At two points $U_{s}$ and $U_{u}$ along the stable/unstable parts of the
branch, invariant manifolds $W^{\Sigma_{1}}(U_{s})$ (slow-stable) and
$W^{u}(U_{u})$ (unstable) were computed via the graph style of
parameterization up to order $7$ – these are shown embedded in $e-d$ space
alongside the branch in the top-right panel of Figure 11. The reduced,
$1$-dimensional dynamics $R$ on the model space $\Theta=[-1,1]$ are shown on
the left panel. The embedding $K$ transfers the dynamics over into the full
phase space, and hence the exact dynamics are obtained along the invariant
manifold in the vicinity of the travelling wave solution – or, in some
subcases, even up to the nearest exact coherent state.

Moreover, the obtained local dynamics about $U_{s}$ (resp. $U_{u}$) are
representative of all nearby solutions, since these are (locally) attracted
towards $W^{\Sigma_{1}}(U_{s})$ (resp. $W^{u}(U_{u})$) along the remaining,
stable directions at some rate $\mathcal{O}(e^{\beta t})$. Locally, around
$U_{s}$ ($U_{u}$), these ’stable directions’ are represented by the foliation
of Theorem 4.21, and the decay rate is specified by Theorem 4.21(ii) as
$\mathcal{O}(e^{\beta t})$ for any $\beta>\lambda_{2}$ with $\lambda_{2}$ the
least stable eigenvalue of
$\Sigma_{2}\subset\sigma(\mathsf{A}_{\mathsf{U}_{s}})$ (resp.
$\Sigma_{2}\subset\sigma(\mathsf{A}_{\mathsf{U}_{u}})$). In particular,
unstable manifold $W_{u}(U_{u})$ isolates the optimal (fastest) way for the
flow state to grow towards more complex solutions (the upper branch) or
descend back towards the laminar state. Its slow-stable counterpart at
$U_{s}$, $W^{\Sigma_{1}}(U_{s})$ isolates the slowest route for trajectories
to approach $U_{s}$.

With $\tilde{U}$ and $\hat{U}$ as in Figure 11, we have that that
$W^{u}(\tilde{U})=W^{\Sigma_{1}}(U_{s})$ on the arc $[U_{s},\tilde{U}]$, based
on the discussion in Section 6.1 (also, $W^{u}(U_{u})=W^{\Sigma_{1}}(\hat{U})$
on $[U_{u},\hat{U}]$). In general, we observe that slow stable manifolds along
stable parts of the branch and unstable manifolds along its unstable segments
contain nearby exact coherent states.

Whether or not these heteroclinic connections are actually observable along
the computed manifolds (c.f. Figure 11) comes down to the accuracy of the
numerical procedure at the distance of the nearby state. A relevant concept is
the fundamental domain of the manifold – the domain on which the power series
expansions satisfy the invariance equation sufficiently well. This can be
quantified upon defining an error over $\theta$, see Section 2.2.5 of [HCF+16]
for a few recommendations (assuming that full trajectories of the system can
be computed). Here, integration of full-fledged solutions would be unfeasible,
and hence we are forced to resort to a simplified approach, and use

$\mathrm{err}(\theta)=\left|\sum_{j\geq 2}\eta_{j}(\theta)\right|$

as the error norm, with $\eta_{j}$ as in (54). The fundamental domain
$\Theta_{f}$ is defined as

$\Theta_{f}:=\left\\{\theta\in\Theta\;\big{|}\;\mathrm{err}(\theta)<\mathrm{tol}\right\\}$

for some user-specified tolerance $\mathrm{tol}$ (set to $1.5\cdot 10^{-2}$
for the manifolds in Figure 11) that represents ’sufficient accuracy’ for the
system in question – usually determined via trial and error161616As the latter
suggests, this approach is rather fiddly, and hence, whenever possible, we
recommend using those outlined in [HCF+16] instead.. The manifolds displayed
in Figure 11 correspond to $K(\Theta_{f})$, and hence may or may not contain
nearby states depending on their distance from the fixed point (about which
the manifold was computed) – for instance, $W^{u}(U_{u})$ does not contain the
laminar state $U_{b}$. Note that this is solely a feature of the numerical
procedure, and not that of the full system.

The fundamental domain can be enlarged to some degree (with $\mathrm{tol}$
fixed), by increasing the order up to which the expansions are computed, but
the improvements will saturate as higher orders are reached. From then on, the
full extent of the manifolds could be recovered via numerical integration
(e.g. $U_{b}$ could be reached upon forward integrating $W^{u}(U_{u})$) –
however, we will not perform this procedure here due to its computational
costs.

###  6.6 Elastic instability of an Oldroyd-B fluid

For the final example, we consider pressure-driven viscoelastic channel flow
(in two spatial dimensions), motivated by the recent advancements in the field
[BPK22]. We assume that the evolution of the fluid flow is governed by the
Oldroyd-B model171717Note that we have not yet shown the existence of spectral
submanifolds for the case of the Oldroyd-B model. We expect them to retain
their existence due to the structure of (81), but this example is more meant
as an extrapolation towards possible future directions.

