# A Computational Study of Cluster Dynamics in Structural Lubricity: Role of
Cluster Rotation

Wai H. Oo  Mehmet Z. Baykara  Department of Mechanical Engineering, University
of California at Merced, Merced, California 95343, USA  Hongyu Gao
Corresponding author<EMAIL_ADDRESS>Department of Materials
Science and Engineering, Saarland University, Saarbrücken, Saarland 66123,
Germany

###### Abstract

We present a computational study of sliding between gold clusters and a highly
oriented pyrolytic graphite substrate, a material system that exhibits ultra-
low friction due to structural lubricity. By means of molecular dynamics, it
is found that clusters may undergo spontaneous rotations during manipulation
as a result of elastic instability, leading to attenuated friction due to
enhanced interfacial incommensurability. In the case of a free cluster, shear
stresses exhibit a non-monotonic dependency on the strength of the tip-cluster
interaction, whereby rigid clusters experience nearly constant shear stresses.
Finally, it is shown that the suppression of the translational degrees of
freedom of a cluster’s outermost-layer can partially annihilate out-of-plane
phonon vibrations, which leads to a reduction of energy dissipation that is in
compliance with Stokesian damping. It is projected that the physical insight
attained by the study presented here will result in enhanced control and
interpretation of manipulation experiments at structurally lubric contacts.

††preprint: APS/123-QED

##  I Introduction

Superlubricity describes a state of ultra-low friction between two surfaces
under relative motion [1], a particularly desirable property in mechanical
systems for reduction of energy consumption [2]. To distinguish the
superlubricity that arises due to structural incommensurability between
contacting surfaces (rather than due to a lubricant film or an externally
applied electric/magnetic field) [3], the term ‘structural lubricity’ was
introduced [4]. The physics underlying structural lubricity involves a
systematic annihilation of mean lateral forces at the interface due to
geometric incommensurability during sliding motion, which manifests in
exceptionally low interaction energy corrugations. A key resulting
characteristic of structural lubricity is that friction scales sub-linearly
with contact area [5, 6], i.e., $F\propto{A}^{\gamma}$, where exponent
$\gamma$=0.5 serves as an upper-bound, representing a scenario where one of
the contacting surfaces is amorphous. Although it has been generally accepted
to be very challenging to implement in practical applications, promising
progresses has been achieved in terms of preserving the structurally lubric
state at macro-scale contacts [7, 8] and under ambient conditions [9, 10].

A particular limitation of structural lubricity arises at extended contact
sizes. Specifically, when the elastic correlation length associated with the
slider is exceeded, interfacial structural defects in the form of dislocations
cause the sliding cluster to ‘pin’, leading to a breakdown of the ultra-low
friction state [11, 12]. In such cases, shear stresses will no longer depend
on interfacial alignment but saturate at a finite value instead, namely the
Peierls stress [11]. Even at small length scales where clusters are nominally
rigid, pinning can still take place at the edges of a cluster, in the absence
of pinning agents such as adsorbed layers [5] and unsaturated chemical bonds
[13]. Not surprisingly, lateral forces will rise at the contact lines where
the stress gradients are the highest, particularly so when those contact lines
are normal to the sliding direction. As a recent proof of this argument, Gao
and Müser [14] demonstrated that one possible origin of instabilities can be
from quasi-discontinuous dynamics of the moiré patterns (manifesting atom
normal displacements) near a propagating contact line, during the sliding of a
semi-infinite gold slab on a graphite substrate. Such sliding discontinuity
can be attributed to the symmetry breaking nature of a multi-layer gold
cluster with its (111) surface parallel to the graphite basal plane.
Consequently, the mean friction forces exhibit only weak dependence on sliding
speed, or namely Coulomb-type friction [15, 16].

It should be noted that variations of mean kinetic friction beyond statistical
errors are routinely observed in nanoscale experiments [17, 18], for which
various origins can exist, e.g. contact aging [19] and varying cluster
morphology [20]. Additionally, at a structurally well-defined interface free
of contaminants, it is well-known that friction can vary as a function of
sliding orientation as well as misfit angles [21, 20]. Finally, along a given
path, a cluster can rotate (i) in response to an external magnetic field [22]
or (ii) in the presence of a driving atomic force microscopy (AFM) tip, due to
weakly constrained in-plane rotational degree of freedom (RDoF), provided that
point-contact is formed between the driving tip and the cluster’s surface
[23]. Here comes a critical question: Does cluster rotation always result in a
lowering of the interfacial lattice mismatch and thus lead to an elevated
energy barrier, which would manifest as increased friction [23]? Despite the
critical nature of this question, discussions about in-plane rotational
friction and its coupling with translational friction at nanoscale contacts
are limited, mainly because the interplay is uneasy to clarify experimentally
or to be properly addressed by means of simplified theoretical models [15,
24].

Motivated by the question above, we present here molecular dynamics (MD)
simulations of nanoscale cluster manipulation performed on the representative
gold-graphite system, which has been shown to exhibit structural lubricity
under both ultra-high vacuum (UHV) and ambient conditions [20, 6, 9, 14]. In
particular, kinetic friction has been studied in the presence of spontaneous
cluster rotations, whereby the tip-cluster interaction has been modeled in the
form of a harmonic spring linked directly to the cluster’s topmost-layer.
Additionally, the effect of boundary control of atom translational degrees of
freedom on interfacial phonon vibrations (and thereby the modeled shear
stresses) has been analyzed via phonon spectra converted from Fourier-
transformed instantaneous forces. Overall, we expect this study to provide
physical insights into friction control at structurally lubric nanoscale
contacts.

##  II Methodology

The MD simulations model the dynamic sliding process of a circular gold
cluster on a highly oriented pyrolytic graphite (HOPG) substrate in a
contaminant-free environment. As the model in Fig. 1 shows, the two surfaces
forming the gold-graphite interface are structurally well-defined. The cluster
moves along the $x$-axis, dragged by a point mass via a spring that connects
to the center-of-mass (COM) of its topmost-layer. The stiffness of the driving
spring ($k_{\rm spr}$) along both the $x$\- and $y$-axis is the same, varying
in a range of 1.6-1600 N/m. The gold cluster (with a radius of 2.85 nm)
consists of six atomic layers in the (111) configuration stacked parallel to
the graphite basal plane. The three-layer graphite substrate has in-plane
dimensions of 11.5×11.6 nm2, by means of which periodic boundary conditions
are applied. The atomic interactions for isolated gold and graphite are
described by an EAM potential [25] and the AIREBO potential [26],
respectively. The interaction between the two materials is described by a
Morse potential [27]. The point mass does not directly interact with any atoms
in the system except by means of the spring attached to the top surface of the
gold cluster.

Figure 1: Snapshots of the MD model at rest. During sliding, the point mass
drags the cluster at a constant speed of 20 m/s along the $x$-axis. The spring
constants, both in the lateral ($x$) and the transverse ($y$) directions, vary
in the range of 1.6-1600 N/m.

The COM of the bottommost-layer of graphite is fixed throughout the
simulations, whereas the related atoms can still move freely according to
Newton’s equation of motion. The temperature of the system is maintained at
0.1 K through a Langevin thermostat applied to the mid-graphite layer in only
the $z$-direction. Such a cryogenic environmental control aims only to get rid
of the thermal noise so as to achieve an improved signal-to-noise ratio in the
simulated values. The topmost-layer of the gold cluster is set to be either
rigid or free: A ‘rigid’ layer signifies that the relative displacements of
the associated atoms are completely suppressed such that the whole layer acts
as a unit; while, a ‘free’ layer, together with the rest of the free atoms in
the system, indicates that no constraints have been put on the motion of the
atoms. In both scenarios, the RDoFs of the cluster are not constrained. The
point mass slides along the armchair direction of the graphite substrate,
i.e., the $x$-axis, at a constant speed of 20 m/s as default, unless specified
otherwise. The total sliding distance is over 500 nm, of which data points in
the steady-state (typically within the latter 400 nm of the total sliding
distance) are used in the calculation of mean values. In the normal direction,
the cluster and the graphite substrate interact only through adhesion without
any external loads applied. The simulation timestep is 1.0 fs. All MD
simulations are carried out using the open-source code LAMMPS [28].

The calculation of shear stresses ($\tau$) is conducted according to the
definition: ${\tau}={\langle}F{\rangle}/A$, where $F$ denotes the friction
force, $A$ the contact area, and $\langle...\rangle$ a time average.
Interfacial forces acting between the cluster and substrate are derived
directly from the interfacial interaction with statistical errors ($\Delta
O^{2}$) obtained from an integration of a time auto-correlation function
$C_{OO}(t)=\langle O(t^{\prime})O(t^{\prime}+t)\rangle$ via $\Delta
O^{2}=\frac{2}{\Gamma}\int_{0}^{\Gamma}C_{OO}(t){\rm d}t$, where $O$ denotes
an observable and $\Gamma$ the simulation time. The mean lateral forces
calculated as above are confirmed to be nearly identical to the mean elastic
restoring forces on the driving spring, as well as the mean resisting forces
from the bottommost-layer of the graphite substrate, thus proving mechanical
consistency. Moreover, mean lateral forces are comparable to those converted
from the dissipated energy adsorbed by the thermal reservoir via $P=\langle
F\rangle\cdot v$, where $P$ is the dissipated friction power that can be
calculated via $P=\gamma\sum_{i\in{\rm mid}}(m\langle v_{iz}^{2}\rangle-k_{\rm
B}T)$. In the latter expression, $\gamma$ is the damping coefficient and
$v_{iz}$ is the thermal velocity of atom $i$ in the $z$-direction. The contact
area is a summation of per-atom areas of gold atoms in direct contact with
graphite, which is estimated in terms of an average nearest-neighbor distance
($\sim$2.8 Å). Within the elastic limit, the contact areas vary only
marginally between different simulation runs.

##  III Results and discussion

###  III.1 Role of driving spring stiffness

When the driving spring stiffness $k_{\rm spr}$ is much stronger than the
interfacial interaction between the cluster and the substrate, the latter
dominates the effective contact stiffness ($k_{\rm eff}$) assuming that (i)
the interfacial interaction can also be represented by a virtual spring and
(ii) the two springs are connected in series [29]. The model spring constants
in the simulations span a wide range, with the lower bound in the proximity of
typical experimental values [30, 31], i.e., $\mathcal{O}(10~{}{\rm N/m})$. As
shown in Fig. 2, for the case of a rigid topmost cluster layer, MD simulations
reveal that the mean shear stresses are independent of the driving spring
stiffness. Within the framework of single-asperity contact, this result is
consistent with what is predicted by the Prandtl model in the regime
$\eta\leq$ 1 ($\eta=4\pi^{2}E_{0}/k_{\rm eff}a^{2}$), where the overdamped
point mass slides smoothly with ultra-low energy dissipation [31]. Please note
that detailed calculations based on the Prandtl model can be found in the
Appendix. In terms of this model, further increasing $k_{\rm spr}$ should have
a marginal effect on the mean shear stresses; whereas further decreasing
$k_{\rm spr}$ may cause mean shear stresses to rise at some point due to
enhanced elastic instability, provided that the corrugation potential
landscape of the substrate remains unchanged.

Figure 2: Shear stress of a gold cluster sliding on graphite as a function of
spring constant $k_{\rm spr}$. The cluster’s topmost layer is either free (red
squares) or constrained (black circles) to a rigid, yet freely rotating plane.
The driving spring moves along the armchair direction of graphite at a
constant speed of 20 m/s. Error bars smaller than the signal size are not
shown, same as in the following figures.

Unless the cluster grows epitaxially from the substrate such that the
interfacial structural mismatch is fairly small, shear stresses between a
physisorbed gold cluster and graphite depend only weakly on the sliding path.
Note that this should not be confused with those cases in which the angular
misfit between the two sliding objects varies while the sliding orientation
(associating with the wavelength of the substrate surface potential) does not.
In the latter scenario, the six-fold symmetry still holds. It should also be
discriminated from (quasi-)static calculations where atoms are in thermal
equilibrium at any given moment, from which the static friction and the
potential effect of contact aging can be deduced [32, 19], which are out of
the scope of this study. Although nominally rigidity holds the upper hand
against elastic deformation and dislocations, sliding-induced vibration of
individual atoms about their non-equilibrium steady-state positions
annihilates the accumulation of instantaneous forces systematically, thereby a
vanishing of mean lateral force (i.e., structural lubricity) arises. For a
finite-sized geometry, instability can be introduced from a propagating
contact line normal to the sliding direction as a result of the generically
asymmetric structure of the slider [14], although such an effect will decay
with cluster size in terms of the scaling argument [5, 6].

When the cluster is free, i.e., no constraints are applied except for the
springs, non-monotonic dependency of shear stresses on $k_{\rm spr}$ is
observed in the MD simulations, with a turning point located in the proximity
of $k_{\rm spr}$=500 N/m, as shown in Fig. 2. Please note that this particular
value itself may not be practically significant, but regardless, it brings
qualitative discernment into the effect of driving spring stiffness on shear
stresses in the structurally lubric regime from a computational point of view.
Beyond this critical value, shear stress decreases gradually with increasing
driving spring stiffness, which appears to be in line with predictions from
the Prandtl model in the underdamped regime ($\eta\gtrsim$ 1). In this case,
do the results presented in Fig. 2 mean that a ‘free’ cluster implies a
transition of the model driving spring system from an overdamped to an
underdamped regime? This will be discussed in a bit. Regardless, higher shear
stresses compared to those obtained from the rigid model are observed for free
clusters, indicating an increase in the amount of dissipated energy. Given
that the rigid cluster can be simply viewed as a free cluster with extended
dimensions, this observation obeys the scaling argument. Below the transition
value for the driving spring stiffness shown in Fig. 2, shear stress decreases
with decreasing driving spring stiffness, until becoming comparable to the
rigid model. Such a propensity is however counter-intuitive since weak springs
are more compliant and thereby supposedly lead to higher levels of energy
dissipation (due to increased elastic instability), in the absence of thermal
activation. Now, the question is why?

