with frequencies going to zero or infinity, into an error term which is small
in $Y$ (which suffices for our purposes as we have now developed a robust
small data theory in $Y$!).

A precise formulation of the profile decomposition we require is as follows.

######  Theorem 6.1 (Profile decomposition).

Let $(f_{n},g_{n})\in H^{1}\times L^{2}$ be a bounded sequence. Then after
passing to a subsequence if necessary, there exists
$J^{*}\in\mathbb{N}\cup\\{\infty\\}$, non-zero profiles $(f^{(j)},g^{(j)})\in
H^{1}\times L^{2}$, and group elements
$\mathfrak{g}_{n}^{(j)}=\mathfrak{g}[t_{n}^{(j)},x_{n}^{(j)}]$ such that if we
define $(w_{n}^{(J)},e_{n}^{(J)})\in H^{1}\times L^{2}$ as

$(f_{n},g_{n})=\sum_{j=1}^{J}\mathfrak{g}_{n}^{(j)}(f^{(j)},g^{(j)})+(w_{n}^{(J)},e_{n}^{(J)})$

then we have the properties:

  1. (i)

The energy of the profiles decouples, thus for any $J\leqslant J^{*}$

$\lim_{n\to\infty}\Big{(}\|f_{n}\|_{H^{1}}^{2}-\sum_{j=1}^{J}\|f^{(j)}\|_{H^{1}}^{2}-\|w^{(J)}_{n}\|_{H^{1}}^{2}\Big{)}=0=\lim_{n\to\infty}\Big{(}\|g_{n}\|_{L^{2}}^{2}-\sum_{j=1}^{J}\|g^{(j)}\|_{L^{2}}^{2}-\|e^{(J)}_{n}\|_{L^{2}}^{2}\Big{)}$

and

$\lim_{n\to\infty}\Big{(}\mathcal{E}_{Z}(f_{n},g_{n})-\sum_{j=1}^{J}\mathcal{E}_{Z}\big{(}\mathfrak{g}^{(j)}_{n}(f^{(j)},g^{(j)})\big{)}-\mathcal{E}_{Z}(w^{(J)}_{n},e^{(J)}_{n})\Big{)}=0.$

  2. (ii)

The free evolution of the error $(w^{(J)}_{n},e^{(J)}_{n})$ goes to zero in
$Y\times Y$, thus

$\lim_{J\to
J^{*}}\limsup_{n\to\infty}\big{(}\|e^{it\Delta}w^{(J)}_{n}\|_{Y}+\|e^{it|\nabla|}e^{(J)}_{n}\|_{Y}\big{)}=0.$

  3. (iii)

For any $j\not=k$, the group elements
$\mathfrak{g}_{n}^{(j)}=\mathfrak{g}[t^{(j)}_{n},x^{(j)}_{n}]$ satisfy the
asymptotic orthogonality property

$\lim_{n\to\infty}\big{(}|t_{n}^{(j)}-t_{n}^{(k)}|+|x^{(j)}_{n}-x^{(k)}_{n}|\big{)}=0.$

Moreover, we have the normalisation condition that either $t_{n}^{(j)}=0$ for
all $n\in\mathbb{N}$, or $|t_{n}^{(j)}|\to\infty$ as $n\to\infty$.

###### Proof.

The proof is a minor modification of the profile decomposition of Bahouri-
Gérard [2], and thus we only give a sketch of the proof. A standard argument,
see for instance [23], shows that it suffices to prove that if

$\lim_{n\to\infty}\|(f_{n},g_{n})\|_{H^{1}\times
L^{2}}=A,\qquad\lim_{n\to\infty}\|(e^{it\Delta}f_{n},e^{it|\nabla|}g_{n})\|_{Y\times
Y}=\epsilon>0,$ (6.1)

then there exits a profile $(f^{(1)},g^{(1)})\in H^{1}\times L^{2}$ and a
sequence of group elements $\mathfrak{g}_{n}=\mathfrak{g}[t_{n},x_{n}]$ such
that, after potentially taking a subsequence, the sequence
$(\mathfrak{g}_{n})^{-1}(f_{n},g_{n})$ converges weakly to $(f^{(1)},g^{(1)})$
in $H^{1}\times L^{2}$, we have the lower bound

$\|(f^{(1)},g^{(1)})\|_{H^{1}\times L^{2}}\gtrsim\epsilon,$ (6.2)

either $t_{n}=0$ for all $n$ or $|t_{n}|\to\infty$, and the energy (and
$L^{2}$ and $H^{1}$ norms) decouples

$\begin{split}\lim_{n\to\infty}\Big{(}\mathcal{E}_{Z}[(f_{n},g_{n})]-\mathcal{E}_{Z}\big{[}\mathfrak{g}_{n}(f^{(1)},g^{(1)})\big{]}-\mathcal{E}_{Z}\big{[}(f_{n},g_{n})-\mathfrak{g}_{n}(f^{(1)},g^{(1)})\big{]}\Big{)}&=0,\\\
\lim_{n\to\infty}\Big{(}\|f_{n}\|_{H^{1}}^{2}-\|f^{(1)}\|_{H^{1}}^{2}-\|f_{n}-e^{-it_{n}\Delta}\tau_{x_{n}}f^{(1)}\|_{H^{1}}^{2}\Big{)}&=0,\\\
\lim_{n\to\infty}\Big{(}\|g_{n}\|_{L^{2}}^{2}-\|g^{(1)}\|_{L^{2}}^{2}-\|g_{n}-e^{-it_{n}|\nabla|}\tau_{x_{n}}g^{(1)}\|_{L^{2}}^{2}\Big{)}&=0.\end{split}$
(6.3)

To construct the profile $(f^{(1)},g^{(1)})$ we observe that by definition
there exists $(s_{n},x_{n})\in\mathbb{R}\times\mathbb{R}^{4}$ and
$\lambda_{n}\in 2^{\mathbb{N}}$ such that

$\lambda_{n}^{-4}\big{|}\big{(}e^{is_{n}\Delta}P_{\lambda_{n}}f_{n},e^{is_{n}|\nabla|}P_{\lambda_{n}}g_{n}\big{)}(x_{n})\big{|}\geqslant\frac{1}{2}\epsilon.$

An application of Bernstein’s inequality gives
$\lambda_{n}^{-2}\|(f_{n},g_{n})\|_{H^{1}\times L^{2}}\gtrsim\epsilon$, and
hence $\limsup_{n\to\infty}\lambda_{n}^{2}\lesssim\epsilon^{-1}A$. In
particular, as $\lambda_{n}\in 2^{\mathbb{N}}$, there exists $\lambda_{*}\in
2^{\mathbb{N}}$ and a subsequence such that $\lambda_{n}=\lambda_{*}$ for all
$n\in\mathbb{N}$. Define the group element
$\mathfrak{g}_{n}=\mathfrak{g}_{n}[t_{n},x_{n}]$, where $t_{n}=s_{n}$ if
$|s_{n}|\to\infty$, and otherwise $t_{n}=0$ for all $n\in\mathbb{N}$. As
$(\mathfrak{g}_{n})^{-1}(f_{n},g_{n})$ is a bounded sequence in $H^{1}\times
L^{2}$, after potentially taking a further subsequence, there exists
$(f^{(1)},g^{(1)})\in H^{1}\times L^{2}$ such that
$(\mathfrak{g}_{n})^{-1}(f_{n},g_{n})$ converges weakly to $(f^{(1)},g^{(1)})$
in $H^{1}\times L^{2}$. Letting $K_{\lambda_{*}}$ denote the kernel of the
Fourier multiplier $P_{\lambda_{*}}$, we have by the continuity of the flow
$t\mapsto(e^{it\Delta},e^{it|\nabla|})$ on $H^{1}\times L^{2}$,

$\displaystyle|(P_{\lambda_{*}}f^{(1)},P_{\lambda_{*}}g^{(1)})(0)|$
$\displaystyle=\limsup_{n\to\infty}\Big{|}\int_{\mathbb{R}^{4}}K_{\lambda_{*}}(-y)(\mathfrak{g}_{n})^{-1}(f_{n},g_{n})(y)dy$
$\displaystyle=\lambda_{*}^{4}\limsup_{n\to\infty}\lambda_{n}^{-4}\big{|}\big{(}e^{is_{n}\Delta}P_{\lambda_{n}}f_{n},e^{is_{n}|\nabla|}P_{\lambda_{n}}g_{n}\big{)}(x_{n})\big{|}\geqslant\frac{1}{2}\lambda_{*}^{4}\epsilon$

and hence Bernstein’s inequality and fact that $\lambda_{*}\in 2^{\mathbb{N}}$
gives the lower bound (6.2). The $L^{2}$ and $H^{1}$ decoupling in (6.3)
follows immediately from the fact that $e^{it_{n}|\nabla|}\tau_{-x_{n}}f_{n}$
converges weakly in $H^{1}$ to $f^{(1)}$, while
$e^{it_{n}|\nabla|}\tau_{-x_{n}}g_{n}$ converges weakly to $g^{(1)}$, and
noting that both the $H^{1}$ and $L^{2}$ norms are invariant under the action
of $\mathfrak{g}_{n}$. To verify the energy decoupling in (6.3) we have to
work slightly harder. In view of the $H^{1}$ and $L^{2}$ decoupling, it
suffices to prove that

$\lim_{n\to\infty}\Big{|}\int_{\mathbb{R}^{4}}g_{n}|f_{n}|^{2}-e^{-it_{n}|\nabla|}\tau_{x_{n}}g^{(1)}\big{|}e^{-it_{n}\Delta}\tau_{x_{n}}f^{(1)}\big{|}^{2}-\big{(}g_{n}-e^{-it_{n}|\nabla|}\tau_{x_{n}}g^{(1)}\big{)}\big{|}f_{n}-e^{-it_{n}\Delta}\tau_{x_{n}}f^{(1)}\big{|}^{2}dx\Big{|}=0.$

If $|t_{n}|\to\infty$, then after approximating by smooth functions the limit
follows from the dispersive decay of the wave and Schrödinger propagators.
Thus we may assume $t_{n}=0$ and after a short computation via translation
invariance and weak convergence, our goal is to now prove that

$\lim_{n\to\infty}\int_{\mathbb{R}^{4}}2|\tau_{-x_{n}}g_{n}||f^{(1)}||\tau_{-x_{n}}f_{n}-f^{(1)}|+|g^{(1)}||\tau_{-x_{n}}f_{n}+f^{(1)}||\tau_{-x_{n}}f_{n}-f^{(1)}|dx=0.$
(6.4)

But this follows by noting that since $\tau_{-x_{n}}f_{n}$ bounded in $H^{1}$,
by the Rellich-Kondrachov Theorem, we have
$\|\tau_{-x_{n}}f_{n}-f^{(1)}\|_{L^{2}(\Omega)}\to 0$ for any compact
$\Omega\subset\mathbb{R}^{4}$. Hence limit follows by localising in space. ∎

Later we exploit the fact that asymptotically orthogonal nonlinear profiles
only interact weakly.