(81a) $\displaystyle
Re\Big{[}\partial_{t}u+\left(u\cdot\nabla\right)u\Big{]}+\nabla p=\beta\Delta
u+\nabla\cdot T+\begin{pmatrix}f\\\ 0\end{pmatrix},$ (81b)
$\displaystyle\nabla\cdot u=0,$ (81c)
$\displaystyle\partial_{t}T+\left(u\cdot\nabla\right)T+\frac{1}{Wi}T=T\cdot\nabla
u+\left(\nabla u\right)^{T}\cdot T+\frac{(1-\beta)}{Wi}\left((\nabla
u)^{T}+\nabla u\right)+\varepsilon\Delta T.$

Here $T$ is the ($2\times 2$) tensor representing the polymer stress field and
$\beta:=\nu_{s}/\nu\in[0,1]$ denotes the viscosity ratio where $\nu_{s}$ and
$\nu_{p}=\nu-\nu_{s}$ are the solvent and polymer contributions to the total
kinematic viscosity $\nu$. The viscoelastic fluid is assumed to occupy the
same (periodic) domain $\Omega=\left(0,2\pi/k\right)\times\left(-1,1\right)$
as in the prior examples. The dimensionless form of (81) was achieved via the
same route as in Section 6.2, upon implementing the Reynolds number (67) and
the Weissenberg number

$Wi=\tau Q,$

where $\tau$ is the polymer relaxation time and $Q$ is the volume flux (66).

Equation (81) is supplemented with non-slip boundary conditions on the
velocity field. To treat the stress tensor $T$, we set $\varepsilon=0$ at the
walls of the channel (i.e. $x_{2}=\pm 1$). In the streamwise ($x_{1}$)
direction, periodic boundary conditions are imposed on both $u$ and $T$, as in
Section 6.2.

The study of viscoelastic flows forms a very active field within the applied
fluids literature, which hence pose a multitude of novel challenges. In
particular, there are two new linear instabilities that persist even in the
inertialess ($Re=0$) limit: the center-mode instability discovered by Garg et
al. [GCK+18] and the wall-mode instability of Beneitez et al. [BPK22]. The
central question puzzling the field is whether these instabilities could be
dynamically connected to nearby (in parameter space) turbulent states known as
elastic [Ste21] and elasto-inertial turbulence [SDH+13]. While this question
remains inconclusive for the case of the center-mode instability, the wall-
mode has already produced promising results. In the work of Beneitez et al.
[BPK22], it is shown that this latter instability eventually evolves into what
is perceived to be elastic turbulence in plane-Couette flows, and similar
results have been confirmed for the case of channel flows by the same authors
[BPK23].

If this is indeed the case, the exact coherent states determined by the branch
emanating from the laminar bifurcation become of particular interest, as they
would serve to explain the (laminar-turbulent) transition process. The initial
segment of this branch, however, behaves in a very stiff manner as parameters
are varied, and hence standard techniques to initialize the branch
continuation (e.g., weakly nonlinear analysis) become almost impossible to
use.

Figure 12: The laminar state $\Psi_{b}$ of an Oldroyd-B fluid at
$(Wi,\beta,\varepsilon,Re)=(13.6,0.9,10^{-2},0)$. Colored contours correspond
to $\mathrm{tr}\,T_{b}$. The base velocity field $U_{b}$ remains almost equal
to that of Figure 5, with only minor adjustments due to the $\varepsilon\Delta
T$ term.

Here, we use spectral submanifolds (as in Section 6.1) as an alternative route
of initialization, which does not even rely on the existence of a nearby
laminar bifurcation. While doing so, we also explore an elementary transition
process between two segments of the branch connected to the wall-mode
instability of [BPK22]. Let us fix $(\beta,\varepsilon,Re)=(0.9,10^{-2},0)$, a
fairly standard set of parameters to be considered in the inertialess limit.
For the discretization, we choose $N_{2}=50$ Chebyshev and $N_{1}=10$ Fourier
modes. The laminar state $\Psi_{b}=(U_{b},T_{b})$ (Figure 12) bifurcates at
$Wi\approx 13.5$ for a critical streamwise wavenumber of $k=2.3$. The spectrum
of the linearized operator about $\Psi_{b}$ is shown in Figure 13.

Figure 13: Spectrum of the differential operator corresponding to (81)
linearized about the base state $\Psi_{b}$ shown on the complex plane,
partitioned according to (8). As in Figure 7, the spectrum obtained via the
standard, single Fourier mode eigenvalue problem is highlighted in orange.

We compute a parameterization $K$ for the unstable manifold of $\Psi_{b}$
slightly beyond the bifurcation, at $Wi=13.6$, to sixth order (Figure 14). The
resulting reduced model on $\Theta$ reads

(82a) $\displaystyle\dot{r}$
$\displaystyle=10^{-3}\left(0.249r-0.160r^{3}+0.0147r^{5}\right),$ (82b)
$\displaystyle\dot{\vartheta}$
$\displaystyle=0.119+0.0005r^{2}+0.000006r^{4}.$

The two nontrivial invariant sets of the reduced dynamics (82) are
$\Gamma_{1}(t)=\\{(r,\vartheta(t))\;|\;r=r_{1}\\}$ and
$\Gamma_{2}(t)=\\{(r,\vartheta(t))\;|\;r=r_{2}\\}$, with $r_{1}\approx 1.37$
and $r_{2}\approx 3.01$. The corresponding limit cycles of the full system are
obtained as $\Psi_{1}=K(\Gamma_{1})$ (stable) and $\Psi_{2}=K(\Gamma_{2})$
(unstable), shown via blue and red (respectively) in Figure 14. In place of
the usual measure of kinetic energy, we shall use

(83)
$\mathcal{T}:=\frac{\int_{\Omega}\mathrm{tr}\,T}{\int_{\Omega}\mathrm{tr}\,T_{b}}$

instead to visualize the embedding (Figure 14), representing the relative
elastic energy of the fluid state.