###  III.2 Spontaneous cluster rotation

From a fundamental point of view, individual atoms at a contact interface can
be viewed as the smallest possible units that form a multi-asperity contact,
which is analogous to what is depicted in a high-dimensional Frenkel-Kontorova
model [24]. As long as the perturbation of each atom at the sliding interface
is small enough (typically less than a unit lattice), the ensuing dissipation
can fall into the description of linear-response theory [33]. Along this line,
an interesting observation is that springs with high compliance promote
oscillatory cluster rotation. This is evidenced by the distribution of
relative rotation angles ($\Delta\alpha$) of a sliding cluster, as shown in
Fig. 3a, whose in-plane RDoF is highly inhibited when it is linked to a stiff
spring as opposed to a compliant one. It has been already known that even a
small angular mismatch between the sliding surfaces can alter the interfacial
commensurability [20]. This results in the damping coefficient (and thus, the
shear stress) being a rotation-angle-dependent function. The lower lateral
forces observed at a larger $|\Delta\alpha|$, as shown in Fig. 3b, can be
attributed to spontaneous, rotation-induced depinning of the cluster subject
to torques ($T$) that arise from the interfacial interaction, which is also in
alignment with the Arrhenius equation such that the interplay may change under
the effect of sliding speed [34].

Figure 3:  (a) Probability of in-plane rotation angles $\Delta\alpha$ of a
free cluster dragged by springs with stiffness values of $k_{\rm spr}$=16
(red) and 160 (blue) N/m. The inset shows a re-scaled view of the blue curve
in (a). (b) A representative relationship between lateral force $F$ and
instantaneous torque $T$ at $k_{\rm spr}$=16 N/m. Here,
$T\propto|\Delta\alpha|$ given that the magnitude of $\Delta\alpha$ is always
less than $\pi/6$.

It is important to note that experiments involving ‘tip-on-top’ manipulations
of gold clusters on graphite often result in spontaneous rotations of the
cluster (see Fig. 4 for two representative examples). Specifically, for this
structurally superlubric material system [9], the static friction between the
gold cluster and graphite substrate is significantly smaller than the static
friction between the tip apex and the cluster. Thus, when the AFM tip is
placed on top of a cluster (hence the phrase ‘tip-on-top’ [35]) and scans a
small area in contact mode, the tip will effectively drag the cluster to slide
laterally on the substrate. As described in Fig. 4, spontaneous rotations of
the cluster within the range of 0 – 75° are often observed during the
manipulations.

Figure 4:  AFM topography images of two representative gold clusters (a, c)
before and (b, d) after manipulation, showing spontaneous rotations of (b) 42°
and (d) 11°, respectively. The experiments were conducted with an applied
normal load of 0.0 nN, i.e., under adhesive interaction only.

The experimental observations of spontaneous rotations can be attributed to
two main reasons: (i) Experimentally, it is highly unlikely for the tip to be
placed directly on top of the cluster’s in-plane COM. Thus, a torque will
inevitably be introduced during the manipulation process and (ii) the adhesion
force between the tip and the cluster is not strong (mainly due to small
contact areas), such that the tip may occasionally slip during the reciprocal
sliding action. These slips may introduce torsional forces as well. These
experimental observations support the simulation results, where clusters are
spontaneously rotated and depinned due to torques. However, the range of in-
plane rotation angles for the experiments are significantly larger compared to
the simulations. This result is expected because the experimental lateral
stiffness of the tip-cluster contact $k_{\rm spr}$ is expected to be
significantly smaller (on the order of only a few N/m [31]) than the simulated
lateral stiffness $k_{\rm spr}$ values of 16 N/m and 160 N/m. It is important
to note here that the experimental shear stresses are within the range of 0.27
– 20 kPa (overlapping with the range of simulated shear stresses from Fig.
2.).

Theoretically, the dependence of friction on rotation angle can be
rationalized in terms of the moiré patterns formed at the contact interface,
which quantify the out-of-plane displacement (puckering) of the topmost
graphite layer underneath the sliding cluster. Characterization in terms of
atom normal displacements rather than an in-plane description [36] complies
with the idea from Tomlinson [16] such that the discontinuous dynamics of the
cluster are captured fairly well. This is also in agreement with the nature of
graphene as a 2D material, whereby the out-of-plane stiffness is much lower
than the in-plane stiffness. Structurally, the second-to-the-bottom atomic
layer of a gold (111) cluster breaks the mirror symmetry generically, which
would intensify interfacial atom vibration at the propagating contact line,
especially when it is normal to the sliding direction [14]. Compared to a
semi-infinite slab, instability of a circular cluster is located primarily at
its circumference, where the highest contrast in the moiré patterns are
visualized as shown in Fig. 5. This can be a reason why quasi-discontinuous
motion of the cluster is not distinctively observed in the current
simulations. Besides, rotated Moiré patterns can effectively disturb the
contact asperities from advancing in a collective manner.

Figure 5: Atomic normal displacement of the topmost-layer of the graphite
substrate at three consecutive instances during sliding motion from (a) to
(c). The relative shift of the cluster with respect to graphite in panels (b)
and (c), with respect to panel (a), are 1.7 and 2.6 Å, respectively. The in-
plane positions of the cluster are centered for straightforward comparison

###  III.3 Energy dissipation via phonons

Friction describes an energy dissipation process, in which mechanical energy
is converted into thermal energy via phonons as the main carrier. In a non-
equilibrium state, excessive phonons can be excited at the corresponding
washboard frequency ($\nu_{0}$) [37, 38] such that more energy is dissipated
via interfacial lattice vibration and thereby causes friction to rise [39].
Overall, Stokesian resistance results from the current model, as shown in Fig.
6, which manifests in quasi-elastic deformations of the contact bodies as well
as inevitable hysteresis at the sliding interface. However, outliers can also
be found at some particular sliding speeds with high magnitudes that exceed
typical statistical errors, e.g., the data point visualized by the red square
at 20 m/s in Fig. 6. To rationalize such an inconsistency, a vibrational
spectrum $g(\nu)$ of atoms at the contact interface is deducted from MD-
calculated instantaneous forces by means of fast Fourier transform (FFT) [40].
An equivalent result should also be obtained from the vibrational density of
states converted from a normalized mass-weighted velocity auto-correlation
function as demonstrated in an earlier simulation work [41].

Figure 6: Velocity-dependent shear stresses ($\tau$) of a free (red squares)
and a rigid (black circles) cluster. The dashed line is a linear fit to the
black data points. For both cases, $k_{\rm spr}$=1600 N/m.

In a phonon spectrum, the excited peaks, as shown in Fig. 7, indicate at what
frequencies the force signals are correlated, or in other words, energy is
dissipated. Discrete modes arise for sliding speeds such as 10 and 20 m/s,
with the corresponding major peaks located in the vicinity of the fractures of
the washboard frequency, i.e., $\frac{1}{3}\nu_{0}$ and $\frac{2}{3}\nu_{0}$.
As indicated by the dashed lines in Fig. 7, the washboard frequency is
calculated as $\nu_{0}=v/a$ [38], where $a$ = 1.42 Å is the side length of a
hexagonal graphite unit cell. The correlation lengths converted from those
peak frequencies, i.e., $2a$ and $3a$, reflect precisely the substrate lattice
spacing, which depicts a repetitive underdamped single-slip [42] sliding
motion of the cluster along the armchair direction of graphite. It suggests
that, although the sliding mode is ‘discontinuous’ by definition, structural
lubricity can be retained at a multi-asperity contact due to lattice
vibration. Resonance does not occur under such circumstances given the higher
natural frequency of an isolated cluster $\nu_{\rm nat}=1/2\pi\sqrt{k_{\rm
spr}/m}=0.244$ THz as well as an even higher resonant frequency of a sliding
cluster of $\nu_{\rm res}=1/2\pi\sqrt{(k_{\rm spr}+k_{\rm con})/m}$, where
$k_{\rm con}$ is the contact stiffness (an unknown that is much smaller than
$k_{\rm spr}$) and $m$ the mass of the cluster.

Figure 7: Fourier transfer of the instantaneous lateral forces ($F_{x,y,z}$)
for a free/rigid cluster sliding at 20 m/s (a-c) and a free cluster sliding at
10 or 20 m/s (d-f). The blue dashed lines represent the corresponding
washboard frequency $\nu_{0}$=0.141 THz at 20 m/s. The frequencies from 10 m/s
sliding are doubled to enable a direct comparison with those from 20 m/s
sliding. For all cases, $k_{\rm spr}$=1600 N/m.

While the phonon modes corresponding to the lateral forces ($F_{x}$) or the
transverse forces ($F_{y}$) appear similar between the free (red) and the
rigid (black) cluster, as shown in Fig. 7a and 7b, the partially differing
spectra of the normal forces ($F_{z}$) can be the origin of the friction
discrepancy. As shown in Fig. 7c, the missing frequency spike of a rigid
cluster (black) at around 0.19 THz in contrast to a free cluster (red)
indicates a lack of energy dissipated at this particular frequency. Within
this context, the overdamped normal forces in a rigid cluster may likely lead
to a reduction of friction forces due to the partly suppressed atomic DoFs. To
this end, one can interestingly conclude that forces in the normal direction
are inherently correlated with the lateral forces not only via their
mathematical mean (Amontons’ law) but also their vibrational frequencies.
Moreover, as shown in Fig. 7f, the spectrum intensity of the normal forces
from a faster sliding cluster (red) is more pronounced than a slower one
(black) across a certain range, which is an indication of enhanced energy
dissipation in line with the Stokesian correlation between friction and
sliding velocity.

##  IV Conclusions

In this work, MD simulations have been carried out to investigate kinetic
friction between a crystalline gold cluster and a graphite substrate in the
absence of contaminants. The gold cluster, as a nominally rigid body, slides
under dragging forces imposed by a virtual spring that models the tip-cluster
interaction in an AFM-based manipulation experiment. Beyond a critical value,
shear stresses of a free cluster decrease with increasing spring stiffness,
which is consistent with theoretical predictions from the Prandtl model in the
underdamped regime ($\eta\gtrsim$ 1). On the other hand, below the critical
value, shear stresses decrease with decreasing spring stiffness. This is
explained by spontaneous rotations of the cluster induced by elastic
instability, which effectively reduces the energy barrier to sliding via a
commensurability change; this is evidenced by an inverse friction-torque
relationship. Interestingly, such a non-monotonic dependency of shear stresses
on driving spring stiffness is not observed when the translational DoFs of
atoms in the cluster’s topmost-layer are suppressed (thus resulting in a more
‘rigid’ cluster or equivalently, a cluster with extended dimensions). In this
case, shear stresses are independent of driving spring stiffness, with values
less than one half of the maximum shear stress for a free cluster. Contour of
moiré patterns indicates that instability (in terms of normal displacements of
atoms in the topmost-layer of graphite) at the circumference of a round
cluster may not necessarily cause distinct discontinuous stick-slip motion.
Instead, a Stokesian relationship between shear stress and sliding speed
arises, whereby shear stresses increase linearly with increasing sliding
speed. Having said this, the asymmetric nature of the gold (111) surface
parallel to the graphite basal plane can occasionally lead to the observation
of high friction values in a free cluster that are beyond the range of
statistical errors. Approaching this observation by means of phonon
vibrational spectra, one can find that, unlike free clusters, rigid clusters
can suppress out-of-plane phonon excitation at certain frequencies, thus
leading to suppressed shear stresses.

It is the hope of the authors that this work will bring new physical insights
into friction variations observed during manipulation experiments performed on
nanoscale clusters on atomically flat substrates, and also inspire new ideas
with respect to friction control at small-scale, structurally lubric contacts.

###### Acknowledgements.

H.G. thanks Martin H. Müser for useful discussions. W.H.O. acknowledges
support from the National Science Foundation (NSF) in the form of a Graduate
Research Fellowship (GRF). This research was supported by the German Research
Foundation (DFG) under grant number GA 3059/2-1 and the NSF under award number
2131976.

*

##  Appendix A Predictions from the Prandtl model

Manipulation of a nanoscale cluster using an AFM tip [43, 9] can be idealized
within the framework of the Prandtl model [15, 44] provided that the rigidity
of the cluster exceeds the interfacial interaction strength such that the
cluster can be treated as a point mass. As a whole, the cluster slides over a
1D periodic potential landscape dragged by a virtual spring that models the
tip-cluster interaction. The associated effective stiffness ($k_{\rm eff}$)
integrates three contributions in series, namely: (i) the torsional stiffness
of the cantilever, (ii) the interaction strength between the tip apex and the
cluster, and (iii) the interaction strength between the cluster and the
substrate, in which the weakest one plays the predominant role [1]. Here (i)
and (ii) are combined as one assuming that the tip and the cluster are firmly
attached. In order to achieve ultra-low friction, $k_{\rm eff}$ should be
large enough such that ${\eta}=4{\pi}^{2}V_{0}/k_{\rm eff}a^{2}$ ($V_{0}$:
energy barrier, and $a$: lattice spacing) is less than 1, at which sliding is
smooth with extremely low energy dissipation [31]. If $k_{\rm eff}$ is too
small, the point mass will be underdamped due to elastic instability and
thereby single/multi-slip can occur under the influence of, e.g., normal load
and thermal activation [45, 31, 42].

The Prandtl model is a reduced-order model that provides qualitative insights
into small-scale friction, where the total potential energy of the system can
be expressed as

$V(x,t)=V_{0}{\rm cos}(\frac{2\pi{x}}{a})-\frac{1}{2}k_{\rm eff}(vt-x)^{2},$
(1)

where $x$ denotes the position of the point mass, $v$ the speed of the
dragging support, and $t$ the time. Without thermal activation, the point mass
starts to jump into the next potential well at a saddle point
$V^{\prime\prime}(x,t)=0$, where the position of the support relative to the
point mass determines the amount of energy to be dissipated. Dynamics of the
point mass can be described by a Langevin equation, written as

$m\ddot{x}+m{\gamma}\dot{x}+k(x-vt)={\rm sin}x,$ (2)

where $m$ is the mass and $\gamma$ the damping coefficient. To simplify the
problem, parameters in this calculation are with reduced units with assigned
values, i.e., $m=V_{0}=2\pi/a=1$ and $\gamma=0.1$, referring to a recent work
by Müser [46] where more detailed explanations shall be found. Kinetic
friction $F_{\rm kin}$ reported in the following denotes a mean value of
lateral forces over numerous stick-slip periods at steady-state.