######  Lemma 6.2 (Orthogonal profiles interact weakly).

Let $(t_{n},x_{n}),(t_{n}^{\prime},x_{n}^{\prime})\in\mathbb{R}^{1+4}$ be
sequences such that

$\lim_{n\to\infty}\big{(}|t_{n}-t_{n}^{\prime}|+|x_{n}-x_{n}^{\prime}|\big{)}=\infty.$

Then for any $V\in\underline{W}^{0}$ and $u,w\in S^{\frac{1}{2}}$ we have

$\lim_{n\to\infty}\big{\|}\mathcal{I}_{0}\big{[}\Re(V_{n})u_{n}\big{]}\big{\|}_{S^{\frac{1}{2}}}=\lim_{n\to\infty}\big{\|}\mathcal{J}_{0}\big{[}|\nabla|(\overline{w}_{n}u_{n})\big{]}\big{\|}_{W^{0}}=\lim_{n\to\infty}\|\overline{w}_{n}u_{n}\|_{L^{1}_{t}L^{2}_{x}}=0$
(6.5)

where we take

$u_{n}(t,x)=u(t-t_{n},x-x_{n}),\qquad
V_{n}(t,x)=V(t-t_{n}^{\prime},x-x_{n}^{\prime}),\qquad
w_{n}(t,x)=w(t-t_{n}^{\prime},x-x_{n}^{\prime}).$

###### Proof.

The proof is essentially the standard approximation argument to reduce to the
$C^{\infty}_{0}$ case. We begin by observing that after translating in space-
time, for all limits in (6.5) it is enough to consider the case
$t_{n}^{\prime}=x_{n}^{\prime}=0$. For the first limit, as

$\lim_{R\to\infty}\|P_{\geqslant
R}V\|_{\underline{W}^{0}}=\lim_{R\to\infty}\|P_{\geqslant
R}u\|_{S^{\frac{1}{2}}}=0$

an application of Theorem 2.4 shows that we only have to consider bounded
frequencies. In particular, since

$\big{\|}\mathcal{I}_{0}\big{[}\Re(V_{<R})P_{<R}u_{n}\big{]}\big{\|}_{S^{\frac{1}{2}}}\lesssim
R^{\frac{1}{2}}\|V_{<R}(P_{<R}u)_{n}\|_{L^{2}_{t}L^{\frac{4}{3}}_{x}}$

it suffices to prove that for the bounded frequency contribution we have

$\lim_{n\to\infty}\|V_{<R}P_{<R}u_{n}\|_{L^{2}_{t}L^{\frac{4}{3}}_{x}}=0.$

This can be reduced further after noting that as $u\in
L^{2}_{t}L^{4}_{x}\subset S^{\frac{1}{2}}$ we have

$\lim_{R^{\prime}\to\infty}\big{\|}\mathbbold{1}_{\\{|t|+|x|\geqslant
R^{\prime}\\}}P_{<R}u\big{\|}_{L^{2}_{t}L^{4}_{x}}=0.$

Letting $B_{R^{\prime}}=\\{|t|+|x|<R^{\prime}\\}$ we have

$\displaystyle\big{\|}V_{\leqslant
R}\big{(}\mathbbold{1}_{B_{R^{\prime}}}P_{\leqslant
R}u\big{)}_{n}\big{\|}_{L^{2}_{t}L^{\frac{4}{3}}_{x}}$
$\displaystyle\lesssim\big{\|}V_{\leqslant
R}(t,x)\mathbbold{1}_{B_{R^{\prime}}}(t-t_{n},x-x_{n})\big{\|}_{L^{2}_{t}L^{\frac{4}{3}}_{x}}\|P_{\leqslant
R}u\|_{L^{\infty}_{t,x}}$ $\displaystyle\lesssim
R^{\frac{3}{2}}(R^{\prime})^{\frac{7}{3}}\|V_{\leqslant
R}(t,x)\mathbbold{1}_{B_{R^{\prime}}}(t-t_{n},x-x_{n})\|_{L^{2}_{t}L^{6}_{x}}\|u\|_{L^{\infty}_{t}H^{\frac{1}{2}}}$

and hence it is enough to prove that

$\lim_{n\to\infty}\|V_{\leqslant
R}(t,x)\mathbbold{1}_{B_{R^{\prime}}}(t-t_{n},x-x_{n})\|_{L^{2}_{t}L^{6}_{x}}=0.$

But this is immediate via the dominated convergence theorem since
$\|V_{\leqslant R}\|_{L^{2}_{t}L^{6}_{x}}\lesssim
R^{\frac{5}{6}}\|V\|_{\underline{W}^{0}}$ which is a consequence of (2.2).
Hence the first limit in (6.5) follows.

To prove the second limit in (6.5), an analogous argument via Theorem 2.4
shows that again is suffices to consider bounded frequencies. After observing
that an application of Bernstein’s inequality gives

$\|\big{\|}\mathcal{J}_{0}\big{[}|\nabla|(\overline{w_{\leqslant
R}}P_{\leqslant R}u_{n})\big{]}\big{\|}_{W^{0}}\lesssim
R\|\overline{w_{\leqslant R}}P_{\leqslant R}u_{n}\|_{L^{1}_{t}L^{2}_{x}\cap
L^{2}_{t}L^{\frac{4}{3}}_{x}},$

and noting that $u,w\in L^{2}_{t}L^{4}_{x}\subset S^{\frac{1}{2}}$, it only
remains to prove that

$\lim_{n\to\infty}\|\overline{w}u_{n}\|_{L^{1}_{t}L^{2}_{x}\cap
L^{2}_{t}L^{\frac{4}{3}}_{x}}=0.$

However this is a consequence of the fact that $w,u\in L^{2}_{t}L^{4}_{x}$
together with the argument used to prove the first limit in (6.5). This also
completes the proof of the final limit in (6.5). ∎

##  7\. The Ground State Constraint

Define the functional

$\mathcal{K}(f)=\|f\|_{\dot{H}^{1}}^{2}-\|f\|_{L^{4}}^{4}=\frac{d}{d\lambda}\mathcal{E}_{NLS}(\lambda
f)\big{|}_{\lambda=1}.$

This functional can be seen as the scaling derivative of the NLS energy
$\mathcal{E}_{NLS}(f)$ and appears in the Virial identity for both the NLS and
Zakharov equations, see for instance the discussion in [16, 17]. The
functional $\mathcal{K}$ is also closely related to the Zakharov energy, for
instance we have the identity

$\frac{d}{d\lambda}\mathcal{E}_{Z}(\lambda
f,\lambda^{2}g)=\lambda^{-1}\mathcal{K}(\lambda
f)+\lambda^{3}\big{\|}g+|f|^{2}\big{\|}_{L^{2}_{x}}^{2}.$ (7.1)

This identity is particularly useful as it shows that provided
$\mathcal{K}(\lambda f)>0$, the energy increases along the curve
$\lambda\mapsto(\lambda f,\lambda^{2}g)$ in $\dot{H}^{1}\times L^{2}$.

An application of Hölder and Sobolev embedding gives

$4\mathcal{E}_{Z}(f,g)\leqslant
2\|f\|_{\dot{H}^{1}}^{2}+\|g\|_{L^{2}}^{2}+2\|Q\|_{\dot{H}^{1}}^{-1}\|g\|_{L^{2}}\|f\|_{\dot{H}^{1}}^{2}\lesssim\|f\|_{\dot{H}^{1}}^{2}(1+\|g\|_{L^{2}})+\|g\|_{L^{2}}^{2}$

and thus the energy is always finite provided $(f,g)\in\dot{H}^{1}\times
L^{2}$. The reverse inequality is false in general, in particular the energy
is not necessarily positive. However, if we impose a size constraint on the
functions $(f,g)\in\dot{H}^{1}\times L^{1}$, then the energy is always
positive. A natural condition to ensure that the energy is coercive can be
phrased in terms of the ground state (or Aubin-Talenti function)
$Q(x)=(1+\frac{|x|^{2}}{8})^{-1}$. Recall that the ground state
$Q\in\dot{H}^{1}$ satisfies the properties

$\Delta
Q=-Q^{3},\qquad\|Q\|_{L^{4}}^{2}=\|Q\|_{\dot{H}^{1}},\qquad\|Q\|_{\dot{H}^{1}}^{-\frac{1}{2}}=\sup_{\|f\|_{\dot{H}^{1}}=1}\|f\|_{L^{4}},\qquad\mathcal{E}_{Z}(Q,-Q^{2})=\mathcal{E}_{NLS}(Q)=\frac{1}{4}\|Q\|_{\dot{H}^{1}}^{2}.$

The properties of the ground state quickly give the implications

$\|g\|_{L^{2}}\leqslant\|Q\|_{\dot{H}^{1}}\qquad\Longrightarrow\qquad\|g\|_{L^{2}}^{2}\leqslant
4\mathcal{E}_{Z}(f,g)$ (7.2)

and

$\|f\|_{\dot{H}^{1}}\leqslant\|Q\|_{\dot{H}^{1}}\qquad\Longrightarrow\qquad\|f\|_{\dot{H}^{1}}^{2}\leqslant
4\mathcal{E}_{Z}(f,g).$ (7.3)

More precisely, we simply note that the (sharp) Sobolev embedding gives

$\displaystyle 4\mathcal{E}_{Z}(f,g)$ $\displaystyle\geqslant
2\|f\|_{\dot{H}^{1}}^{2}+\|g\|_{L^{2}}^{2}-2\|g\|_{L^{2}}\|f\|_{L^{4}}^{2}\geqslant
2\|f\|_{\dot{H}^{1}}^{2}+\|g\|_{L^{2}}^{2}-2\|Q\|_{\dot{H}^{1}}^{-1}\|g\|_{L^{2}}\|f\|_{\dot{H}^{1}}^{2}$

and hence rearranging we have the lower bound

$\begin{split}4\mathcal{E}_{Z}(f,g)&\geqslant\|g\|_{L^{2}}^{2}+2\|Q\|_{\dot{H}^{1}}^{-1}\|f\|_{\dot{H}^{1}}^{2}\big{(}\|Q\|_{\dot{H}^{1}}-\|g\|_{L^{2}}\big{)}\\\
&=\|f\|_{\dot{H}^{1}}^{2}+\big{(}\|g\|_{L^{2}}-\|Q\|_{\dot{H}^{1}}^{-1}\|f\|_{\dot{H}^{1}}^{2}\big{)}^{2}+\|Q\|_{\dot{H}^{1}}^{-2}\|f\|_{\dot{H}^{1}}^{2}\big{(}\|Q\|_{\dot{H}^{1}}^{2}-\|f\|_{\dot{H}^{1}}^{2}\big{)}.\end{split}$
(7.4)

from which the implications (7.2) and (7.3) easily follow. These implications
can be improved if we assume that the energy is at most the energy
$\mathcal{E}_{Z}(Q,-Q^{2})=\frac{1}{4}\|Q\|_{\dot{H}^{1}}^{2}$ of the ground
state solution $(Q,-Q^{2})$.

