouville equations, and secondly, solving the uniqueness and existence questions for the latter. As an outcome, we formulate a hierarchy equation starting from any initial value problem which is U(1)-invariant and prove a general principle which can be stated formally as follows:

• (i) The uniqueness of weak solutions of an initial value problem implies the uniqueness of solutions for the related hierarchy equation.

• (ii) The existence of solutions for an initial value problem implies the existence of solutions for the related hierarchy equation.

In particular, several new well-posedness results, as well as a counterexample to uniqueness for the Gross–Pitaevskii hierarchy equation, are proved. The novelty in our work lies in the aforementioned duality and the use of Liouville equations with powerful transport techniques extended to infinite dimensional functional spaces.

Appendices

Appendix

In this appendix we recall a few useful known results concerning the topology of the space of trace class operators and the space of probability measures as well as some useful arguments for measurable sets and maps.

Kadec–Klee Property

A dual Banach space $(E,||\cdot ||)$ has the Kadec–Klee property (KK*) if any sequence $(x_n)_{n\in {\mathbb {N}}}$ which converges with respect to the weak-$*$ topology to a limit $x\in E$ with $\lim _n ||x_n||=||x||$, satisfies $x_n\rightarrow x$ in the norm topology of E (see for example  [72, 85]).

Theorem A.1

The space of trace class-operators ${\mathscr {L}}^1({\mathfrak {H}})$, over a separable Hilbert space ${\mathfrak {H}}$, satisfies the Kadec–Klee property (KK*).

Weak and Strong Narrow Convergence

Let ${\mathfrak {H}}$ be a separable Hilbert space and $X_1$ its closed unit ball $B_{\mathfrak {H}}(0,1)$. The set of Borel probability measures ${\mathfrak {P}}(X_1)$ can be endowed with the weak and strong narrow convergence topology defined according to (20)–(21). Let $(e_i)_{i\in {\mathbb {N}}}$ be an O.N.B of the Hilbert space ${\mathfrak {H}}$. We have the following two useful results:

Theorem B.1

Let $(\mu ,(\mu _j)_{j\in {\mathbb {N}}})$ be a sequence in ${\mathfrak {P}}(X_1)$. Then the pointwise convergence of the characteristic functions

$$\begin{aligned} {\hat{\mu }}_j(y)=\int _{X_1} e^{-i \mathrm{Re}\langle x,y\rangle } \,\mathrm{d}\mu _j \;\rightarrow {\hat{\mu }}(y)=\int _{X_1} e^{-i \mathrm{Re}\langle x,y\rangle } \,\mathrm{d}\mu , \quad \forall y\in {\mathfrak {H}} \end{aligned}$$

implies the weak narrow converges of the measures $\mu _j\rightharpoonup \mu$.

Proof

Since the closed unit ball $X_1$ is compact and separable with respect to the weak topology (induced for instance by the metric $d_w$ in (22)), then by Prokhorov’s theorem the sequence $(\mu _j)_{j\in {\mathbb {N}}}$ is relatively sequentially compact in ${\mathfrak {P}}(X_1)$ with respect to the weak narrow topology. Moreover, $(\mu _j)_{j\in {\mathbb {N}}}$ admits a unique cluster point, because otherwise, the characteristic functions ${\hat{\mu }}_j$ would converge pointwisely to two distinct limits. Thus, the sequence $(\mu _j)_{j\in {\mathbb {N}}}$ should converge weakly narrowly to the measure $\mu$. $\quad \square$

Theorem B.2

Let $(\mu ,(\mu _j)_{j\in {\mathbb {N}}})$ be a sequence in ${\mathfrak {P}}(X_1)$. Then

$$\begin{aligned} \left( \mu _j\rightharpoonup \mu \text { and } \quad \forall \varepsilon >0, \quad \lim _{N\rightarrow \infty } \sup _{j\in {\mathbb {N}}} \int _{X_1}\; 1_{\{\sum _{i=N}^\infty |\langle \varphi , e_i\rangle |^2\geqq \varepsilon \}} \; \mathrm{d}\mu _j=0 \right) \;\Longleftrightarrow \; \mu _j\rightarrow \mu . \end{aligned}$$

