# Evaporating primordial black holes, the string axiverse, and hot dark
radiation

Marco Calzà<EMAIL_ADDRESS>Univ Coimbra, Faculdade de Ciências e Tecnologia
da Universidade de Coimbra and CFisUC, Rua Larga, 3004-516 Coimbra, Portugal
John March-Russell  Rudolf Peierls Centre for Theoretical Physics, University
of Oxford, Oxford OX1 3PU, United Kingdom  João G. Rosa<EMAIL_ADDRESS>Univ
Coimbra, Faculdade de Ciências e Tecnologia da Universidade de Coimbra and
CFisUC, Rua Larga, 3004-516 Coimbra, Portugal

###### Abstract

We show that primordial black holes (PBHs) develop non-negligible spins
through Hawking emission of the large number of axion-like particles
generically present in string theory compactifications. This is because
scalars can be emitted in the monopole mode ($l=0$), where no angular momentum
is removed from the BH, so a sufficiently large number of scalars can
compensate for the spin-down produced by fermion, gauge boson, and graviton
emission. The resulting characteristic spin distributions for
$10^{8}$-$10^{12}$ kg PBHs could potentially be measured by future gamma-ray
observatories, provided that the PBH abundance is not too small. This yields a
unique probe of the total number of light scalars in the fundamental theory,
independent of how weakly they interact with known matter. The present local
energy density of hot, MeV-TeV, axions produced by this Hawking emission can
possibly exceed $\rho_{\rm CMB}$. Evaporation constraints on PBHs are also
somewhat weakened.

Superstring theory is one of the leading candidates for a fundamental theory
combining quantum gravity and the Standard Model (SM) of particle physics. The
energy scale at which ‘stringy’ effects become relevant is, however, unknown,
and it may well be too high for these effects to be tested in the foreseeable
future. The need for six additional compact spatial dimensions, with unknown
size and geometry, and (broken) supersymmetry also leads to a plethora of
different low energy theories, making it hard to identify specific signatures
of the underlying fundamental theory that could be tested in the laboratory or
with astrophysical observations.

There is, however, a fairly generic prediction of string compactifications if
the strong CP problem is solved by the Peccei-Quinn mechanism Peccei:1977hh ;
Weinberg:1977ma ; Wilczek:1977pj , namely the existence of a large number of
light pseudoscalar particles in the effective four-dimensional theory
Arvanitaki:2009fg . These axion-like particles (ALPs) arise as the zero-modes
in the Kaluza-Klein expansion of antisymmetric tensor fields in the underlying
string theory, including the Neveu-Schwarz 2-form $B_{2}$, existent in all
closed string theories, and the Ramond-Ramond $p$-forms of Type II and Type 1
string theories Svrcek:2006yi . (There are also other ways that ALPs can
arise.) Each $p$-form can give rise to a multitude of ALPs in the 4d theory,
as many as the number of homologically inequivalent $p$-cycles (closed
surfaces) in the extra-dimensional manifold. This number is typically large,
of the order of $10^{2}$ or even $10^{5}$, simply due to the number of
different ways in which a closed surface can be embedded within a six-
dimensional compact manifold.

One of these particles can be the strong-CP-problem solving QCD axion, and
much like the latter a large fraction of the ALPs are protected at all loop
orders by shift symmetries inherited from the gauge symmetries of their parent
tensor fields. The generic expectation is therefore that realistic string
scenarios exhibit a large number of light or even ultra-light axions, whose
exponentially small masses are generated solely by non-perturbative effects.

Light string axions can have a wide range of cosmological and astrophysical
effects, e.g. steps in the matter power-spectrum, rotation of the CMB
polarization and black hole superradiant instabilities Arvanitaki:2009fg ;
Arvanitaki:2010sy . These effects can yield interesting observational
signatures of individual string axions in particular mass ranges, but not of
the whole ‘string axiverse’. The number of relativistic degrees of freedom
during cosmological nucleosynthesis or recombination could, in principle, be
sensitive to the total number of light axions below a certain mass scale, but
this depends crucially on whether they have been in thermal equilibrium with
SM particles in the early Universe, which given the feebleness of their
expected interactions with the latter is rather unlikely.