######  Lemma 7.1 (Energy coercive below ground state [16]).

Let $(f,g)\in\dot{H}^{1}\times L^{2}$ and suppose that

$\min\\{\|f\|_{\dot{H}^{1}},\|g\|_{L^{2}}\\}\leqslant\|Q\|_{\dot{H}^{1}}\qquad\text{
and }\qquad 4\mathcal{E}_{z}(f,g)\leqslant\|Q\|_{\dot{H}^{1}}^{2}.$

Then we have the improved bounds

$\max\\{\|f\|_{\dot{H}^{1}}^{2},\|g\|_{L^{2}}^{2}\\}\leqslant
4\mathcal{E}_{Z}(f,g)\qquad\text{ and
}\qquad\mathcal{K}(f)+\big{\|}g+|f|^{2}\big{\|}_{L^{2}_{x}}\geqslant
4\mathcal{E}_{Z}(f,g)\frac{\|Q\|_{\dot{H}^{1}}^{2}-4\mathcal{E}_{Z}(f,g)}{2\|Q\|_{\dot{H}^{1}}^{2}}.$

###### Proof.

This is essentially contained in [16, Lemma 6.1], but we give a slightly more
direct proof by arguing directly from the properties of the ground state
function $Q$. More precisely, by rearranging the lower bound (7.4) and
applying the assumption
$4\mathcal{E}_{Z}(f,g)\leqslant\|Q\|_{\dot{H}^{1}}^{2}$ we have

$\big{(}\|g\|_{L^{2}}-\|Q\|_{\dot{H}^{1}}^{-1}\|f\|_{\dot{H}^{1}}^{2}\big{)}^{2}\leqslant\|Q\|_{\dot{H}^{1}}^{-2}\big{(}\|Q\|_{\dot{H}^{1}}^{2}-\|f\|_{\dot{H}^{1}}^{2}\big{)}^{2}$

and

$2\|f\|_{\dot{H}^{1}}^{2}\|Q\|_{\dot{H}^{1}}^{-1}\big{(}\|Q\|_{\dot{H}^{1}}-\|g\|_{L^{2}}\big{)}\leqslant\|Q\|_{\dot{H}^{1}}^{2}-\|g\|_{L^{2}}^{2}.$

In particular, a short computation shows provided
$4\mathcal{E}_{Z}(f,g)\leqslant\|Q\|_{\dot{H}^{1}}^{2}$ we have the
implication

$\min\\{\|f\|_{\dot{H}^{1}},\|g\|_{L^{2}}\\}\leqslant\|Q\|_{\dot{H}^{1}}\qquad\Longrightarrow\qquad\max\\{\|f\|_{\dot{H}^{1}},\|g\|_{L^{2}}\\}\leqslant\|Q\|_{\dot{H}^{1}}.$

In view of the implications (7.2) and (7.3), this completes the proof of the
first bound.

Finally, the second bound in the statement of the lemma follows by observing
that (7.4) and $\|f\|_{\dot{H}^{1}}\leqslant 4\mathcal{E}_{Z}(f,g)$ in fact
implies the slightly sharper bound

$\|f\|_{\dot{H}^{1}}^{2}\leqslant\frac{\|Q\|_{\dot{H}^{1}}^{2}}{2\|Q\|_{\dot{H}^{1}}^{2}-\|f\|_{\dot{H}^{1}}^{2}}4\mathcal{E}_{Z}(f,g)\leqslant\frac{\|Q\|_{\dot{H}^{1}}^{2}}{2\|Q\|_{\dot{H}^{1}}^{2}-4\mathcal{E}_{Z}(f,g)}4\mathcal{E}_{Z}(f,g)$

and hence

$\displaystyle\mathcal{K}(f)+\big{\|}g+|f|^{2}\big{\|}_{L^{2}_{x}}=4\mathcal{E}_{Z}(f,g)-\|f\|_{\dot{H}^{1}}^{2}$
$\displaystyle\geqslant
4\mathcal{E}_{Z}(f,g)\Big{(}\frac{\|Q\|_{\dot{H}^{1}}^{2}-4\mathcal{E}_{Z}(f,g)}{2\|Q\|_{\dot{H}^{1}}^{2}-4\mathcal{E}_{Z}(f,g)}\Big{)}$

which clearly suffices as $\mathcal{E}_{Z}(f,g)\geqslant 0$. ∎

######  Remark 7.2 (Characterising ground state condition).

As observed in [16], there are a number of ways to characterise the ground
state condition in Lemma 7.1. For instance, as long as the energy is below the
energy of the ground state, the sign of $\mathcal{K}(f)$ can be used to
determine the coercivity of the energy for both $\mathcal{E}_{Z}$ and
$\mathcal{E}_{NLS}$. This is well known in the case of the NLS [20]. In fact,
for any $f\in\dot{H}^{1}$ with $4\mathcal{E}_{NLS}(f)<\|Q\|_{\dot{H}^{1}}$ we
have the implications

$\mathcal{K}(f)>0\,\,\,\,\Longleftrightarrow\,\,\,\,0<\|f\|_{L^{4}}<\|Q\|_{L^{4}}\,\,\,\,\Longleftrightarrow\,\,\,\,0<\|f\|_{\dot{H}^{1}}<\|Q\|_{\dot{H}^{1}}\,\,\,\,\Longleftrightarrow\,\,\,\,\|f\|_{\dot{H}^{1}}<4\mathcal{E}_{NLS}(f)$
(7.5)

and

$\mathcal{K}(f)=0\qquad\Longleftrightarrow\qquad f=0.$ (7.6)

As in the proof of Lemma 7.1, the implications (7.5) and (7.6) follow from the
properties of the ground state solution, together with the identity

$4\mathcal{E}_{NLS}(f)=2\|f\|_{\dot{H}^{1}}^{2}-\|f\|_{L^{4}}^{4}=\|f\|_{\dot{H}^{1}}^{2}+\mathcal{K}(f).$

A similar characterisation holds in the case of the Zakharov equation. In fact
since $\mathcal{E}_{NLS}(f)\leqslant\mathcal{E}_{Z}(f,g)$, somewhat trivially,
the implications (7.5) and (7.6) also hold for any $(f,g)\in\dot{H}^{1}\times
L^{2}$ with $4\mathcal{E}_{Z}(f,g)<\|Q\|_{\dot{H}^{1}}^{2}$.

On the other hand, the ground state condition can also be characterised using
the wave data $g\in L^{2}$ [16, Lemma 6.1]. More precisely for any $(f,g)\in
H^{1}\times L^{2}$ with $4\mathcal{E}_{Z}(f,g)<\|Q\|_{\dot{H}^{1}}^{2}$ we
have

$\mathcal{K}(f)\geqslant
0,\qquad\Longleftrightarrow\qquad\|g\|_{L^{2}}<\|Q\|_{\dot{H}^{1}}\quad\Longleftrightarrow\quad\|g\|_{L^{2}}\leqslant\mathcal{E}_{Z}(f,g)$
(7.7)

and

$\mathcal{K}(f)=0\qquad\Longleftrightarrow\qquad f=0.$ (7.8)

The implications (7.7) and (7.8) follow directly from Lemma 7.1 together with
the Schrödinger counterparts (7.5) and (7.6). Note that the lack of strict
inequalities in (7.7) is a consequence of the fact that

$4\mathcal{E}_{Z}(0,g)=\|g\|_{L^{2}},\qquad
4\mathcal{E}_{Z}(f,0)=2\|f\|_{\dot{H}^{1}}^{2}$

and in particular, $f=0$ (or $g=0$) does not necessarily imply that
$\mathcal{E}_{Z}(f,g)=0$.

##  8\. A Palais-Smale Type Condition

In this section our goal is to apply the results obtained in the previous
section to show that show that any bounded sequence of solutions to the
Zakharov equation for which the dispersive norm $\|\cdot\|_{D}\to\infty$ and
lie below the ground state solution, must by precompact modulo translations.
This type of result is the key step in the proof of Theorem 1.6. The arguments
used in this section are largely adapted from [22, 20, 23]. The key point is
that if a mass/energy threshold exists (see Definition 1.5), then via the
profile decomposition in Theorem 6.1, together with the well-posedness theory
in Section 5, we can extract some compactness from any sequence of solutions
approaching the critical threshold.

######  Theorem 8.1 (Palais-Smale type condition).

Let $(M_{c},E_{c})$ be a mass/energy threshold with
$4E_{c}<\|Q\|_{\dot{H}^{1}}^{2}$. Suppose that $(u_{n},V_{n})\in
C(\mathbb{R},H^{1}\times L^{2})$ is a sequence of global solutions to (1.2)
such that

$u_{n}\in
L^{2}_{t,loc}W^{\frac{1}{2},4}_{x},\qquad\lim_{n\to\infty}\mathcal{E}_{Z}(u_{n},V_{n})=E_{c},\qquad\lim_{n\to\infty}\mathcal{M}(u_{n})=M_{c},\qquad\sup_{n\in\mathbb{N}}\|V_{n}(0)\|_{L^{2}_{x}}\leqslant\|Q\|_{\dot{H}^{1}}$
(8.1)

and

$\lim_{n\to\infty}\|u_{n}\|_{D([0,\infty))}=\lim_{n\to\infty}\|u_{n}\|_{D((-\infty,0])}=\infty.$
(8.2)

Then there exists $x_{n}\in\mathbb{R}$ such that the translated sequence
$(u_{n},V_{n})(0,x+x_{n})$ has a convergent subsequence in $H^{1}\times
L^{2}$.

###### Proof.

Define $(f_{n},g_{n})=(u_{n},V_{n})(0)\in H^{1}\times L^{2}$. The first step
is to verify that the sequence $(f_{n},g_{n})$ is bounded in $H^{1}\times
L^{2}$. Since
$\lim_{n\to\infty}4\mathcal{E}_{Z}(f_{n},g_{n})=4E_{c}<\|Q\|_{\dot{H}^{1}}$,
the assumption (8.1) implies that for all sufficiently large $n$ we have

$4\mathcal{E}_{Z}(f_{n},g_{n})<\|Q\|_{\dot{H}^{1}}^{2},\qquad\|g_{n}\|_{L^{2}}\leqslant\|Q\|_{\dot{H}^{1}}.$

Consequently, the variational properties of the ground state (see Lemma 7.1)
give the upper bounds

$\limsup_{n\to\infty}\|f_{n}\|_{\dot{H}^{1}}^{2}\leqslant
4E_{c},\qquad\limsup_{n\to\infty}\|g_{n}\|_{L^{2}}^{2}\leqslant 4E_{c}.$ (8.3)

Together with the assumed boundedness of the mass, we conclude that
$\sup_{n\in\mathbb{N}}\|(f_{n},g_{n})\|_{H^{1}\times L^{2}}<\infty$. In other
words the sequence $(f_{n},g_{n})$ is a bounded sequence in $H^{1}\times
L^{2}$.

