# The vertical-velocity skewness in the inertial sublayer of turbulent wall
flows

Elia Buono1,2∗ Gabriel Katul1 Michael Heisel3 Davide Vettori2 Davide Poggi2
Cosimo Peruzzi4 Costantino Manes2 1Department of Civil and Environmental
Engineering, Duke University, Durham, NC, USA  2Dipartimento di Ingegneria
dell’Ambiente, del Territorio e delle Infrastrutture, Politecnico di Torino,
Torino, Italia  3School of Civil Engineering, University of Sydney, Sydney,
Australia  4Area for Hydrology, Hydrodynamics, Hydromorphology and Freshwater
Ecology, Italian Institute for Environmental Protection and Research, Rome,
Italy  Elia Buono<EMAIL_ADDRESS>
###### Abstract

We provide empirical evidence that within the inertial sub layer of adiabatic
turbulent flows over smooth walls, the skewness of the vertical velocity
component $Sk_{w}$ displays universal behaviour, being constant and
constrained within the range $Sk_{w}\approx 0.1-0.16$, regardless of flow
configuration and Reynolds number. A theoretical model is proposed to explain
the observed behaviour, including the observed range of variations of
$Sk_{w}$. The model clarifies why $Sk_{w}$ cannot be predicted from down-
gradient closure approximations routinely employed in meteorological and
climate models whereby $Sk_{w}$ impacts cloud formation and dispersion
processes. The model also offers an alternative and implementable approach.

††preprint: APS/123-QED

z

##  I Introduction

Much of the effort devoted to the study of adiabatic and hydrodynamically
smooth wall turbulence has focused on the characterization of velocity
statistics within the so-called logarithmic or inertial sublayer (ISL). The
Attached Eddy Model (AEM), which is probably the most cited model for ISL-
turbulence, predicts that first and second order velocity statistics can be
described as [1, 2, 3]:

$\overline{u}^{+}=\frac{1}{\kappa}\log\left({z^{+}}\right)+A;~{}\sigma_{u}^{2+}=A_{u}-B_{u}\log\left(\frac{z}{\delta}\right);$
(1)

and, a less studied outcome, $\sigma_{w}^{2+}=A_{w}^{2}$, where $u$ and $w$
are the longitudinal and wall-normal velocity component, respectively; $z$ is
the wall normal coordinate; $\sigma_{u}=\sqrt{\overline{u^{\prime 2}}}$ and
$\sigma_{w}=\sqrt{\overline{w^{\prime 2}}}$ are the standard deviation of $u$
and $w$ respectively; primes identify fluctuations due to turbulence around
the mean; the overline represents averaging over coordinates of statistical
homogeneity; the plus index indicates classical inner scaling whereby
velocities and lengths are normalized with the friction velocity $u_{*}$ and
viscous length scale $\nu/u_{*}$, respectively, with $\nu$ being the kinematic
viscosity of the fluid; $\delta$ is the outer length scale of the flow;
$\kappa$, $A$, $A_{u}$, $A_{w}$, $B_{u}$ are coefficients that are thought to
attain asymptotic constant values at very large Reynolds numbers
$Re_{\tau}=u_{*}\delta/\nu$ [4, 2, 5]. The AEM has been extended to velocity
moments of any order as well as cross-correlations between different velocity
components thereby providing an expanded picture of ISL flow statistics [6].
However, convincing empirical support for the aforementioned theoretical
predictions is limited to the statistics of $u$ [2, 4, 7, 8, 9]. In contrast,
the statistics of $w$ have been much less reported and investigated, partly
because of the technical difficulties associated with accurately measuring $w$
in the near wall region of laboratory flows at high $Re_{\tau}$. As a result,
theoretical predictions of $w-$statistics have received mixed support from the
literature [10, 11, 12] and higher order moments of $w^{\prime}$ are rarely
reported but with few notable exceptions [13, 14, 15, 16, 17]. We argue that
this overlook contributed to hide a universal property of ISL turbulence,
which is herein reported and discussed. The aim of this letter is to
demonstrate that the skewness of $w^{\prime}$, $Sk_{w}=\overline{w^{\prime
3}}/\sigma_{w}^{3}$, is constant (i.e. $z-$independent) and robust to
variations in $Re_{\tau}$ within the ISL. Moreover, a theoretical model that
explains this observed behaviour and links $Sk_{w}$ to established turbulence
constants is proposed, leading to satisfactory predictions. Finally, we note
that the asymmetry in the probability density function of $w^{\prime}$, as
quantified by $Sk_{w}$, cannot be accounted for with gradient-diffusion
representations routinely employed in meteorological and climate models [18].
Rectifying this limitation is of significance because $Sk_{w}$ was shown to
impact cloud formation [19, 20, 21] and dispersion processes [22, 23, 24, 25].

Figure 1 reports the variations of $Sk_{w}$ with normalized wall-normal
distance ($z/\delta$) using data from Direct Numerical Simulations (DNS) [26]
and laboratory experiments [17] pertaining to flat plate turbulent boundary
layers (TBLs). Data from experiments on uniform [27] and weakly non-uniform
turbulent open channel flows [15, 16], whereby accurate measurements of $w$
are available are also included. This data refers to flows within the low to
medium range of $Re_{\tau}$ (Table 1). A reference value of $Sk_{w}=0.1$ is
added to the figure as often reported for atmospheric surface layers in
adiabatic conditions across multiple heights and surface cover [28]. This
value is representative of flows at extremely high $Re_{\tau}$. A region of
constant $Sk_{w}$ weakly varying between 0.1 and 0.16 (here weakly means that
variations are much smaller than those displayed by $Sk_{w}$ over the entire
flow domain) is evident in all profiles within the range
$2.6\sqrt{Re_{\tau}}\nu/u_{*}$ up to $0.15-0.25\delta$, which is often
associated with the ISL [29, 30, 11]. This finding is remarkable given the
large differences in $Re_{\tau}$, measurement techniques and experimental
facilities used. In what follows, a theoretical model that predicts and
explains such a behaviour is provided.

##  II Theory

To explain the observed behaviour of $Sk_{w}$, a stationary and planar
homogeneous incompressible flow in the absence of subsidence is considered for
$\overline{w^{\prime 3}}$. For these conditions, the model can be derived from
the Reynolds averaged Navier-Stokes equations and is given as [31, 32]

$\displaystyle\frac{\partial\overline{w^{\prime 3}}}{\partial t}=0$
$\displaystyle=+\overbrace{3\sigma_{w}^{2}\frac{\partial\sigma_{w}^{2}}{\partial
z}}^{{\rm Source/Sink}}-\overbrace{\frac{\partial\overline{w^{\prime}w^{\prime
3}}}{\partial z}}^{{\rm Turbulent~{}transport}}$ (2)
$\displaystyle-\underbrace{3\left(\overline{w^{\prime}w^{\prime}\frac{\partial
p^{\prime}}{\partial z}}\right)}_{{\rm
Pressure~{}velocity~{}destruction}}-\underbrace{2\nu\left(3\overline{w^{\prime}\frac{\partial
w^{\prime}}{\partial z}\frac{\partial w^{\prime}}{\partial z}}\right)}_{{\rm
Viscous~{}destruction}},$

where $t$ is time and $p^{\prime}$ is the pressure deviation from the mean or
hydrostatic state normalized by a constant fluid density $\rho$.

Figure 1: Variation of the vertical velocity skewness $Sk_{w}$ with normalized
wall-normal distance $z/\delta$. The dashed line is $Sk_{w}=0.16$ and the
dotted line is $Sk_{w}=0.10$. Data sources and references are summarized in
Table 1. Red symbols and lines identify the ISL range. For HL1 and HL2, near
wall data are not reported due to issues with velocity measurements associated
with an excessive sampling volume of the x-probe employed in the experiments
[17].

The first two terms on the right-hand side of equation 2 arise from inertial
effects or convective acceleration, the third and fourth terms arise due to
interactions between $w^{\prime}w^{\prime}$ and the forces acting on a fluid
element ($p^{\prime}/\rho$ and viscous stresses). A quasi-normal approximation
for the fourth moment is used [33] so that $FF_{w}=\overline{w^{\prime
4}}/(\sigma_{w})^{4}=3+a$ and the overall inertial term simplifies to

$-\frac{\partial\overline{w^{\prime 4}}}{\partial
z}+3\sigma_{w}^{2}\frac{\partial\sigma_{w}^{2}}{\partial
z}=-(3+2a)\sigma_{w}^{2}\frac{\partial\sigma_{w}^{2}}{\partial z},$ (3)

where $a\neq 0$ allows for deviations from Gaussian tails ($a=0$ recovers a
Gaussian flatness factor). Usage of a quasi-Gaussian approximation to close a
fourth (and even) moment budget makes no statement on the asymmetry (or odd
moments) of the $w^{\prime}$ probability density function, only that large-
scale intermittency is near-Gaussian, a finding well supported in the
literature [34] and many phenomenological approaches [6]. Models for the
pressure velocity and viscous destruction terms are now needed to integrate
equation 2. A return to isotropy (or Rotta) model [35] given by

$\displaystyle-2\overline{w^{\prime}\frac{\partial p^{\prime}}{\partial
z}}=\frac{C_{R}}{\tau}\left(\frac{\overline{q}}{3}-\sigma_{w}^{2}\right),$ (4)

may be used to derive an expression for the pressure-velocity destruction term
in in equation 2 where
$q=u^{\prime}u^{\prime}+v^{\prime}v^{\prime}+w^{\prime}w^{\prime}$ is twice
the instantaneous turbulent kinetic energy, $\overline{q}=2K$, $K$ is the
averaged turbulent kinetic energy, $v^{\prime}$ is the lateral turbulent
velocity, and $C_{R}=1.8$ is a well established constant, the Rotta constant
[36]. The $C_{R}$ relates the so-called relaxation time
$\tau=\overline{q}/\overline{\epsilon}$ to the time it takes for isotropy to
be attained at the finest scales, where $\overline{\epsilon}$ is the mean
turbulent kinetic energy dissipation rate. In deriving such an extension, the
non-averaged form of equation 4 is first multiplied by $(3/2)w^{\prime}$ and
then averaged to yield