In this Letter, we propose a new way to probe the total number of ALPs with
mass $m<{\rm few}$ MeV through the spin distribution of primordial black holes
(PBHs) that are evaporating today. PBHs Hawking:1971ei  can be formed in the
early Universe through the gravitational collapse of large overdensities once
their scale becomes smaller than the Hubble horizon Carr:1974nx ; Carr:1975qj
. These could be generated by a plethora of different mechanisms, eg, non-
standard inflation scenarios, curvaton models, or bubble collisions in
cosmological phase transitions, spanning a wide range of masses. PBHs may
account for at least a fraction of the dark matter in the Universe, which has
sparked a renewed interest in this subject, alongside the recent detection of
gravitational waves from a population of astrophysical BH binaries (see e.g.
Carr:2020xqk  for a recent review).

Here we will be interested in small PBHs born with masses around
$10^{11}$-$10^{12}$ kg, which are evaporating today. Although Hawking
evaporation generically spins down a BH, an exception is the emission of light
scalar particles, as first pointed out by Taylor, Chambers and Hiscock (TCH)
Chambers:1997ai ; Taylor:1998dk , since these are the only type of particle
that can be emitted in the monopole ($l=0$) mode. Scalar emission may
therefore reduce a BH’s mass without reducing its angular momentum, therefore
increasing its dimensionless spin parameter $a_{*}=J/M^{2}$ (we set
$G=\hbar=c=k_{B}=1$). In fact, TCH showed a BH evaporating solely through
scalar emission would asymptote to a configuration with $a_{*}\simeq 0.555$,
and that at least 32 light scalars could compensate the BH spin down through
the emission of photons, neutrinos and gravitons.

Given the large number of ALPs expected in realistic string compactifications,
this motivates exploring the spin evolution of PBHs in this context. We will
assume that Hawking emission is the only mechanism that affects the PBH spin,
an assumption that we will critically review at the end of our discussion. In
the simplest scenario of PBH formation where large density fluctuations are
generated on super-horizon scales and reenter the Hubble horizon during the
radiation era, the nearly spherical collapse endows the BHs with only a small
spin, at the few percent level Chiba:2017rvs ; Mirbabayi:2019uph ;
DeLuca:2019buf . There are, however, several possible scenarios where PBHs
could be born with much larger spins, e.g. formation during an early matter-
dominated epoch Harada:2017fjm . We will be agnostic about such initial
conditions, since as we will show any natal spin is erased once a PBH has
evaporated significantly unless a substantial number of light scalar fields is
emitted.

To determine the evolution of PBHs we follow the formalism of Page Page:1976df
; Page:1976ki , where the dynamics of the BH mass and spin is determined by
the functions $\mathcal{F}\equiv-M^{2}dM/dt$ and
$\mathcal{G}\equiv-(M/a_{*})dJ/dt$, which removes the dependence on the BH
mass. These are given by:

$\begin{pmatrix}\mathcal{F}\\\
\mathcal{G}\end{pmatrix}=\sum_{i,l,m}\frac{1}{2\pi}\int_{0}^{\infty}dx\frac{\Gamma_{i,l,m}}{e^{2\pi
k/\kappa}\pm 1}\begin{pmatrix}x\\\ ma_{*}^{-1}\end{pmatrix}$ (1)

where the sum is taken over all particle species $i$ and angular momentum
quantum numbers $(l,m)$, $x=\omega M$, $k=\omega-m\Omega$ and
$\kappa=\sqrt{1-a_{*}^{2}}/2r_{+}$ is the surface gravity of the Kerr BH, with
$\Omega$ denoting its angular velocity at the event horizon, $r_{+}$. The
upper (lower) sign in the denominator corresponds to fermion (boson) fields.
The greybody factors $\Gamma=1-|Z_{\mathrm{out}}/Z_{\mathrm{in}}|^{2}$, where
$Z_{\mathrm{in}}$ and $Z_{\mathrm{out}}$ denote the amplitudes of the ingoing
and outgoing wave modes, respectively. For massless fields, these can be
computed by solving the Teukolsky equation for the radial wave function, upon
separation of variables of the underlying wave equation(s), imposing ingoing
boundary conditions at the horizon Teukolsky:1973 ; Press:1973zz ;
Teukolsky:1974yv . This can also be done for massive fields, but given that
the emission of particles with masses, $m$, above the Hawking temperature
$T_{H}=\kappa/2\pi\simeq 1\,{\rm GeV}(10^{10}{\rm kg}/M)$ is exponentially
suppressed we work in the approximation where particles are considered
massless for $T_{H}>m$ and are otherwise absent from the emission spectrum. As
the methodology for computing these factors is well known, we refer the
interested reader to e.g. Taylor:1998dk  for more details.