Proof

We refer, for instance, to [79, Theorem 1]. $\quad \square$

Measurable Maps and Sets

Let I be a bounded open interval and consider the space ${\mathfrak {X}}={\mathscr {Z}}_{-\sigma }\times {\mathscr {C}}({\bar{I}},{\mathscr {Z}}_{-\sigma })$, introduced in Section 4.1, and endowed with the two norms

$$\begin{aligned} ||(x,\varphi )||_{{\mathfrak {X}}}= & {} ||x||_{{\mathscr {Z}}_{-\sigma }}+\sup _{t\in {\bar{I}}}||\varphi (t)||_{{\mathscr {Z}}_{-\sigma }}, \\ ||(x,\varphi )||_{{\mathfrak {X}}_{w}}= & {} ||x||_{{\mathscr {Z}}_{-\sigma ,w}}+\sup _{t\in {\bar{I}}}||\varphi (t)||_{{\mathscr {Z}}_{-\sigma ,w}}, \end{aligned}$$

where

$$\begin{aligned} ||x||^2_{{\mathscr {Z}}_{-\sigma ,w}}=\sum _{n\in {\mathbb {N}}} \frac{1}{2^n} |\langle e_n, x\rangle _{{\mathscr {Z}}_{-\sigma }}|^2 ,\qquad ||\varphi (t)||_{{\mathscr {Z}}_{-\sigma ,w}}^2=\sum _{n\in {\mathbb {N}}} \frac{1}{2^n} |\langle e_n, \varphi (t)\rangle _{{\mathscr {Z}}_{-\sigma }}|^2, \end{aligned}$$

with $(e_n)_{n\in {\mathbb {N}}}$ is an O.N.B. of ${\mathscr {Z}}_{-\sigma }$.

Lemma C.1

The $\sigma$-algebras of Borel sets of $({\mathfrak {X}},||\cdot ||_{{\mathfrak {X}}} )$ and $({\mathfrak {X}}, ||\cdot ||_{{\mathfrak {X}}_{w}})$ coincide.

Proof

Let $\imath : ({\mathfrak {X}}, ||\cdot ||_{{\mathfrak {X}}}) \rightarrow ({\mathfrak {X}}, ||\cdot ||_{{\mathfrak {X}}_{w}})$ be the identity map. It is clear that $\imath$ is continuous and hence it is measurable. This implies the following inclusion of $\sigma$-algebras:

$$\begin{aligned} {\mathscr {T}}({\mathfrak {X}}, ||\cdot ||_{{\mathfrak {X}}_{w}})\subset {\mathscr {T}}({\mathfrak {X}}, ||\cdot ||_{{\mathfrak {X}}}). \end{aligned}$$

To show the opposite inclusion, one uses an approximation of identity:

$$\begin{aligned} \Psi _{\varepsilon }: ({\mathfrak {X}}, ||\cdot ||_{{\mathfrak {X}}_{w}})\longrightarrow & {} ({\mathfrak {X}}, ||\cdot ||_{{\mathfrak {X}}})\\ (x,\varphi )\longrightarrow & {} \sum _{n\in {\mathbb {N}}} \frac{1}{1+n\varepsilon } \, \big (\langle x, e_n\rangle _{{\mathscr {Z}}_{-\sigma }} e_n ; \langle \varphi , e_n\rangle _{{\mathscr {Z}}_{-\sigma }} e_n\big ), \end{aligned}$$

for some $\varepsilon >0$. It is clear that $\Psi _\varepsilon$ is continuous. Furthermore, for x and $\varphi$ fixed,

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \sum _{n\in {\mathbb {N}}} \frac{1}{1+n\varepsilon } \,\langle x, e_n\rangle _{{\mathscr {Z}}_{-\sigma }} e_n =x\quad \text { in } ({\mathscr {Z}}_{-\sigma },||\cdot ||_{{\mathscr {Z}}_{-\sigma }}), \end{aligned}$$

and

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \sum _{n\in {\mathbb {N}}} \frac{1}{1+n\varepsilon } \langle \varphi , e_n\rangle _{{\mathscr {Z}}_{-\sigma }} e_n=\varphi \quad \text { in } {\mathscr {C}}({\bar{I}},{\mathscr {Z}}_{-\sigma }). \end{aligned}$$

The last limit follows using the fact that $\varphi ({\bar{I}})$ is compact and the operator $\sum _{n\in {\mathbb {N}}} \frac{1}{1+n\varepsilon } |e_n\rangle \langle e_n|$ converges strongly, as $\varepsilon \rightarrow 0$, to the identity on ${\mathscr {Z}}_{-\sigma }$. Since one can show that $\Psi _\varepsilon$ converges pointwisely to the identity map $\imath ^{-1}: ({\mathfrak {X}}, ||\cdot ||_{{\mathfrak {X}}_{w}}) \rightarrow ({\mathfrak {X}}, ||\cdot ||_{{\mathfrak {X}}})$, the opposite inclusion holds true. $\quad \square$