By substituting the chosen values into the expression of $\eta$, one can
readily clarify the simple inverse relationship between $\eta$ and $k_{\rm
eff}$, i.e., $\eta$ = 1/$k_{\rm eff}$. Note, the $ad~{}hoc$ damping
coefficient $\gamma=0.1$ is chosen to ensure that the sliding object is in an
underdamped regime with single stick-slip motion being captured. To this end,
the more compliant the spring, the higher the elastic instability of the point
mass and thereby more elastic energy is stored during the ‘stick’ phase, which
will be released subsequently while the object slips into the next potential
well. As shown in Fig. 8, kinetic friction increases with $\eta$ across a wide
range of sliding speeds. Given the current range of spring stiffnesses, a
plateau that corresponds to a continuous sliding in an overdamped regime
(i.e., when $\eta\leq$1) has not yet been reached. In an athermal environment,
resistance of the point mass in response to the dragging forces falls in the
linear response regime [33], which leads to a linear dependence of kinetic
friction on the sliding velocity, or in other words, Stokesian friction. At
low sliding velocities, the point mass is in thermal equilibrium at a given
moment, beyond which a transition from a linear to a logarithmic dependence
can take place when thermal effects kick in [46].

Figure 8: Mean kinetic friction $F_{\rm kin}$ predicted from the Prandtl model
as functions of $\eta$ (inversely related to the effective spring stiffness
$k_{\rm eff}$) and sliding velocity $v$.

## References

  * Baykara  _et al._ [2018] M. Z. Baykara, M. R. Vazirisereshk, and A. Martini, Appl. Phys. Rev. 5, 041102 (2018).
  * Holmberg and Erdemir [2017] K. Holmberg and A. Erdemir, Friction 5, 263 (2017).
  * Shinjo and Hirano [1993] K. Shinjo and M. Hirano, Surf. Sci. 283, 473 (1993).
  * Müser [2004] M. H. Müser, EPL 66, 97 (2004).
  * Müser  _et al._ [2001] M. H. Müser, L. Wenning, and M. O. Robbins, Phys. Rev. Lett. 86, 1295 (2001).
  * Dietzel  _et al._ [2013] D. Dietzel, M. Feldmann, U. D. Schwarz, H. Fuchs, and A. Schirmeisen, Phys. Rev. Lett. 111, 235502 (2013).
  * Liu  _et al._ [2012] Z. Liu, J. Yang, F. Grey, J. Z. Liu, Y. Liu, Y. Wang, Y. Yang, Y. Cheng, and Q. Zheng, Phys. Rev. Lett. 108, 205503 (2012).
  * Berman  _et al._ [2015] D. Berman, S. A. Deshmukh, S. K. R. S. Sankaranarayanan, A. Erdemir, and A. V. Sumant, Science 348, 1118 (2015).
  * Cihan  _et al._ [2016] E. Cihan, S. İpek, E. Durgun, and M. Z. Baykara, Nat. Commun. 7, 12055 (2016).
  * Deng  _et al._ [2018] H. Deng, M. Ma, Y. Song, Q. He, and Q. Zheng, Nanoscale 10, 14314 (2018).
  * Sharp  _et al._ [2016] T. A. Sharp, L. Pastewka, and M. O. Robbins, Phys. Rev. B 93, 121402 (2016).
  * Monti and Robbins [2020] J. M. Monti and M. O. Robbins, ACS Nano 14, 16997 (2020).
  * Dietzel  _et al._ [2017] D. Dietzel, J. Brndiar, I. Štich, and A. Schirmeisen, ACS Nano 11, 7642 (2017).
  * Gao and Müser [2022] H. Gao and M. H. Müser, Front. Chem. 10, 935008 (2022).
  * Prandtl [1928] L. Prandtl, J. Appl. Math. Mech. 8, 85 (1928).
  * Tomlinson [1929] G. A. Tomlinson, Lond. Edinb. Dublin Philos. Mag. J. Sci. 7, 905 (1929).
  * Dietzel  _et al._ [2008] D. Dietzel, C. Ritter, T. Mönninghoff, H. Fuchs, A. Schirmeisen, and U. D. Schwarz, Phys. Rev. Lett. 101, 125505 (2008).
  * Dietzel  _et al._ [2018] D. Dietzel, A. S. de Wijn, M. Vorholzer, and A. Schirmeisen, Nanotechnology 29, 155702 (2018).
  * Feldmann  _et al._ [2014] M. Feldmann, D. Dietzel, H. Fuchs, and A. Schirmeisen, Phys. Rev. Lett. 112, 155503 (2014).
  * de Wijn [2012] A. S. de Wijn, Phys. Rev. B 86, 085429 (2012).
  * Dienwiebel  _et al._ [2004] M. Dienwiebel, G. S. Verhoeven, N. Pradeep, J. W. M. Frenken, J. A. Heimberg, and H. W. Zandbergen, Phys. Rev. Lett. 92, 126101 (2004).
  * Cao  _et al._ [2022] X. Cao, A. Silva, E. Panizon, A. Vanossi, N. Manini, E. Tosatti, and C. Bechinger, Phys. Rev. X 12, 021059 (2022).
  * Filippov  _et al._ [2008] A. E. Filippov, M. Dienwiebel, J. W. M. Frenken, J. Klafter, and M. Urbakh, Phys. Rev. Lett. 100, 046102 (2008).
  * Frenkel and Kontorova [1939] J. Frenkel and T. Kontorova, Izv. Akad. Nauk, Ser. Fiz. 1, 137 (1939).
  * Zhou  _et al._ [2004] X. W. Zhou, R. A. Johnson, and H. N. G. Wadley, Phys. Rev. B 69, 144113 (2004).
  * Stuart  _et al._ [2000] S. J. Stuart, A. B. Tutein, and J. A. Harrison, J. Chem. Phys. 112, 6472 (2000).
  * de la Rosa-Abad  _et al._ [2016] J. A. de la Rosa-Abad, G. J. Soldano, S. J. Mejía-Rosales, and M. M. Mariscal, RSC Adv. 6, 77195 (2016).
  * Plimpton [1995] S. Plimpton, J. Comput. Phys. 117, 1 (1995).
  * Carpick  _et al._ [1997] R. W. Carpick, D. F. Ogletree, and M. Salmeron, Appl. Phys. Lett. 70, 1548 (1997).
  * Dong  _et al._ [2013] Y. Dong, Q. Li, and A. Martini, J. Vac. Sci. Technol. A 31, 030801 (2013).
  * Socoliuc  _et al._ [2004] A. Socoliuc, R. Bennewitz, E. Gnecco, and E. Meyer, Phys. Rev. Lett. 92, 134301 (2004).
  * Li  _et al._ [2011] Q. Li, T. E. Tullis, D. Goldsby, and R. W. Carpick, Nature 480, 233 (2011).
  * Adelman and Doll [1976] S. A. Adelman and J. D. Doll, J. Chem. Phys. 64, 2375 (1976).
  * Guerra  _et al._ [2010] R. Guerra, U. Tartaglino, A. Vanossi, and E. Tosatti, Nat. Mater. 9, 634 (2010).
  * Dietzel  _et al._ [2009] D. Dietzel, M. Feldmann, H. Fuchs, U. D. Schwarz, and A. Schirmeisen, Appl. Phys. Lett. 95, 053104 (2009).
  * He  _et al._ [1999] G. He, M. H. Müser, and M. O. Robbins, Science 284, 1650 (1999).
  * Sokoloff [1984] J. Sokoloff, Surf. Sci. 144, 267 (1984).
  * Braun  _et al._ [2005] O. M. Braun, M. Peyrard, V. Bortolani, A. Franchini, and A. Vanossi, Phys. Rev. E 72, 056116 (2005).
  * Duan  _et al._ [2021] Z. Duan, Z. Wei, S. Huang, Y. Wang, C. Sun, Y. Tao, Y. Dong, J. Yang, Y. Zhang, Y. Kan, D. Li, and Y. Chen, Nano Lett. 21, 4615 (2021).
  * Heideman  _et al._ [1985] M. T. Heideman, D. H. Johnson, and C. S. Burrus, Arch. Hist. Exact Sci. 34, 265 (1985).
  * Gao  _et al._ [2021] H. Gao, T. P. W. Menzel, M. H. Müser, and D. Mukherji, Phys. Rev. Mater. 5, 065605 (2021).
  * Medyanik  _et al._ [2006] S. N. Medyanik, W. K. Liu, I.-H. Sung, and R. W. Carpick, Phys. Rev. Lett. 97, 136106 (2006).
  * Dietzel  _et al._ [2007] D. Dietzel, T. Mönninghoff, L. Jansen, H. Fuchs, C. Ritter, U. D. Schwarz, and A. Schirmeisen, J. Appl. Phys. 102, 084306 (2007).
  * Popov and Gray [2012] V. Popov and J. Gray, J. Appl. Math. Mech. 92, 683 (2012).
  * Sang  _et al._ [2001] Y. Sang, M. Dubé, and M. Grant, Phys. Rev. Lett. 87, 174301 (2001).
  * Müser [2020] M. H. Müser, Lubricants 8, 38 (2020).
  * Chen and Qian [2021] L. Chen and L. Qian, Friction 9, 1 (2021).


# Hardy’s inequality and (almost) the Landau equation

Maria Gualdani and Nestor Guillen  The University of Texas at Austin
Mathematics Department 2515 Speedway Stop C1200 Austin, Texas 78712-1202
<EMAIL_ADDRESS>Texas State University Department of Mathematics 601
University Drive San Marcos, Texas 78666-4684<EMAIL_ADDRESS>
###### Abstract.

In this manuscript we establish an $L^{\infty}$ estimate for the isotropic
analogue of the homogeneous Landau equation. This is done for values of the
interaction exponent $\gamma$ in (a part of) the range of very soft
potentials. The main observation in our proof is that the classical weighted
Hardy inequality leads to a weighted Poincaré inequality, which in turn
implies the propagation of some $L^{p}$ norms of solutions. From here, the
$L^{\infty}$ estimate follows from certain weighted Sobolev inequalities and
De Giorgi-Nash-Moser theory.

MPG is supported by DMS-2019335 and would like to thank NCTS Mathematics
Division Taipei for their kind hospitality.

##  1\. Introduction and main result

We consider the equation

(1.1) $\displaystyle\partial_{t}f$
$\displaystyle=c_{d,\gamma}\textnormal{div}\left(\int_{\mathbb{R}{{}^{d}}}\frac{1}{|v-v_{*}|^{-2-\gamma}}\left[f(v_{*},t)\nabla
f(v,t)-f(v,t)\nabla_{*}f(v_{*},t)\right]\;dv_{*}\right),$

where $f(v,t)$ is a non-negative function and the constant $c_{d,\gamma}$ is
positive and only depends on the dimension and $\gamma$. The constant $\gamma$
belongs to the range of very soft potential, $\gamma\in[-d,-2]$, $d\geq 3$.
Our chief interest in (1.1) comes from its apparent similarity to the
homogeneous Landau equation

(1.2)
$\displaystyle\partial_{t}f=c_{d,\gamma}\textnormal{div}\left(\int_{\mathbb{R}{{}^{d}}}\frac{\mathbb{P}(v-v_{*})}{|v-v_{*}|^{-2-\gamma}}\left[f(v_{*},t)\nabla
f(v,t)-f(v,t)\nabla_{*}f(v_{*},t)\right]\;dv_{*}\right),$

where $\mathbb{P}(z)$ is the matrix kernel given by the projection into the
orthogonal complement to $z$

$\displaystyle\mathbb{P}(z)=\mathbb{Id}-\frac{z\otimes z}{|z|^{2}},\quad
z\in\mathbb{R}^{d}\setminus\\{0\\}.$

The term $\mathbb{P}(z)$ in (1.2) is an anisotropy, and it is not present in
(1.1). Therefore one could consider (1.1) as an isotropic version of the
Landau equation (1.2).

Equation (1.1) was first introduced by Krieger and Strain in [14], motivated
precisely by this relationship, with the hope that an understanding of (1.1)
might lead to new insights on (1.2). In [14] the authors considered a
modification of (1.1), for $\gamma=-d=-3$,

(1.3) $\displaystyle\partial_{t}f=a[f]\Delta f+\alpha f^{2},\;\alpha\in[0,1],$

where $a[f]:=f\ast|v|^{-1}$. Here, (1.3) with $\alpha=1$ corresponds to (1.1).
In [14], the authors manage to show that, for $\alpha<3/4$, spherically
symmetric and radially decreasing solutions to (1.3) do not form singularities
in finite time, and in fact become smooth. Shortly after, in [11], Gressman,
Krieger, and Strain improved this result to the range $\alpha<74/75$, thanks
to a new nonlocal inequality for $\gamma=-d$,

$\displaystyle\int_{\mathbb{R}^{d}}f^{p+1}\;dv\leq\frac{(p+1)^{2}}{p^{2}}\int_{\mathbb{R}^{d}}a[f]|\nabla
f^{\frac{p}{2}}|\;dv.$

The authors of the present manuscript showed later in [12] that for $\alpha=1$
spherically symmetric and radially decreasing solutions to (1.3) regularize
instantaneously and remain smooth for all times. Moreover, the approach in
[12] covers (1.3) for all $\alpha\in[0,1]$ and also provides an $L^{\infty}$
result for spherically symmetric solutions of (1.2), conditioned on a certain
spectral bound holding (for more on this, see the discussion about (2.6)
below). Later, in [13], we make use of the fact that $a[f]$ is an $A_{1}$
weight and prove an $L^{\infty}$-estimate for weak solutions of (1.1) and
(1.2) for $\gamma\geq-2$ and a conditional $L^{\infty}$-estimate for solutions
of the equation for $\gamma\in[-d,-2)$.