We include in our numerical calculation an arbitrary number, $N_{a}$, of
scalars which, for simplicity, we assume to have masses below the initial
$T_{H}$, which is around the 10MeV scale for PBHs with a lifetime comparable
to the age of the Universe. These are emitted alongside photons, gravitons,
neutrinos and electrons/positrons from the start of our calculation. As a PBH
evaporates, its $T_{H}$ increases, and we eventually include muons, taus and
QCD degrees of freedom in the evolution at the corresponding mass thresholds.
Pions ($\pi^{0}$ and $\pi^{\pm}$) are the only hadrons with masses well below
the QCD scale, and so the only hadronic states included directly in the BH
emission spectrum. For $T_{H}$ above the QCD scale a PBH emits elementary
quarks and gluons that subsequently hadronize, and this can be taken into
account by considering their effective QCD masses MacGibbon:1990zk . The
effective masses given in ParticleDataGroup:2006fqo  are used; we have checked
that our results do not change significantly if we use other values proposed
in the literature, such as Iritani:2009mp .

In Figure 1, we give our result for the present spin of PBHs, $a_{*0}$, as a
function of their present mass, $M_{0}$, for different numbers of emitted
ALPs, assuming an initial spin $a_{*}=0.01$ corresponding to PBH formation in
the radiation era. Note that BHs in the given mass range correspond to the
evaporated remnants of an initial population of PBHs with comparable masses
and which are presently at different stages of their evolution. This justifies
assuming a common initial spin. We also note that the critical PBH mass for a
lifetime matching the age of the Universe of 13.8 Gyr exhibits a weak
dependence on the number of different axions emitted, ranging from $5\times
10^{11}\,$kg in the absence of light axions to $\sim 2.7\times 10^{12}\,$kg
for $N_{a}=1000$, with a $N_{a}^{1/3}$ scaling for $N_{a}\gtrsim 10$.

Figure 1: Present PBH spin, $a_{*0}$, as a function of their present mass,
$M_{0}$, for an initial population with spin $a_{*}=0.01$ and varying mass.
Curves labelled by number of light ALPs.

While Figure 1 shows the current snapshot of the PBH Regge plot, it can also
be viewed as a time evolution, since heavier BHs have evaporated little and
lighter BHs have already lost a significant fraction of their original mass.
For $N_{a}=0$, PBHs quickly lose their initial spin as they evaporate, as has
been the standard lore in the literature. But the inclusion of many ALPs
changes this picture considerably, with $a_{*}$ increasing initially due to
the emission of the light scalars. For $N_{a}\lesssim 400$, the spin increases
until $T_{H}$ reaches the threshold for quark and gluon emission, at which
point the BH starts to spin down due to the large number of colored spin-1/2
and spin-1 degrees of freedom. For yet larger $N_{a}$, which is still
plausible in string theories, the BH spin parameter stabilizes at values
$a_{*0}\gtrsim 0.2$ and tending to the critical value $a_{*0}\simeq 0.555$
found by TCH for pure scalar emission. Note that, for $N_{a}\gtrsim 400$, PBHs
never spin down completely even including all SM degrees of freedom, since
$d\ln a_{*}/d\ln M=\mathcal{G}/\mathcal{F}-2$ has a non-trivial zero.

For comparison, we show in Figure 2 the results for an initial population of
near-extremal PBHs. Again, we find that even in this case PBHs with masses
below $\sim 10^{11}\,$kg should have a presently negligible spin for
$N_{a}=0$, but the inclusion of many light axions halts the BH spin-down or at
least delays it considerably until QCD degrees of freedom can be efficiently
emitted. Once more we find that for $N_{a}\gtrsim 400$, $a_{*0}\gtrsim 0.2$
for light BHs. We note that the slightly oscillatory behaviour that can be
observed in some of the curves in Figure 2 in the mass range
$10^{10}$-$10^{11}$kg is due to the opposing effects of $\pi$’s and $\mu$’s on
the spin evolution, with the former contributing to the BH spin-up and the
latter spinning the BH down.