$\displaystyle-3\left(\overline{w^{\prime}w^{\prime}\frac{\partial
p^{\prime}}{\partial
z}}\right)=\frac{3}{2}\frac{C_{R}}{\tau_{s}}\left(\frac{\overline{w^{\prime}q}}{3}-\overline{w^{\prime}w^{\prime
2}}\right),$ (5)

where $\tau_{s}$ is another decorrelation time that can differ from $\tau$.
The difference between $\tau$ and $\tau_{s}$ arises because when multiplying
the instantaneous form of equation 4 by $w^{\prime}$ to arrive at the
instantaneous form of equation 5, the correlation between $q$ and $w^{\prime}$
must be considered after averaging. While expected to be small relative to the
pressure-velocity interaction term, the viscous destruction contribution is
herein retained and represented as [32]

$-2\nu\left(3\overline{w^{\prime}\frac{\partial w^{\prime}}{\partial
z}\frac{\partial w^{\prime}}{\partial
z}}\right)=-2\overline{\epsilon^{\prime}w^{\prime}}=-c_{2}\frac{\overline{w^{\prime}q}}{\tau_{s}},$
(6)

where $c_{2}$ is a similarity constant, and $\epsilon^{\prime}\sim q/\tau_{s}$
is the fluctuating dissipation rate around $\overline{\epsilon}$. Inserting
these approximations into equation 2 yields,

$\overline{w^{\prime
3}}=-\frac{2}{3}\frac{(3+2a)\tau_{s}\sigma_{w}^{2}}{C_{R}}\frac{\partial\sigma_{w}^{2}}{\partial
z}+\overline{w^{\prime}q}\left(\frac{1}{3}-\frac{2c_{2}}{3C_{R}}\right).$ (7)

Table 1:  Overview of smooth-wall boundary layer experiments (OC = open channel/flumes, WT = wind tunnel) and direct numerical simulation* in Figure 1. The $Re_{\tau}=\delta u_{*}/\nu$ is the friction Reynolds number, $B_{u}$ and $A_{w}$ were computed from data using AEM. For the DNS, the highest and lowest $Re_{\tau}$ are shown given the small variability in $B_{u}$ (0.91-0.94) and $A_{w}$ (1.15-1.17). The computed $Sk_{w}$ using equation 11 is also presented. Source | Data set | Flow | $Re_{\tau}$ | $B_{u}$ | $A_{w}$ | $Sk_{w}$
---|---|---|---|---|---|---
Manes  _et al._ [15] | MN | OC | 2160 | 0.58 | 1.06 | 0.11
Sillero  _et al._ [26] | DNS* |  | 1307 | 0.85 | 1.15 | 0.13
|  |  | 2000 | 0.86 | 1.17 | 0.12
Heisel  _et al._ [17] | HL1 | WT | 3800 | 0.85 | 0.96 | 0.21
| HL2 | WT | 4700 | 0.63 | 1.00 | 0.15
Poggi  _et al._ [27] | PG1 | OC | 1232 | 0.73 | 0.90 | 0.23
| PG2 | OC | 1071 | 0.78 | 1.02 | 0.17
| PG3 | OC | 845 | 1.03 | 0.90 | 0.33
Peruzzi  _et al._ [16] | PR1 | OC | 2240 | 0.78 | 1.12 | 0.13
| PR2 | OC | 999 | 0.63 | 1.06 | 0.12
| PR3 | OC | 1886 | 0.85 | 1.06 | 0.16

A model for $\overline{w^{\prime}q}$ is further needed to infer $Sk_{w}$. To
arrive at this model, the $K$ budget for the same flow conditions leading to
equation 2 are employed. When mechanical production is balanced by
$\overline{\epsilon}$ as common to the ISL, the $K$ budget leads to two
outcomes [37]

$u_{*}^{2}\frac{\partial\overline{u}}{\partial
z}-\overline{\epsilon}=0;~{}~{}-\frac{1}{2}\frac{\partial\overline{w^{\prime}q}}{\partial
z}=0.$ (8)

The height-independence of $\overline{w^{\prime}q}$ is suggestive that it must
be controlled by local conditions and a down-gradient approximation is
justified given by [37]

$-\frac{1}{2}\overline{w^{\prime}q}=\kappa zu_{*}\frac{\partial K}{\partial
z}.$ (9)

The model in equation 9 has received experimental support even for rough-wall
turbulent boundary layers and across a wide range of Reynolds numbers and
surface roughness values [37]. Noting that $K\approx\sigma_{u}^{2}$ yields

$\displaystyle\overline{w^{\prime 3}}=$
$\displaystyle-\frac{2}{3}\left[K_{t,w}\frac{\partial\sigma_{w}^{2}}{\partial
z}+K_{t,u}\frac{\partial\sigma_{u}^{2}}{\partial z}\right];$ (10)
$\displaystyle K_{t,w}=$
$\displaystyle\frac{(3+2a)\tau_{s}\sigma_{w}^{2}}{C_{R}};~{}K_{t,u}=\kappa
zu_{*}\left(1-\frac{2c_{2}}{C_{R}}\right),$

where $K_{t,w}$ and $K_{t,u}$ are eddy viscosity terms. These two eddy
viscosity values become comparable in magnitude when setting $\tau_{s}=\kappa
z/u_{*}$ (i.e. following classical ISL scaling) and $C_{R}=1.8$ \- its
accepted value [36] as expected in the ISL. To determine
$\partial{\sigma_{w}^{2}}/{\partial z}$, the mean vertical velocity equation
is considered for the same idealized flow conditions as equation 2. This
consideration results in ${\partial\sigma_{w}^{2}}/{\partial
z}=-({1}/{\rho})({\partial\overline{P}}/{\partial z})-g$, where $g$ is the
gravitational acceleration. When $\overline{P}=-\rho gz$ (i.e. hydrostatic),
$\partial\sigma_{w}^{2}/\partial z=0$ or $A_{w}$ is constant in $z$ within the
ISL. That is, the AEM requires $\overline{P}$ to be hydrostatic. However, the
AEM precludes a $\partial\sigma_{u}^{2}/\partial z=0$ in the ISL. In fact, the
AEM predicts a $\partial\sigma_{u}^{2}/\partial z=-u_{*}^{2}B_{u}/z$ when the
$Re_{\tau}$ is very large as expected in the ISL of an adiabatic atmosphere.
Inserting this estimate into equation 10, setting $u_{*}=\sigma_{w}/A_{w}$ and
momentarily ignoring $\partial{\sigma_{w}^{2}}/{\partial z}$ relative to
$\partial\sigma_{u}^{2}/\partial z$ as a simplification consistent with the
AEM, leads to

$Sk_{w}=\frac{\overline{w^{\prime
3}}}{\sigma_{w}^{3}}=\frac{2}{3}\left(1-\frac{2c_{2}}{C_{R}}\right)\frac{\kappa
B_{u}}{A_{w}^{3}}.$ (11)

This equation is the sought outcome. The term $2c_{2}/C_{R}$ reflects the
relative importance of the pressure-velocity to viscous destruction terms.
Pressure-velocity destruction effects are far more efficient than viscous
effects supporting the argument that $2c_{2}/C_{R}<<1$ at very high
$Re_{\tau}$ [38] such as the atmosphere. This implies that the numerical value
of $Sk_{w}$, as obtained from equation 11, depends on three well established
phenomenological constants, namely $\kappa$, $A_{w}$, and $B_{u}$ [8, 3, 9],
which, in turn, may depend weakly on $Re_{\tau}$ and the flow type. Equation
11 is also insensitive to the choices made for $\tau_{s}$, because the AEM
requires $\partial\sigma_{w}^{2}/\partial z=0$.

##  III Discussion and Conclusion

Equation 11 demonstrates two inter-related aspects about $Sk_{w}$ in the ISL:
(i) why $Sk_{w}$ is positive and constant with $z$, and (ii) why conventional
gradient diffusion approximations fail to predict $\overline{w^{\prime 3}}$
from $\partial\sigma_{w}^{2}/\partial z$. Regarding the first, equation 11
predicts that $Sk_{w}>0$ consistent with the paradigm that ejective eddy
motions ($w^{\prime}>0,u^{\prime}<0$) are more significant in momentum
transfer than sweeping motions ($w^{\prime}<0,u^{\prime}>0$) within the ISL.
This assertion is supported by numerous experiments and simulations [39, 40,
17] and adds further confidence in the physics associated with the derivation
of equation 11. Moreover, values of the constants in equation 11 for flat
plate turbulent boundary layers (TBL) at $Re_{\tau}\to\infty$ correspond to
$\kappa=0.39$, $A_{w}=1.33$, and $B_{u}=1.26$ [2, 9]. Upon further setting
$c_{2}=0.1$ and $C_{R}=1.8$ (conventional values), leads to $Sk_{w}=0.12$.
This estimate compares well with $Sk_{w}=0.1$ reported for the ISL in the
adiabatic atmosphere [28, 41]. At the low to moderate $Re_{\tau}$ pertaining
to the experiments and DNS associated with Figure 1, equation 11 cannot be
used to estimate $Sk_{w}$ using asymptotic values of $A_{w}$ and $B_{u}$.
However, Figure 1 shows that these flows attain similar (i.e. slightly higher)
and reasonably z-independent values of $Sk_{w}$. To explain this behaviour,
the DNS data are used as they allow exploring theoretical predictions offered
by equation 10 with reliable estimations of $\sigma_{w}^{2}$ and
$\sigma_{u}^{2}$ vertical gradients. Figure 2 indicates that, for mostof the
ISL, the first term on the right hand side of the equation is an order of
magnitude smaller than the second and can be discarded as predicted by the AEM
and advocated in the proposed theory. Predictions of $Sk_{w}$ obtained from
the second term are excellent in the ISL and resemble the observed
z-independent behaviour. Besides providing further confidence on the proposed
theory, this result indicates that, since $K_{t,u}$ is directly proportional
to $z$, $\partial\sigma_{u}^{2}/\partial z$ must overall scale as $\sim 1/z$,
as predicted by the AEM. Hence, we argue that the AEM represents a reasonable
approximation provided $B_{u}$ and $A_{w}$ are adjusted to accommodate for low
$Re_{\tau}$ effects. As shown in Figure 3, this is the case for both DNS and
laboratory data.