The aim of this paper is to show a new $L^{\infty}$-estimate for solutions to
(1.1) when $\gamma\in(\gamma_{*},-2]$, $\gamma_{*}$ being specified bellow.
Here is the main theorem:

######  Theorem 1.1.

Let $f:\mathbb{R}^{d}\times[0,T]\to\mathbb{R}$ be a non-negative classical
solution of (1.1) with $\gamma\in\left[-1-\frac{d}{2},-2\right]$ and initial
data $f_{\textnormal{in}}$ that belongs to $L_{2}^{1}\cap
L^{1}(\mathbb{R}^{d})$.

  1. (1)

For any $p$ with $1\leq p\leq\frac{d+\gamma}{-2-\gamma}$, the norm
$\|f(t)\|_{L^{p}(\mathbb{R}^{d})}$ is non-increasing in time.

  2. (2)

Let $\gamma_{*}\in\left(-1-\frac{d}{2},-2\right)$ be the unique solution to

$\displaystyle\tfrac{d}{d+\gamma+2}=\tfrac{d+\gamma}{-\gamma-2}.$

If $f_{\textnormal{in}}$ also belongs to $L^{p}(\mathbb{R}^{d})$ for some
$p>\tfrac{d}{d+\gamma+2}$, any solution to (1.1) for
$\gamma\in(\gamma_{*},-2]$ is uniformly bounded for times away from zero, and

$\displaystyle\sup\limits_{B_{R}\times[\tau,T]}f(v,t)\leq
C(d,\gamma,f_{\textnormal{in}},R,\tau,T).$

In particular, for $\gamma\in(\gamma_{*},-2]$, classical solutions of (1.1)
remain smooth for every finite time.

To illustrate what portion of the “very soft potential” range covered in the
above theorem, observe that for $d=3$ we have $\gamma_{*}\approx-2.458$ while
for $d=4$ we have $\gamma_{*}\approx-2.87$. The value $\gamma_{*}$ is uniquely
defined in the range $(-d,-2)$.

The proof of Theorem 1.1 is relatively straightforward. For that, we set up
first some notation. Let $h[f]$ be the Riesz potential of $f$,

$\displaystyle
h[f]:=(-\Delta)^{-\frac{d+\gamma}{2}}f=C(d,d+\gamma)\int_{\mathbb{R}^{d}}\frac{f(v_{*})}{|v-v_{*}|^{-\gamma}}\;dv_{*},$

where $C(d,d+\gamma)$ is the normalization constant arising in the operator
$(-\Delta)^{-\frac{d+\gamma}{2}}$, namely

$\displaystyle
C(d,\alpha)=\pi^{-\tfrac{d}{2}}2^{-\alpha}\frac{\Gamma\left(\frac{d-\alpha}{2}\right)}{\Gamma\left(\frac{\alpha}{2}\right)},\quad
0<\alpha<d.$

In particular, $h[f]\to f$ as $\gamma\to-d$. Next, we shall denote by $a[f]$
the convolution

$\displaystyle
a[f]:=c_{d,\gamma}\int_{\mathbb{R}^{d}}\frac{f(v_{*})}{|v-v_{*}|^{-2-\gamma}}\;dv_{*}.$

With this notation, we rewrite (1.1) as

$\displaystyle\partial_{t}f=\textrm{div}\left(a[f]\nabla f-f\nabla
a[f]\right),$

or, in a non-divergence form,

$\displaystyle\partial_{t}f=a[f]\Delta f-(2+\gamma)fh[f],$

provided $c_{d,\gamma}$ is chosen as

$\displaystyle c_{d,\gamma}:=\frac{C(d,d+\gamma)}{d+\gamma},\quad\textrm{or
equivalently}\quad c_{d,\gamma}:=(2+\gamma)C(d,d+2+\gamma),$

so that

(1.4) $\displaystyle\Delta
a[f]=c_{d,\gamma}(d+\gamma)(\gamma+2)\int_{\mathbb{R}^{d}}\frac{f(v_{*})}{|v-v_{*}|^{-\gamma}}\;dv_{*}=(2+\gamma)h[f].$

One can also define $a[f]$ to be the operator
$(2+\gamma)(-\Delta)^{-\frac{d+2+\gamma}{2}}f$. Note that our choice of
$c_{d,\gamma}$ (which is well defined for any $\gamma\in[-d,-2]$), does not
interfere with the structure of $\eqref{eqn:isotropic equation}$: one can
eliminate such constant with a simple time rescaling.

For $\gamma=-2$, equation (1.1) reduces substantially: for sufficiently smooth
functions $f$, it simplifies to a heat equation,

$\displaystyle\partial_{t}f=c_{d,-2}\textnormal{div}\left(\int_{\mathbb{R}{{}^{d}}}f(v_{*},t)\nabla
f(v,t)\;dv_{*}\right)=c_{d,-2}\|f_{\textnormal{in}}\|_{L^{1}}\Delta f,$

thanks to the conservation of the $L^{1}$ norm. This is different from the
classical Landau equation (1.2) when $\gamma=-2$, which has an additional
reaction term arising from the derivatives of $\mathbb{P}(v)$.

Now we explain the basic idea of the proof. For a classical solution of (1.1)
we have

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}^{d}}f^{p}\;dv=-\frac{4(p-1)}{p}\int_{\mathbb{R}^{d}}a[f]|\nabla
f^{p/2}|^{2}\;dv+(p-1)(2+\gamma)\int_{\mathbb{R}^{d}}h[f]f^{p}\;dv.$

To bound the right hand side, we estimate the second integral by the first
one. This is the key point for Theorem 1.1, and it is achieved through the
weighted Hardy inequality [9]

$\displaystyle(d+\gamma)^{2}\int_{\mathbb{R}^{d}}|v|^{\gamma}\phi^{2}\;dv\leq
4\int_{\mathbb{R}^{d}}|v|^{2+\gamma}|\nabla\phi|^{2}\;dv,\quad\gamma>-d.$

This inequality implies, via convolution, the following weighted Poincare’s
inequality

(1.5) $\displaystyle(d+\gamma)\int_{\mathbb{R}^{d}}h[f]\phi^{2}\;dv\leq
4\int_{\mathbb{R}^{d}}a[f]|\nabla\phi|^{2}\;dv.$

For $\gamma$ belonging to the interval $(\gamma_{*},-2]$, thanks to (1.5), we
can show that

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}^{d}}f^{p}\;dv\leq 0,$

for some $p>\frac{d}{d+2+\gamma}$. This is all we need to conclude that $f$
belongs to $L^{\infty}$ for positive times.

The proof of this last statement exists in the literature. See for example
Theorem 3.8 in [16] or Theorem 2.9 in [13]. For completeness, we report the
proof here, following the method in [13]. We will show that a bound for $f$ in
$L^{\infty}(0,T,L^{p}(\mathbb{R}^{d}))$ for some $p>\frac{d}{d+2+\gamma}$
yields an $\varepsilon$-Poincare’s inequality of the form

(1.6)
$\displaystyle\int_{\mathbb{R}^{d}}h[f]\;\phi^{2}\;dv\leq\varepsilon\int_{\mathbb{R}^{d}}a[f]|\nabla\phi|^{2}\;dv+C(f_{\textnormal{in}},R)\varepsilon^{-\frac{2}{\eta}}\int_{\mathbb{R}^{d}}a[f]\phi^{2}\;dv.$

This last inequality, combined with a Moser’s iteration, yields the desired
$L^{\infty}$-bound for $f$.

###  1.1. Short background on the homogeneous Landau equation

The main result in this note states that for the isotropic equation (1.1)
there is a nontrivial portion of the “very soft potentials range”,
$\gamma\in[-d,-2]$, for which one can rule out the formation of singularities
in finite time. This is of interest given the similarity of (1.1) with the
homogeneous Landau equation (1.2), and the unresolved question of $L^{\infty}$
estimates for the latter equation when $\gamma\in[-d,-2]$. We briefly discuss
some of the literature on the Landau equation in order to illustrate what is
presently understood about (1.2) and what makes the equation in the very soft
potentials range much harder to analyze.

The problem of $L^{\infty}$-estimates for solutions to the Landau equation
(1.2) stands as a difficult unresolved question when $\gamma<-2$. Indeed, as
$\gamma$ decreases, the kernel in $h[f]$ becomes more and more singular, and
one requires more integrability for $f$ to control $h[f]$. On the contrary, in
the range $\gamma\geq-2$, the regularity theory is well developed. For the
hard potentials case ($\gamma\geq 0$) the works of Desvillettes and Villani
[5, 6] do not only answer the question of regularity of solutions, but also
cover existence and long time convergence to equilibrium. For soft potentials
($\gamma\in(-2,0)$) and $d=3$, Alexandre, Liao, and Lin [1] obtain the
propagation of $L^{2}$ estimates for solutions (potentially growing with
time), from where higher $L^{\infty}$ estimates and higher regularity can be
obtained. For $\gamma=-2$ we refer to [17]. Silvestre [16] obtains a priori
$L^{\infty}$ estimates for solutions when $\gamma\in(-2,0)$ that are fully
determined from the mass, energy, and entropy of the solution (and accordingly
are not growing with time). The results in [16] also say that for
$\gamma\in[-d,-2)$ the $L^{\infty}$ norm is controlled once the $L^{p}$-norm
of $f$, with $p>\frac{d}{d+2+\gamma}$, is bounded uniformly in time. Similar
results, proven with different method, appear in [13]. The estimates in [13]
are proved for weak solutions and use the divergence structure of the
equation, but yield an $L^{\infty}$ estimate for $f(v,t)$ that deteriorates as
$v$ grows. In contrast, the work [16] uses non-divergence techniques and
yields bounds that are global in space.

For very soft potentials there have been several recent works about the nature
of potential singularities, which narrow down the ways in which blow up may
occur. In [10], Golse, Gualdani, Imbert and Vasseur proved that the set of
singular times for weak solutions to (1.2) (with $\gamma=-3$) has Hausdorff
dimension at most 1/2. Very recently, Desvillettes, He, and Jiang provided in
[4] new insides about the behavior of solutions to (1.2) (with $\gamma=-3$) in
$\dot{H}^{1}$-norm near the blow-up time. Most importantly, they show that
even if there is blow-up, after that solutions become smooth again and remain
that way for all later times. In [2], Bedrossian, Gualdani and Snelson rule
out type I self similar blow-up for solutions to (1.2) for any $\gamma<-2$.

There is an important connection between $L^{\infty}$ bounds and uniqueness.
Fournier and Guérin proved a uniqueness result for bounded weak solutions in
[7], this being for $\gamma\in(-d,-2)$. In fact, the work [7] guarantees
uniqueness of solutions with $f\in L^{1}L^{\infty}$ and in particular to
bounded solutions. The work [7] was followed by Fournier’s work in [8] with a
corresponding uniqueness result for $\gamma=-d$. Later in [3], Chern and
Gualdani proved a uniqueness result for sufficiently integrable solutions for
the Landau equation with Coulomb interactions.

###  1.2. Outline

The rest of the paper is organized as follows. In Section 2 we recall Hardy’s
inequality and make the main new observation of the paper, the inequality in
Lemma 2.1. The new inequality is immediately put to use in Section 3 to prove
the first part of Theorem 1.1. Then, Sections 5 and 6 deal with weighted
normed inequalities and a Moser iteration leading to the $L^{\infty}$ estimate
that makes the second half of Theorem 1.1.

##  2\. Hardy’s inequality

The main ingredient in this note is the classical Hardy inequality,

(2.1)
$\displaystyle(d+\gamma)^{2}\int_{\mathbb{R}^{d}}|v|^{\gamma}\phi^{2}\;dv\leq
4\int_{\mathbb{R}^{d}}|v|^{2+\gamma}|\nabla\phi|^{2}\;dv,\quad\gamma>-d.$

Let us review one elementary way of proving (2.1), a deeper and broader
discussion on Hardy’s inequality can be found in the book by Ghoussoub and
Moradifam [9]. First notice that

$\displaystyle-\Delta|v|^{2+\gamma}=(d+\gamma)(-\gamma-2)|v|^{\gamma}.$

Multiply both sides of this equation by $\phi$. Integration by parts and
Cauchy-Schwarz yield

$\displaystyle(d+\gamma)(-\gamma-2)\int\phi^{2}|v|^{\gamma}\;dv$
$\displaystyle=2\int\nabla|v|^{2+\gamma}\phi\nabla\phi\;dv$ $\displaystyle\leq
2\left(\int\alpha^{2}(v)|\nabla|v|^{2+\gamma}|^{2}\phi^{2}\;dv\right)^{1/2}\left(\int\frac{|\nabla\phi|^{2}}{\alpha^{2}(v)}\;dv\right)^{1/2}.$

We pick now the best weight $\alpha(v)$ such that

$\displaystyle\alpha^{2}(v)|\nabla|v|^{2+\gamma}|^{2}\leq|v|^{\gamma}.$

or equivalently

$\displaystyle\alpha^{2}(v)=\frac{1}{|v|^{2+\gamma}(2+\gamma)^{2}}.$

With this choice of $\alpha^{2}(v)$, we obtain (2.1).