Figure 14: Parameterization of the unstable manifold
$W^{u}(\Psi_{b})=W^{\Sigma_{1}}(\Psi_{b})$ about the laminar state. (Left) The
model space $\Theta$ along with its reduced dynamics described by $R$. (Right)
The manifold $W^{u}(\Psi_{b})=\mathrm{im}(K)$ embedded in a three-dimensional
phase space spanned by $\mathcal{T}$ (83), $\mathrm{mwnv}$ (60) and
$\mathrm{svf}$ (61). In this example, we choose $\hat{x}_{2}$ in the
definition of $\mathrm{svf}$ to be the $10-$th Gauss-Lobatto point $x_{2,10}$,
as it shows far larger fluctuations than the centerline velocity. A single
trajectory traversing between the two limit cycles $\Gamma_{1}$ and
$\Gamma_{2}$ (resp. $\Psi_{1}$ and $\Psi_{2}$) is shown in orange on both
panels.

Next, we trace down the evolution of a single trajectory describing the
transition between the two limit cycles. For this, we take an initial
condition on $\Theta$ just inside $\Gamma_{2}$, with initial radius $r_{1}\ll
r_{0}<r_{2}$ and initial angle $\vartheta_{0}$, then integrate (82) to obtain
its orbit $\mathcal{O}^{+}(\\{r_{0},\vartheta_{0}\\})$. The resulting
trajectory $\mathcal{O}^{+}(\\{r_{0},\vartheta_{0}\\})$ and its counterpart
$K(\mathcal{O}^{+}(\\{r_{0},\vartheta_{0}\\}))$ in the full system are
highlighted in orange on the left and right panels of Figure 14 respectively.
Snapshots of its evolution, taken at four points denoted via distinctly
colored dots in Figure 14, are shown in Figure 15.

Figure 15: Snapshots along the evolution of the orange trajectory of Figure
14. Colored contours show $\mathrm{tr}\,(T-T_{b})$, the trace of the
perturbation polymer stress tensor. Lines correspond to level sets of the
perturbation stream function.

With the nontrivial states in hand, we are now in a position to initiate the
branch continuation routine from $Wi=13.6$. After fixing a Poincaré section
and imposing condition (69e) onto the states $\Psi_{1}$ or $\Psi_{2}$, we
reach the travelling wave formulation of Section 6.3. Figure 16 shows the
branch launched from $\Psi_{1}$ in the direction of increasing $Wi$. The shape
of the branch confirms the initial supercriticality of the Hopf bifurcation of
the laminar state at $Wi\approx 13.5$. A saddle-node bifurcation follows
shortly thereafter, resulting in an unstable branch of edge states reaching to
significantly lower $Wi$. Note that, due to its adequate convergence (see
caption of Figure 16), we could have initiated the branch continuation routine
directly from $\Psi_{2}$ as well to reach the same conclusions.

Figure 16:  Bifurcation diagram in the close vicinity of where the wall-mode
instability of [BPK22] emerges. The point at which the laminar state undergoes
a (supercritical Hopf) bifurcation is denoted by a purple square symbol
($Wi\approx 13.5$ for a critical wavenumber of $k=2.3$). Circles are centered
at the states predicted on the unstable manifold $W^{u}(\Psi_{b})$ (also shown
via the same colors in Figure 14); full dots denote the corresponding
converged states via the Newton-Raphson iteration of the continuation routine.

##  7 Discussion

In this work, we have shown the existence of a generalized class of invariant
manifolds (termed spectral submanifolds) about fixed points and periodic
orbits in the phase space $V$ of the Navier-Stokes equations. Notably, for any
$\beta\in\mathbb{R}$ partitioning the spectrum into two disjoint components
$\Sigma_{1}$ (containing elements with real parts larger than $\beta$) and
$\Sigma_{2}$ (resp. smaller), we have a local nonlinear continuation of the
spectral subspace $P_{\Sigma_{1}}V$ invariant under the semiflow, and a
foliation identifying synchronized trajectories that attract each other at a
rate of $O(e^{\beta t})$ (Theorem 4.18). Moreover, we have classified the
subscenarios in which uniqueness and a higher degree of smoothness of these
structures is to be expected.

Alongside this paper, a computational tool was developed to accommodate
spectral submanifold computations for the case of the Navier-Stokes and
Oldroyd-B systems. The algorithm obtains said manifolds via power series
expansions along the lines of the parameterization method [CFDlL03a, HCF+16],
built on the well-established and by now properly optimized code of [JTLH21,
JH22] from the engineering mechanics literature. Our adapted version is
available at https://github.com/gergelybuza/SSM_fluids, written in Matlab.

Through the course of a few examples within the framework of channel flows, we
have demonstrated some avenues in which spectral submanifolds could serve as
useful tools aiding our understanding of the dynamical structure of phase
space. The range of applications is by no means exhausted, and we anticipate
that further directions will be explored in the near future. Even if we merely
utilize spectral submanifolds for the computational benefits they provide as
reduced order models, the above examples already show promising qualities.
Indeed, all computations shown herein were done in a matter of minutes, with
peak memory consumption kept below $32$GB. The latter only occurs during the
assembly of the bilinear term (which could be stored as it is independent of
the problem at hand), otherwise the memory usage remained below $16$GB,
suitable for use on an average laptop. We acknowledge, nonetheless, that there
is still a lot to be improved.