Figure 2: Panel a): Variation of the vertical velocity skewness $Sk_{w}$ with
normalized wall-normal distance $z/\delta$ from DNS [26]; panel b): $Sk_{w,m}$
is the modelled vertical velocity skewness using the first term (blue line)
and second term (black line) on the right hand side of equation 10 both scaled
with $\sigma_{w}^{3}$. In both panels, red lines identify the ISL range. The
dashed line is $Sk_{w}=0.16$ and the dotted line is $Sk_{w}=0.10$ Figure 3:
Panel a): difference between $\sigma_{u}^{2+}$ and estimations obtained from
the AEM, $\sigma_{u,m}^{2+}=A_{u}-B_{u}\log(z/\delta)$ using values of $A_{u}$
and $B_{u}$ obtained from regression of data within the ISL range (identified
by red symbols and lines); panel b): non dimensional vertical velocity
variance $\sigma_{w}^{2}$ normalized with $A_{w}$ obtained from data fitting
within the ISL vs wall-normal distance $z/\delta$. Data sources and references
are summarized in Table 1.

For the DNS, appropriate values of $A_{w}$(=1.15-1.17) and $B_{u}$(=0.85-0.86)
were estimated by fitting the AEM to the available data for all available
$Re_{\tau}$. The constant $\kappa$=0.39 was assumed as reported in the
literature [4, 16]. When inserting these choices of $A_{w}$ and $B_{u}$ from
the DNS into equation 11, the computed $Sk_{w}=0.13$, which is close to
reported values in Figure 1c. The same approach was used for all laboratory
studies. When combining all the runs together (wind tunnel and open channel
experiments), ensemble-averaged $A_{w}=1.01\pm 0.074$ and the ensemble-
averaged $B_{u}=0.76\pm 0.14$ were obtained across runs within an experiment
and across experiments. These values result in an ensemble-averaged
$Sk_{w}=0.18\pm 0.07$ and agrees with the measurements reported in Figure 1.

This analysis and Figure 1 suggest that $Sk_{w}$ for DNS and experiments is
higher than $0.12$, as estimated for $Re_{\tau}\to\infty$. This is probably
because of deviations in the ISL variance statistics from their asymptotic
constants in Equation 1. The effects of such deviations on $Sk_{w}$ are
however modest because, while low-$Re_{\tau}$ flows are characterized by
values of $A_{w}$ and $B_{u}$ that are significantly lower than their
counterparts at $Re_{\tau}\to\infty$ (i.e. $A_{w}=1.33$, and $B_{u}=1.26$, see
Table 1), equation 11 indicates that $Sk_{w}$ is dictated by the ratio
$B_{u}/A_{w}^{3}$, and hence the effect of such deviations are in good part
compensated.

Regarding the second feature, equation 10 offers an explanation as to why
conventional down-gradient closure models with eddy viscosity
$K_{t}\propto\overline{q}l_{m}$ ($l_{m}$ is a ’master’ mixing length)
expressed in general index notation ($[x_{1},x_{2},x_{3}]=[x,y,z]$ and
$[u^{\prime}_{1},u^{\prime}_{2},u^{\prime}_{3}]=[u^{\prime},v^{\prime},w^{\prime}]$)
as [42]

$\overline{u_{i}^{\prime}u_{j}^{\prime}u_{k}^{\prime}}=-K_{t}\left[\frac{\partial\overline{u_{i}^{\prime}u_{j}^{\prime}}}{\partial
x_{k}}+\frac{\partial\overline{u_{i}^{\prime}u_{k}^{\prime}}}{\partial
x_{j}}+\frac{\partial\overline{u_{j}^{\prime}u_{k}^{\prime}}}{\partial
x_{i}}\right]$ (12)

spectacularly fail when $i=j=k=3$ and when $A_{w}$ is approximately constant
in the ISL (as in the AEM). Yet, the derived equation here also offers a
rectification based on the AEM. This rectification accommodates the role of
finite $\partial\sigma_{u}^{2}/\partial z$ on $\overline{w^{\prime 3}}$ that
cannot arise from equation 12. While studies of the failure of gradient-
diffusion models have a long tradition in turbulence research [43], the mode
of failure these studies identify differ from the standard one offered here.
The classical failure of the so-called gradient diffusion phenomenology is the
lack of accommodation of flux transport terms [44, 43]. The origin of the
failure of equation 12 here is attributed to a return-to-isotropy mechanism
(i.e. pressure-velocity destruction) that requires a model for the vertical
transport of $K$, which is dominated by $\sigma_{u}^{2}$ [37]. To be clear,
equation 2 does maintain the turbulent flux transport term of
$\overline{w^{\prime 3}}$ through $\partial\overline{w^{\prime 4}}/\partial
z$, which is modeled using a quasi-normal approximation. However, this term is
not the main reason why equation 12 fails. In fact, the final $Sk_{w}$
expression is independent of $a$ when $A_{w}$ is constant (as in AEM)
irrespective of whether $a=0$ or not. What was earlier assumed was $a$ not
vary appreciably with $z$ within the ISL.

In conclusion, this letter demonstrates that, within the ISL of turbulent and
adiabatic smooth wall flows, $Sk_{w}$ attain z-independent values that are
predictable from well known turbulence constants relating to the AEM. This
behavior is reported for a variety of different wall flows and is independent
of variations in $Re_{\tau}$, hence universal and robust.