######  Lemma 2.1.

Let $\gamma\in(-d,-2)$. Fix a non-negative $f\in L^{1}(\mathbb{R}^{d})$ and
let $a[f]$ and $h[f]$ be as in Section 1, then the following inequality holds
for all $\phi\in C^{1}_{c}(\mathbb{R}^{d})$ (and limits of such functions)

(2.2) $\displaystyle(d+\gamma)\int_{\mathbb{R}^{d}}h[f]\phi^{2}\;dv\leq
4\int_{\mathbb{R}^{d}}a[f]|\nabla\phi|^{2}\;dv.$

###### Proof.

Fix $\phi\in C^{1}_{c}(\mathbb{R}^{d})$. By a change of variables, we see that
(2.1) is equivalent to the inequalities (with $w\in\mathbb{R}^{d}$)

(2.3)
$\displaystyle(d+\gamma)^{2}\int_{\mathbb{R}^{d}}|v-w|^{\gamma}\phi^{2}\;dv\leq
4\int_{\mathbb{R}^{d}}|v-w|^{2+\gamma}|\nabla\phi|^{2}\;dv.$

Let us multiply (2.3) by $f(w)\geq 0$ and integrate the resulting expression
in $w$, we obtain

(2.4)
$\displaystyle(d+\gamma)^{2}\int_{\mathbb{R}^{d}}(f*|v|^{\gamma})\phi^{2}\;dv\leq
4\int_{\mathbb{R}^{d}}(f*|v|^{2+\gamma})|\nabla\phi|^{2}\;dv.$

Substituting in (2.4) the expression for $\Delta a[f]$ and making use of
(1.4), the lemma is proved.

∎

Although elementary, Lemma 2.1 is key, as it leads to the propagation of
$L^{p}$ bounds for solutions to (1.1), proven in the next section. The range
of $p$’s is limited by the constants appearing in (2.2), and this is the sole
limitation on the range of $\gamma$’s covered by Theorem 1.1. This motivates
the following (admittedly open ended) eigenvalue problem.

######  Problem.

Fix $d$ and $\gamma\in[-d,-2]$. Let $f\in L^{1}(\mathbb{R}^{d})$ be non-
negative, and let

(2.5)
$\displaystyle\Lambda_{\textnormal{iso}}(f):=\inf\limits_{\phi}\frac{\int_{\mathbb{R}^{d}}a[f]|\nabla\phi|^{2}\;dv}{\int_{\mathbb{R}^{d}}h[f]\phi^{2}\;dv}.$

Determine under what circumstances can we say that

$\displaystyle\Lambda_{\textnormal{iso}}(f)>\frac{d+\gamma}{4}.$

If $f$ is just a generic function in $L^{1}$, then one cannot do better than
inequality (2.2). To see this, take a sequence of functions $f_{n}$ which are
converging as $n\to\infty$ to a Dirac delta at $0$. For this sequence, (2.2)
converges to (2.1), which is known to be sharp [9, Section 4.3].

The corresponding problem for the Landau equation would be,

(2.6)
$\displaystyle\Lambda_{\textnormal{Landau}}(f):=\inf\limits_{\phi}\frac{\int_{\mathbb{R}^{d}}(A[f]\nabla\phi,\nabla\phi)\;dv}{\int_{\mathbb{R}^{d}}h[f]\phi^{2}\;dv},$

where
$A[f]:=\int_{\mathbb{R}{{}^{d}}}\frac{\mathbb{P}(v-v_{*})}{|v-v_{*}|^{-2-\gamma}}f(v_{*})\;dv_{*}$.
The significance of this eigenvalue problem is well known in the Landau and
Boltzmann literature. We do not know whether an elementary argument as in
Lemma 2.1 yields a similar bound for (2.6). If one argues by direct analogy
with Lemma 2.1, one would have to contend with the projection term
$\mathbb{P}(v)$ appearing in $A[f]$, and it is not immediately clear how this
can be done.

The theory of weighted normed inequalities can yield certain estimates for
$\Lambda_{\textnormal{iso}}$, or $\Lambda_{\textnormal{Landau}}$. For
instance, the value in (2.5) is directly related to the quantity

(2.7)
$\displaystyle\sup\limits_{B}|B|^{\frac{2}{d}}\frac{\int_{B}h[f]\;dv}{\int_{B}a[f]\;dv},$

which happens to be bounded from above for all non-negative $f\in
L^{1}(\mathbb{R}^{d})$ by a universal constant (see [13] for further
discussion). Finally, it is worth mentioning that for any spherically
symmetric and radially decreasing $f$ solving (1.2) ($\gamma=-d$), the
$L^{\infty}$ norm of $f$ cannot blow up at a finite time $T$ if, for this $f$,
the quantity (2.7) remains bounded by $1/96$ (this is likely a non-sharp
estimate). This result was shown in [12].

##  3\. Propagation of $L^{p}$ bounds

In this section we shall make use of Lemma 2.1 to show that various $L^{p}$
norms propagate forward in time, at least for some range of $\gamma$’s.

######  Lemma 3.1.

(Propagation of integrability.) Let $f$ be a nonnegative solution to (1.1)
with initial data $f(v,0)=f_{\textnormal{in}}$. For every $p$ such that

$\displaystyle 1\leq p\leq\frac{d+\gamma}{-2-\gamma}$

the norm $\|f(t)\|_{L^{p}}$ is non-increasing in $t$. In particular, for every
$t\in[0,T]$ we have

$\displaystyle\|f(t)\|_{L^{p}}\leq\|f_{\textnormal{in}}\|_{L^{p}}.$

###### Proof.

Multiply (1.1) by $f^{p-1}$ for some $p\geq 1$ and integrate over
$\mathbb{R}{{}^{d}}$. We obtain

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}^{d}}f^{p}\;dv=-\frac{4(p-1)}{p}\int_{\mathbb{R}^{d}}a[f]|\nabla
f^{p/2}|^{2}\;dv+(p-1)(-2-\gamma)\int_{\mathbb{R}^{d}}h[f]f^{p}\;dv.$

To estimate the last term $\int_{\mathbb{R}^{d}}h[f]f^{p}\;dv$ we use Lemma
2.1 with $\phi=f^{p/2}$. One gets

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}^{d}}f^{p}\;dv$
$\displaystyle\leq-4(p-1)\left[\frac{1}{p}-\frac{(-2-\gamma)}{(d+\gamma)}\right]\int_{\mathbb{R}^{d}}a[f]|\nabla
f^{p/2}|^{2}\;dv.$

It follows that $\|f(t)\|_{p}^{p}$ is non-increasing whenever the expression
in the brackets is non-positive, which is the case given the assumption on
$p$. This concludes the proof of the lemma, and of the first part of Theorem
1.1.

∎

######  Remark 3.2.

For there to be any $p$ such that $1\leq p\leq(d+\gamma)/(-2-\gamma)$ it must
be that

$\displaystyle\gamma\geq-1-\frac{d}{2}.$

It follows that Lemma 3.1 is of no use for values of $\gamma$ close to $-d$.

##  4\. Controlling the second moment

Solutions to (1.1) conserve mass and first moment, but not second moment. In
the next lemma we show that second moments grow linearly in time, provided
$a[f]$ is uniformly bounded.

######  Lemma 4.1.

The second moment of $f$, solution to (1.1), evolves according to the formula

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}^{d}}f(v,t)|v|^{2}\;dv=2(d+2+\gamma)\int_{\mathbb{R}^{d}}f(v,t)a[f](v,t)\;dv.$

In particular, for all $t\in[0,T]$ and $p>\frac{d}{d+2+\gamma}$ we have

$\displaystyle\int_{\mathbb{R}^{d}}f(T,v)|v|^{2}\;dv\leq\int_{\mathbb{R}^{d}}f_{\textnormal{in}}(v)|v|^{2}\;dv+TC_{d,\gamma,p}\|f\|_{L^{\infty}(0,T,L^{p})}^{1-\theta}\|f_{\textnormal{in}}\|_{L^{1}}^{1+\theta},$

where $\theta:=\frac{|2+\gamma|p^{\prime}}{d}$.

###### Proof.

Integration by parts yields

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^{d}}f(t,v)|v|^{2}\;dv=-\int_{\mathbb{R}^{d}}(v,a[f]\nabla
f-f\nabla a[f])\;dv.$

Using the integral form for $a[f]$ we rewrite the expression on the right,
leading to

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^{d}}f(t,v)|v|^{2}\;dv=$
$\displaystyle-
c_{d,\gamma}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}{v}\cdot\frac{f(w)\nabla
f(v)-f(v)\nabla f(w)}{|v-w|^{-2-\gamma}}\;dwdv$
$\displaystyle-\frac{c_{d,\gamma}}{2}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}{v-w}\cdot\frac{f(w)\nabla
f(v)-f(v)\nabla f(w)}{|v-w|^{-2-\gamma}}\;dwdv.$

Integration by parts in both $v$ and $w$ yields

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}^{d}}f(t,v)|v|^{2}\;dv$
$\displaystyle=2(d+2+\gamma)\int_{\mathbb{R}^{d}}fa[f]\;dv,$

since

$\textrm{div}\left(\frac{z}{|z|^{-2-\gamma}}\right)=\frac{d+2+\gamma}{|z|^{-2-\gamma}}.$

This proves the first part of the lemma. For the second part, it is clear that

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^{d}}f(t,v)|v|^{2}\;dv$
$\displaystyle\leq(d+2+\gamma)\|f(t)\|_{L^{1}}\|a(t)\|_{L^{\infty}}.$

Then, integrating the resulting inequality in time, the estimate follows
thanks to an elementary interpolation argument (see Remark 4.2)

$\displaystyle\|a\|_{L^{\infty}(0,T,L^{\infty}(\mathbb{R}^{d}))}\leq
C_{d,\gamma,p}\|f(t)\|_{L^{\infty}(0,T,L^{p}(\mathbb{R}^{d}))}^{\theta}\|f_{\textnormal{in}}\|_{L^{1}}^{1-\theta}.$

∎

######  Remark 4.2.

The following estimate is well known and we recall it here for completeness:
let $p>\frac{d}{d+\gamma+2}$, for every $s>0$ we have that

$\displaystyle a[f]$
$\displaystyle=c_{d,\gamma}\int_{B_{s}(v)}f(w)|v-w|^{2+\gamma}\;dw+c_{d,\gamma}\int_{\mathbb{R}^{d}\setminus
B_{s}(v)}f(w)|v-w|^{2+\gamma}\;dw$ $\displaystyle\leq
c_{d,\gamma}\|f\|_{L^{p}}\left(\int_{B_{s}}|w|^{(2+\gamma)p^{\prime}}\;dw\right)^{1/p^{\prime}}+c_{d,\gamma}s^{2+\gamma}\|f\|_{L^{1}}.$

Optimizing the right hand side with respect to $s$, the following estimate
follows

$\displaystyle\|a[f]\|_{L^{\infty}}\leq
C_{d,\gamma,p}\|f\|_{L^{p}}^{\theta}\|f\|_{L^{1}}^{1-\theta},\;\textnormal{
where }\theta=\frac{|2+\gamma|p^{\prime}}{d}.$

######  Corollary 4.3.

Let $p>\frac{d}{d+2+\gamma}$. For all $v\in\mathbb{R}^{d}$ and $t\in[0,T]$,
the following inequality holds

$\displaystyle a[f](v,t)\geq\ell\langle v\rangle^{2+\gamma},$

where

$\displaystyle\ell:=c_{d,\gamma}\frac{\|f_{\textnormal{in}}\|_{L^{1}}^{-1-\gamma}}{\left(c_{1}+c_{2}T\|f\|_{L^{\infty}(0,T,L^{p}(\mathbb{R}^{d}))}^{1-\theta}\|f_{\textnormal{in}}\|_{L^{1}}^{1+\theta}\right)^{-2-\gamma}}.$

###### Proof.

For any $r>0$ we have

$\displaystyle\int_{\mathbb{R}^{d}\setminus B_{r}}f(v,t)\;dv\leq
r^{-2}\int_{\mathbb{R}^{d}\setminus B_{r}}f(v,t)|v|^{2}\;dv\leq
r^{-2}\|f(t)\|_{L^{1}_{2}}.$

From here, taking $r=\frac{2\|f(t)\|_{L^{1}_{2}}}{\|f(t)\|_{L^{1}}}$, we get

$\displaystyle\int_{B_{r}}f(v,t)\;dv\geq\tfrac{1}{2}\|f_{\textnormal{in}}\|_{L^{1}}.$

Then, since $|v-w|\leq|v|+r$ whenever $w\in B_{r}$,

$\displaystyle a(v,t)$ $\displaystyle\geq
c_{d,\gamma}\int_{B_{r}}f(w,t)(|v|+r)^{2+\gamma}\;dw\geq\tfrac{1}{2}c_{d,\gamma}\|f_{\textnormal{in}}\|_{L^{1}}(|v|+r)^{2+\gamma}.$

In particular

$\displaystyle a(v,t)$
$\displaystyle\geq\tfrac{1}{2}c_{d,\gamma}\|f_{\textnormal{in}}\|_{L^{1}}r^{2+\gamma}\langle
v\rangle^{2+\gamma}$ $\displaystyle\geq
c_{d,\gamma}\frac{\|f_{\textnormal{in}}\|_{L^{1}}^{-1-\gamma}}{\left(c_{1}+c_{2}T\|f(t)\|_{L^{p}}^{1-\theta}\|f_{\textnormal{in}}\|_{L^{1}}^{1+\theta}\right)^{-2-\gamma}}\langle
v\rangle^{2+\gamma},$

using Lemma 4.1 to bound $r^{2+\gamma}$ from below. ∎

##  5\. Some weighted inequalities

In this section we invoke a few results from the theory of integral
inequalities with $A_{p}$ weights. The result will be integral inequalities of
the form

$\displaystyle\left(\int_{\mathbb{R}^{d}}\omega_{1}|\phi|^{r}\;dv\right)^{\frac{1}{r}}\leq
C\left(\int_{\mathbb{R}^{d}}\omega_{2}|\nabla\phi|^{2}\;dv\right)^{\frac{1}{2}},$

for various choices of the exponent $r$, weights $\omega_{i}$, and constant
$C$ which are pertinent to obtaining estimates a la De Giorgi-Nash-Moser for
solutions of (1.1). For a more complete discussion we refer the reader to [13,
Section 3.2].

A central object in these inequalities is the following product of averages of
the weights, taken over an arbitrary cube $Q\subset\mathbb{R}^{d}$, (here,
“$\fint$” denotes average over the set of integration)

$\displaystyle\sigma_{r,s}(Q,\omega_{1},\omega_{2}):=|Q|^{\frac{1}{d}-\frac{1}{2}+\frac{1}{r}}\left(\fint_{Q}\omega_{1}^{s}\;dv\right)^{\frac{1}{rs}}\left(\fint_{Q}\omega_{2}^{-s}\;dv\right)^{\frac{1}{2s}}.$

The significance of $\sigma_{r,s}(Q,\omega_{1},\omega_{2})$ is captured by the
following theorem (see [15, Theorem 1]).