The bottleneck, for the most part, is not computational speed, but rather the
immense memory requirements posed by storing all necessary polynomial
coefficients. There are a few recently-emerged techniques that attempt to
mitigate these costs via reducing the number of coefficients needed to be
stored. For instance, an alternative form of discretization is encouraged in
[GMJT22] for parabolic PDEs, the finite element method. While this seems
advantageous for most applications due to the sparsity of coefficient
matrices, finite elements generally require significantly higher resolutions
when compared to Chebyshev-Fourier methods in the specific context of fluid
flows (especially wall-bounded flows), which would likely diminish the
advantages of sparsity. Alternatively, differing routes to solving the
invariance equation could serve advantageous, as demonstrated by the works of
Roberts et al. [RMB14, Rob14] and Vizzaccaro and Opreni et al. [VOS+22,
OVTF23]. Another, perhaps more enticing route is to employ data-driven
techniques. These have proven to be incredibly efficient in engineering
mechanics [CAY+22, CAB+22], and now in plane-Couette flows as well [KCH22]. In
particular, the authors in [KCH22] were able to tackle systems even larger
than those considered here, facilitating a possible transition to the realm of
industrial-scale applications.

##  Appendix A Proof of Proposition 3.3

In this appendix, we show the existence and smoothness of a local semiflow
generated by the (perturbation) Navier-Stokes system (3) on $V$. We reiterate
that these results are not novel and the majority of them are already given in
the works of Weissler [Wei79, Wei80b].

We first recall a local existence result which in essence is given in [Wei79]
(Theorem 2.1; see also [Wei80b]). In anticipation of the smoothness
considerations in the latter half of this appendix, we shall state this result
in a more general setting, allowing for non-autonomous generators.

Non-autonomous semiflows are defined similarly to (16), with the dependence on
the initial time $t_{0}\geq 0$ included, i.e., (16a) and (16b) are replaced
with

$\displaystyle\varphi_{t_{0}}^{t_{0}}$ $\displaystyle=I,$
$\displaystyle\varphi_{s}^{t}\circ\varphi^{s}_{t_{0}}$
$\displaystyle=\varphi_{t_{0}}^{t},\qquad\forall t\geq s\geq t_{0}.$

######  Proposition A.1.

Let $\mathcal{A}_{U}$ be as in (4) and let $J_{t}:V\to L_{\sigma}^{p}$ be a
locally Lipschitz map with Lipschitz constants locally uniform with respect to
$t$, for $t\in[0,t^{*})$ ($t^{*}$ possibly infinite) and
$\frac{2d}{d+2}<p<\infty$. Assume further that $(t,v)\mapsto J_{t}(v)$ is
continuous and that $J_{t}(0)=0$. Then there exists a unique local semiflow
$\varphi_{t_{0}}^{t}$ on $V$ over the interval $[0,t^{*})$ defined by

(84)
$\varphi_{t_{0}}^{t}(u_{0})=e^{(t-t_{0})\mathcal{A}_{U}}u_{0}+\int_{t_{0}}^{t}e^{(t-s)\mathcal{A}_{U}}J_{s}\circ\varphi_{t_{0}}^{s}(u_{0})\,ds,\qquad
0\leq t_{0}<t<t^{*},$

such that the following hold:

  1. (i)

For each $T>0$ and $r>0$, there exists a $\delta>0$ such that
$\mathrm{Lip}_{V}\big{(}\varphi_{t_{0}}^{t}\,|\,\overline{B_{r}(0)}\big{)}\leq
C_{r,T}$ holds uniformly over $t_{0}\in[0,T]$ and $t\in[t_{0},t_{0}+\delta]$.

  2. (ii)

Fix $u_{0}\in V$. If
$\sup\\{t\,|\,u_{0}\in\mathrm{dom}(\varphi_{t_{0}}^{t})\\}=T<\infty$, then
$\lim_{t\to T}\|\varphi_{t_{0}}^{t}(u_{0})\|_{V}=\infty$.

  3. (iii)

$\mathrm{dom}(\varphi_{t_{0}}^{t})$ is open in $V$. Moreover, if
$u_{0}\in\mathrm{dom}(\varphi_{t_{0}}^{T})$, there is a neighborhood $O$ of
$u_{0}$ such that $O\subset\mathrm{dom}(\varphi_{t_{0}}^{T})$ and the
$\varphi_{t_{0}}^{t}$, for $t\in[t_{0},T]$, are uniformly Lipschitz on $O$.