Figure 2: As Figure 1 except initial spin parameter $a_{*}=0.99$.

Although not shown explicitly, we have found similar spin distributions for
intermediate values of the initial spin, with either a ‘peak’- or
‘plateau’-like shape in the $10^{10}$-$10^{11}\,$kg (present) BH mass range,
where $T_{H}\sim 0.1$-$1\,$GeV. These are the PBHs that have evaporated
essentially through the emission of non-colored degrees of freedom and for
which the effect of the additional axions is more pronounced. A generic
feature is thus that PBHs in this mass range should have present spin
parameters of at least $a_{*0}\sim{\rm few}\times 10^{-2}$, and up to or
slightly over $\sim 0.5$ in the presence of hundreds of light axions, so that
detecting BHs with such properties would constitute a ‘smoking-gun’ for the
string axiverse.

This conclusion is, of course, based on the assumption that evaporation is the
main mechanism driving spin evolution. This is justified in the relevant PBH
mass range as, firstly, accretion has been shown to be irrelevant for small
PBHs with a lifetime close to the age of the Universe Rice:2017avg  and is
also expected to occur in a quasi-spherical regime where a BH cannot
efficiently spin up Ali-Haimoud:2016mbv . Secondly, mergers should also be
rare events for PBHs in this mass range. Following e.g. Sasaki:2016jop , two
BHs can become gravitationally bound and decouple from the Hubble flow if
their physical separation at matter-radiation equality $x<f^{1/3}\bar{x}$,
where $\bar{x}$ is the average BH separation at this redshift and $f$ is the
PBH fraction of the dark matter density. Given the existing constraints on $f$
for BHs that are evaporating today from the contribution of Hawking radiation
to the extra-galactic diffuse gamma-ray background, $f\lesssim 10^{-7}$
Carr:2009jm ; Arbey:2019vqx , we conclude that only a small fraction of such
BHs can become gravitationally bound and subsequently merge. Furthermore,
mergers occurring early in the cosmic history will only affect the ‘initial’
conditions for the subsequent evolution driven essentially by evaporation.

Finally, PBHs in the interesting mass range also suffer from pion superradiant
instabilities, where dense clouds of $\pi^{0}$’s are produced around the PBHs
at the expense of their rotational energy Ferraz:2020zgi . However, such
instabilities are only triggered for large initial spins and remain active for
less than $\sim 1\,$Gyr, before the PBHs have evaporated significantly.
Hawking evaporation will thus still determine the final BH spins as assumed in
our computation.

The PBH spin distribution found here is, in any case, characteristic of the
evaporation process in the presence of many light scalar fields and cannot, to
our knowledge, be mimicked by other processes. Thus, observing such a
distribution would constitute a unique signature of an underlying theory with
a large number of light scalars.

Evaporating PBHs in the interesting mass range may in principle be detected
through their Hawking photon emission, which includes both primary and
secondary photons, i.e. those produced by the decays of the Hawking-emitted
particles. The latter dominate the flux for PBHs in the
$10^{10}$-$10^{11}\,$kg mass range MacGibbon:2015mya . According to Fermi-
LAT:2018pfs , a $10^{10}\,$kg PBH could be detectable with Fermi-LAT up to a
distance of $\sim 200\,$AU. This compares to the mean distance between PBHs of
present mass $M_{0}$ accounting for a present fraction $f_{0}$ of the local
dark matter density $d\simeq 40\,{\rm AU}(M_{0}/10^{10}{\rm
kg})^{1/3}(10^{-7}/f_{0})^{1/3}$.

Note the presence of many ALPs weakens the limits on the initial $f$ (and
somewhat $f_{0}$) as, for a given present $M_{0}$ a reduced fraction of the
total lifetime-integrated BH luminosity goes into photons. Moreover the past-
emitted photons are less energetic than they would have been in the absence of
the ALPs as the past BH must be heavier, and thus colder, than in the
$N_{a}=0$ case. Precise evaluation of the new limits requires a dedicated
study which we leave to future work. (For related studies of modified PBH
constraints in the context of extra-dimensional and other non-SM theories see
Argyres:1998qn ; Green:1999yh ; Johnson:2020tiw ; Baker:2021btk ;
Arbey:2021yke .) We take $f_{0}\sim 10^{-7}$ to roughly saturate current
bounds Carr:2009jm ; Arbey:2019vqx ; footnote .