######  Theorem 5.1.

Let $Q\subset\mathbb{R}^{d}$, $r\geq 2$, and let $\omega_{1},\omega_{2}$ be
two weights. Define, for some $C(d,s,r)$,

$\displaystyle\mathcal{C}_{r,s}(Q,\omega_{1},\omega_{2}):=C(d,s,r)\sup\limits_{Q^{\prime}\subset
8Q}\sigma_{r,s}(Q^{\prime},\omega_{1},\omega_{2}).$

Then, for any $\phi$ supported in $Q$ or any $\phi$ such that
$\fint_{Q}\phi\;dv=0$, we have

$\displaystyle\left(\int_{Q}\omega_{1}|\phi|^{r}\;dv\right)^{\frac{1}{r}}\leq\mathcal{C}_{r,s}(Q,\omega_{1},\omega_{2})\left(\int_{Q}\omega_{2}|\nabla\phi|^{2}\;dv\right)^{\frac{1}{2}}.$

The next two propositions give estimates on
$\sigma_{r,s}(Q,\omega_{1},\omega_{2})$ for two combination of weights, namely
$\omega_{1}=a[f]^{m},\;\omega_{2}=a[f]$ and
$\omega_{1}=h[f],\;\omega_{2}=a[f]$.

There are two exponents that will be appearing repeatedly in what follows:

(5.1) $\displaystyle m:=\frac{d}{d-2},\;\;q:=2\left(1+\frac{2}{d}\right).$

######  Proposition 5.2.

There exists $s>1$ and $C>0$ depending only on $d$ and $\gamma$ such that for
non-negative $f\in L^{1}(\mathbb{R}^{d})$ and any cube
$Q\subset\mathbb{R}^{d}$,

$\displaystyle\sigma_{2m,s}(Q,a[f]^{m},a[f])\leq C.$

###### Proof.

For $\gamma>-d$, Lemma 3.5 from [13, Section 3] says there is some $s>1$ such
that

$\displaystyle\left(\fint_{Q}a[f]^{ms}\;dv\right)^{\frac{1}{ms}}\leq
C\fint_{Q}a[f]\;dv,\quad C=C(d,\gamma,ms).$

As it was also noted in [13, Section 3], there is a universal constant such
that

$\displaystyle\fint_{Q}a[f]\;dv\leq C\inf\limits_{Q}a[f],$

which means also that

$\displaystyle\sup\limits_{Q}a[f]^{-1}\leq
C\left(\fint_{Q}a[f]\;dv\right)^{-1}.$

Putting these two observations together it follows that

$\displaystyle\left(\fint_{Q^{\prime}}a[f]^{ms}\;dv\right)^{\frac{1}{2ms}}\left(\fint_{Q^{\prime}}(a[f])^{-s}\;dv\right)^{\frac{1}{2s}}\leq
C.$

Lastly, $m$ solves $\frac{1}{d}-\frac{1}{2}+\frac{1}{2m}=0$ (it is its
determining property), and the proposition is proved. ∎

The next one if the key proposition for the proof of (1.6):

######  Proposition 5.3.

There is $s>\tfrac{d}{2}$ such that given a cube $Q\subset B_{R}$ with
$|Q|\leq 1$ we have

$\displaystyle\left[\sigma_{2,s}(Q,h[f],a[f])\right]^{2}\leq\theta(|Q|^{\tfrac{1}{d}}):=C(\|f\|_{L^{\infty}(L^{p(\gamma,d,s)})},\|f\|_{L^{\infty}(L^{1})})\langle
R\rangle^{|2+\gamma|}|Q|^{\tfrac{\eta}{d}}.$

Here $\eta:=2-\tfrac{d}{s}>0$ and $p(\gamma,d,s):=\tfrac{ds}{d+s(d+\gamma)}$.

In particular, one can chose $s$ infinitesimally close to $\tfrac{d}{2}$,
resulting in $p(\gamma,d,s)$ to be greater, but as close as one wishes to
$\tfrac{d}{d+\gamma+2}$.

###### Proof.

Classical fractional integral estimates say that

$\displaystyle\|h[f]\|_{L^{\frac{dp}{d-p(d+\gamma)}}}\leq
C_{d,d+\gamma}\|f\|_{L^{p}},\;\textnormal{ provided }p<\frac{d}{d+\gamma}.$

We want to choose $p$ so that $s=\frac{dp}{d-p(d+\gamma)}$, which results in
$p(\gamma,d,s)=\frac{ds}{d+s(d+\gamma)}$. Therefore,

$\displaystyle|Q^{\prime}|^{\frac{2}{d}}\left(\fint_{Q^{\prime}}h[f]^{s}\;dv\right)^{\frac{1}{s}}\leq
C|Q^{\prime}|^{\frac{2}{d}-\frac{1}{s}}\|f\|_{L^{p(\gamma,d,s)}}.$

We can take $s$ larger but arbitrarily close to $\tfrac{d}{2}$ (to have
$\frac{2}{d}-\tfrac{1}{s}$ positive) which results in $p(\gamma,d,s)$ be
strictly greater, but arbitrarily close to, $d/(d+2+\gamma)$. Hence,

$\displaystyle|Q|^{\frac{2}{d}}\left(\fint_{Q^{\prime}}h[f]^{s}\;dv\right)^{\frac{1}{s}}\leq
C|Q^{\prime}|^{\frac{\eta}{d}}\|f\|_{L^{p(\gamma,d,s)}}.$

Thanks to the bound from below for $a[f]$ from Corollary 4.3, we have

$\displaystyle\left(\fint_{Q^{\prime}}a[f]^{-s}\;dv\right)^{\frac{1}{s}}\leq
C\langle R\rangle^{|2+\gamma|},\;\forall\;Q^{\prime}\subset 8Q.$

We work towards estimating the other factor.

It follows that

$\displaystyle|Q^{\prime}|^{\frac{2}{d}}\left(\fint_{Q^{\prime}}h[f]^{s}\;dv\right)^{\frac{1}{s}}\left(\fint_{Q^{\prime}}a[f]^{-s}\;dv\right)^{\frac{1}{s}}\leq
C\langle
R\rangle^{|2+\gamma|}|Q^{\prime}|^{\frac{\eta}{d}}\|f\|_{L^{p(\gamma,s)}}.$

This estimate is for all cubes $Q^{\prime}$ such that $Q^{\prime}\subset 8Q$,
which proves the proposition.

∎

An immediate consequence of Theorem 5.1 and Proposition 5.2 is the following
inequality.

######  Corollary 5.4.

There is a universal constant $C$ such that for all $\phi$ we have

$\displaystyle\left(\int_{\mathbb{R}^{d}}\;a^{m}[f]\phi^{2m}\;dv\right)^{\frac{1}{m}}\leq
C\int_{\mathbb{R}^{d}}a[f]\;|\nabla\phi|^{2}\;dv.$

Corollary 5.4 implies, via an elementary interpolation argument, a space-time
integral inequality for functions
$\phi:\mathbb{R}^{d}\times[0,T]\to\mathbb{R}$.

######  Corollary 5.5.

There is a universal constant $C$ such that

(5.2) $\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{d}}\;a[f]\phi^{q}\;dvdt\leq
C\left(\int_{0}^{T}\int_{\mathbb{R}^{d}}a[f]\;|\nabla\phi|^{2}\;dvdt+\sup\limits_{(0,T)}\int_{\mathbb{R}^{d}}\phi^{2}\;dv\right)^{q/2}.$

###### Proof.

We follow the standard proof of this space-time inequality (see proof of
Theorem 2.12 and 2.13 in [13]). First, we estimate the integral of
$|\phi|^{q}$ with weight $a[f]$ by interpolation

$\displaystyle\int_{\mathbb{R}^{d}}a[f]|\phi|^{q}\;dv$
$\displaystyle=\int_{\mathbb{R}^{d}}a[f]\phi^{q\theta+q(1-\theta)}\;dv$
$\displaystyle\leq\left(\int_{\mathbb{R}^{d}}a[f]^{\frac{2m}{(1-\theta)q}}|\phi|^{2m}\;dv\right)^{\frac{(1-\theta)q}{2m}}\left(\int_{\mathbb{R}^{d}}\phi^{2}\;dv\right)^{\frac{q\theta}{2}}.$

The exponent $\theta\in(0,1)$ is determined from the relation
$\tfrac{1}{q}=\tfrac{1-\theta}{2m}+\tfrac{\theta}{2}$. Simplifying, we obtain

$\displaystyle\int_{\mathbb{R}^{d}}a[f]|\phi|^{q}\;dv$
$\displaystyle\leq\left(\int_{\mathbb{R}^{d}}a[f]^{m}|\phi|^{2m}\;dv\right)^{\frac{1}{m}}\left(\int_{\mathbb{R}^{d}}\phi^{2}\;dv\right)^{\frac{m-1}{m}}.$

Now, by Corollary 5.4

$\displaystyle\int_{\mathbb{R}^{d}}a[f]|\phi|^{q}\;dv\leq
C\left(\int_{\mathbb{R}^{d}}a[f]|\nabla\phi|^{2}\;dv\right)\left(\int_{\mathbb{R}^{d}}\phi^{2}\;dv\right)^{\frac{m-1}{m}}.$

Integrating this over time we have

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{d}}a[f]|\phi|^{q}\;dvdt\leq
C\left(\int_{0}^{T}\int_{\mathbb{R}^{d}}a[f]|\nabla\phi|^{2}\;dvdt\right)\left(\sup\limits_{(0,T)}\int_{\mathbb{R}^{d}}\phi^{2}\;dv\right)^{\frac{m-1}{m}}.$

From this last inequality it follows trivially that

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{d}}a[f]|\phi|^{q}\;dvdt\leq
C\left(\int_{0}^{T}\int_{\mathbb{R}^{d}}a[f]|\nabla\phi|^{2}\;dvdt+\sup\limits_{(0,T)}\int_{\mathbb{R}^{d}}\phi^{2}\;dv\right)^{2-\frac{1}{m}}.$

Noting that $2-\frac{1}{m}=2-\frac{d-2}{d}=1+\frac{2}{d}=\frac{q}{2}$, the
corollary is proved. ∎

The other important use of Theorem 5.1 is in proving a $\varepsilon$-Poincaré
inequality, which also relies crucially on Proposition 5.3 and the $L^{p}$
bound on $f$.

######  Corollary 5.6.

Let $R>0$ and $\varepsilon\in(0,\varepsilon_{0})$. For any $\phi$ supported in
$B_{R}(0)$ we have

$\displaystyle\int_{\mathbb{R}^{d}}h[f]\;\phi^{2}\;dv\leq\varepsilon\int_{\mathbb{R}^{d}}a[f]|\nabla\phi|^{2}\;dv+C(f_{\textnormal{in}},R)\varepsilon^{-\frac{2}{\eta}}\int_{\mathbb{R}^{d}}a[f]\phi^{2}\;dv.$

Here, $\varepsilon_{0}:=\theta(1)/C(d,\gamma)$, $\theta(r)$ and $\eta$ are as
in Proposition 5.3, and

$\displaystyle C(f_{\textnormal{in}},R)=(4C)^{\frac{2}{\eta}}\langle
R\rangle^{\frac{2|2+\gamma|}{\eta}}\|f_{\textnormal{in}}\|_{L^{p(\gamma,d,s)}}^{\frac{2}{\eta}}.$

###### Proof.

Let $Q$ be any cube in $\mathbb{R}^{d}$ with $|Q|\leq 1$. Since
$\phi=\phi-(\phi)_{Q}+(\phi)_{Q}$ it is elementary that

(5.3) $\displaystyle\int_{Q}h[f]\phi^{2}\;dv\leq
4\int_{Q}h[f](\phi-(\phi)_{Q})^{2}\;dv+4(\phi)_{Q}^{2}\int_{Q}h[f]\;dv,$

where $(\phi)_{Q}$ denotes the average over $Q$,

$\displaystyle(\phi)_{Q}=\fint_{Q}\phi\;dv.$

Applying Hölder’s inequality to
$\int_{Q}a^{-\frac{1}{2}}(a^{\frac{1}{2}}|\phi|)\;dv$, it follows that

$\displaystyle(\phi)_{Q}^{2}\leq\left(\fint_{Q}|\phi|\;dv\right)^{2}\leq\left(\fint_{Q}a\phi^{2}\;dv\right)\left(\fint_{Q}a^{-1}\;dv\right).$

Therefore,

$\displaystyle 4(\phi)_{Q}^{2}\int_{Q}h\;dv$ $\displaystyle\leq
4\left(\fint_{Q}h\;dv\right)\left(\fint_{Q}a^{-1}\;dv\right)\left(\int_{Q}a\phi^{2}\;dv\right)\leq
4C|Q|^{-\frac{2}{d}}\int_{Q}a\phi^{2}\;dv.$

Now, we bound the first term on the right of (5.3) by means of Theorem 5.1, so

$\displaystyle\int_{Q}h\phi^{2}\;dv$ $\displaystyle\leq
4\mathcal{C}_{2,2}(Q,h,a)\int_{Q}a|\nabla\phi|^{2}\;dv+4C|Q|^{-\frac{2}{d}}\int_{Q}a\phi^{2}\;dv.$

Then, by Proposition 5.3, we conclude that

$\displaystyle\int_{Q}h\phi^{2}\;dv$ $\displaystyle\leq
4C\theta(r)\int_{Q}a|\nabla\phi|^{2}\;dv+4C|Q|^{-\frac{2}{d}}\int_{Q}a\phi^{2}\;dv,\;r:=|Q|^{\tfrac{1}{d}}.$

where $C=C(d,\gamma)$ and $\theta$ is as in Proposition 5.3. Adding up these
inequalities for each $Q$ of the form $r[0,1]^{d}+rz$, $z\in\mathbb{Z}^{d}$,

$\displaystyle\int_{\mathbb{R}^{d}}h\phi^{2}\;dv$ $\displaystyle\leq
4\theta(r)\int_{\mathbb{R}^{d}}a|\nabla\phi|^{2}\;dv+4Cr^{-2}\int_{\mathbb{R}^{d}}a\phi^{2}\;dv.$

Let $\varepsilon\in(0,\eta(1)/(4C))$, then there is some $r\in(0,1)$ such that
$4\theta(r)=\varepsilon$, namely

$\displaystyle\varepsilon=4C\langle
R\rangle^{|2+\gamma|}\|f_{\textnormal{in}}\|_{L^{p(\gamma,d,s)}}r^{\eta}.$

Indeed, this $r=r(\varepsilon)$ is such that

$\displaystyle r^{-2}=(4C)^{\frac{2}{\eta}}\langle
R\rangle^{\frac{2|2+\gamma|}{\eta}}\|f_{\textnormal{in}}\|_{L^{p(\gamma,d,s)}}^{\frac{2}{\eta}}\varepsilon^{-\frac{2}{\eta}}.$

Thus,

$\displaystyle\int_{\mathbb{R}^{d}}h\phi^{2}\;dv$
$\displaystyle\leq\varepsilon\int_{\mathbb{R}^{d}}a|\nabla\phi|^{2}\;dv+C(f_{\textnormal{in}},R)\varepsilon^{-\frac{2}{\eta}}\int_{\mathbb{R}^{d}}a\phi^{2}\;dv,$

and the corollary is proved.