######  Proof A.2.

Under these assumptions the map

$t\mapsto\int_{t_{0}}^{t}e^{(t-s)\mathcal{A}_{U}}J_{s}(u(s))\,ds$

is well defined and continuous for functions $u$ that are continuous.
Existence/uniqueness follows from a standard contraction mapping argument. Fix
$u_{0}\in V$, and pick $r_{0}>0$ such that
$u_{0}\in\overline{B_{r_{0}}(0)}\subset V$; $r>C_{0}r_{0}$ with $C_{0}$ as in
(13). Define

$X:=\left\\{\,u\in
C^{0}\big{(}[t_{0},T],\overline{B_{r}(0)}\big{)}\;\big{|}\;u(t_{0})=u_{0}\,\right\\},$

a closed subset of $C^{0}([t_{0},T],V)$ and hence a complete metric space with
the inherited (standard) metric. Define $\mathcal{F}:X\to C^{0}([t_{0},T],V)$
by

$(\mathcal{F}u)(t):=e^{(t-t_{0})\mathcal{A}_{U}}u_{0}+\int_{t_{0}}^{t}e^{(t-s)\mathcal{A}_{U}}J_{s}(u(s))\,ds,\qquad
t\in[t_{0},T].$

With $p$ as in the hypothesis of the theorem, we may choose $\alpha<1$ in
accordance with (12). Then, using (15) and the assumption that $J_{t}(0)=0$,
we have that

$\displaystyle\sup_{t\in[t_{0},T]}\|\mathcal{F}u(t)\|_{V}$ $\displaystyle\leq
C_{0}\max\\{e^{\omega(T-t_{0})},1\\}r_{0}$ (85)
$\displaystyle+C_{\alpha}\frac{(T-t_{0})^{1-\alpha}}{1-\alpha}\max\\{e^{\omega(T-t_{0})},1\\}\sup_{t\in[t_{0},T]}\mathrm{Lip}_{V\to
L^{p}_{\sigma}}\big{(}J_{t}\,|\,\overline{B_{r}(0)}\big{)}r.$

The assumptions ensure that $0<T<t^{*}$ may be chosen such that the right hand
side of (85) exists and is bounded by $r$, i.e. $\mathcal{F}X\subset X$.
Similarly,

$\sup_{t\in[t_{0},T]}\|\mathcal{F}u(t)-\mathcal{F}v(t)\|_{V}\leq\\\
C_{\alpha}\frac{(T-t_{0})^{1-\alpha}}{1-\alpha}\max\\{e^{\omega(T-t_{0})},1\\}\sup_{t\in[t_{0},T]}\mathrm{Lip}_{V\to
L^{p}_{\sigma}}\big{(}J_{t}\,|\,\overline{B_{r}(0)}\big{)}\sup_{t\in[t_{0},T]}\|u(t)-v(t)\|_{V},$

so the same choice of $T$ guarantees that $\mathcal{F}$ is a contraction.
Thus, there exists a unique fixed point of $\mathcal{F}$ in $X$, which, if
denoted by $\varphi^{t}_{t_{0}}(u_{0})$, constitutes our local semiflow on
$V$.

Statement (i) follows from an argument entirely analogous to Lemma 4.1; its
proof is therefore omitted.

Statement (ii) follows from standard ODE arguments. Suppose, conversely, that

$\sup_{t\in[t_{0},T)}\|\varphi_{t_{0}}^{t}(u_{0})\|_{V}<\infty.$

Then, using the integral equation, $\lim_{t\to T}\varphi^{t}_{t_{0}}(u_{0})$
exists and is bounded (by continuity). Repeating the argument above, we may
now find a solution on $[T,T+\varepsilon]$ with initial condition
$\varphi^{T}_{t_{0}}(u_{0})$, for some $\varepsilon>0$ depending solely on
$\|\varphi^{T}_{t_{0}}(u_{0})\|_{V}$. Since
$\varphi_{T}^{T+\varepsilon}\circ\varphi^{T}_{t_{0}}=\varphi_{t_{0}}^{T+\varepsilon}$,
this contradicts the maximality of $T$.

To see (iii), let $u_{0}\in\mathrm{dom}(\varphi_{t_{0}}^{T})$ and choose $r>0$
such that $\|\varphi^{t}_{t_{0}}(u_{0})\|_{V}\leq r/2$ for $t\in[t_{0},T]$.
Statement (i) gives $\delta>0$ such that
$\mathrm{Lip}_{V}\big{(}\varphi_{t_{0}}^{t}\,|\,\overline{B_{r}(0)}\big{)}\leq
C_{r,T}$ on $t\in[t_{0},t_{0}+\delta]$, with $t_{0}\in[0,T]$. Choose
$k\in\mathbb{N}$ such that $k\delta>T$. Then the set

$O:=\left\\{\,v\in V\;|\;C_{r,T}^{k}\|u_{0}-v\|_{V}<r/2\,\right\\}$

has the required properties with
$\mathrm{Lip}_{V}\big{(}\varphi_{t_{0}}^{t}\,|\,O\big{)}\leq C_{r,T}^{k}$ for
$t\in[t_{0},T]$.

######  Remark A.3.

If $J_{t}$ in Proposition A.1 is globally Lipschitz for all $t\geq 0$, then
combining statement (ii) with standard ODE arguments yields a globally defined
semiflow.

Next we show that under the additional hypothesis that $J_{t}\in
C^{\infty}(V,L^{p}_{\sigma})$, we obtain smoothness for the semiflow itself.
If we formally differentiate the equation defining $\varphi_{t_{0}}^{t}$ (84),
we get

(86)
$D\varphi_{t_{0}}^{t}(u_{0})[v_{0}]=e^{(t-t_{0})\mathcal{A}_{U}}v_{0}+\int_{t_{0}}^{t}e^{(t-s)\mathcal{A}_{U}}DJ_{s}(\varphi_{t_{0}}^{s}(u_{0}))\circ
D\varphi_{t_{0}}^{s}(u_{0})[v_{0}]\,ds,$

for some $v_{0}\in V$. The next proposition confirms that a semiflow
satisfying (86) exists, and moreover, that it coincides with the Fréchet
derivative of $\varphi_{t_{0}}^{t}$ at $u_{0}$.