## References

  * Townsend [1976] A. Townsend,  _The Structure of Turbulent Shear Flow_ (Cambridge University Press, 1976).
  * Smits  _et al._ [2011] A. J. Smits, B. J. McKeon, and I. Marusic, High-Reynolds number wall turbulence, Annual Review of Fluid Mechanics 43 (2011).
  * Marusic and Monty [2019] I. Marusic and J. P. Monty, Attached eddy model of wall turbulence, Annual Review of Fluid Mechanics 51, 49 (2019).
  * Marusic  _et al._ [2013] I. Marusic, J. P. Monty, M. Hultmark, and A. J. Smits, On the logarithmic region in wall turbulence, Journal of Fluid Mechanics 716, R3 (2013).
  * Stevens  _et al._ [2014] R. J. Stevens, M. Wilczek, and C. Meneveau, Large-eddy simulation study of the logarithmic law for second-and higher-order moments in turbulent wall-bounded flow, Journal of Fluid Mechanics 757, 888 (2014).
  * Woodcock and Marusic [2015] J. Woodcock and I. Marusic, The statistical behaviour of attached eddies, Physics of Fluids 27 (2015).
  * Meneveau and Marusic [2013] C. Meneveau and I. Marusic, Generalized logarithmic law for high-order moments in turbulent boundary layers, Journal of Fluid Mechanics 719, R1 (2013).
  * Banerjee and Katul [2013] T. Banerjee and G. G. Katul, Logarithmic scaling in the longitudinal velocity variance explained by a spectral budget, Physiscs of Fluids 25, 125106 (2013).
  * Huang and Katul [2022] K. Y. Huang and G. G. Katul, Profiles of high-order moments of longitudinal velocity explained by the random sweeping decorrelation hypothesis, Physical Review Fluids 7, 044603 (2022).
  * Morrill-Winter  _et al._ [2015] C. Morrill-Winter, J. Klewicki, R. Baidya, and I. Marusic, Temporally optimized spanwise vorticity sensor measurements in turbulent boundary layers, Experiments in Fluids 56, 216 (2015).
  * Örlü  _et al._ [2017] R. Örlü, T. Fiorini, A. Segalini, G. Bellani, A. Talamelli, and P. H. Alfredsson, Reynolds stress scaling in pipe flow turbulence—first results from CICLoPE, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160187 (2017).
  * Zhao and Smits [2007] R. Zhao and A. J. Smits, Scaling of the wall-normal turbulence component in high-Reynolds-number pipe flow, Journal of Fluid Mechanics 576, 457 (2007).
  * Flack  _et al._ [2007] K. Flack, M. Schultz, and J. Connelly, Examination of a critical roughness height for outer layer similarity, Physics of Fluids 19 (2007).
  * Schultz and Flack [2007] M. Schultz and K. Flack, The rough-wall turbulent boundary layer from the hydraulically smooth to the fully rough regime, Journal of Fluid Mechanics 580, 381 (2007).
  * Manes  _et al._ [2011] C. Manes, D. Poggi, and L. Ridolfi, Turbulent boundary layers over permeable walls: scaling and near-wall structure, Journal of Fluid Mechanics 687, 141 (2011).
  * Peruzzi  _et al._ [2020] C. Peruzzi, D. Poggi, L. Ridolfi, and C. Manes, On the scaling of large-scale structures in smooth-bed turbulent open-channel flows, Journal of Fluid Mechanics 889, A1 (2020).
  * Heisel  _et al._ [2020] M. Heisel, G. G. Katul, M. Chamecki, and M. Guala, Velocity asymmetry and turbulent transport closure in smooth-and rough-wall boundary layers, Physical Review Fluids 5, 104605 (2020).
  * Mellor and Yamada [1982] G. L. Mellor and T. Yamada, Development of a turbulence closure model for geophysical fluid problems, Reviews of Geophysics 20, 851 (1982).
  * Bogenschutz  _et al._ [2012] P. Bogenschutz, A. Gettelman, H. Morrison, V. Larson, D. Schanen, N. Meyer, and C. Craig, Unified parameterization of the planetary boundary layer and shallow convection with a higher-order turbulence closure in the Community Atmosphere Model: Single-column experiments, Geoscientific Model Development 5, 1407 (2012).
  * Huang  _et al._ [2020] M. Huang, H. Xiao, M. Wang, and J. D. Fast, Assessing CLUBB PDF closure assumptions for a continental shallow-to-deep convective transition case over multiple spatial scales, Journal of Advances in Modeling Earth Systems 12, e2020MS002145 (2020).
  * Li  _et al._ [2022] T. Li, M. Wang, Z. Guo, B. Yang, Y. Xu, X. Han, and J. Sun, An updated CLUBB PDF closure scheme to improve low cloud simulation in CAM6, Journal of Advances in Modeling Earth Systems 14, e2022MS003127 (2022).
  * Bærentsen and Berkowicz [1984] J. H. Bærentsen and R. Berkowicz, Monte Carlo simulation of plume dispersion in the convective boundary layer, Atmospheric Environment 18, 701 (1984).
  * Luhar and Britter [1989] A. K. Luhar and R. E. Britter, A random walk model for dispersion in inhomogeneous turbulence in a convective boundary layer, Atmospheric Environment (1967) 23, 1911 (1989).
  * Wyngaard and Weil [1991] J. C. Wyngaard and J. C. Weil, Transport asymmetry in skewed turbulence, Physics of Fluids A: Fluid Dynamics 3, 155 (1991).
  * Maurizi and Tampieri [1999] A. Maurizi and F. Tampieri, Velocity probability density functions in lagrangian dispersion models for inhomogeneous turbulence, Atmospheric Environment 33, 281 (1999).
  * Sillero  _et al._ [2013] J. A. Sillero, J. Jiménez, and R. D. Moser, One-point statistics for turbulent wall-bounded flows at Reynolds numbers up to $\delta$\+ = 2000, Physics of Fluids 25 (2013).
  * Poggi  _et al._ [2002] D. Poggi, A. Porporato, and L. Ridolfi, An experimental contribution to near-wall measurements by means of a special laser doppler anemometry technique, Experiments in Fluids 32, 366 (2002).
  * Chiba [1978] O. Chiba, Stability dependence of the vertical wind velocity skewness in the atmospheric surface layer, Journal of the Meteorological Society of Japan. Ser. II 56, 140 (1978).
  * Zhou and Klewicki [2015] A. Zhou and J. Klewicki, Properties of the streamwise velocity fluctuations in the inertial layer of turbulent boundary layers and their connection to self-similar mean dynamics, International Journal of Heat and Fluid Flow 51, 372 (2015).
  * Örlü  _et al._ [2016] R. Örlü, A. Segalini, J. Klewicki, and P. H. Alfredsson, High-order generalisation of the diagnostic scaling for turbulent boundary layers, Journal of Turbulence 17, 664 (2016).
  * Canuto  _et al._ [1994] V. Canuto, F. Minotti, C. Ronchi, R. Ypma, and O. Zeman, Second-order closure pbl model with new third-order moments: Comparison with les data, Journal of the Atmospheric Sciences 51, 1605 (1994).
  * Zeman and Lumley [1976] O. Zeman and J. L. Lumley, Modeling buoyancy driven mixed layers, Journal of Atmospheric Sciences 33, 1974 (1976).
  * André  _et al._ [1976] J. André, G. De Moor, P. Lacarrere, and R. Du Vachat, Turbulence approximation for inhomogeneous flows: Part I. the clipping approximation, Journal of Atmospheric Sciences 33, 476 (1976).
  * Meneveau [1991] C. Meneveau, Analysis of turbulence in the orthonormal wavelet representation, Journal of Fluid Mechanics 232, 469 (1991).
  * Rotta [1951] J. Rotta, Statistische Theorie nichthomogener Turbulenz, Zeitschrift für Physik 129, 547 (1951).
  * Bou-Zeid  _et al._ [2018] E. Bou-Zeid, X. Gao, C. Ansorge, and G. G. Katul, On the role of return to isotropy in wall-bounded turbulent flows with buoyancy, Journal of Fluid Mechanics 856, 61 (2018).
  * Lopez and García [1999] F. Lopez and M. H. García, Wall similarity in turbulent open-channel flow, Journal of Engineering Mechanics 125, 789 (1999).
  * Katul  _et al._ [2013] G. G. Katul, A. Porporato, C. Manes, and C. Meneveau, Co-spectrum and mean velocity in turbulent boundary layers, Physics of Fluids 25 (2013).
  * Nakagawa and Nezu [1977] H. Nakagawa and I. Nezu, Prediction of the contributions to the Reynolds stress from bursting events in open-channel flows, Journal of Fluid Mechanics 80, 99 (1977).
  * Raupach [1981] M. Raupach, Conditional statistics of Reynolds stress in rough-wall and smooth-wall turbulent boundary layers, Journal of Fluid Mechanics 108, 363 (1981).
  * Barskov  _et al._ [2023] K. Barskov, D. Chechin, I. Drozd, A. Artamonov, A. Pashkin, A. Gavrikov, M. Varentsov, V. Stepanenko, and I. Repina, Relationships between second and third moments in the surface layer under different stratification over grassland and urban landscapes, Boundary-Layer Meteorology 187, 311 (2023).
  * Launder  _et al._ [1975] B. E. Launder, G. J. Reece, and W. Rodi, Progress in the development of a Reynolds-stress turbulence closure, Journal of Fluid Mechanics 68, 537 (1975).
  * Corrsin [1975] S. Corrsin, Limitations of gradient transport models in random walks and in turbulence, in  _Advances in Geophysics_ , Vol. 18 (Elsevier, 1975) pp. 25–60.
  * Cava  _et al._ [2006] D. Cava, G. Katul, A. Scrimieri, D. Poggi, A. Cescatti, and U. Giostra, Buoyancy and the sensible heat flux budget within dense canopies, Boundary-Layer Meteorology 118, 217 (2006).

###### Acknowledgements.

EB acknowledges Politecnico di Torino (Italy) for supporting the visit to Duke
University. GK acknowledges support from the U.S. National Science Foundation
(NSF-AGS-2028633) and the U.S. Department of Energy (DE-SC0022072). DV and CM
acknowledge European Union’s Horizon 2020 research and innovation programme
under the Marie Sklodowska-Curie grant agreement No 101022685 (SHIEELD). DP
acknowledges support from Fondo europeo di sviluppo regionale (FESR) for
project Bacini Ecologicamente sostenibili e sicuri, concepiti per
l’adattamento ai Cambiamenti ClimAtici (BECCA) in the context of Alpi Latine
COoperazione TRAnsfrontaliera (ALCOTRA) and project Nord Ovest Digitale e
Sostenibile - Digital innovation toward sustainable mountain (Nodes - 4).


# MATHER measures FOR space-time periodical nonconvex Hamiltonians

Eddaly Guerra- Velasco  CONACYT-Universidad Autónoma de Chiapas, Facultad de
Ciencias en Física y Matemáticas, México<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In [G] Diogo Gomes developed techniques and tools with the purpose of
extending the Aubry-Mather theory in a stochastic setting, namely he proved
the existence of stochastic Mather measures and their properties. These
results were also extended in the time-dependent setting in the doctoral
thesis the author [GV]. However to construct analogs to the Aubry–Mather
measures for nonconvex Hamiltonians it is necessary to use the adjoint method
introduced by Evans [E1] and H. V. Tran [T], the construction of the measures
is in [CGT]. The main goal of this paper is to construct Mather measures for
space-time periodical nonconvex Hamiltonians using the techniques in [E1], [T]
and [CGT] .

###### Key words and phrases:

Hamilton-Jacobi, non-convex, periodic Hamiltonians

###### 1991 Mathematics Subject Classification:

37J50, 49L25,

##  1\. Introduction.

###  1.1. The Convex Case.

Let $\mathbb{T}^{d+1}$ be the $d+1$ \- torus and consider a smooth periodic
Tonelli Hamiltonian $H:\mathbb{T}^{d+1}\times\mathbb{R}^{d}\to\mathbb{R}$.

Let $L:\mathbb{T}^{d+1}\times\mathbb{R}^{d}\to\mathbb{R}$ be the Lagrangian
associated to the Hamiltonian:

(1) $L(x,v,t)=\max_{p}\,pv-H(x,p,t),$

for every $(x,v,t)\in\mathbb{T}^{d+1}\times\mathbb{R}^{d}$.

Now we consider the corresponding flow of the time dependent Hamiltonian:

(2) $\displaystyle\dot{x}=D_{p}H(x,p,t),$
$\displaystyle\dot{p}=-D_{x}H(x,p,t).$

Now the dynamics transforms to

(3) $\displaystyle\dot{X}=D\bar{H}(P),$ $\displaystyle\dot{P}=0.$

under the change of variables

$(p,x)\to(P,X)$

where $p=P+D_{x}u(x,P,t)$, $X=x+D_{P}u(x,P,t)$ if we suppose that both
$u(x,P,t)$ and $\bar{H}(P)$ are smooth functions and $u(x,P,t)$ satisfies the
time dependent Hamilton-Jacobi equation

(4) $u_{t}+H(x,D_{x}u,t)=\bar{H}(P).$

######  Definition 1.

A continuous function $u:\mathbb{T}^{d+1}\to\mathbb{R}$ is called a forward
viscosity solution of (4) if it satisfies the two properties.

  1. (1)

If $v$ is a $C^{1}$ function and $u-v$ has a local maximum at $(x,t)$, then

$v_{t}+H(x,D_{x}v(x,t),t)\geq\bar{H}(P),$

  2. (2)

If $v$ is a $C^{1}$ function and $u-v$ has a local minimum at $(x,t)$, then

$v_{t}+H(x,D_{x}v(x,t),t)\leq\bar{H}(P).$

Backward viscosity solutions are defined by reversing both inequalities.

It is known ([CIS], [EG]) that there is only one value $\bar{H}(P)$, such that
(4) has a time periodic viscosity solution.

As a consequence of the semilinearity and convexity there is a consequence map
$\Phi:\mathbb{T}^{d+1}\times\mathbb{R}^{d}\to\mathbb{T}^{d+1}\times\mathbb{R}^{d},$
given by $\Phi(x,v,t)=(x,D_{v}L(x,v,t),t),$ wich is well defined and one-to-
one.