∎

##  6\. Moser’s iteration

A $\varepsilon$-Poincare, inequality like the one obtained in Corollary 5.6,
when valid, yields an $L^{\infty}$ estimate for the respective solution $f$ of
(1.1). This is the case regardless of the value of $\gamma\in[-d,-2]$, as was
observed for the Landau equation in [13].

In Proposition 5.3 and Corollary 5.6 we have proved that the
$\varepsilon$-Poincaré inequality holds if $f\in
L^{\infty}(0,T,L^{p}(\mathbb{R}^{d}))$ for $p>\frac{d}{d+2+\gamma}$. In view
of Lemma 3.1, solutions to (1.1) belong to $L^{p}(\mathbb{R}^{d})$ with
$p>\frac{d}{d+2+\gamma}$ if the initial data belong to the same
$L^{p}(\mathbb{R}^{d})$ space and, most importantly, if
$\frac{d}{d+2+\gamma}\leq\frac{d+\gamma}{-\gamma-2}$. This last inequality
holds true for $\gamma\in(\gamma_{*},-2]$, with $\gamma_{*}$ the unique
solution to

$\frac{d}{d+2+\gamma}=\frac{d+\gamma}{-\gamma-2}.$

Observe that for $\gamma\in(-d,-2)$ the function $\gamma\mapsto
d/(d+\gamma+2)$ is strictly decreasing, while
$\gamma\mapsto(d+\gamma)/(-\gamma-2)$ is strictly increasing. At $\gamma=-d$
they are equal to $\frac{d}{2}$ and $0$, respectively and at $\gamma=-2$ they
are equal to $1$ and $+\infty$, respectively. It follows there is exactly one
$\gamma_{*}\in(-d,-2)$ where they agree. Alternatively, after solving the
respective quadratic equation one can see that $\gamma_{*}$ is given by the
formula

$\displaystyle\gamma_{*}=-1-\tfrac{3}{2}d+\tfrac{1}{2}\sqrt{5d^{2}-4d+4}.$

So, if $\gamma\in(\gamma_{*},-2]$, it should be no surprise that the
$L^{\infty}$ estimate for $f$ follows. As mentioned earlier, there are several
ways how to show that. We will follow the Moser’s approach introduced in [13],
with consists on estimating the norms

$\displaystyle\left(\int_{0}^{T}\int_{\mathbb{R}^{d}}a[f]\rho^{2}f^{p}\;dvdt\right)^{\frac{1}{p}},$

for ever increasing powers of $p$ and varying cut off functions $\rho$.
Although the arguments are very similar to those in [13], we go in detail over
their derivation here. This will complete the proof of Theorem 1.1.

We start by deriving an energy identity, then we will use the
$\varepsilon$-Poincaré inequality to control the most problematic term (the
integral involving a $h[f]f^{p}$ term), and arrive at an energy inequality.
This energy inequality, together with the space-time weighted inequality
(5.2), will be repeatedly used to bound the $L^{p}$ norms as $p\to\infty$.

######  Proposition 6.1.

Let $p>1$ and let $\rho\in C^{2}_{c}(\mathbb{R}^{d})$, then

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}^{d}}\rho^{2}f^{p}\;dv$
$\displaystyle=-\frac{4(p-1)}{p}\int_{\mathbb{R}^{d}}a[f]|\nabla(\rho
f^{p/2})|^{2}\;dv+\int_{\mathbb{R}^{d}}(c_{1}(p)|\nabla\rho|^{2}-\Delta\rho^{2})a[f]f^{p}\;dv$
$\displaystyle\;\;\;\;+(p-1)\int_{\mathbb{R}^{d}}(-\Delta
a[f])\rho^{2}f^{p}\;dv-c_{2}(p)\int_{\mathbb{R}^{d}}a[f]f^{p/2}(\nabla(\rho
f^{p/2}),\nabla\rho)\;dv,$

where $c_{1}(p)=4(p-\frac{(p-1)}{p})$, $c_{2}(p)=4(p-2\frac{p-1}{p})$.

###### Proof.

For simplicity we shall write $a$ instead of $a[f]$. From the equation and
integration by parts, we have

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}^{d}}\rho^{2}f^{p}\;dv$
$\displaystyle=p\int_{\mathbb{R}^{d}}\rho^{2}f^{p-1}\partial_{t}f\;dv=-p\int_{\mathbb{R}^{d}}(\nabla(\rho^{2}f^{p-1}),a\nabla
f-f\nabla a)\;dv.$

The integral on the right is equal to the sum of four terms, which we denote
$(\textnormal{I}),(\textnormal{II}),(\textnormal{III})$, and
$(\textnormal{IV})$, and which we now analyze one by one.

First, note that $(\nabla f^{p-1},\nabla f)=(p-1)f^{p-2}|\nabla
f|^{2}=(p-1)(4/p^{2})|\nabla f^{\frac{p}{2}}|$, therefore

$\displaystyle(\textnormal{I})=\int_{\mathbb{R}^{d}}\rho^{2}(\nabla
f^{p-1},a\nabla
f)\;dv=\frac{4(p-1)}{p^{2}}\int_{\mathbb{R}^{d}}\rho^{2}a|\nabla
f^{\frac{p}{2}}|^{2}\;dv.$

Next, we rewrite each of the other three terms using integration by parts, as
follows

$\displaystyle(\textnormal{II})=\int_{\mathbb{R}^{d}}f^{p-1}(\nabla\rho^{2},a\nabla
f)\;dv$
$\displaystyle=\frac{p-1}{p}\int_{\mathbb{R}^{d}}(\nabla\rho^{2},a\nabla
f^{p})\;dv$
$\displaystyle=-\frac{p-1}{p}\int_{\mathbb{R}^{d}}f^{p}(\nabla\rho^{2},\nabla
a)+f^{p}a\Delta\rho^{2}\;dv,$
$\displaystyle(\textnormal{III})=\int_{\mathbb{R}^{d}}\rho^{2}(\nabla
f^{p-1},-f\nabla a)\;dv$
$\displaystyle=\frac{p-1}{p}\int_{\mathbb{R}^{d}}\rho^{2}(\nabla f^{p},-\nabla
a)\;dv$ $\displaystyle=\frac{p-1}{p}\int_{\mathbb{R}^{d}}\rho^{2}f^{p}\Delta
a+f^{p}(\nabla\rho^{2},\nabla a)\;dv,$
$\displaystyle(\textnormal{IV})=\int_{\mathbb{R}^{d}}f^{p-1}(\nabla\rho^{2},-f\nabla
a)\;dv$ $\displaystyle=-\int_{\mathbb{R}^{d}}f^{p}(\nabla\rho^{2},\nabla
a)\;dv$ $\displaystyle=2\int_{\mathbb{R}^{d}}af^{p/2}(\nabla
f^{p/2},\nabla\rho^{2})\;dv+\int_{\mathbb{R}^{d}}af^{p}\Delta\rho^{2}\;dv.$

Adding these identities up, we have

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}^{d}}\rho^{2}f^{p}\;dv$
$\displaystyle=-\frac{4(p-1)}{p}\int_{\mathbb{R}^{d}}\rho^{2}a|\nabla
f^{p/2}|^{2}\;dv-\int_{\mathbb{R}^{d}}(\Delta\rho^{2})af^{p}\;dv$
$\displaystyle\;\;\;\;+(p-1)\int_{\mathbb{R}^{d}}(-\Delta
a)\rho^{2}f^{p}\;dv-2p\int_{\mathbb{R}^{d}}af^{p/2}(\nabla
f^{p/2},\nabla\rho^{2})\;dv.$

We use the elementary identity $\rho\nabla f^{p/2}=\nabla(\rho
f^{p/2})-f^{p/2}\nabla\rho$ and rewrite further,

$\displaystyle\int_{\mathbb{R}^{d}}\rho^{2}a|\nabla f^{p/2}|^{2}\;dv=$
$\displaystyle\int a|\nabla(\rho f^{p/2})|^{2}\;dv+\int
f^{p}a|\nabla\rho|^{2}\;dv$ $\displaystyle\;\;\;\;-2\int f^{p/2}a(\nabla(\rho
f^{p/2}),\nabla\rho)\;dv,$ $\displaystyle\int_{\mathbb{R}^{d}}af^{p/2}(\nabla
f^{p/2},\nabla\rho^{2})\;dv$
$\displaystyle=2\int_{\mathbb{R}^{d}}af^{p/2}(\nabla(\rho
f^{p/2}),\nabla\rho)\;dv-2\int_{\mathbb{R}^{d}}af^{p}|\nabla\rho|^{2}\;dv.$

In conclusion,

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}^{d}}\rho^{2}f^{p}\;dv$
$\displaystyle=-\frac{4(p-1)}{p}\int_{\mathbb{R}^{d}}a|\nabla(\rho
f^{p/2})|^{2}\;dv$
$\displaystyle\;\;\;\;+\int_{\mathbb{R}^{d}}(4(p-\frac{(p-1)}{p})|\nabla\rho|^{2}-\Delta\rho^{2})af^{p}\;dv$
$\displaystyle\;\;\;\;+(p-1)\int_{\mathbb{R}^{d}}(-\Delta
a)\rho^{2}f^{p}\;dv-4(p-2\frac{p-1}{p})\int_{\mathbb{R}^{d}}af^{p/2}(\nabla(\rho
f^{p/2}),\nabla\rho)\;dv.$

∎

Since for Theorem 1.1 we only consider $p$’s with $p>\frac{d}{d+\gamma+2}$,
for the rest of this section we will always assume that
$p>\frac{d}{d+\gamma+2}$. This has the added benefit of simplifying some of
the constants since we are now bounded away from $1$ (note that for
$\gamma<-2$, we have $\frac{d}{d+\gamma+2}>1$).

######  Proposition 6.2.

Let $\rho\in C^{2}_{c}(B_{R})$. Given any three times $T_{1}<T_{2}<T_{3}$ in
$[0,T]$ the quantity

$\displaystyle\sup\limits_{(T_{2},T_{3})}\int_{\mathbb{R}^{d}}\rho^{2}f^{p}\;dv\;$
$\displaystyle+\;\frac{(p-1)}{p}\int_{T_{2}}^{T_{3}}\int_{\mathbb{R}^{d}}a[f]|\nabla(\rho
f^{p/2})|^{2}\;dvdt$

is not greater than

(6.1)
$\displaystyle\left(\frac{1}{T_{2}-T_{1}}+C_{1}\right)\int_{T_{1}}^{T_{3}}\int\rho^{2}f^{p}\;dvdt+C(d,\gamma)p^{2}\int_{T_{1}}^{T_{3}}\int_{\mathbb{R}^{d}}a[f]f^{p}(|\nabla\rho|^{2}+|\Delta\rho^{2}|)\;dvdt,$

where $C_{1}=C_{1}(f_{\textnormal{in}},R,p)$.