######  Proposition A.4 (Theorem 2.2, [Wei79]).

Let $\mathcal{A}_{U}$ and $J_{t}$ be as in Proposition A.1 and let
$\varphi^{t}_{t_{0}}$ denote the resulting semiflow. Suppose further that
$J_{t}$ is continuously Fréchet differentiable infinitely many times, i.e.
$J_{t}\in C^{\infty}(V,L^{p}_{\sigma})$ for all $t$, with derivatives
depending continuously on $t$. Then each $\varphi^{t}_{t_{0}}$ is $C^{\infty}$
on its domain, with derivatives depending continuously on $t$.

######  Proof A.5.

Fix $u_{0}\in V$ and take
$K_{t}=DJ_{t}(\varphi^{t}_{t_{0}}(u_{0}))\in\mathcal{L}(V,L^{p}_{\sigma})$,
defined for $t\in[t_{0},T_{u_{0}})$, where $T_{u_{0}}$ is the maximal time of
existence associated to $u_{0}$. Then, Proposition A.1 applied with $K_{t}$
gives a family of linear maps $\theta_{t_{0}}^{t}$ that are defined by

$\theta_{u_{0};t_{0}}^{t}(v_{0})=e^{(t-t_{0})\mathcal{A}_{U}}v_{0}+\int_{t_{0}}^{t}e^{(t-s)\mathcal{A}_{U}}DJ_{s}(\varphi_{t_{0}}^{s}(u_{0}))[\theta_{u_{0};t_{0}}^{s}(v_{0})]\,ds$

for all times in the original interval $t\in[t_{0},T_{u_{0}})$, by linearity
of $K_{t}$.

Let $T\in(t_{0},T_{u_{0}})$ and choose $r>0$ such that
$\sup_{t\in[t_{0},T]}\|\varphi^{t}_{t_{0}}(u_{0})\|_{V}\leq r$. Pick
$\alpha<1$ according to (12) again and denote by

(87)
$k_{r}(t_{0},t):=C_{\alpha}\frac{(t-t_{0})^{1-\alpha}}{1-\alpha}\max\\{e^{\omega(t-t_{0})},1\\}\sup_{s\in[t_{0},t]}\mathrm{Lip}_{V\to
L^{p}_{\sigma}}\big{(}J_{s}\,|\,\overline{B_{r}(0)}\big{)},$

so that

(88)
$\left\|\int_{t_{0}}^{t}e^{(t-s)\mathcal{A}_{U}}DJ_{s}(\varphi_{t_{0}}^{s}(u_{0}))ds\right\|_{\mathcal{L}(V)}\leq
k_{r}(t_{0},t)\qquad\text{for }t\in[t_{0},T]$

with the above choice of $r$. Now choose $\tau>t_{0}$ such that
$k_{r}(t_{0},\tau)\leq k_{2r}(t_{0},\tau)<1$.

We next show Gateaux differentiability. Using Proposition A.1(iii), we have a
neighborhood $O$ of $u_{0}$ in $V$ such that
$O\subset\mathrm{dom}(\varphi_{t_{0}}^{\tau})$ and
$\sup_{t\in[t_{0},\tau]}\mathrm{Lip}_{V}\big{(}\varphi^{t}_{t_{0}}\,|\,O\big{)}\leq
C$. We claim that $\theta_{u_{0};t_{0}}^{t}=D^{G}\varphi_{t_{0}}^{t}(u_{0})$
for $t\in[t_{0},\tau]$, i.e. that

$\Delta_{h}(t):=\frac{\varphi^{t}_{t_{0}}(u_{0}+hv_{0})-\varphi^{t}_{t_{0}}(u_{0})}{h}-\theta_{u_{0};t_{0}}^{t}(v_{0})$

(with $h>0$ small enough such that $u_{0}+hv_{0}\in O$) converges to $0$ as
$h\to 0$ for any $v_{0}\in V$.

Writing out $\theta_{u_{0};t_{0}}^{t}$ and $\varphi^{t}_{t_{0}}$ using the
integral equations that define them, we get

(89)
$\Delta_{h}(t)=\int_{t_{0}}^{t}e^{(t-s)\mathcal{A}_{U}}\mu_{h}(s)ds+\int_{t_{0}}^{t}e^{(t-s)\mathcal{A}_{U}}DJ_{s}(\varphi^{s}_{t_{0}}(u_{0}))[\Delta_{h}(s)]ds,$

where

(90)
$h\mu_{h}(s)=J_{s}\circ\varphi^{s}_{t_{0}}(u_{0}+hv_{0})-J_{s}\circ\varphi^{s}_{t_{0}}(u_{0})-DJ_{s}(\varphi^{s}_{t_{0}}(u_{0}))[\varphi^{s}_{t_{0}}(u_{0}+hv_{0})-\varphi^{s}_{t_{0}}(u_{0})]\\\
=\int_{0}^{1}\Big{(}DJ_{s}(\varphi^{s}_{t_{0}}(u_{0})+\lambda(\varphi^{s}_{t_{0}}(u_{0}+hv_{0})-\varphi^{s}_{t_{0}}(u_{0})))-DJ_{s}(\varphi^{s}_{t_{0}}(u_{0}))\Big{)}[\varphi^{s}_{t_{0}}(u_{0}+hv_{0})-\varphi^{s}_{t_{0}}(u_{0})]d\lambda.$