We now apply the profile decomposition in Theorem 6.1 to the sequence
$(f_{n},g_{n})$, and obtain $J^{*}\in\mathbb{N}\cup\\{\infty\\}$ and for each
$1\leqslant j\leqslant J^{*}$ group elements
$\mathfrak{g}^{(j)}_{n}=\mathfrak{g}[t_{n}^{(j)},x_{n}^{(j)}]$ and profiles
$(f^{(j)},g^{(j)})\not=(0,0)$ such that (after replacing $(f_{n},g_{n})$ with
a suitable subsequence) for any $0\leqslant J\leqslant J^{*}$ we can write

$(f_{n},g_{n})=\sum_{j=1}^{J}\mathfrak{g}^{(j)}_{n}(f^{(j)},g^{(j)})+(w^{(J)}_{n},e^{(J)}_{n})$

where the profiles and errors $(w^{(J)}_{n},e^{(J)}_{n})$ satisfy the
conditions (i), (ii), and (iii) in the statement of Theorem 6.1. In
particular, we have the $L^{2}$ decoupling of the wave profiles

$\sum_{j=1}^{J}\|g^{(j)}\|_{L^{2}}^{2}+\limsup_{n\to\infty}\|e^{(J)}_{n}\|_{L^{2}}^{2}\leqslant\limsup_{n\to\infty}\|g_{n}(0)\|_{L^{2}}^{2}\leqslant
4E_{c}$ (8.4)

the decoupling of the energy

$\limsup_{n\to\infty}\Big{[}\sum_{j=1}^{J}\mathcal{E}_{Z}\big{(}\mathfrak{g}^{(j)}_{n}(f^{(j)},g^{(j)})\big{)}+\mathcal{E}_{Z}(w^{(J)}_{n},e^{(J)}_{n})\Big{]}\leqslant\lim_{n\to\infty}\mathcal{E}_{Z}(f_{n},g_{n})(0)=E_{c}$
(8.5)

and the decoupling of the Schrödinger mass

$\sum_{j=1}^{J}\mathcal{M}(f^{(j)})+\limsup_{n\to\infty}\mathcal{M}(w^{(J)}_{n})\leqslant\lim_{n\to\infty}\mathcal{M}(f_{n})=M_{c}.$
(8.6)

The above limits quickly imply that each profile has energy below the ground
state. More precisely, as $4E_{c}<\|Q\|_{\dot{H}^{1}}^{2}$, the $L^{2}$
decoupling (8.4) implies that for every $1\leqslant j\leqslant J$ we have
$\|g^{(j)}\|_{L^{2}}<\|Q\|_{\dot{H}^{1}}$, and for all sufficiently large $n$,
the error satisfies $\|e^{(J)}_{n}\|_{L^{2}}<\|Q\|_{\dot{H}^{1}}$. Hence the
implication (7.2) implies that

$\limsup_{n\to\infty}4\mathcal{E}_{Z}\big{(}\mathfrak{g}^{(j)}_{n}(f^{(j)},g^{(j)})\big{)}\geqslant\|g^{(j)}\|_{L^{2}}^{2}\qquad\text{
and
}\qquad\limsup_{n\to\infty}4\mathcal{E}_{Z}(w^{(J)}_{n},e^{(J)}_{n})\geqslant\limsup_{n\to\infty}\|e^{(J)}_{n}\|_{L^{2}}^{2}.$

Consequently, as each profile is non-zero, energies of each of the
(translated) profiles $\mathfrak{g}^{(j)}_{n}(f^{(j)},g^{(j)})$ must be
strictly positive. Together with (8.6), we conclude that both the profiles and
the error term have energy below the threshold. Namely we have the bounds

$0\leqslant\mathcal{M}(f^{(j)})\leqslant M_{c},\qquad
0<\limsup_{n\to\infty}\mathcal{E}_{Z}\big{(}\mathfrak{g}^{(j)}_{n}(f^{(j)},g^{(j)})\big{)}\leqslant
E_{c},\qquad\|g^{(j)}\|_{L^{2}}\leqslant 4E_{c},$ (8.7)

and

$\limsup_{n\to\infty}\mathcal{E}_{Z}\big{(}w^{(J)}_{n},e^{(J)}_{n}\big{)}\leqslant
E_{c},\qquad\limsup_{n\to\infty}\|e^{(J)}_{n}\|_{L^{2}}\leqslant 4E_{c}.$
(8.8)

Moreover, the $H^{1}$ decoupling of the profiles together with (8.3) gives the
upper bound

$\sum_{j=1}^{J}\|f^{(j)}\|_{H^{1}}^{2}+\limsup_{n\to\infty}\|w^{(J)}_{n}\|_{H^{1}}^{2}\leqslant\limsup_{n\to\infty}\|f_{n}\|_{H^{1}}^{2}\leqslant
M_{c}+4E_{c}.$ (8.9)

The next step is to evolve the profiles $(f^{(j)},g^{(j)})$ via the Zakharov
equation. More precisely, in view of the energy constraint (8.7) and the
assumption $4E_{c}<\|Q\|_{\dot{H}^{1}}^{2}$, we can apply Theorem 1.1 (if
$t_{n}=0$) or Theorem 3.5 (if $t_{n}\to\pm\infty$, potentially after
reflecting in time) and obtain global solutions $(u^{(j)},V^{(j)})\in
C(\mathbb{R};H^{1}\times L^{2})$ to (1.2) satisfying

$\lim_{n\to\infty}\big{\|}(u^{(j)},V^{(j)})(-t_{n})-\big{(}e^{-it_{n}\Delta}f^{(j)},e^{-it_{n}|\nabla|}g^{(j)}\big{)}\big{\|}_{H^{1}\times
L^{2}}=0$

with mass $\mathcal{M}(u^{(j)})=\mathcal{M}(f^{(j)})\leqslant M_{c}$ and
energy

$\begin{split}\mathcal{E}_{Z}(u^{(j)},V^{(j)})=\lim_{n\to\infty}\mathcal{E}_{Z}\big{(}e^{-it_{n}\Delta}f^{(j)},e^{-it_{n}|\nabla|}g^{(j)}\big{)}&=\lim_{n\to\infty}\mathcal{E}_{Z}\big{(}\mathfrak{g}^{(j)}_{n}(f^{(j)},g^{(j)})\big{)}\leqslant
E_{c},\\\
\max\big{\\{}\|u^{(j)}\|_{L^{\infty}_{t}\dot{H}^{1}_{x}},\|V^{(j)}\|_{L^{\infty}_{t}L^{2}_{x}}\big{\\}}&\leqslant
4\mathcal{E}_{Z}(u^{(j)},V^{(j)}).\end{split}$ (8.10)

We now translate in space-time, and define the translated solutions

$u_{n}^{(j)}(t,x)=u^{(j)}(t-t_{n},x-x_{n}),\qquad
V^{(j)}_{n}(t,x)=V^{(j)}(t-t_{n},x-x_{n})$

the key point being that we now have

$\lim_{n\to\infty}\big{\|}\big{(}u_{n}^{(j)},V_{n}^{(j)}\big{)}(0)-\mathfrak{g}^{(j)}_{n}(f^{(j)},g^{(j)})\|_{H^{1}\times
L^{2}}=0.$ (8.11)

We now consider two cases, either there exists a profile with energy precisely
$E_{c}$, or all profiles have energy strictly below $E_{c}$. In the former
case, the decoupling inequalities (8.4), (8.5), and (8.6), imply that there is
only one profile and moreover the error goes to zero in $\dot{H}^{1}\times
L^{2}$. Upgrading the convergence to $H^{1}\times L^{2}$ relies on the fact
that $(M_{c},E_{c})$ is a mass/energy threshold together with a short argument
via Theorem 5.2. In the later case where the initial profile has energy
strictly less than $E_{c}$, we show that each profile is uniformly bounded in
the dispersive norm $\|\cdot\|_{D}$, and moreover only interacts weakly.
Applying the stability result in Theorem 5.2 we eventually conclude that
$\limsup_{n\to\infty}\|u_{n}\|_{D}<\infty$ which contradicts our initial
assumption (8.2).

Case 1: $\mathcal{E}_{Z}(u^{(1)},V^{(1)})=E_{c}$.

Our goal is to show that if $\mathcal{E}_{Z}(u^{(1)},V^{(1)})=E_{c}$, then we
have

$\lim_{n\to\infty}\big{\|}(f_{n},g_{n})(x)-(f^{(1)},g^{(1)})(x-x_{n})\big{\|}_{H^{1}\times
L^{2}}=0.$ (8.12)

We start by observing that the decoupling of the energy (8.5) together with
(8.10) implies that for all $1<j\leqslant J$

$\mathcal{E}_{Z}(u^{(j)},V^{(j)})=0=\limsup_{n\to\infty}\mathcal{E}_{Z}(w^{(J)}_{n},e^{(J)}_{n}).$

Hence, as energies of all profiles lies below the ground state solution
(namely (8.7) holds), an application of Lemma 7.1 gives

$\big{\|}\big{(}u^{(j)},V^{(j)}\big{)}\big{\|}_{\dot{H}^{1}\times
L^{2}}^{2}\leqslant 4\mathcal{E}_{Z}(u^{(j)},V^{(j)})=0$

and

$\limsup_{n\to\infty}\|(w^{(J)}_{n},e^{(J)}_{n})\|_{\dot{H}^{1}\times
L^{2}}^{2}\leqslant
4\limsup_{n\to\infty}\mathcal{E}_{Z}(w^{(J)}_{n},e^{(J)}_{n})=0.$

In other words there is only one profile, and moreover the error goes to zero
in $\dot{H}^{1}\times L^{2}$.

This is almost what we want, but to obtain (8.12) we need to upgrade the
convergence to $H^{1}\times L^{2}$. To this end, suppose for the moment we
have $\mathcal{M}(u^{(1)})\leqslant M_{c}-\delta$ for some $\delta>0$. Since
$(M_{c},E_{c})$ is a mass/energy threshold, we then see that

$\|u^{(1)}\|_{D}\leqslant L(M_{c}-\delta,E_{c})<\infty.$

In particular, applying translation invariance, Theorem 3.1, and collecting
the above bounds/limits, we have

$\limsup_{n\to\infty}\Big{(}\|f_{n}\|_{H^{1}}+\|u^{(1)}_{n}\|_{S^{\frac{1}{2}}}\Big{)}\lesssim_{L,M_{c},E_{c}}1,\qquad\limsup_{n\to\infty}\|V^{(1)}_{n}\|_{L^{\infty}_{t}L^{2}_{x}}\leqslant
4E_{c}<\|Q\|_{\dot{H}^{1}}^{2}$

and

$\lim_{n\to\infty}\Big{(}\big{\|}f_{n}-u_{n}^{(1)}(0)\big{\|}_{\dot{H}^{1}}+\big{\|}g_{n}-V^{(1)}_{n}(0)\big{\|}_{L^{2}}\Big{)}=0.$

Thus in view of (5.3), an application of Theorem 5.2 with $F=G=0$ implies the
uniform dispersive bound $\limsup_{n\to\infty}\|u_{n}\|_{D}<\infty$ which
clearly contradicts the assumption (8.2). Therefore it is not possible for
both the dispersive norm to blow-up and the mass of the profile
$(u^{(1)},V^{(1)})$ to remain strictly less than $M_{c}$. In other words, we
must have the mass constraint $\mathcal{M}(u^{(1)})=M_{c}$. In view of (8.6),
we then conclude that $\limsup_{n\to\infty}\mathcal{M}(w^{(J)}_{n})=0$.
Consequently, unpacking the definition of $u^{(1)}_{n}$, we have the
$H^{1}\times L^{2}$ limit