Recall the Poisson bracket,

$\\{F\,,G\\}:=D_{p}F\cdot D_{x}G-D_{x}F\cdot D_{p}G.$

In Hamiltonian coordinates, the property of invariance for a probability
measure $\nu$ can be written as

$\int_{\mathbb{T}^{d+1}\times\mathbb{R}^{d}}\phi_{s}+\\{H,\phi\\}\,d\mu=0$

for every $\phi\in C_{c}^{1}(\mathbb{T}^{d+1}\times\mathbb{R}^{d})$, where
$\mu=\Phi_{\\#}\nu$ is the push-forward of the measure $\nu$ with respect to
the map $\Phi$, i.e. the measure $\mu$ such that

$\int_{\mathbb{T}^{d+1}\times\mathbb{R}^{d}}\phi(x,p,t)d\mu(x,p,t)=\int_{\mathbb{T}^{d+1}\times\mathbb{R}^{d}}\phi(x,D_{v}L(x,v,t),t)d\nu(x,v,t)$

for every $\phi\in C_{c}^{1}(\mathbb{T}^{d+1}\times\mathbb{R}^{d})$.

Denoting by $\mathcal{P}(\mathbb{T}^{d+1}\times\mathbb{R}^{d})$ the class of
probability measures on $\mathbb{T}^{d+1}\times\mathbb{R}^{d}$, and taking
$\Omega=\mathbb{T}^{d+1}\times\mathbb{R}^{d}$, where $(x,v,t)=z$ represents a
generic point $z\in\Omega$ with $(x,t)\in\mathbb{T}^{d+1}$ and
$v\in\mathbb{R}^{d}$.

Now let $\mathcal{D}$ be the class of probability measures in $\Omega$ that
are invariant under the Euler-Lagrange flow, so we have

$\mathcal{D}=\Big{\\{}\nu\in\mathcal{P}(\Omega):\int_{\Omega}\phi_{t}+\\{H,\phi\\}d\Phi_{\\#}\nu(x,p,t)=0\text{
for every }\phi\in C_{c}^{1}(\Omega)\Big{\\}},$

and the set of holonomic measures

$\mathcal{F}=\\{\nu\in\mathcal{P}(\Omega):\int_{\Omega}\psi_{t}+v\cdot\,D\psi(x,t)\,d\nu(x,v,t)=0,\,\text{
for every }\psi\in C^{1}(\mathbb{T}^{d+1})\\}.$

We recall the Mather problem

(5) $\min_{v\in\mathcal{F}}\int_{\Omega}L(x,v,t)\,d\nu(x,v,t),$

a more general version of (5) consists in studying for each
$P\in\mathbb{R}^{d}$ fixed

(6) $\min_{\nu\in\mathcal{F}}\int_{\Omega}(L(x,v,t)-P\cdot v)d\nu.$

Any minimizer of (6) is a Mather measure, now the following proposition will
be helpful to prove an important result.

######  Proposition 2.

Let $H:\mathbb{T}^{d+1}\times\mathbb{R}^{d}\to\mathbb{R}$ be a smooth function
that satisfies the classical hypotheses. Let $P\in\mathbb{R}^{d}$,
$\nu\in\mathcal{P}(\Omega)$ be a minimizer of (6) and set $\mu=\Phi_{\\#}\nu$.
Then

  1. (i)

$\mu$ is invariant under the Hamiltonian dynamics, i.e.,

$\int_{\Omega}\phi_{t}+\\{H,\phi\\}d\mu(x,p,t)=0,\,\text{\,\, for every
}\phi\in C^{1}_{c}(\Omega)$

  2. (ii)

$\mu$ is supported on the graph

$\Sigma:=\\{(x,p,t)\in\mathbb{T}^{d+1}\times\mathbb{R}^{d}:p=P+D_{x}u(x,P,t)\\}$

where $u$ is any viscosity solution of (4).

The proof of the proposition is a consequence of results in [B] and [CIS].

As in [CGT] the following theorem gives a characterization of Mather measures
in the time dependent convex case.

######  Theorem 3.

Assume $H:\mathbb{T}^{d+1}\times\mathbb{R}^{d}\to\mathbb{R}$ is a smooth
function that satisfies the classical hypotheses of convexity, superlinearity,
and periodicity and let $P\in\mathbb{R}$. Then $\nu\in\mathcal{P}(\Omega)$ is
a solution of

$-\min_{\nu\in\mathcal{F}}\int_{\Omega}(L(x,v,t)-P\cdot v)d\nu(x,v,t),$

if and only if

  1. (a)

$\int_{\Omega}\,\phi_{t}+H(x,p,t)d\mu=\bar{H}(P)=H(x,p,t)\,$ $\,\mu$ a.e.,

  2. (b)

$\int_{\Omega}\phi_{t}+(p+P)\cdot\,D_{p}H(x,p,t)\,d\mu(x,p,t)=0$,

  3. (c)

$\int_{\Omega}\phi_{t}+D_{p}H(x,p,t)\cdot D\phi(x,p,t)=0$, for every $\phi\in
C^{1}(\mathbb{T}^{d+1})$.

where $\mu=\Phi_{\\#}\nu$ and $\bar{H}(P)$ is the unique value such that (4)
has a time periodic viscosity solution.

###### Proof.

To simplify, we will assume $P=0$. Let us prove that $\mu=\Phi_{\\#}\nu$
satisfies (a)-(c). From (ii) of the last proposition, and (1), we have that

$\int_{\Omega}\phi_{t}+H(x,p,t)d\mu=\bar{H}(0),$

so (a) holds.

Now, we know that

$H(x,p,t)=p\cdot D_{p}H(x,p,t)-L(x,D_{p}H(x,p,t),t),$

and from (a) it follows that

$\int_{\Omega}\phi_{t}+p\cdot D_{p}H(x,p,t)\,d\mu=0.$

Finally (c) follows from that $\nu\in\mathcal{F}$.

Reciprocally let $\mu\in\mathcal{P}({\Omega})$ such that (a), (b) and (c)
holds, and we will show that $\nu=\Phi_{\\#}\mu$ is a minimizer of (6).

Now observe that $\nu\in\mathcal{F}$, then

$\int_{\Omega}\psi_{t}+v\cdot
D\psi(x,t)\,d\nu=\int_{\Omega}\psi_{t}+D_{p}H(x,p,t)\cdot D\psi(x,t)=0,$

for every $\psi\in C^{1}(\mathbb{T}^{d+1})$.

The fact that $\nu$ is a minimizer is obtained by using (a) and (b)

$\bar{H}(0)=\int_{\Omega}\phi_{t}+H(x,p,t)d\mu=\int_{\Omega}\phi_{t}+p\cdot
D_{p}H-Ld\mu=\int_{\Omega}-Ld\mu.$

∎

The previous characterization will help us to define Mather measures in the
nonconvex case.

###  1.2. The Nonconvex Case

Throughout the paper, we will assume that

  1. i.

$H$ is smooth,

  2. ii.

$H(\cdot,p,t)$ is $\mathbb{Z}^{d+1}$-periodic for $(p,t)\in\mathbb{R}^{d+1}$,

  3. iii.

There exists a continuous function $\chi:[0,+\infty)\to\mathbb{R}$ such that

(7) $\int_{0}^{\infty}\chi(u)^{-1}du=\infty\text{ and
}|H(x,p,t)|\leq\chi(|p|).$

######  Example 1.

Consider

$H(x,p,t)=|p_{1}|-|p_{2}|+V(x,t)$

If we take $\chi(u)=2u+C$,

$\int_{0}^{\infty}\chi(u)^{-1}du=\infty$

and $|H(x,p,t)|=||p_{1}|-|p_{2}|+V(x,t)|\leq\chi(|p|)$.

We extend the definition of Mather measure in the nonconvex and time dependent
setting:

######  Definition 4.

We say that a measure $\mu\in\mathcal{P}(\Omega)$ is a Mather measure if there
exists $P\in\mathbb{R}^{d}$ such that properties (a)-(c) in Theorem 3 are
satisfied.

Our main result is:

######  Theorem 5.

Assume that the Hamiltonian is a smooth function that satisfies the conditions
(i.)-(iii.) and let $\\{\mu^{\varepsilon}\\}_{\varepsilon>0}$ be the family of
measures defined in (16). Then there exist a Mather measure $\mu$ and a
nonnegative and symmetric $d\times d$ matrix $(m_{kj})_{k,j=1,\dots d}$ of
Borel measures called the dissipation measure, such that:

  1. (1)

$\mu^{\varepsilon}\overset{*}{\rightharpoonup}\mu$ in the sense of measures up
to subsequences,

  2. (2)

$\int_{\Omega}\varphi_{t}+\\{H,\varphi\\}d\mu+\int_{\Omega}\varphi_{p_{k}p_{j}}dm_{kj}=0\text{
for all }\varphi\in C_{c}(\Omega),$

  3. (3)

$\operatorname{supp}\mu$ and $\operatorname{supp}m$ are compact.

##  2\. Uniform Derivate Bounds

Let us consider the equation:

(8)
$\phi_{t}^{\varepsilon}+\varepsilon\Delta\phi^{\varepsilon}+H(x,P+D\phi^{\varepsilon}(x,t),t)=\bar{H}(P).$

######  Lemma 6.

The periodic solutions of (8) have first derivatives, uniformly bounded in
$\varepsilon$.

###### Sketch of the proof..

For every $\varepsilon>0$ let us consider the following problem

(9)
$\phi_{t}^{\varepsilon}+\Delta\phi^{\varepsilon}+H(x,D\phi^{\varepsilon},t)-\varepsilon\phi^{\varepsilon}=0$

The above equation has a unique smooth solution $\phi^{\varepsilon}$ in
$\mathbb{R}^{d+1}$ which is $\mathbb{Z}^{d+1}$ periodic [GV]. First, we proved
that $D\phi^{\varepsilon}$ is uniformly bounded, by following [BS] we proved
that there exists $K>0$ depending only on $H$ such that

(10) $\sup\|D\phi^{\varepsilon}(\cdot,t)\|_{\infty}\leq K$

Finally if we take $g=d_{t}^{2}phi^{\varepsilon}+|D\phi^{\varepsilon}|^{2}$and
using the Bernstein’s method we prove that $d_{t}\phi^{\varepsilon}$ is
uniformly bounded.