Taking the supremum of (89) over $t\in[t_{0},\tau]$ and using (88), we obtain

$\sup_{t\in[t_{0},\tau]}\|\Delta_{h}(t)\|_{V}\leq(1-k_{r}(t_{0},\tau))^{-1}\sup_{t\in[t_{0},\tau]}\left\|\int_{t_{0}}^{t}e^{(t-s)\mathcal{A}_{U}}\mu_{h}(s)ds\right\|_{V}.$

That the right hand side converges to $0$ as $h\to 0$ follows from the latter
representation of $\mu_{h}$ in (90) and the dominated convergence theorem.

To conclude that $\varphi_{t_{0}}^{t}$ is continuously Fréchet differentiable
at $u_{0}$ (for $t\in[t_{0},\tau]$), it suffices to check that
$u\mapsto\theta_{u;t_{0}}^{t}$ is norm continuous (as a map
$V\to\mathcal{L}(V)$) on some open neighbourhood $O_{2}$ of $u_{0}$. Using
Proposition A.1(iii), take $O_{2}\subset O$ such that
$\sup_{t\in[t_{0},\tau]}\|\varphi_{t_{0}}^{t}(u)\|_{V}\leq 2r$ for all $u\in
O_{2}$. Now let $u_{n}\to u_{0}$ in $O_{2}$, and consider

$\displaystyle\theta_{u_{n};t_{0}}^{t}(v_{0})-\theta_{u_{0};t_{0}}^{t}(v_{0})$
$\displaystyle=\int_{t_{0}}^{t}e^{(t-s)\mathcal{A}_{U}}DJ_{s}(\varphi^{s}_{t_{0}}(u_{n}))[\theta_{u_{n};t_{0}}^{s}(v_{0})-\theta_{u_{0};t_{0}}^{s}(v_{0})]ds$
$\displaystyle+\int_{t_{0}}^{t}e^{(t-s)\mathcal{A}_{U}}\left(DJ_{s}(\varphi^{s}_{t_{0}}(u_{n}))[\theta_{u_{0};t_{0}}^{s}(v_{0})]-DJ_{s}(\varphi^{s}_{t_{0}}(u_{0}))[\theta_{u_{0};t_{0}}^{s}(v_{0})]\right)ds$

for any $v_{0}\in V$. Repeating the procedure above, we obtain

$(1-k_{2r}(t_{0},\tau))\sup_{t\in[t_{0},\tau]}\|\theta_{u_{n};t_{0}}^{t}(v_{0})-\theta_{u_{0};t_{0}}^{t}(v_{0})\|_{V}\leq\\\
\sup_{t\in[t_{0},\tau]}\left\|\int_{t_{0}}^{t}e^{(t-s)\mathcal{A}_{U}}\left(DJ_{s}(\varphi^{s}_{t_{0}}(u_{n}))[\theta_{u_{0};t_{0}}^{s}(v_{0})]-DJ_{s}(\varphi^{s}_{t_{0}}(u_{0}))[\theta_{u_{0};t_{0}}^{s}(v_{0})]\right)ds\right\|_{V},$

where again, the right hand side converges to $0$ by the dominated convergence
theorem, as $n\to\infty$.

So far, we have proven that $\varphi_{t_{0}}^{t}$ is continuously Fréchet
differentiable in a neighbourhood of $u_{0}$ for $t\in[t_{0},\tau]$. To extend
the result to the full time of existence, choose a sequence $\tau_{n}$
inductively such that $k_{2r}(\tau_{n-1},\tau_{n})<1$. By the local (in time)
uniformity assumption on the Lipschitz constant of $J_{t}$, the sequence
$\tau_{n}$ may be chosen such that $\tau_{N}\geq T$ for some finite $N$. Up to
possibly shrinking $\tau_{N}$, we may write
$\varphi_{t_{0}}^{T}=\varphi_{\tau_{N-1}}^{\tau_{N}}\circ\cdots\circ\varphi_{t_{0}}^{\tau_{1}}$
and therefore

$D\varphi_{t_{0}}^{T}(u_{0})=D\varphi_{\tau_{N-1}}^{\tau_{N}}(\varphi_{t_{0}}^{\tau_{N-1}}(u_{0}))\circ\cdots\circ
D\varphi_{t_{0}}^{\tau_{1}}(u_{0}),$

where the right hand side is continuous in $u_{0}$. Since $u_{0}$ was
arbitrary, the result follows for all of $V$.

The same argument applies to the semiflow given by

$(u_{0},v_{0})\mapsto(\varphi_{t_{0}}^{t}(u_{0}),D\varphi_{t_{0}}^{t}(u_{0})[v_{0}])$

on $V\times V$. Therefore, the above procedure can be iterated to yield
$C^{\infty}$ smoothness of the semiflow $\varphi_{t_{0}}^{t}$.