$\lim_{n\to\infty}\big{\|}(u_{n},V_{n})(0,x)-\big{(}e^{-it_{n}^{(1)}\Delta}f^{(1)}(x-x_{n}^{(1)}),e^{-it_{n}^{(1)}|\nabla|}g^{(1)}(x-x_{n}^{(1)})\big{)}\big{\|}_{H^{1}\times
L^{2}}=0.$

To complete the proof of (8.12), it only remains to rule out the case
$t_{n}^{(1)}\to-\infty$ (the case $t_{n}^{(1)}\to\infty$ would then also be
excluded by time reversibility). We start by noting that since
$t_{n}^{(1)}\to-\infty$, the dispersive decay of the free Schrödinger and wave
evolutions implies that

$\lim_{n\to\infty}\big{\|}\mathfrak{g}^{(1)}_{n}\big{(}e^{it\Delta}f,e^{it|\nabla|}g\big{)}\big{\|}_{Y\times
Y([0,\infty))}=\lim_{n\to\infty}\big{\|}\big{(}e^{it\Delta}f,e^{it|\nabla|}g\big{)}\big{\|}_{Y\times
Y([-t_{n},\infty))}=0.$

Therefore, applying the bound (2.4), we have

$\displaystyle\limsup_{n\to\infty}\big{\|}\big{(}e^{it\Delta}f_{n},e^{it|\nabla|}g_{n}\big{)}\big{\|}_{Y\times
Y([0,\infty))}$
$\displaystyle\leqslant\limsup_{n\to\infty}\big{\|}\big{(}e^{it\Delta}f_{n},e^{it|\nabla|}g_{n}\big{)}-\mathfrak{g}^{(1)}_{n}(e^{it\Delta}f^{(1)},e^{it|\nabla|}g^{(1)})\big{\|}_{Y\times
Y([0,\infty))}$
$\displaystyle\qquad+\limsup_{n\to\infty}\|\mathfrak{g}^{(1)}_{n}(e^{it\Delta}f^{(1)},e^{it|\nabla|}g^{(1)})\|_{Y\times
Y([0,\infty))}$
$\displaystyle\lesssim\limsup_{n\to\infty}\|(f_{n},g_{n})-\mathfrak{g}^{(1)}_{n}(f^{(1)},g^{(1)})\|_{H^{1}\times
L^{2}}=0.$

Consequently, we can apply the small data theory in Theorem 1.7 to the
interval $[0,\infty)$, and conclude that
$\limsup_{n}\|u_{n}\|_{D([0,\infty))}<\infty$. But this contradicts the
assumption (8.2) and hence we cannot have $t_{n}^{(1)}\to\pm\infty$.

Case 2: $\mathcal{E}_{Z}(u^{(1)},V^{(1)})<E_{c}$.

As all profiles are non-zero and have positive energy, we conclude from (8.5),
(8.7), and (8.8) that we have the mass/energy bounds

$\sup_{1\leqslant j\leqslant J^{*}}\mathcal{M}(u^{(j)})\leqslant
M_{c},\qquad\sup_{1\leqslant j\leqslant
J^{*}}\mathcal{E}_{Z}\big{(}u^{(j)},V^{(j)}\big{)}\leqslant E_{c}-\delta$

for some $\delta>0$. In particular, in view of the definition of $E_{c}$, we
have the global dispersive bound

$\sup_{1\leqslant j\leqslant J^{*}}\|u^{(j)}\|_{D}\leqslant
L=L(M_{c},E_{c}-\delta)<\infty.$

To upgrade this bound to control over the $\underline{S}^{s}$ norm, we first
note that the bound (8.10) together with the conservation of mass gives

$\|u^{(j)}\|_{L^{\infty}_{t}H^{1}_{x}}\leqslant
M_{c}+4E_{c},\qquad\|V^{(j)}\|_{L^{\infty}_{t}L^{2}_{x}}\leqslant
4E_{c}<\|Q\|_{\dot{H}^{1}}.$

Consequently an application of Theorem 3.1 implies that for any
$\frac{1}{2}\leqslant s<1$ we have

$\|u^{(j)}\|_{\underline{S}^{s}}\lesssim_{s,L,E_{c},M_{c}}\|f^{(j)}\|_{H^{s}},\qquad\|V^{(j)}\|_{\underline{W}^{0}}\lesssim_{s,L,E_{c},M_{c}}\|g^{(j)}\|_{L^{2}}+\|f^{(j)}\|_{H^{\frac{1}{2}}}^{2}$
(8.13)

where the implied constants depend only on $s$, $L$, $E_{c}$, and $M_{c}$. We
now define

$\Psi^{(J)}_{n}=\sum_{j=1}^{J}u_{n}^{(j)}+e^{it\Delta}w^{(J)}_{n},\qquad\Phi^{(J)}_{n}=\sum_{j=1}^{J}V^{(j)}_{n}+e^{it|\nabla|}e^{(J)}_{n}.$

We claim that we have the properties:

  1. (i)

(Data agrees asymptotically) For every $0\leqslant J\leqslant J^{*}$ we have

$\lim_{n\to\infty}\big{\|}(f_{n},g_{n})-(\Psi^{(J)}_{n},\Phi_{n}^{(J)})(0)\big{\|}_{H^{1}\times
L^{2}}=0,\qquad\limsup_{n\to\infty}\|\Phi^{(J)}_{n}\|_{L^{\infty}_{t}L^{2}_{x}}\leqslant
4E_{c}<\|Q\|_{\dot{H}^{1}}.$

  2. (ii)

(Uniformly bounded in $\underline{S}^{s}\times\underline{W}^{0}$) For every
$\frac{1}{2}\leqslant s<1$ we have

$\sup_{0\leqslant J\leqslant
J^{*}}\limsup_{n\to\infty}\big{\|}(\Psi^{(J)}_{n},\Phi^{(J)}_{n})\big{\|}_{\underline{S}^{s}\times\underline{W}^{0}}\lesssim_{s,L,E_{c},M}1.$

  3. (iii)

(Approximate solution) We have

$\lim_{J\to
J^{*}}\limsup_{n\to\infty}\Big{\|}\mathcal{I}_{0}\Big{[}\Re(\Phi^{(J)}_{n})\Psi^{(J)}_{n}-\sum_{j=1}^{J}\Re(V^{(j)}_{n})u^{(j)}_{n}\Big{]}\Big{\|}_{S^{\frac{1}{2}}}=0$

and

$\lim_{J\to
J^{*}}\limsup_{n\to\infty}\Big{\|}|\nabla|\mathcal{J}_{0}\Big{[}|\Psi^{(J)}_{n}|^{2}-\sum_{j=1}^{J}|u_{n}^{(j)}|^{2}\Big{]}\Big{\|}_{W^{0}}=0.$

Assuming these properties hold for the moment, an application of the stability
result in Theorem 5.2 together with (5.3) implies that we have the global
bound $\limsup_{n\to\infty}\|u_{n}\|_{D}<\infty$. But this contradicts the
assumption that $\|u_{n}\|_{D([0,\infty))}\to\infty$ as $n\to\infty$. Hence
Case 2 cannot occur.

It only remains to verify the properties (i), (ii), and (iii). The property
(i) follows immediately from the construction of the profiles
$(f^{(j)},g^{(j)})$. To prove (ii), we start by observing that (8.13) together
with (8.4) and (8.9) implies that

$\sup_{0\leqslant J\leqslant
J^{*}}\Big{(}\sum_{j=1}^{J}\|u^{(j)}\|_{S^{s}}^{2}\Big{)}^{\frac{1}{2}}\lesssim_{s,L,E_{c},M_{c}}\sup_{0\leqslant
J\leqslant
J^{*}}\Big{(}\sum_{j=1}^{J}\|f^{(j)}\|_{H^{s}}^{2}\Big{)}^{\frac{1}{2}}\lesssim_{s,L,E_{c},M_{c}}1$

and

$\sup_{0\leqslant J\leqslant
J^{*}}\Big{(}\sum_{j=1}^{J}\|V^{(j)}\|_{W^{0}}^{2}\Big{)}^{\frac{1}{2}}\lesssim_{s,L,E_{c},M_{c}}\sup_{0\leqslant
J\leqslant
J^{*}}\Big{[}\Big{(}\sum_{j=1}^{J}\|g^{(j)}\|_{L^{2}}^{2}\Big{)}^{\frac{1}{2}}+\sum_{j=1}^{J}\|f^{(j)}\|_{H^{s}}^{2}\Big{]}\lesssim_{s,L,E_{c},M_{c}}1.$

Therefore, (i) and the bound (8.3), together with an application of the energy
inequality (2.5) and the bilinear estimate in Theorem 2.4 gives

$\displaystyle\sup_{0\leqslant J\leqslant
J^{*}}\limsup_{n\to\infty}\big{\|}\Psi^{(J)}_{n}\big{\|}_{\underline{S}^{s}}$
$\displaystyle\lesssim\sup_{0\leqslant J\leqslant
J^{*}}\Big{[}\limsup_{n\to\infty}\big{\|}\Psi^{(J)}_{n}(0)\big{\|}_{H^{s}}+\limsup_{n\to\infty}\Big{\|}\sum_{j=1}^{J}\Re(V^{(j)}_{n})u^{(j)}_{n}\Big{\|}_{N^{s}}\Big{]}$
$\displaystyle\lesssim\sup_{0\leqslant J\leqslant
J^{*}}\Big{[}\limsup_{n\to\infty}\big{\|}\Psi^{(J)}_{n}(0)\big{\|}_{H^{s}}+\Big{(}\sum_{j=1}^{J}\|V^{(j)}\|_{W^{0}}^{2}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{j=1}^{J}\|u^{(j)}\|_{S^{s}}^{2}\Big{)}^{\frac{1}{2}}\Big{]}$
$\displaystyle\lesssim_{s,L,E_{c},M_{c}}1$

where we used the fact that the norms $\|\cdot\|_{W^{0}}$ and
$\|\cdot\|_{\underline{S}^{s}}$ are translation invariant. Similarly, to bound
the wave contribution we have

$\displaystyle\sup_{0\leqslant J\leqslant
J^{*}}\limsup_{n\to\infty}\big{\|}\Phi^{(J)}_{n}\big{\|}_{\underline{W}^{0}}$
$\displaystyle\lesssim\sup_{0\leqslant J\leqslant
J^{*}}\Big{[}\limsup_{n\to\infty}\big{\|}\Phi^{(J)}_{n}(0)\big{\|}_{L^{2}}+\sum_{j=1}^{J}\Big{\|}\mathcal{J}_{0}\big{[}|\nabla||u^{(j)}|^{2}\big{]}\Big{\|}_{\underline{W}^{0}}\Big{]}$
$\displaystyle\lesssim\sup_{0\leqslant J\leqslant
J^{*}}\Big{[}\limsup_{n\to\infty}\big{\|}\Phi^{(J)}_{n}(0)\big{\|}_{L^{2}_{x}}+\sum_{j=1}^{J}\|u^{(j)}\|_{S^{s}}^{2}\Big{]}$
$\displaystyle\lesssim_{s,L,E_{c},M_{c}}1$

and hence (ii) follows.