∎

######  Theorem 7.

For every $\varepsilon>0$ and every $P\in\mathbb{R}^{d}$, there exists a
unique number $H^{\varepsilon}(P)\in\mathbb{R}$ such that the equation (8)
admits a unique (up to constants) $\mathbb{Z}^{d+1}$ periodic viscosity
solution. Moreover, for every $P\in\mathbb{R}^{d}$ $\underset{\varepsilon\to
0^{+}}{\lim}\bar{H}^{\varepsilon}(P)\to\bar{H}(P)$ and
$\phi^{\varepsilon}\to\phi_{0}$ uniformly (up to subsequences), where
$\phi_{0}:\mathbb{T}^{d+1}\to\mathbb{R}$ is that (4) is satisfied in the
viscosity sense.

###### Proof.

The theorem follows by Lemma 6, the stability theorem for viscosity solutions
and the Arzela-Ascoli Theorem. ∎

##  3\. Stochastic Measures

######  Definition 8.

Let $\varepsilon>0$ and $P\in\mathbb{R}^{d}$. The linearized operator
associated to (8) is defined as $L_{\varepsilon,P}:C^{2}(\mathbb{T}^{d+1})\to
C(\mathbb{T}^{d+1}):$

(11)
$L_{\varepsilon,P}\,\psi=\psi_{t}+\varepsilon\Delta\psi+D_{p}H(x,P+D\phi^{\varepsilon}(x,t),t)D\psi,$

for every $\psi\in C^{2}(\mathbb{T}^{d+1}).$

As in [CGT], we denote by $\beta$ either a direction in $\mathbb{R}^{d}$
(i.e., $\beta\in\mathbb{R}^{d}$ with $|\beta|=1$) or a parameter (for example
$\beta=P_{i}$ for some $i\in\\{1,\dots,d\\}$). When $\beta=P_{i}$ for some
$i\in\\{1,\dots,d\\}$ the symbols $H_{\beta}$ and $H_{\beta\beta}$ have to be
understood as $H_{p_{i}}$ and $H_{p_{i}p_{i}}$ respectively. If we derive (8)
with respect to $\beta$ and recalling (11) we get

$L_{\varepsilon,P}\,\phi_{\beta}^{\varepsilon}=\phi_{t\beta}^{\varepsilon}+\varepsilon\Delta\phi_{\beta}^{\varepsilon}+D_{p}H(x,P+D\phi^{\varepsilon}(x,t),t)D\phi_{\beta}^{\varepsilon}+H_{\beta}=\bar{H}^{\varepsilon},$

so

(12)
$L_{\varepsilon,P}\,\phi_{\beta}^{\varepsilon}=\bar{H}^{\varepsilon}-H_{\beta}.$

As before, let $\Omega=\mathbb{T}^{d+1}\times\mathbb{R}^{d}$, where $(x,v,t)$
represents a generic point with $(x,t)\in\mathbb{T}^{d+1}$ and
$v\in\mathbb{R}^{d}$. We need to introduce a probability space
$(\Omega,{\mathcal{B}},\mathbb{P})$ endowed with a Brownian motion
$W(t):\Omega\to\mathbb{T}^{d}$ on the flat $d$-torus. Let $\varepsilon>0$, to
simplify we set $P=0$ and we introduce the time dependent vector field [Fl],
$U_{\varepsilon}(x,t)=D_{p}H(x,D\phi_{\varepsilon}(x,t),t)$ and consider the
solution $X_{\varepsilon}(s)$ of the stochastic differential equation

(13)
$\begin{cases}dX^{\varepsilon}(s)&=U_{\varepsilon}(X_{\varepsilon}(s),s)ds+\sqrt{2\varepsilon}\,dW(s),\\\
X^{\varepsilon}(t)&=x.\end{cases}$

And the momentum variable is defined as

$p^{\varepsilon}(t)=D\phi^{\varepsilon}(X^{\varepsilon}(t),t).$

Now suppose $\mathbf{z}:[0,+\infty)\to\mathbb{R}^{d}$ is a solution to the
stochastic differential equation

$d\mathbf{z}_{i}=a_{i}\,ds+b_{ij}dW(s)$

with $a_{i}$ and $b_{ij}$ bounded and progressively measurable processes. Let
$\varphi:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}$ be a smooth function
where $\varphi(\mathbf{z},t)$ satisfies the Itô formula:

(14)
$d\varphi=\varphi_{z_{i}}dz_{i}+(\varphi_{t}+\frac{1}{2}b_{ij}b_{jk}\,\varphi_{z_{i}z_{k}})dt.$

From hereafter, we will use Einstein’s convention for repeated indices in a
sum. Here, we have $a_{i}=D_{p}H(x,D\phi^{\varepsilon}(x,t),t)$ and
$b_{ij}=\sqrt{2\varepsilon}\delta_{ij}$.

Therefore, from (13), (14) and (12),

$\displaystyle dp_{i}$
$\displaystyle=\phi^{\varepsilon}_{x_{i}x_{j}}[D_{p}H(X^{\varepsilon},D\phi^{\varepsilon}(x,t),t)dt+\sqrt{2\varepsilon}dW^{j}]+\phi_{x_{i}t}^{\varepsilon}(X^{\varepsilon}(x),t)+\varepsilon\Delta\phi_{x_{i}}^{\varepsilon}\,dt,$
$\displaystyle=L_{\varepsilon,P}\,\phi_{x_{i}}^{\varepsilon}\,dt+\sqrt{2\varepsilon}\,\phi_{x_{i}x_{j}}dW^{j}$
$\displaystyle=H_{x_{i}}\,dt+\sqrt{2\varepsilon}\,\phi_{x_{i}x_{j}}dW^{j}.$

Thus $(X_{\varepsilon},p_{\varepsilon})$ satisfies the following stochastic
version of the Hamiltonian dynamics

(15)
$\begin{cases}dX^{\varepsilon}(s)&=U_{\varepsilon}(X_{\varepsilon}(s),s)ds+\sqrt{2\varepsilon}\,dW(s),\\\
dp^{\varepsilon}(s)&=-DH(X^{\varepsilon},p^{\varepsilon},s)ds+\sqrt{2\varepsilon}\,D^{2}\phi^{\varepsilon}dW(s).\end{cases}$

Now we are going to study the solution $\phi^{\varepsilon}$ of (8) along the
trajectory $X_{\varepsilon}(s)$. Due to the Itô formula, and the equations (8)
and (13).

$\displaystyle d\phi^{\varepsilon}(X_{\varepsilon}(s))$
$\displaystyle=D\phi^{\varepsilon}dX_{\varepsilon}+(\phi_{t}^{\varepsilon}+\varepsilon\Delta\phi^{\varepsilon})ds,$
$\displaystyle=D\phi^{\varepsilon}[D_{p}H(x,D\phi^{\varepsilon}(x,s),s)ds+\sqrt{2\varepsilon}dW(s)]+(\phi_{t}^{\varepsilon}+\varepsilon\Delta\phi^{\varepsilon})ds,$
$\displaystyle=(\phi_{t}^{\varepsilon}+\varepsilon\Delta\phi^{\varepsilon}+D_{p}H(x,D\phi^{\varepsilon}(x,s),s)\cdot
D\phi^{\varepsilon})ds+\sqrt{2\varepsilon}D\phi^{\varepsilon}dW(s),$
$\displaystyle=L_{\varepsilon,P}\phi^{\varepsilon}\,ds+\sqrt{2\varepsilon}\,dW(s)=(\bar{H}_{\varepsilon}(P)-H+D_{p}H\cdot
D\phi^{\varepsilon})ds+\sqrt{2\varepsilon}dW(s).$

And using the Dynkin formula, we obtain

$\displaystyle
E[\phi^{\varepsilon}(X_{\varepsilon}(T))-\phi^{\varepsilon}(X_{\varepsilon}(0))]$
$\displaystyle=E\Big{[}\int_{0}^{T}(D_{p}H(X_{\varepsilon},D\phi^{\varepsilon}(x,t),t)D\phi^{\varepsilon}+\varepsilon\Delta\phi^{\varepsilon})\Big{]}dt$
$\displaystyle=E\Big{[}\int_{0}^{T}(D_{p}H(X_{\varepsilon},D\phi^{\varepsilon}(x,t),t)D\phi^{\varepsilon}+\bar{H}_{\varepsilon}(P)-H-\phi_{t}^{\varepsilon})dt\Big{]}$

Now we will associate to each trajectory $(X_{\varepsilon},p_{\varepsilon},t)$
of (15) a probability measure
$\mu_{\varepsilon}\in\mathcal{P}(\mathbb{T}^{d+1}\times\mathbb{R}^{d})$
defined by

(16)
$\int_{\mathbb{T}^{d+1}\times\mathbb{R}^{d}}\phi(x,p,t)d\mu_{\varepsilon}(x,p,t):=\lim_{T\to\infty}\frac{1}{T}E\Big{[}\int_{0}^{T}\phi(X_{\varepsilon}(t),p^{\varepsilon}(t),t)\,dt\Big{]}$

for every $\phi\in C_{c}(\mathbb{T}^{d+1}\times\mathbb{R}^{d}).$ Here, the
definition makes sense provided the limit is taken over an appropiate
subsequence. Then using Dynkin’s formula, we have that for every $\varphi\in
C_{c}(\mathbb{T}^{d+1}\times\mathbb{R}^{d}),$

(17) $\displaystyle
E\Big{[}\varphi(X^{\varepsilon}(T),p^{\varepsilon}(T),T)-\varphi(X^{\varepsilon}(0),p^{\varepsilon}(0),0)\Big{]}$
$\displaystyle=E\Big{[}\int_{0}^{T}\varphi_{t}+(D_{x}\varphi\cdot
D_{p}H-D_{p}\varphi\cdot D_{x}H)dt\Big{]}$
$\displaystyle+E\Big{[}\int_{0}^{T}(\varepsilon\varphi_{x_{i}x_{i}}+2\varepsilon\,\phi^{\varepsilon}_{x_{i}x_{j}}\varphi_{x_{i}p_{j}}+\varepsilon\phi^{\varepsilon}_{x_{i}x_{k}}\phi^{\varepsilon}_{x_{i}x_{j}}\varphi_{p_{k}p_{j}})\,dt\Big{]}.$