######  Corollary A.6.

In the setting of Proposition A.4, suppose that $J_{t}$ depends on a set of
$m$ parameters $\mu\in O\subset\mathbb{R}^{m}$ such that $J_{t}:V\times O\to
L^{p}_{\sigma},$ $(u,\mu)\mapsto J_{t}(u,\mu)$ is jointly smooth. Then the
resulting semiflow $\varphi_{t_{0}}^{t}$ determined by

$\varphi_{t_{0}}^{t}(u_{0},\mu)=e^{(t-t_{0})\mathcal{A}_{U}}u_{0}+\int_{t_{0}}^{t}e^{(t-s)\mathcal{A}_{U}}J_{s}(\varphi_{t_{0}}^{s}(u_{0},\mu),\mu)\,ds$

is also jointly smooth on $V\times O$, for each $t$ where defined. If
$J:(t,u,\mu)\mapsto J_{t}(u,\mu)$ is jointly $C^{\infty}$, then
$\varphi_{t_{0}}$ is jointly $C^{\infty}$ on
$(\mathcal{D}\cap((t_{0},\infty)\times V))\times O$.

######  Proof A.7.

The first assertion follows from the same argument as in Proposition A.4.

For the second part, we employ the same trick as [Hen81], Corollary 3.4.6. For
any $k>0$, define

$\tilde{\varphi}_{0}^{\tau}(u_{0},\tilde{\mu}):=e^{\tau\frac{1}{k}\mathcal{A}_{U}}u_{0}+\int_{0}^{\tau}e^{(\tau-s)\frac{1}{k}\mathcal{A}_{U}}\tilde{J}_{s}(\tilde{\varphi}_{0}^{s}(u_{0},\tilde{\mu}),\tilde{\mu})\,ds$

with $\tilde{\mu}=(\mu,k,t_{0})$ and
$\tilde{J}_{s}(u,\tilde{\mu}):=\frac{1}{k}J_{t_{0}+\frac{s}{k}}(u,\mu)$. We
have that

$\varphi_{t_{0}}^{t}(\,\cdot,\mu)=\tilde{\varphi}_{0}^{k(t-t_{0})}(\,\cdot,\tilde{\mu})$

by the uniqueness property of Proposition A.1. If $t>t_{0}$, we may take
$k:=(t-t_{0})^{-1}$ and obtain

$\varphi_{t_{0}}^{t}(\,\cdot,\mu)=\tilde{\varphi}_{0}^{1}(\,\cdot,(\mu,(t-t_{0})^{-1},t_{0})),$

the latter of which is jointly $C^{\infty}$ over $(\mu,t,t_{0})$ for $t>t_{0}$
by the first assertion.

The content of Lemma 3.1 was to verify that there exists $p$ for which $F:V\to
L^{p}_{\sigma}$ satisfies the assumptions of Propositions A.1 and A.4
($p\in(1,2)$ for $d=2$, $p\in(6/5,3/2]$ for $d=3$), so that there exists a
local (autonomous) semiflow $\varphi$ defined on an open subset of
$\mathbb{R}^{\geq 0}\times V$, such that $\varphi_{t}\in C^{\infty}(V)$ for
each $t$, where defined. Moreover, by Corollary A.6,
$\varphi:(t,u_{0})\mapsto\varphi_{t}(u_{0})$ is jointly $C^{\infty}$ over
$\mathcal{D}\cap(\mathbb{R}^{>0}\times V)$.

Trajectories of the above semiflow are solutions of the integral equation

(91)
$\varphi_{t}(u_{0})=e^{t\mathcal{A}_{U}}u_{0}+\int_{0}^{t}e^{(t-s)\mathcal{A}_{U}}F\circ\varphi_{s}(u_{0})\,ds,$

and their relation to the original equation (3) is apriori unknown. Results of
Kato and Fujita [KF62] lifted to a somewhat more general setting [Wei80a]
(permitting the use of operator $\mathcal{A}_{U}$ in place of
$\mathcal{A}_{0}$) yield the following. If $u_{0}\in V$, then the solution
constructed via the integral equation satisfies the original equation (3) as
an equation on $H$ for $t\in(0,T_{u_{0}})$. In particular,
$\varphi_{t}(u_{0})\in\mathrm{dom}_{2}(\mathcal{A}_{U})$ for
$t\in(0,T_{u_{0}})$. For $u_{0}\in\mathrm{dom}_{2}(\mathcal{A}_{U})$,
$\varphi_{t}(u_{0})$ is differentiable in time at $t=0$ and solves (3) over
$t\in[0,T_{u_{0}})$ in $H$.

In the two-dimensional case ($d=2$), global existence of solutions is well
known (see, for instance, [Rob01]) and therefore $\varphi:\mathcal{D}\to V$
extends to define a global semiflow on $\mathbb{R}^{\geq 0}\times V$.

If equation (17) is considered instead, we obtain a global semiflow
$\varphi^{\rho}:\mathbb{R}^{\geq 0}\times V\to V$ in both the two- and three-
dimensional cases, with $\varphi^{\rho}\in C^{\infty}(V)$ for $t\geq 0$
(jointly smooth for $t>0$), upon applying the preceding propositions with
$J_{t}\equiv F_{\rho}$, considering that $F_{\rho}:V\to L^{p}_{\sigma}$ is
globally Lipschitz (see Lemma 3.1 and Remark A.3).

## Acknowledgments

The author thanks Rich Kerswell and Edriss Titi for the useful discussions.
The author gratefully acknowledges the support of the Harding Foundation
through a PhD scholarship (https://www.hardingscholars.fund.cam.ac.uk).

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