Lemma C.2

Let (M, d) be a metric space. Then for any bounded measurable function $f:[a,b]\times M\rightarrow {\mathbb {R}}$, the mapping

$$\begin{aligned} \begin{aligned} M&\longrightarrow {\mathbb {R}}\\ x&\longrightarrow \int _a^b f(\tau ,x) \,\mathrm{d}\tau \end{aligned} \quad \text { is measurable}. \end{aligned}$$

(84)

Proof

It suffices to use the monotone class theorem. Consider the set

$$\begin{aligned} {\mathscr {F}}:=\{f:[a,b]\times M\rightarrow {\mathbb {R}}\text { bounded measurable and satisfying } (84)\}. \end{aligned}$$

Then one checks that

• 1. ${\mathscr {F}}$ is stable with respect to addition and scalar multiplication.

• 2. ${\mathscr {F}}$ is stable with respect to monotone convergence, that is: if $(f_n)_{n\in {\mathbb {N}}}$ is a bounded sequence of non-negative function in ${\mathscr {F}}$ such that $(f_n)_{n\in {\mathbb {N}}}$ is a non-decreasing sequence of functions which converges pointwisely to a function f, as $n\rightarrow \infty$, then $f\in {\mathscr {F}}$.

• 3. The set ${\mathscr {A}}:=\{ F\subset [a,b]\times M , F \text { closed } \}$ is a $\pi$-system and for any $F\in {\mathscr {A}}$ the indicator function $1_{F}$ belongs to ${\mathscr {F}}$.

Consequently, ${\mathscr {F}}$ contains all real-valued bounded measurable functions on $[a,b]\times M$. $\quad \square$

The following observation is useful in Propositions 1.8–1.9 where we consider the uniqueness of hierarchy equations related to the NLS equation (33) with a nonlinearity g given by (41).

Lemma C.3

The space $L^2({\mathbb {R}}^d)\cap L^r({\mathbb {R}}^d)$ is a Borel subset of $(L^2({\mathbb {R}}^d),||\cdot ||_{L^2({\mathbb {R}}^d)})$. Furthermore, there exists a Borel extension $G:L^2({\mathbb {R}}^d)\rightarrow H^{-1}({\mathbb {R}}^d)$ of the mapping

$$\begin{aligned} g:L^2({\mathbb {R}}^d)\cap L^r({\mathbb {R}}^d)\rightarrow L^{r'}({\mathbb {R}}^d) \end{aligned}$$

given in (41) and satisfying (42).

Proof

Let $\varphi _n$ be a mollifier (that is, $\varphi \in {\mathscr {C}}_0^\infty ({\mathbb {R}}^d)$, $\varphi \geqq 0$, $\int _{{\mathbb {R}}^d} \varphi (x) \mathrm{d}x=1$ and $\varphi _n(x):=n^d \varphi (nx)$). Then the mapping

$$\begin{aligned} T_n: L^2({\mathbb {R}}^d)\longrightarrow & {} {\mathbb {R}}_+ \\ f\longrightarrow & {} ||\varphi _n*f||_{L^r} \end{aligned}$$

is well defined and continuous thanks to Young’s inequality. Moreover, one easily checks that

$$\begin{aligned} \lim _{n\rightarrow \infty } T_n(f)=\left\{ \begin{aligned} ||f||_{L^r}&\quad \text { if } f\in L^2({\mathbb {R}}^d)\cap L^r({\mathbb {R}}^d)&\\ \infty&\quad \text { ifnot.}&\end{aligned} \right. \end{aligned}$$

Hence, $L^2({\mathbb {R}}^d)\cap L^r({\mathbb {R}}^d)$ is a measurable subset of $L^2({\mathbb {R}}^d)$. Consider the mapping

$$\begin{aligned} G: L^2({\mathbb {R}}^d)\longrightarrow & {} L^{r'}({\mathbb {R}}^d)\subset H^{-1}({\mathbb {R}}^d) \\ u\longrightarrow & {} 1_{L^2\cap L^r}(u) \,g(u). \end{aligned}$$

Then G is a Borel map extending g. Indeed, the sequences of mapping $g_n$, defined as

$$\begin{aligned} g_n: L^2({\mathbb {R}}^d)\longrightarrow & {} H^{-1}({\mathbb {R}}^d) \\ u\longrightarrow & {} 1_{L^2\cap L^r}(u) \,g(\varphi _n*u), \end{aligned}$$

is measurable, since $u\rightarrow g(\varphi _n*u)$ are continuous using the assumption (42) and $\lim _{n\rightarrow \infty }g_n(u)=G(u)$ for any $u\in L^2({\mathbb {R}}^d)$. $\quad \square$