Dividing the equation (17) by $T$ and taking the limit when $T\to\infty$ along
a suitable subsequence we obtain:

(18)
$\int_{\mathbb{T}^{d+1}\times\mathbb{R}^{d}}\varphi_{t}+\\{H,\varphi\\}d\mu^{\varepsilon}+\int_{\mathbb{T}^{d+1}\times\mathbb{R}^{d}}(\varepsilon\varphi_{x_{i}x_{i}}+2\varepsilon\,\phi^{\varepsilon}_{x_{i}x_{j}}\varphi_{x_{i}p_{j}}+\varepsilon\phi^{\varepsilon}_{x_{i}x_{k}}\phi^{\varepsilon}_{x_{i}x_{j}}\varphi_{p_{k}p_{j}})\,d\mu^{\varepsilon}$

Let us define the projected measure
$\theta_{\mu^{\varepsilon}}\in\mathcal{P}(\mathbb{T}^{d+1})$ as follows

$\int_{\mathbb{T}^{d+1}}\varphi(x,t)d\theta_{\mu^{\varepsilon}}(x,t):=\int_{\mathbb{T}^{d+1}\times\mathbb{R}^{d}}\varphi(x,t)d\mu^{\varepsilon}(x,p,t)$

for all $\varphi\in C(\mathbb{T}^{d+1})$. And using test functions that do not
depend on $p$ in the last definition:

(19) $-\int_{\mathbb{T}^{d+1}}D_{p}H\cdot
D_{x}\varphi\,d\theta_{\mu^{\varepsilon}}=\int_{\mathbb{T}^{d+1}}(\varphi_{t}+\varepsilon\Delta\varphi)d\theta_{\mu^{\varepsilon}}$

for all $\varphi\in C^{2}(\mathbb{T}^{d+1})$.

Given $\phi^{\varepsilon}$, let us consider the partial differential equation

$\theta_{t}^{\varepsilon}-\varepsilon\Delta\theta^{\varepsilon}+\operatorname{div}(\theta^{\varepsilon}\cdot
D_{p}H(x,D\phi^{\varepsilon},t))=0,$

From lemma 32 and lemma 33 in [GV], we have that 0 is the principal value of
Fokker-Planck operator

$N(\theta^{\varepsilon})=-\theta^{\varepsilon}_{t}+\Delta\theta^{\varepsilon}-\operatorname{div}(\theta^{\varepsilon}\cdot
D_{p}H(x,D\phi^{\varepsilon},t))$

and so $\mu^{\varepsilon}$ can be defined as a unique measure such that

$\int_{\mathbb{T}^{d+1}\times\mathbb{R}^{d}}\psi(x,p,t)d\mu^{\varepsilon}(x,p,t)=\int_{\mathbb{T}^{d+1}}\psi(x,D\phi^{\varepsilon}(x,t),t)d\theta^{\varepsilon}(x,t),$

for every $\psi\in C_{c}(\mathbb{T}^{d+1}\times\mathbb{R}^{d})$.

###  3.1. Uniform Estimates.

######  Lemma 9.

We have the following estimates:

(20)
$\varepsilon\int_{\mathbb{T}^{d+1}}|D\phi^{\varepsilon}_{\beta}|^{2}d\theta_{\mu^{\varepsilon}}=2\int_{\mathbb{T}^{d+1}}\phi_{\beta}^{\varepsilon}(H_{\beta}-\bar{H}^{\varepsilon}_{\beta})d\theta_{\mu^{\varepsilon}}=-2\int_{\mathbb{T}^{d+1}}\phi_{\beta}^{\varepsilon}(\bar{H}^{\varepsilon}_{\beta}-H_{\beta})d\theta_{\mu^{\varepsilon}},$
(21)
$\int_{\mathbb{T}^{d+1}}(\bar{H}_{\beta}^{\varepsilon}-H_{\beta\beta}-2D_{p}H_{\beta}\cdot
D\phi_{\beta}^{\varepsilon}-D^{2}_{pp}HD\phi_{\beta}^{\varepsilon}\cdot
D\phi_{\beta}^{\varepsilon})d\theta_{\mu^{\varepsilon}}=0,$ (22)
$\varepsilon\int_{\mathbb{T}^{d+1}}|D\phi^{\varepsilon}_{\beta\beta}|^{2}\,d\theta_{\mu^{\varepsilon}}=-2\int_{\mathbb{T}^{d+1}}\phi^{\varepsilon}_{\beta\beta}(\bar{H}_{\beta}^{\varepsilon}-H_{\beta\beta}-2D_{p}H_{\beta}\cdot
D\phi_{\beta}^{\varepsilon}-D^{2}_{pp}H:D\phi_{\beta}^{\varepsilon}\otimes
D\phi_{\beta}^{\varepsilon})\,d\theta_{\mu^{\varepsilon}}.$

###### Proof.

Recalling (11), we obtain

$\displaystyle L_{\varepsilon,P}|\phi_{\beta}^{\varepsilon}|^{2}$
$\displaystyle=\frac{\partial}{\partial
t}<\phi_{\beta}^{\varepsilon},\phi_{\beta}^{\varepsilon}>+\varepsilon\Delta<\phi_{\beta}^{\varepsilon},\phi_{\beta}^{\varepsilon}>+D_{p}H(x,P+D<\phi_{\beta}^{\varepsilon},\phi_{\beta}^{\varepsilon}>,t)+H_{\beta}=\bar{H}_{\beta}^{\varepsilon}$
$\displaystyle=2\phi_{t\beta}^{\varepsilon}\cdot\phi_{\beta}^{\varepsilon}+2\varepsilon
D\phi_{\beta}^{\varepsilon}\cdot\phi_{\beta}^{\varepsilon}+2\varepsilon\Delta\phi_{\beta}^{\varepsilon}\cdot\phi_{\beta}^{\varepsilon}+D_{p}H(x,P+D\phi^{\varepsilon}(x,t),t)\cdot
2D\phi_{\beta}^{\varepsilon}\cdot\phi_{\beta}^{\varepsilon}.$

Thus

$L_{\varepsilon,P}|\phi_{\beta}^{\varepsilon}|^{2}=2L_{\varepsilon,P}\,\phi_{\beta}^{\varepsilon}\cdot\phi_{\beta}^{\varepsilon}+\varepsilon|D\phi_{\beta}^{\varepsilon}|^{2}=2\phi_{\beta}^{\varepsilon}(\bar{H}_{\beta}^{\varepsilon}-H_{\beta})+\varepsilon|D\phi_{\beta}^{\varepsilon}|^{2}.$

Integrating with respect to $\theta_{\mu^{\varepsilon}}$ and using (19), we
get (20).

To obtain (21) we differentiate (12) with respect to $\beta$, we have:

(23)
$L_{\varepsilon,P}\,\phi^{\varepsilon}_{\beta\beta}=\bar{H}_{\beta}^{\varepsilon}-H_{\beta\beta}-2D_{p}H_{\beta}\cdot
D\phi_{\beta}^{\varepsilon}-D^{2}_{pp}H:D\phi_{\beta}^{\varepsilon}\otimes
D\phi_{\beta}^{\varepsilon},$

Integrating again with respect to $\theta_{\mu^{\varepsilon}}$ and using (19)
we get (21).

On the other hand

$L_{\varepsilon,P}|\phi^{\varepsilon}_{\beta\beta}|^{2}=2L_{\varepsilon,P}\phi^{\varepsilon}_{\beta\beta}\cdot\phi^{\varepsilon}_{\beta\beta}+\varepsilon|D\phi^{\varepsilon}_{\beta\beta}|^{2},$

using (23) we obtain

$\frac{1}{2}L_{\varepsilon,P}|\phi^{\varepsilon}_{\beta\beta}|^{2}=\phi^{\varepsilon}_{\beta\beta}(\bar{H}_{\beta}^{\varepsilon}-H_{\beta\beta}-2D_{p}H_{\beta}\cdot
D\phi_{\beta}^{\varepsilon}-D^{2}_{pp}H:D\phi_{\beta}^{\varepsilon}\otimes
D\phi_{\beta}^{\varepsilon})+\frac{\varepsilon}{2}|D\phi^{\varepsilon}_{\beta\beta}|^{2},$

once again, integrating with respect to $\theta_{\mu^{\varepsilon}}$ and by
(19) we get (22). ∎

Following the techniques of [CGT], [E1] and [T], we will obtain several
estimates that will be useful in the future.

######  Proposition 10.

We have the following

(24)
$\varepsilon\int_{\mathbb{T}^{d+1}}|D^{2}_{xx}\phi^{\varepsilon}|^{2}d\theta_{\mu^{\varepsilon}}\leq
C$ (25)
$\varepsilon\int_{\mathbb{T}^{d+1}}|D^{2}_{Px}\phi^{\varepsilon}|^{2}d\theta_{\mu^{\varepsilon}}\leq\int_{\mathbb{T}^{d+1}}|D_{P}\phi^{\varepsilon}|^{2}d\theta_{\mu^{\varepsilon}}+\int_{\mathbb{T}^{d+1}}|D_{P}H-D_{P}\bar{H}^{\varepsilon}|^{2}d\theta_{\mu^{\varepsilon}}.$
(26)
$\varepsilon\int_{\mathbb{T}^{d+1}}|D\phi_{x_{i}x_{i}}^{\varepsilon}|^{2}\leq
C(1+\int_{\mathbb{T}^{d+1}}|D^{2}_{xx}\phi^{\varepsilon}|^{3}d\theta_{\mu^{\varepsilon}})$

###### Proof.

Taking $\beta=x_{1},x_{2},\dots,x_{d+1}$ respectively in (20), we have

$\varepsilon\int_{\mathbb{T}^{d+1}}|D\phi^{\varepsilon}_{x_{i}}|^{2}d\theta_{\mu^{\varepsilon}}=2\int_{\mathbb{T}^{d+1}}\phi^{\varepsilon}_{x_{i}}H_{x_{i}}d\theta_{\mu^{\varepsilon}},$

and adding these $d+1$ identities we obtain

$\varepsilon\int_{\mathbb{T}^{d+1}}|D^{2}_{xx}\phi^{\varepsilon}|^{2}d\theta_{\mu^{\varepsilon}}=2\int_{\mathbb{T}^{d+1}}D_{x}\phi^{\varepsilon}\cdot
D_{x}Hd\theta_{\mu^{\varepsilon}},$

now, due to Lemma. 6, $|D_{x}\phi^{\varepsilon}|$, $|D_{x}H|$ are uniformly
bounded, thus we get (24).