Finally to prove (iii), we note that provided $s>\frac{1}{2}$, (ii) together
with (8.4), (8.9), and an application of Theorem 4.1 gives $\theta>0$ such
that

$\displaystyle\big{\|}\mathcal{I}_{0}\big{[}\Re(\Phi^{(J)}_{n}$
$\displaystyle-e^{it|\nabla|}e^{(J)}_{n})e^{it\Delta}w^{(J)}_{n}\big{]}\big{\|}_{S^{\frac{1}{2}}}+\big{\|}\mathcal{I}_{0}\big{[}\Re(e^{it|\nabla|}e^{(J)}_{n})\Psi^{(J)}_{n}\big{]}\big{\|}_{S^{\frac{1}{2}}}$
$\displaystyle\lesssim\Big{(}\|\Phi^{(J)}_{n}\|_{\underline{W}^{0}}+\|e^{(J)}_{n}\|_{L^{2}}\Big{)}\|e^{it\Delta}w^{(J)}_{n}\|_{Y}^{\theta}\|w^{(J)}_{n}\|_{H^{1}}^{1-\theta}+\|e^{it|\nabla|}e^{(J)}_{n}\|_{Y}^{\theta}\|e^{(J)}_{n}\|_{L^{2}}^{1-\theta}\|\Psi^{(J)}_{n}\|_{\underline{S}^{s}}$
$\displaystyle\lesssim_{s,L,E_{c},M_{c}}\|e^{it\Delta}w^{(J)}_{n}\|_{Y}^{\theta}+\|e^{it|\nabla|}e^{(J)}_{n}\|_{Y}^{\theta}.$

Therefore, the asymptotic decoupling provided by Lemma 6.2 and the fact that
the error vanishes in the dispersive norm $Y$ implies that

$\displaystyle\limsup_{J\to J^{*}}$
$\displaystyle\limsup_{n\to\infty}\Big{\|}\mathcal{I}_{0}\Big{[}\Re(\Phi^{(J)}_{n})\Psi^{(J)}_{n}-\sum_{j=1}^{J}\Re(V^{(j)}_{n})u^{(j)}_{n}\Big{]}\Big{\|}_{S^{\frac{1}{2}}}$
$\displaystyle\lesssim\limsup_{J\to
J^{*}}\limsup_{n\to\infty}\Big{(}\sum_{\begin{subarray}{c}1\leqslant
j,k\leqslant J\\\
j\not=k\end{subarray}}\big{\|}\mathcal{I}_{0}\big{[}\Re(V^{(j)}_{n})u^{(k)}_{n}\big{]}\big{\|}_{S^{\frac{1}{2}}}+\big{\|}\mathcal{I}_{0}\big{[}\Re(\Phi^{(J)}_{n}-e^{it|\nabla|}e^{(J)}_{n})e^{it\Delta}w^{(J)}_{n}\big{]}\big{\|}_{S^{\frac{1}{2}}}$
$\displaystyle\qquad\qquad\qquad\qquad+\big{\|}\mathcal{I}_{0}\big{[}\Re(e^{it|\nabla|}e^{(J)}_{n})\Psi^{(J)}_{n}\big{]}\big{\|}_{S^{\frac{1}{2}}}\Big{)}$
$\displaystyle\lesssim_{s,L,E_{c},M_{c}}\limsup_{J\to
J^{*}}\limsup_{n\to\infty}\Big{(}\sum_{\begin{subarray}{c}1\leqslant
j,k\leqslant J\\\
j\not=k\end{subarray}}\big{\|}\mathcal{I}_{0}\big{[}\Re(V^{(j)}_{n})u^{(k)}_{n}\big{]}\big{\|}_{S^{\frac{1}{2}}}+\|e^{it\Delta}w^{(J)}_{n}\|_{Y}^{\theta}+\|e^{it|\nabla|}e^{(J)}_{n}\|_{Y}^{\theta}\Big{)}=0.$

Similarly, by another application of Theorem 4.1 we see that

$\displaystyle\big{\|}|\nabla|\mathcal{J}_{0}\big{[}\Re\big{(}\overline{\Psi}^{(J)}_{n}e^{it\Delta}w^{(J)}_{n}\big{)}\big{]}\big{\|}_{W^{0}}+$
$\displaystyle\big{\|}|\nabla|\mathcal{J}_{0}\big{[}|e^{it\Delta}w^{(J)}_{n}|^{2}\big{]}\big{\|}_{W^{0}}$
$\displaystyle\lesssim\|e^{it\nabla}w^{(J)}_{n}\|_{Y}^{\theta}\|w^{(J)}_{n}\|_{H^{1}}^{1-\theta}\Big{(}\|\Psi^{(J)}_{n}\|_{\underline{S}^{s}}+\|w^{(J)}_{n}\|_{H^{1}}\Big{)}\lesssim_{s,L,E_{c},M_{c}}\|e^{it\nabla}w^{(J)}_{n}\|_{Y}^{\theta}$

and hence

$\displaystyle\limsup_{J\to J^{*}}$
$\displaystyle\limsup_{n\to\infty}\Big{\|}|\nabla|\mathcal{J}_{0}\Big{[}|\Psi^{(J)}_{n}|^{2}-\sum_{j=1}^{J}|u_{n}^{(j)}|^{2}\Big{]}\Big{\|}_{W^{0}}$
$\displaystyle\lesssim\limsup_{J\to
J^{*}}\limsup_{n\to\infty}\Big{(}\sum_{\begin{subarray}{c}1\leqslant
j,k\leqslant J\\\
j\not=k\end{subarray}}\big{\|}|\nabla|\mathcal{J}_{0}\big{[}\Re\big{(}\overline{u}^{(j)}_{n}u^{(k)}_{n}\big{)}\big{]}\big{\|}_{W^{0}}+\big{\|}|\nabla|\mathcal{J}_{0}\big{[}\Re\big{(}\overline{\Psi}^{(J)}_{n}e^{it\Delta}w^{(J)}_{n}\big{)}\big{]}\big{\|}_{W^{0}}$
$\displaystyle\qquad\qquad\qquad\qquad\big{\|}|\nabla|\mathcal{J}_{0}\big{[}|e^{it\Delta}w^{(J)}_{n}|^{2}\big{]}\big{\|}_{W^{0}}\Big{)}$
$\displaystyle\lesssim_{s,L,E_{c},M_{c}}\limsup_{J\to
J^{*}}\limsup_{n\to\infty}\Big{(}\sum_{\begin{subarray}{c}1\leqslant
j,k\leqslant J\\\
j\not=k\end{subarray}}\big{\|}|\nabla|\mathcal{J}_{0}\big{[}\Re\big{(}\overline{u}^{(j)}_{n}u^{(k)}_{n}\big{)}\big{]}\big{\|}_{W^{0}}+\|e^{it\nabla}w^{(J)}_{n}\|_{Y}^{\theta}\Big{)}=0.$

Consequently (iii) also holds. ∎

##  9\. Almost Periodic Solutions

In this section we give the construction of the critical elements (or almost
periodic solutions) in Theorem 1.6. The first step is to show that if
Conjecture 1.4 failed, then there must exist a mass/energy threshold with
energy below the ground state.

######  Lemma 9.1 (Existence of a mass/energy threshold).

Suppose Conjecture 1.4 failed. Then there exists a mass/energy threshold
$(M_{c},E_{c})$ with $4E_{c}<\|Q\|_{\dot{H}^{1}}^{2}$.

###### Proof.

Given $M>0$ we let

$E^{*}(M)=\sup\\{E>0\mid L(M,E)<\infty\\}$

where we recall that $L(M,E)$ is defined in (1.7). Note that if
$4E<\|Q\|_{\dot{H}^{1}}^{2}$ and $(u,V)\in\Omega(E)$ (see (1.6)) then Lemma
7.1 gives the $\dot{H}^{1}$ bound

$\|u\|_{L^{\infty}_{t}\dot{H}^{1}_{x}}^{2}\leqslant
4\mathcal{E}_{Z}(u,V)\leqslant 4E.$

In particular, Theorem 1.7 together with (1.8) implies that for fixed $M>0$
and all sufficiently small $E>0$, we have $L(M,E)<\infty$ and hence
$E^{*}(M)>0$. Moreover, by construction we have the implication

$E<E^{*}(M)\qquad\Longrightarrow\qquad L\big{(}M,E\big{)}<\infty.$ (9.1)

If Conjecture (1.4) failed, then there must exist some $M_{0}>0$ such that
$4E^{*}(M_{0})<\|Q\|_{\dot{H}^{1}}$. We now define

$E_{c}=E^{*}(M_{0}),\qquad M_{c}=\inf\\{M>0\mid E^{*}(M)=E^{*}(M_{0})\\}.$

In view of (1.8) and Theorem 1.7, we have global well-posedness and scattering
whenever the initial data satisfies

$\min\\{\|f\|_{L^{2}},\|f\|_{\dot{H}^{1}}\\}\ll_{\|f\|_{H^{1}}}1$

and hence $E_{c},M_{c}>0$. Moreover, as $L(M,E)$ is increasing in both $E$ and
$M$, the definition of $E^{*}$ together with (9.1) implies that if
$E<E_{c}=E^{*}(M_{0})$ then $L(M_{c},E)\leqslant L(M_{0},E)<\infty$. On the
other hand, if $M<M_{c}$, then by construction $E^{*}(M)$ is decreasing in $M$
and hence $E^{*}(M)>E^{*}(M_{0})=E_{c}$ and so another application of the
implication (9.1) gives $L(M,E_{c})<\infty$.

Therefore, to show that $(M_{c},E_{c})$ is a mass/energy threshold, it only
remains to prove that $L(M_{c},E_{c})=\infty$. As usual, this is consequence
of the (right) continuity of $L(M,E)$. More precisely, we claim that for any
$0<4E<\|Q\|_{\dot{H}^{1}}^{2}$ and $M>0$, there exists $C=C(M,E)>0$,
$\theta=\theta(M,E)>0$, and $\epsilon_{0}=\epsilon_{0}(M,E)>0$ such that for
any $0<\epsilon<\epsilon_{0}$ we have

$L(M,E)\leqslant L(M+\epsilon,E+\epsilon)\leqslant L(M,E)+C\epsilon^{\theta}.$
(9.2)

Clearly (9.2) implies that $L(M_{c},E_{c})=\infty$. The right continuity of
$L$ is a consequence of the stability result in Theorem 5.2 together with the
observation in [16] that $\lambda\mapsto\mathcal{E}_{Z}(\lambda
f,\lambda^{2}g)$ is an increasing function under the assumption that $(f,g)$
lie below the ground state $Q$. Roughly the point is that if we are at energy
$E+\epsilon$, then flowing back in $\lambda$ reduces the energy at which point
we can apply the definition of $L(M,E)$. Provided $\epsilon>0$ is sufficiently
small, applying Theorem 5.2 then bounds $L(M+\epsilon,E+\epsilon)$ in terms of
$L(M,E)$. Making this argument precise relies on the variational properties of
the ground state contained in Lemma 7.1.