###### Proof.

Take the identity in Proposition 6.1. Per Young’s inequality, for every
$\varepsilon>0$ we have

$\displaystyle 2af^{p/2}|(\nabla(\rho f^{p/2}),\nabla\rho)|\leq\varepsilon
a|\nabla(\rho f^{p/2})|^{2}+\varepsilon^{-1}af^{p}|\nabla\rho|^{2}.$

For $\varepsilon=\frac{2(p-1)}{p}\frac{1}{2(p-2\frac{p-1}{p})}$ in particular,
it follows that

$\displaystyle-4(p-2\frac{p-1}{p})af^{p/2}(\nabla(\rho
f^{p/2}),\nabla\rho)\leq\frac{2(p-1)}{p}a|\nabla(\rho
f^{p/2})|^{2}+2(p-2\frac{p-1}{p})^{2}\frac{p-1}{p}af^{p}|\nabla\rho|^{2}.$

Therefore,

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}^{d}}\rho^{2}f^{p}\;dv$
$\displaystyle\leq-\frac{2(p-1)}{p}\int_{\mathbb{R}^{d}}a|\nabla(\rho
f^{p/2})|^{2}\;dv$
$\displaystyle\;\;\;\;+\int_{\mathbb{R}^{d}}(4(p-\frac{(p-1)}{p})|\nabla\rho|^{2}-\Delta\rho^{2})af^{p}\;dv$
$\displaystyle\;\;\;\;+(p-1)\int_{\mathbb{R}^{d}}(-\Delta
a)\rho^{2}f^{p}\;dv+2(p-2\frac{p-1}{p})^{2}\frac{p}{2(p-1)}\int_{\mathbb{R}^{d}}af^{p}|\nabla\rho|^{2}\;dv.$

Combining various terms, this inequality can be written more succinctly as

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}^{d}}\rho^{2}f^{p}\;dv+\frac{2(p-1)}{p}\int_{\mathbb{R}^{d}}a|\nabla(\rho
f^{p/2})|^{2}\;dv$
$\displaystyle\;\;\;\;\leq\int_{\mathbb{R}^{d}}(C(p)|\nabla\rho|^{2}-\Delta\rho^{2})af^{p}\;dv+(p-1)\int_{\mathbb{R}^{d}}(-\Delta
a)\rho^{2}f^{p}\;dv,$

where $C(p)=2(p-2\frac{p-1}{p})^{2}\frac{p}{2(p-1)}+4(p-\frac{p-1}{p})$. Since
$p\geq\frac{d}{d+\gamma+2}>2$, it is elementary that

$\displaystyle C(p)\leq C(d,\gamma)p^{2}.$

Now we apply Corollary 5.6 with $\varepsilon=\min\\{\frac{1}{p},\eta(1)\\}$.
this yields

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}^{d}}\rho^{2}f^{p}\;dv$
$\displaystyle+\frac{p-1}{p}\int_{\mathbb{R}^{d}}a|\nabla(\rho
f^{p/2})|^{2}\;dv$
$\displaystyle\leq\int_{\mathbb{R}^{d}}(C(d,\gamma)p^{2}|\nabla\rho|^{2}-\Delta\rho^{2})af^{p}\;dv+C(f_{\textnormal{in}},\gamma,p_{0},R)p^{m}\int_{\mathbb{R}^{d}}\rho^{2}f^{p}\;dv.$

Integrate now in $t\in(t_{1},t_{2})$ and take, on both sides, the sup with
respect to $t_{2}$ in $(T_{2},T_{3})$ and average with respect to $t_{1}$ in
$(T_{1},T_{2})$. From here it results that

$\displaystyle\sup\limits_{(T_{2},T_{3})}\int_{\mathbb{R}^{d}}\rho^{2}f^{p}\;dv\;+\;\frac{(p-1)}{p}\int_{T_{2}}^{T_{3}}\int_{\mathbb{R}^{d}}a|\nabla(\rho
f^{p/2})|^{2}\;dvdt$

is no larger than

$\displaystyle\left(\frac{1}{T_{2}-T_{1}}+{C(f_{\textnormal{in}},\gamma,p_{0},R)p^{m}}\right)\int_{T_{1}}^{T_{3}}\int_{\mathbb{R}^{d}}\rho^{2}f^{p}\;dvdt+C(d,\gamma)p^{2}\int_{T_{1}}^{T_{3}}\int_{\mathbb{R}^{d}}af^{p}(|\nabla\rho|^{2}+|\Delta\rho^{2}|)\;dvdt.$

∎

All that remains of the proof of Theorem 1.1 is covered by the following
lemma. This lemma amounts to an $L^{p}\to L^{\infty}$ estimate in the spirit
of De Giorgi-Nash-Moser theory, following Moser’s approach. In what follows,
recall that the exponent $q=2(1+\tfrac{2}{d})$ was defined in (5.1) and that
$p>\frac{d}{d+\gamma+2}$.

######  Lemma 6.3.

Let $R\geq 1$ and $T>0$, then for any $v\in B_{R}$ and $t\in(T/2,T)$ we have

$\displaystyle f(v,t)\leq
C(f_{\textnormal{in}},p,\gamma,R,T)\left(1+\frac{1}{T}\right)^{\frac{1}{p}\frac{q}{q-2}}\left(\int_{T/4}^{T}\int_{B_{2R}}a[f]f^{p}\;dvdt\right)^{\frac{1}{p}}.$

###### Proof.

We introduce the sequences

$\displaystyle
T_{n}:=\frac{1}{4}\left(2-\frac{1}{2^{n}}\right)T,\;\;R_{n}:=\left(1+\frac{1}{2^{n}}\right)R,\;\;p_{n}:=p\left(\frac{q}{2}\right)^{n},\;\;\forall\;n\in\mathbb{N}.$

We also pick a sequence of functions $\rho_{n}\in C^{2}_{c}(B_{R_{n}}(0))$
such that

(6.5) $\displaystyle\left\\{\begin{array}[]{ll}&0\leq\rho_{n}\leq
1\textnormal{ in }B_{R_{n}}(0)\setminus B_{R_{n+1}}(0),\;\;\rho_{n}\equiv
1\textnormal{ in }B_{R_{n+1}}(0),\\\ &|\nabla\rho_{{n+1}}|\leq
C(d)R^{-1}2^{n},\\\ &|\Delta\rho^{2}_{n+1}|\leq
C(d)R^{-2}4^{n}.\end{array}\right.$

Now, for each $n\geq 0$, let $E_{n}$ denote the quantity,

$\displaystyle
E_{n}:=\left(\int_{T_{n}}^{T}\int_{\mathbb{R}^{d}}\rho_{n}^{q}a[f]f^{p_{n}}\;dvdt\right)^{\frac{1}{p_{n}}}.$

As it is standard for divergence elliptic equations, we are going to derive a
recursive relation for $E_{n}$. To do this, first note $E_{n+1}$ may be
written as,

$\displaystyle
E_{n+1}^{p_{n}}=\left(\int_{T_{n+1}}^{T}\int_{\mathbb{R}^{d}}a[f]\left(\rho_{{n+1}}f^{\frac{p_{n}}{2}}\right)^{q}\;dvdt\right)^{\frac{2}{q}}.$

Thanks to the space-time inequality (5.2) we have

$\displaystyle
C(d,\gamma)^{-1}E_{n+1}^{p_{n}}\leq\sup\limits_{(T_{n+1},T)}\left\\{\int_{\mathbb{R}^{d}}\rho_{n+1}^{2}f^{p_{n}}(t)\;dv\right\\}+\frac{(p_{n}-1)}{p_{n}}\int_{T_{n+1}}^{T}\int_{\mathbb{R}^{d}}a[f]|\nabla(\rho_{n+1}f^{\frac{p_{n}}{2}})|^{2}\;dvdt.$

Then, the energy inequality from Proposition 6.2 says that
$C(d,\gamma)^{-1}E_{n+1}^{p_{n}}$ is no larger than

$\displaystyle\left(\frac{2^{n+2}}{T}+C(f_{\textnormal{in}},\gamma,R)p_{n}^{m}\right)\int_{T_{n}}^{T}\int_{\mathbb{R}^{d}}\rho_{n+1}^{2}f^{p_{n}}\;dvdt$
$\displaystyle+C(d,\gamma)p_{n}^{2}\int_{T_{n}}^{T}\int_{\mathbb{R}^{d}}a[f]f^{p_{n}}(|\nabla\rho_{n+1}|^{2}+|\Delta\rho_{n+1}^{2}|)\;dvdt.$

Note that the first integral is no larger than

$\displaystyle\left(\frac{2^{n+2}}{T}+C(f_{\textnormal{in}},\gamma,p_{0},R)p_{n}^{m}\right)\frac{1}{\min\limits_{B_{R}\times(0,T)}a[f]}\int_{T_{n}}^{T}\int_{\mathbb{R}^{d}}\rho_{n}^{q}af^{p_{n}}\;dv.$

Next, again thanks to $\rho_{n}\equiv 1$ in the support of $\rho_{n+1}$, and
in particular

$\displaystyle|\nabla\rho_{n+1}|^{2}\leq
CR^{-2}4^{n}\rho_{n}^{q},\;|\Delta\rho_{n+1}^{2}|\leq
CR^{-2}4^{n}\rho_{n}^{q},$

so the second integral above is no larger than

$\displaystyle
C(d,\gamma)p_{n}^{2}R^{-2}4^{n}\int_{T_{n}}^{T}\int_{\mathbb{R}^{d}}a[f]\rho_{n}^{q}f^{p_{n}}\;dvdt.$

In conclusion,

$\displaystyle E_{n+1}\leq
C\left[\left(\frac{2^{n+2}}{T}+C(f_{\textnormal{in}},\gamma,R)p_{n}^{m}\right)C(f_{\textnormal{in}},T)R^{-2-\gamma}+C(d,\gamma)p_{n}^{2}R^{-2}4^{n}\right]^{\frac{1}{p_{n}}}E_{n}.$

Set

$\displaystyle b$ $\displaystyle=\max\\{4q,(q/2)^{m}\\},$ $\displaystyle m$
$\displaystyle=C\left[(T^{-1}+C(f_{\textnormal{in}},\gamma,R)p^{m})C(f_{\textnormal{in}},R^{|2+\gamma|})+C(d,\gamma)p^{2}R^{-2}\right].$

Then

$\displaystyle E_{n+1}\leq b^{\frac{n}{p_{n}}}m^{\frac{1}{p_{n}}}E_{n}.$

This recursive relation, and a straightforward induction argument, yield that

$\displaystyle E_{n}\leq
b^{\sum\limits_{k=0}^{n-1}\frac{k}{p_{k}}}m^{\sum\limits_{k=0}^{n-1}\frac{1}{p_{k}}}E_{0}.$

Since

$\displaystyle\sum\limits_{k=0}^{n-1}\frac{k}{p_{k}}=\frac{1}{p}\sum\limits_{k=0}^{n-1}k\left(\frac{2}{q}\right)^{k}\leq\frac{1}{p}\sum\limits_{k=0}^{\infty}k\left(\frac{2}{q}\right)^{k}=\frac{1}{p}\frac{2q}{(q-2)^{2}},$
$\displaystyle\sum\limits_{k=0}^{n-1}\frac{1}{p_{k}}=\frac{1}{p}\sum\limits_{k=0}^{n-1}\left(\frac{2}{q}\right)^{k}\leq\frac{1}{p}\sum\limits_{k=0}^{\infty}\left(\frac{2}{q}\right)^{k}=\frac{1}{p}\frac{q}{q-2},$

we conclude that

(6.6) $\displaystyle E_{n}\leq
b^{\frac{1}{p}\frac{2q}{(q-2)^{2}}}m^{\frac{1}{p}\frac{q}{q-2}}E_{0}.$

Observe that

$\displaystyle
E_{0}\leq\left(\int_{T/4}^{T}\int_{B_{2R}(0)}a[f]f^{p}\;dvdt\right)^{1/p}.$

Now, since $\eta_{n}\geq 1$ in $B_{R}$ and $T_{n}\leq T/2$ for all $n$, it
follows that

$\displaystyle
E_{n}\geq\left(\int_{T/2}^{T}\int_{B_{2R}}a[f]f^{p_{n}}\;dvdt\right)^{\frac{1}{p_{n}}}.$

Considering that $a>0$ everywhere, it follows that

$\displaystyle\limsup\limits_{n\to\infty}E_{n}\geq\|f\|_{L^{\infty}(B_{R}\times(T/2,T))}.$

This, together with (6.6) concludes the proof of the lemma, and of Theorem
1.1.

∎

## References

  * [1] R. Alexandre, J. Liao, and C. Lin.  Some a priori estimates for the homogeneous Landau equation with soft potentials.  KRM Volume 8, Issue 4, December 2015, 617 - 650, 2013.
  * [2] J. Bedrossian, M. Gualdani, and S. Snelson.  Non-existence of some approximately self-similar singularities for the landau, vlasov-poisson-landau, and boltzmann equations.  Arxiv Preprint.
  * [3] J.L. Chern and M. Gualdani.  Uniqueness of higher integrable solution to the Landau equation with Coulomb interactions.  To appear in Math. Res. Let., 2021.
  * [4] L. Desvillettes, Ling-Bing He, and Jin-Cheng Jiang.  A new monotonicity formula for the spatially homogeneous Landau equation with Coulomb potential and its applications.  Arxiv preprint.
  * [5] L. Desvillettes and C. Villani.  On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness.  Comm. Partial Differential Equations, 25(1-2):179–259, 2000.
  * [6] L. Desvillettes and C. Villani.  On the spatially homogeneous Landau equation for hard potentials. II. $H$-theorem and applications.  Comm. Partial Differential Equations, 25(1-2):261–298, 2000.
  * [7] N. Fournier and H. Guérin.  Well-posedness of the spatially homogeneous Landau equation for soft potentials.  Journal of Functional Analysis, 256(8):2542–2560, 2009.
  * [8] Nicolas Fournier.  Uniqueness of bounded solutions for the homogeneous Landau equation with a Coulomb potential.  Communications in Mathematical Physics, 299(3):765–782, 2010.
  * [9] Nassif Ghoussoub and Amir Moradifam.  Functional Inequalities: New Perspectives and New Applications: New Perspectives and New Applications, volume 187.  American Mathematical Soc., 2013.
  * [10] F. Golse, M. Gualdani, C.Imbert, and A. Vasseur.  Partial regularity in time for the space homogeneous Landau equation with Coulomb potential.  To appear in Annales scientifiques de l’ ENS (2021), 2019.
  * [11] P. Gressman, J. Krieger, and R. Strain.  A non-local inequality and global existence.  Advances in Mathematics, 230(2):642–648, 2012.
  * [12] M. Gualdani and N. Guillen.  Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential.  Analysis and PDE, 9(8):1772–1809, 2016.
  * [13] M. Gualdani and N. Guillen.  On Ap weights and regularization effects for homogeneous Landau equations.  Calc. Var. Partial Differential Equations, 1(58), 2019.
  * [14] J. Krieger and R. Strain.  Global solutions to a non-local diffusion equation with quadratic non-linearity.  Comm. Partial Differential Equations, 37(4):647–689, 2012.
  * [15] E. Sawyer and R. Wheeden.  Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces.  American Journal of Mathematics, 114(4):813 – 874, 1992.
  * [16] L. Silvestre.  Upper bounds for parabolic equations and the Landau equation.  J. Differential Equations, 262 (2017), no. 3, 3034 - 3055., 2017\.
  * [17] K-C. Wu.  Global in time estimates for the spatially homogeneous Landau equation with soft potentials.  J. Funct. Anal. 266, no. 5, 3134-3155, 2014.