Instead, the BH Hawking luminosity is dominated by a population of hot quasi-
blackbody ALPs with mean energy $\sim 2.8T_{H}$, and which are ‘dark’ with
respect to the SM up to possible feeble interactions set by $1/f_{a}$, where
$f_{a}$ is the appropriate axion ‘decay constant’. The integrated flux of ALPs
from a single PBH is $\sim 3\times 10^{22}N_{a}(10^{10}{\rm kg}/M)s^{-1}$ in
the relevant PBH spin range. Unlike the case of greatly-red-shifted dark
matter particles and/or axions produced by the Hawking evaporation of micro
PBHs in the very early pre-BBN universe Fujita:2014hha ; Allahverdi:2017sks ;
Lennon:2017tqq ; Hooper:2020evu  the axions in our case are hardly red-shifted
at all as the relevant Hawking emission is occurring now.

Although extremely challenging to detect, the much higher energies of these
axions (and masses too for some of the ALPs) might enable new detection
strategies as compared to the standard QCD axion solar flux expected for a
given $1/f_{a}$. Also note that the usual $\Delta N_{eff}$ constraints
Lennon:2017tqq ; Hooper:2019gtx ; Hooper:2020evu  on extra relativistic
degrees of freedom from BH evaporation derived from observations of BBN, CMB
or structure-formation epochs do not apply as the Hawking emission conversion
of PBH mass into relativistic ALPs is dominated by late times and the present
local energy density of hot ‘dark-radiation’ ALPs can potentially exceed
$\rho_{CMB}$. Detection of one or more sub-components of this background of
energetic dark axions would be a striking signal for both axiverse physics and
the existence of Hawking evaporating PBHs.

Returning to the photon flux, planned gamma-ray observatories, e.g. HARPO and
e-ASTROGAM, are expected to reach a somewhat better sensitivity than Fermi-LAT
in the energy range of 0.1-1GeV Knodlseder:2016pey , although an improvement
of a few orders of magnitude could be required if $f_{0}\ll 10^{-7}$, or if a
precise measurement of the full photon spectrum, including secondaries, is
required for an accurate determination of the PBH mass and spin.

An interesting possibility is that some ALPs could decay into photon pairs
after emission. Although this requires $f_{a}$’s well below the typical GUT
scale values of string constructions, such reduced decay constants are
possible for some ALPs in warped and large-volume constructions Svrcek:2006yi
; Flacke:2006ad ; Cicoli:2012sz . In this case such ALPs would contribute to
the secondary photon emission, which would encode additional information on
the axiverse. The corresponding contribution to the diffuse gamma-ray
background could also yield additional constraints on PBHs.

In Figure 3 we show the primary photon spectrum for different PBH masses and
spins, given by

${d^{2}N_{\gamma}\over dtdE_{\gamma}}={1\over
2\pi}\sum_{l,m}{\Gamma_{1,l,m}(\omega)\over e^{2\pi\omega/\kappa}-1}~{},$ (2)

with $E_{\gamma}=\omega$. As one can see, the peak emission is quite sensitive
to the PBH spin, while the corresponding energy depends mostly on the PBH
mass. A similar effect is obtained for spin-1/2 particles, so that we expect
both the primary and secondary photon spectra to exhibit non-trivial
dependences on the PBH mass and spin. Although we will leave a detailed
analysis of the full spectrum for a dedicated future work, this suggests that
in principle one can measure PBH spins $a_{*0}\gtrsim 0.1$ with at least a
$\sim 10\%$ precision measurement of the photon spectrum, provided the
distance to the PBH can be measured with comparable accuracy through parallax.
Note that the time evolution of $T_{H}$, and thus photon flux and mean energy,
is faster in the presence of many ALPs, than in the $N_{a}=0$ case. Given
suitable high-precision measurements this can give confirmation of $N_{a}\neq
0$.

Figure 3: PBH primary photon emission rate.

Moreover, halo PBHs velocities relative to the Earth are 2-3$\times
10^{2}\,$km/s, corresponding roughly to 40-60 AU$/{\rm yr}$. Thus the
population of bright (for both photons and hot ALPs) ‘point source’
evaporating PBHs in the solar neighbourhood can potentially change every few
years, so increasing the chance of detecting BHs at different stages of their
evaporation process. This further allows one to fully assess the effects of a
large number of axions.