We now turn to the details. The first inequality in (9.2) is immediate from
the definition. To prove the second, it is enough to consider the case
$L(M,E)<\infty$. Let $(u,V)\in\Omega(E+\epsilon)$ with
$\mathcal{M}(u)\leqslant M+\epsilon$. Our goal is to prove that
$\|u\|_{D}-L(M,E)\lesssim_{E,M}\epsilon^{\theta}$ with the implied constant
only depending on $E$ and $M$. Define

$(f,g)=(\lambda u,\lambda^{2}V)(0)\in H^{1}\times L^{2}$

where $0<\lambda<1$ is to be chosen later. Note that

$\mathcal{M}(u)-\mathcal{M}(f)=(1-\lambda^{2})\mathcal{M}(u)\gtrsim_{M}(1-\lambda).$

On the other hand, for any
$0<\epsilon\leqslant\frac{1}{8}(\|Q\|^{2}_{\dot{H}^{1}}-E)$, as
$4E<\|Q\|_{\dot{H}^{1}}^{2}$ Lemma 7.1 gives for any $0<\lambda\leqslant 1$
the lower bound

$\displaystyle\mathcal{K}\big{(}\lambda
u(0)\big{)}+\lambda^{4}\big{\|}V(0)+|u(0)|^{2}\big{\|}_{L^{2}_{x}}^{2}$
$\displaystyle\geqslant\lambda^{4}\Big{(}\mathcal{K}\big{(}u(0)\big{)}+\big{\|}V(0)+|u(0)|^{2}\big{\|}_{L^{2}_{x}}^{2}\Big{)}$
$\displaystyle\geqslant\lambda^{4}4\mathcal{E}_{Z}(u,V)\frac{\|Q\|_{\dot{H}^{1}}^{2}-4\mathcal{E}_{Z}(u,V)}{2\|Q\|_{\dot{H}^{1}}^{2}}\gtrsim_{E}\lambda^{4}\mathcal{E}_{Z}(u,V).$

Consequently the identity (7.1) implies that

$\displaystyle\mathcal{E}_{Z}(u,V)-\mathcal{E}_{Z}(f,g)=\int_{\lambda}^{1}a^{-1}\mathcal{K}\big{(}au(0)\big{)}+a^{3}\big{\|}V(0)+|u(0)|^{2}\big{\|}_{L^{2}_{x}}^{2}da\gtrsim_{E}(1-\lambda)\mathcal{E}_{Z}(u,V).$

In particular, the energy and mass decrease as $\lambda\to 0$, and hence
choosing $(1-\lambda)\approx_{E}\epsilon$ we have
$\mathcal{E}_{Z}(f,g)\leqslant E$ and $\mathcal{M}(f)\leqslant M$. Therefore,
letting $(\psi,\phi)\in\Omega(E)$ denote the corresponding global solution to
(1.2) with data $(\psi,\phi)(0)=(f,g)$ given by Theorem 1.1, we see that
$\|\psi\|_{D}\leqslant L(M,E)$. To conclude the corresponding dispersive bound
for $u$ we apply the stability result from Theorem 5.2. More precisely,
Theorem 3.1 and Lemma 7.1 implies that

$\|u(0)\|_{H^{1}}+\|\psi\|_{S^{\frac{1}{2}}}\lesssim_{E,M}1,\qquad\|\phi\|_{L^{\infty}_{t}L^{2}_{x}}\leqslant
2E^{\frac{1}{2}}<\|Q\|_{\dot{H}^{1}}$

(strictly speaking the implied constant here also depends on $L(M,E)$). On the
other hand, the choice of $(f,g)$ gives

$\|u(0)-\psi(0)\|_{\dot{H}^{1}}+\|V(0)-\phi(0)\|_{L^{2}}\lesssim_{E,M}1-\lambda\lesssim_{E,M}\epsilon.$

Hence provided $\epsilon>0$ is sufficiently small (depending only on $M$ and
$E$, albeit via $L(M,E)$), (5.3) together with Theorem 5.2 implies that $u\in
S^{\frac{1}{2}}$ with the bound

$\|u-\psi\|_{D}\lesssim\|u-\psi-e^{it\Delta}(u-\psi)(0)\|_{S^{\frac{1}{2}}}+\|(u-\psi)(0)\|_{H^{1}}\lesssim_{E,M}\epsilon^{\theta}.$

In other words, we have a constant $C=C(M,E)>0$ depending only on $E$ and $M$
(and $L(M,E)$) such that $\|u\|_{D}\leqslant L(M,E)+C\epsilon^{\theta}$.
Taking the sup over all $(u,V)\in\Omega(E+\epsilon)$ with $\mathcal{M}(u)=M$
we conclude the required bound. ∎

The proof of Theorem 1.6 is now an application of the Palais-Smale type
property together with the stability result obtained earlier.

###### Proof of Theorem 1.6.

Suppose Conjecture 1.4 failed. Applying Lemma 9.1 we would then conclude that
there exists a mass/energy threshold $(M_{c},E_{c})$ with
$4E_{c}<\|Q\|_{\dot{H}^{1}}^{2}$. In particular, as $L(M_{c},E_{c})=\infty$,
there exists a sequence $(u_{n},V_{n})\in\Omega(E_{c})$ such that

$\lim_{n\to\infty}\mathcal{E}_{Z}(u_{n},V_{n})=E_{c},\qquad\lim_{n\to\infty}\mathcal{M}(u_{n})=M_{c},\qquad\lim_{n\to\infty}\|u_{n}\|_{D}=\infty.$

Choose $t_{n}\in\mathbb{R}$ such that

$\lim_{n\to\infty}\|u_{n}\|_{D((-\infty,t_{n}])}=\lim_{n\to\infty}\|u_{n}\|_{D([t_{n},\infty))}=\infty.$

After replacing $u_{n}(t)$ with $u_{n}(t+t_{n})$, we may assume that $t_{n}=0$
for all $n\in\mathbb{N}$. Theorem 8.1 then implies that, up to a subsequence,
we have $(f_{c},g_{c})\in H^{1}\times L^{2}$ and $x_{n}\in\mathbb{R}^{4}$ such
that

$\lim_{n\to\infty}\big{\|}(u_{n},V_{n})(0,x+x_{n})-(f_{c},g_{c})(x)\big{\|}_{H^{1}\times
L^{2}}=0.$ (9.3)

Note that $\mathcal{E}_{Z}(f_{c},g_{c})=E_{c}$, $\mathcal{M}(f_{c})=M_{c}$,
and $\|g_{c}\|_{L^{2}}\leqslant\|Q\|_{\dot{H}^{1}}$. Hence applying Theorem
1.1 with data $(f_{c},g_{c})\in H^{1}\times L^{2}$ we obtain a global solution
$(\psi,\phi)\in C(\mathbb{R};H^{1}\times L^{2})$ to (1.2) with

$\psi\in
L^{2}_{t,loc}W^{\frac{1}{2},4}_{x},\qquad\mathcal{E}_{Z}(\psi,\phi)=E_{c},\qquad\mathcal{M}(\psi)=M_{c},\qquad\|\phi\|_{L^{\infty}_{t}L^{2}_{x}}\leqslant\|Q\|_{\dot{H}^{1}}.$
(9.4)

Moreover, the limit (9.3) together with the stability result in Theorem 5.2
gives

$\|\psi\|_{D((-\infty,0])}=\|\psi\|_{D([0,\infty))}=\infty.$ (9.5)

It only remains to verify that there exists $x(t):\mathbb{R}\to\mathbb{R}^{4}$
such that the orbit

$\big{\\{}(\psi,\phi)\big{(}t,x+x(t)\big{)}\,\,\big{|}\,\,t\in\mathbb{R}\big{\\}}$
(9.6)

is precompact in $H^{1}\times L^{2}$. To prove the existence of the
translations $x(t)$, one option is to argue abstractly as in [32].
Alternatively, and this is the approach we take here, we can give a concrete
definition111This observation was kindly communicated to us by Kenji
Nakanishi. of the translations by noting that $x(t)$ should essentially be the
“centre of mass” of the solution $(\psi,\phi)(t)$. To this end, we choose the
components $x_{j}(t)\in\mathbb{R}$ of $x(t)\in\mathbb{R}^{4}$ as

$\int_{x_{j}(t)}^{\infty}\int_{\mathbb{R}^{3}}\big{(}|\nabla\psi|^{2}+|\psi|^{2}+|\phi|^{2}\big{)}(t,y)dy^{\prime}dy_{j}=\frac{1}{2}\|(\psi,\phi)(t)\|_{H^{1}\times
L^{2}}^{2}$

where $y^{\prime}\in\mathbb{R}^{3}$ denotes the remaining spatial variables.
In other words, $x(t)$ is roughly the centre of the $H^{1}\times L^{2}$ mass
of $(\psi,\phi)$. Suppose (9.6) is not precompact. Then there exists sequence
$t_{n}\in\mathbb{R}$ and $C>0$ such that for all $n\not=m$ we have

$\big{\|}(\psi,\phi)\big{(}t_{n},x+x_{n}\big{)}-(\psi,\phi)\big{(}t_{m},x+x_{m}\big{)}\big{\|}_{H^{1}\times
L^{2}}\geqslant C$ (9.7)

where for ease of notation we let $x_{n}=x(t_{n})$. Applying Theorem 8.1 to
the sequence $(\psi,\phi)(t+t_{n})$, the properties (9.4) and (9.5) imply that
there exists $\tilde{x}_{n}\in\mathbb{R}^{4}$ and $(\tilde{f},\tilde{g})\in
H^{1}\times L^{2}$ such that up to a subsequence, the translated sequence
$(\psi,\phi)(t_{n},x+\tilde{x}_{n})$ converges to $(\tilde{f},\tilde{g})\in
H^{1}\times L^{2}$. In particular, after translating once more, we have

$\lim_{n\to\infty}\big{\|}(\psi,\phi)(t_{n},x+x_{n})-(\tilde{f},\tilde{g})(x+x_{n}-\tilde{x}_{n})\big{\|}_{H^{1}\times
L^{2}}=0.$ (9.8)

If $\sup_{n}|x_{n}-\tilde{x}_{n}|<\infty$, then after taking a further
subsequence we can assume $x_{n}-\tilde{x}_{n}$ converges. But then (9.8) is
clearly a contradiction to (9.7). Similarly, if $(\tilde{f},\tilde{g})=0$,
then again (9.8) contradicts (9.7). Thus, we may assume that
$(\tilde{f},\tilde{g})\not=0$, and writing the components of the vectors
$x_{n},\tilde{x}_{n}\in\mathbb{R}^{4}$ as $x_{n,j}$ and $\tilde{x}_{n,j}$,
there must exist some $1\leqslant j\leqslant 4$ such that
$|x_{n,j}-\tilde{x}_{n,j}|\to\infty$. Observe that, by our choice of $x(t)$
and (9.8), we have

$\displaystyle\lim_{n\to\infty}\int_{x_{n,j}-\tilde{x}_{n,j}}^{\infty}\int_{\mathbb{R}^{3}}\big{(}|\nabla\tilde{f}|^{2}+|\tilde{f}|^{2}+|\tilde{g}|^{2}\big{)}(y)\,dy^{\prime}dy_{j}$
$\displaystyle=\lim_{n\to\infty}\int_{x_{n,j}}^{\infty}\int_{\mathbb{R}^{3}}\big{(}|\nabla\psi|^{2}+|\psi|^{2}+|\phi|^{2}\big{)}(t_{n},y)dy^{\prime}dy_{j}$
$\displaystyle=\frac{1}{2}\lim_{n\to\infty}\big{\|}(\psi,\phi)(t_{n})\big{\|}_{H^{1}\times
L^{2}}^{2}=\frac{1}{2}\big{\|}(\tilde{f},\tilde{g})\big{\|}_{H^{1}\times
L^{2}}^{2}.$

But this is again a contradiction, as the left hand side converges to either
$0$ or $\|(\tilde{f},\tilde{g})\|_{H^{1}\times L^{2}}^{2}\not=0$. Therefore
(9.7) cannot hold, and hence the orbit (9.6) is precompact as claimed. ∎

## Acknowledgements

The author would like to thank Kenji Nakanishi for many helpful conversations
on the concentration compactness argument for dispersive PDE. Financial
support from the Marsden Fund Council grant 19-UOO-142, managed by Royal
Society Te Apārangi is gratefully acknowledged.