Now, the relation (25) follows by taking $\beta=P_{1},P_{2},\dots,P_{d+1}$ in
(20), adding the $d+1$ identities

$\varepsilon\int_{\mathbb{T}^{d+1}}|D^{2}_{Px}\phi^{\varepsilon}|^{2}d\theta_{\mu^{\varepsilon}}=2\int_{\mathbb{T}^{d+1}}D_{P}\phi^{\varepsilon}[D_{P}H-D_{P}\bar{H}^{\varepsilon}]d\theta_{\mu^{\varepsilon}}$

And using the Young’s inequality.

To obtain (26), taking $\beta=x_{i}$ in (22)

$\varepsilon\int_{\mathbb{T}^{d+1}}|D\phi_{x_{i}x_{i}}^{\varepsilon}|^{2}=2\int_{\mathbb{T}^{d+1}}\phi^{\varepsilon}_{x_{i}x_{i}}(H_{x_{i}x_{i}}+2D_{p}H_{x_{i}}\cdot
D\phi_{x_{i}}^{\varepsilon}-D^{2}_{pp}H:D\phi_{x_{i}}^{\varepsilon}\otimes
D\phi_{x_{i}}^{\varepsilon})d\theta_{\mu^{\varepsilon}}$

Due to Lemma 6, we have that
$|H_{x_{i},x_{i}}|,\,|D_{p}H_{x_{i}}|,\,|D^{2}_{pp}H|\leq C$ on the support of
$d\theta_{\mu^{\varepsilon}}$, so

$\displaystyle\varepsilon\int_{\mathbb{T}^{d+1}}|D\phi_{x_{i}x_{i}}^{\varepsilon}|^{2}$
$\displaystyle\leq
C\Big{(}\int_{\mathbb{T}^{d+1}}|D^{2}_{xx}\phi^{\varepsilon}|^{2}d\theta_{\mu^{\varepsilon}}+\int_{\mathbb{T}^{d+1}}|D^{2}_{xx}\phi^{\varepsilon}|^{2}d\theta_{\mu^{\varepsilon}}+\int_{\mathbb{T}^{d+1}}|D^{2}_{xx}\phi^{\varepsilon}|^{3}d\theta_{\mu^{\varepsilon}}\Big{)}$
$\displaystyle\leq
C\Big{(}1+\int_{\mathbb{T}^{d+1}}|D^{2}_{xx}\phi^{\varepsilon}|^{3}d\theta_{\mu^{\varepsilon}}\Big{)}.$

∎

##  4\. Existence of Mather measures.

Now we are able to prove the existence of Mather measures.

###### Proof of Theorem 5..

The proof straightforward noticing that $\\{\phi^{\varepsilon}\\}$ have a
uniform Lipschitz estimate, therefore there exists a compact set
$K\subset\Omega$ such that $\operatorname{supp}\mu^{\varepsilon}\subset
K\,\forall\varepsilon>0$. Moreover, up to subsequences, we have
$\mu^{\varepsilon}\overset{*}{\rightharpoonup}\mu$, that is

$\lim_{\varepsilon\to
0}\int_{\Omega}\phi\,d\mu^{\varepsilon}\to\int_{\Omega}\phi\,d\mu$

for every function $\phi\in C_{c}(\Omega)$, for some probability measure
$\mu\in\mathcal{P}(\Omega)$, and also it follows that
$\operatorname{supp}\mu\subset K$.

To obtain (2), let us remember (18) particularly the second term

(27)
$\int_{\Omega}(\varepsilon\varphi_{x_{i}x_{i}}+2\varepsilon\phi^{\varepsilon}_{x_{i}x_{j}}\varphi_{x_{i}p_{j}}+\varepsilon\phi^{\varepsilon}_{x_{i}x_{k}}\phi^{\varepsilon}_{x_{i}x_{j}}\varphi_{p_{k}p_{j}})\,d\mu^{\varepsilon}$

But

(28)
$\displaystyle\Big{|}\int_{\Omega}(\varepsilon\varphi_{x_{i}x_{i}}+2\varepsilon\phi^{\varepsilon}_{x_{i}x_{j}}\varphi_{x_{i}p_{j}}\Big{|}\,d\mu^{\varepsilon}$
$\displaystyle\leq
C\varepsilon+C\varepsilon\int_{\Omega}|\phi^{\varepsilon}_{x_{i}x_{j}}|\,d\mu^{\varepsilon}$
(29) $\displaystyle\leq C\varepsilon+C\varepsilon^{\frac{1}{2}}.$

by using the estimates in Proposition 10, so the

$\underset{\varepsilon\to
0}{\lim}\int_{\Omega}(\varepsilon\varphi_{x_{i}x_{i}}+2\varepsilon\phi^{\varepsilon}_{x_{i}x_{j}}\varphi_{x_{i}p_{j}})\,d\mu^{\varepsilon}=0$

Note that
$\varepsilon\int_{\Omega}(\phi^{\varepsilon}_{x_{i}x_{k}}\phi^{\varepsilon}_{x_{i}x_{j}}\varphi_{p_{k}p_{j}})\,d\mu^{\varepsilon}$
does not vanish in the limit, through a subsequence for every $k,j=1,\dots,n$
we have

$\varepsilon\int_{\Omega}(\phi^{\varepsilon}_{x_{i}x_{k}}\phi^{\varepsilon}_{x_{i}x_{j}}\varphi_{p_{k}p_{j}})\,d\mu^{\varepsilon}\to\int_{\Omega}\varphi_{p_{k}p_{j}}\,dm_{kj},\,\,\forall\varphi\in
C_{c}(\Omega)$

for some nonnegative, symmetric $d\times d$ matrix $(m_{kj})\,\,k,j=1,\dots,n$
of Borel measures, so condition 2, follows. To obtain (3), recall that
$\sup|p^{\varepsilon}(t)|<\infty$ and the periodicity in time.

Now we will prove that $\mu$ satisfies the conditions (a)-(c) in Definition 4
with $P=0$. Following [CGT], [E1] and [T] consider

$\int_{\Omega}(\phi_{t}+H(x,p,t)-\bar{H}^{\varepsilon})^{2}\,d\mu^{\varepsilon}=\varepsilon^{2}\int_{\Omega}|\Delta\phi^{\varepsilon}(x,t)|^{2}d\mu^{\varepsilon}\to
0$

when $\varepsilon\to 0$ due to (8) and (24), thus (a) occurs.

Recalling the equation (18) and choosing as a test function
$\varphi=\psi(\phi^{\varepsilon}(x,t))$

$\int_{\Omega}\psi^{\prime}(\phi^{\varepsilon})[\phi_{t}^{\varepsilon}+D_{x}\phi^{\varepsilon}\cdot
D_{p}H]d\mu^{\varepsilon}+\varepsilon\int_{\Omega}\psi^{\prime}(\phi^{\varepsilon})[\phi_{x_{i}x_{i}}^{\varepsilon}+(\phi_{x_{i}}^{\varepsilon})^{2}]d\mu^{\varepsilon}=0,$

if $\varepsilon$ goes to zero, we obtain
$\int_{\Omega}\psi^{\prime}(\phi)[\phi_{t}+p\cdot D_{p}H]d\mu=0$, choosing
$\psi(\phi)=\phi$, we obtain b). Now part c) follows by choosing in (2) test
functions $\varphi$ that do not depend on the variable p. ∎

## References

  * [B] P. Bernard, _Young measures, superposition, and transport_ , Indiana Univ. Math. J. 57 (1), 247–276 (2008)
  * [BS] G. Barles y P. E. Souganidis, _Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations_. SIAM J. Math. Anal., 32, no. 6 (2001) 1311-1323.
  * [CGT] F. Cagnetti, D.Gomes, & H. V. Tran _Aubry-Mather Measures in the Nonconvex Setting_ ; SIAM J. Math. Anal. Vol. 43, No. 6, 2601–262.
  * [CIS] G. Contreras, R. Iturriaga & H. Sánchez-Morgado _Weak solutions of the Hamilton Jacobi equation for Time Periodic Lagrangians_. arXiv:1307.0287.
  * [E] L. C. Evans _Partial Differential Equations_ , AMS, (2000).
  * [E1] L. C. Evans, _Adjoint and compensated compactness methods for Hamilton-Jacobi PDE_ ; Arch.Ration. Mech. Anal., 197 (2010)1053– 1088\.
  * [EG] L. C. Evans & D. Gomes _Effective Hamiltonians and Averaging for Hamiltonian Dynamics II_ ; Arch. Rational Mech. Anal. 161 , (2002) 271-305.
  * [Fl] W. Fleming, M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer 1993\.
  * [G] D. Gomes, _A Stochastic analog of Aubry-Mather Theory_ Nonlinearity,15, no. 3 (2002) 581-603.
  * [GV] E.Guerra-Velasco(2015). The viscous Hamilton-Jacobi equation for space-time periodic Hamiltonians. (Doctoral Thesis). UNAM Database: http://132.248.9.195/ptd2015/abril/506015252/Index.html;
  * [T] H. V. Tran, _Adjoint Methods for Static Hamilton–Jacobi Equations_ , Calc. Var. Partial Differential Equations, 41 (2011) 301–319.