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# WiCV 2022: The Tenth Women In Computer Vision Workshop

Doris Antensteiner1, Silvia Bucci2, Arushi Goel3, Marah Halawa4, Niveditha
Kalavakonda5,
Tejaswi Kasarla6, Miaomiao Liu7, Nermin Samet8, Ivaxi Sheth9

1Austrian Institute of Technology, 2Polytechnic of Turin, 3University of
Edinburgh,
4Technical University of Berlin, 5Univeristy of Washington, 6University of
Amsterdam,
7Australian National University, 8Ecole des Ponts ParisTech, 9Mila-Quebec AI,
ETS Montreal
<EMAIL_ADDRESS>
###### Abstract

In this paper, we present the details of Women in Computer Vision Workshop -
WiCV 2022, organized alongside the hybrid CVPR 2022 in New Orleans, Louisiana.
It provides a voice to a minority (female) group in the computer vision
community and focuses on increasing the visibility of these researchers, both
in academia and industry. WiCV believes that such an event can play an
important role in lowering the gender imbalance in the field of computer
vision. WiCV is organized each year where it provides a) opportunity for
collaboration between researchers from minority groups, b) mentorship to
female junior researchers, c) financial support to presenters to overcome
monetary burden and d) large and diverse choice of role models, who can serve
as examples to younger researchers at the beginning of their careers. In this
paper, we present a report on the workshop program, trends over the past
years, a summary of statistics regarding presenters, attendees, and
sponsorship for the WiCV 2022 workshop.

##  1 Introduction

While excellent progress has been made in a wide variety of computer vision
research areas in recent years, similar progress has not been made in the
increase of diversity in the field and the inclusion of all members of the
computer vision community. Despite the rapid expansion of our field, females
still only account for a small percentage of the researchers in both academia
and industry. Due to this, many female computer vision researchers can feel
isolated in workspaces which remain unbalanced due to the lack of inclusion.

The Women in Computer Vision workshop is a gathering for both women and men
working in computer vision. It aims to appeal to researchers at all levels,
including established researchers in both industry and academia (e.g. faculty
or postdocs), graduate students pursuing a Masters or PhD, as well as
undergraduates interested in research. This aims to raise the profile and
visibility of female computer vision researchers at each of these levels,
seeking to reach women from diverse backgrounds at universities and industry
located all over the world.

There are three key objectives of the WiCV workshop. The first to increase the
WiCV network and promote interactions between members of this network, so that
female students may learn from professionals who are able to share career
advice and past experiences. A mentoring banquet is run alongside the
workshop. This provides a casual environment where both junior and senior
women in computer vision can meet, exchange ideas and even form mentoring or
research relationships.

The workshop’s second objective is to raise the visibility of women in
computer vision. This is done at both the junior and senior levels. Several
senior researchers are invited to give high quality keynote talks on their
research, while junior researchers are invited to submit recently published or
ongoing works with many of these being selected for oral or poster
presentation through a peer review process. This allows junior female
researchers to gain experience presenting their work in a professional yet
supportive setting. We strive for diversity in both research topics and
presenters’ backgrounds. The workshop also includes a panel, where the topics
of inclusion and diversity can be discussed between female colleagues.

Finally, the third objective is to offer junior female researchers the
opportunity to attend a major computer vision conference which they otherwise
may not have the means to attend. This is done through travel grants awarded
to junior researchers who present their work in the workshop via a poster
session. These travel grants allow the presenters to not only attend the WiCV
workshop, but also the rest of the CVPR conference.

##  2 Workshop Program

The workshop program consisted of 4 keynotes, 7 oral presentations, 34 poster
presentations, a panel discussion, and a mentoring session. As with previous
years, our keynote speakers were selected to have diversity among topic,
background, whether they work in academia or industry, as well as their
seniority. It is crucial to provide a diverse set of speakers so that junior
researchers have many different potential role models who they can relate to
in order to help them envision their own career paths.

The workshop schedule was as follows:

  * •

Introduction

  * •

Invited Talk 1: Marina Marie-Claire Höhne (Technische Universität Berlin,
Germany), Improving Explainable AI by using Bayesian Neural Networks

  * •

Oral Session 1

    * –

Jennifer Hobbs, Deep Density Estimation Based on Multi-Spectral Remote Sensing
Data for In-Field Crop Yield Forecasting

    * –

Ranya Almohsen , Generative Probabilistic Novelty Detection with Isometric
Adversarial Autoencoders.

    * –

Sarah A. Schneider , A Comparative Analysis in the Realm of Anomaly Detection

  * •

Invited Talk 2: Tatiana Tommasi(Polytechnic University of Turin, Italy),
Reliable 2D and 3D Models for Open World Applications

  * •

Poster Session (in person)

  * •

Invited Talk 3: Michal Irani ( Weizmann Institute of Science, Israel), “Mind
Reading”: Self-supervised decoding of visual data from brain activity

  * •

Oral Session 2

    * –

Maxine A Perroni-Scharf, Material Swapping for 3D Scenes using a Learnt
Material Similarity Measure

    * –

Mengyuan Zhang , Enriched Robust Multi-View Kernel Subspace Clustering.
Presenter

    * –

Asra Aslam , Detecting Objects in Less Response Time for Processing Multimedia
Events in Smart Cities

    * –

Sonam Gupta , RV-GAN: Recurrent GAN for Unconditional Video Generation

  * •

Invited Talk 4: Angela Yao (School of Computing at the National University of
Singapore), Capturing and Understanding 3D Hands in Action

  * •

Panel Discussion

  * •

Closing Remarks

  * •

Mentoring Session and Dinner (in person)

    * –

Speaker: Angela Dai (Technical University of Munich, Germany)

    * –

Speaker: Djamila Aouada (University of Luxembourg)

###  2.1 Hybrid Setting

This year, the organization has been slightly modified as CVPR 2022 was held
in hybrid setting. We had to make two plans, one for in-person attendance and
one for virtual attendance. We made sure to make the virtual WiCV workshop as
engaging and interactive as possible. Talks, oral sessions, and the panel was
shared via zoom for the virtual attendances. While the poster session was
repeated virtually a week after the conference, similar to the main conference
setting. We also provided online mentoring sessions held via Zoom, for mentors
and mentees that are able to attend only virtually.

##  3 Workshop Statistics

Originally, the first workshop for WiCV was held in conjunction with CVPR
2015. Since then, the participation rate and number quality of submissions to
WiCV have been steadily increasing. Following the examples from the editions
held in previous years [1, 2, 3, 4, 5], we were encouraged to collect the top
quality submissions into workshop proceedings. By providing oral and poster
presenters with the opportunity to publish their work in the conference’s
proceedings, we believe that the visibility of female researchers will be
further increased. This year, the workshop was held as a half day hybrid
gathering, the virtual setting was over gather-town and Zoom, and the in-
person setting was in New Orleans, Louisiana. Senior and junior researchers
were invited to present their work, and poster presentations are included as
already described in the previous Section 2.

The organizers for this year WiCV workshop are working in both academia and
industry from various institutions located in different time zones. Their
miscellaneous backgrounds and research areas have pledged the organizing
committee a diverse perspective. Their research interests in computer vision
and machine learning include video understanding, representation learning, 3D
reconstruction, domain adaptation, domain generalization, vision and language,
and semi/self-supervised learning in different application areas such as
vision for robotics, and healthcare.

Figure 1: WiCV Submissions. The number of submissions over the past years of
WiCV.

This year we had 70 high quality submissions from a wide range of topics and
institutions. It is on par with WiCV@CVPR21. The most popular topics were deep
learning architectures and techniques followed by video action and event
recognition , segmentation and shape analysis, and medical application. Over
all 70 submissions, 64 went into the review process. 7 papers were selected to
be presented as oral talks and appeared into the CVPR22 workshop’s
proceedings, and 34 papers were selected to be presented as posters. Within
the accepted submissions. The comparison with previous years is presented in
Figure 1. With the great effort of an interdisciplinary program committee
consisting of 41 reviewers, the submitted papers were evaluated and received
valuable feedback.

This year we kept WiCV tradition of last year’s workshops [1, 2, 3, 4, 5] in
providing grants to help the authors of accepted submissions participate in
the workshop. The grants covered the conference registration fees itinerary
(two ways flight), and two days accommodation for all the authors of accepted
submissions who requested funding.

The total amount of sponsorship this year is $62,000 USD with 10 sponsors,
reaching a very good target. In Figure 2 you can find the details with respect
to the past years.

Figure 2: WiCV Sponsors. The number of sponsors and the amount of sponsorship
for WiCV. The amount is expressed in US dollar (USD).

##  4 Conclusions

WiCV at CVPR 2022 has continued to be a valuable opportunity for presenters,
participants and organizers in providing a platform to bring the community
together. It continues to overcome the existing issue of gender balance
prevailing around us and we hope that it has played an important part in
making the community even stronger. It provided an opportunity for people to
connect from all over the world from their personal comforts. With a high
number of paper submissions and even higher number of attendees, we foresee
that the workshop will continue the marked path of previous years and foster
stronger community building with increased visibility, providing support, and
encouragement inclusively for all the female researchers in academia and in
industry.

##  5 Acknowledgments

First of all, we would like to thank our sponsors. We are very grateful to our
other Platinum sponsors: Toyota Research Institute, Google, and Apple. We
would also like to thank our Gold sponsor: Microsoft, and DeepMind; Silver
Sponsors: Meta, Disney Research, and Zalando ; Bronze sponsors: Meshcapade,
and Nvidia. We would also like to thank San Francisco Study Center as our
fiscal sponsor, which helped to process our sponsorships and travel awards. We
would also like to thank and acknowledge the organizers of previous WiCV,
without the information flow and support from the previous WiCV organizers,
this WiCV would not have been possible. Finally, we would like to acknowledge
the time and efforts of our program committee, authors, reviewers, submitters,
and our prospective participants for being part of WiCV network community.

##  6 Contact

Website: https://sites.google.com/view/wicvcvpr2022/home
E-mail<EMAIL_ADDRESS>
Facebook: https://www.facebook.com/WomenInComputerVision/
Twitter: https://twitter.com/wicvworkshop
Google group<EMAIL_ADDRESS>

## References

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