In conclusion, detecting evaporating PBHs is of course a worthy endeavour to
pursue on its own, given what it can tell us about both the early history of
the Universe and the nature of semi-classical BHs. Our analysis shows that, in
addition, their mass-spin distribution can give a unique probe of beyond the
SM physics, which we hope may motivate future efforts in PBH detection.

###### Acknowledgements.

M.C. is supported by the FCT doctoral grant SFRH/BD/146700/2019. J.G.R. is
supported by the FCT Grant No. IF/01597/2015 and by the CFisUC project No.
UID/FIS/04564/2019, and partially by the ENGAGE SKA grant
(POCI-01-0145-FEDER-022217).

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# On the energy and helicity conservation of the incompressible Euler
equations

Yanqing Wang111College of Mathematics and Information Science, Zhengzhou
University of Light Industry, Zhengzhou, Henan 450002, P. R. China Email:
<EMAIL_ADDRESS>Wei Wei222School of Mathematics and Center for
Nonlinear Studies, Northwest University, Xi’an, Shaanxi 710127, P. R. China
Email<EMAIL_ADDRESS>Gang Wu333School of Mathematical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, P. R. China Email:
<EMAIL_ADDRESS>and Yulin Ye444School of Mathematics and Statistics,
Henan University, Kaifeng, 475004, P. R. China. Email<EMAIL_ADDRESS>
###### Abstract

In this paper, we are concerned with the minimal regularity of weak solutions
implying the law of balance for both energy and helicity in the incompressible
Euler equations. In the spirit of recent works due to Berselli [5] and
Berselli-Georgiadis [6], it is shown that the energy of weak solutions is
invariant if $v\in L^{p}(0,T;B^{\frac{1}{p}}_{\frac{2p}{p-1},c(\mathbb{N})})$
with $1<p\leq 3$ and the helicity is conserved if $v\in
L^{p}(0,T;B^{\frac{2}{p}}_{\frac{2p}{p-1},c(\mathbb{N})})$ with $2<p\leq 3$
for both the periodic domain and the whole space, which generalizes the
classical work of Cheskidov-Constantin-Friedlander-Shvydkoy in [10]. This
indicates the role of the time integrability, spatial integrability and
differential regularity of the velocity in the conserved quantities of weak
solutions of the ideal fluid.

MSC(2020): 35Q35, 35Q86, 76D03
Keywords: Euler equations; Onsager conjecture; energy conservation; helicity
conservation

##  1 Introduction

The evolution of homogeneous inviscid incompressible flows on $\Omega$ is
described by the following Euler equations

$\left\\{\begin{aligned} &v_{t}+v\cdot\nabla v+\nabla\Pi=0,\\\
&\text{div}\,v=0,\\\ &v|_{t=0}=v_{0}(x),\end{aligned}\right.$ (1.1)

where $\Omega$ is either the whole space $\mathbb{R}^{d}$ or the periodic
domain $\mathbb{T}^{d}$ with $d=2,3$. The unknown vector
$v=(v_{1},\cdots,v_{d})$ is velocity field and $\Pi$ represents the pressure.
The initial velocity $v_{0}$ satisfies $\text{div}\,v_{0}=0$. The vorticity
$\omega=\text{curl\,}v$ of the velocity field is determined by

$\omega_{t}+v\cdot\nabla\omega-\omega\cdot\nabla
v=0,~{}~{}\text{div\,}\omega=0.$ (1.2)

As we know, there exist only two conserved quantities of second order in the
Euler equations: kinetic energy conservation

$\frac{1}{2}\int_{\Omega}|v(x,t)|^{2}dx=\frac{1}{2}\int_{\Omega}|v(x,0)|^{2}dx$
(1.3)

and helicity preservation

$\int_{\Omega}\omega(x,t)\cdot v(x,t)dx=\int_{\Omega}\omega(x,0)\cdot
v(x,0)dx.$ (1.4)

These two conserved quantities attract a lot of attentions and research of
physicists and mathematicians(see e.g. [21, 19, 17, 12, 16, 25, 26, 18, 10,
13, 29, 27]). The celebrated Kolmogorov 4/5 law and Yaglom 4/3 law are in
terms of energy of the Euler equations. The study of energy and helicity plays
an important role in the theory of turbulence (see e.g. [12, 21, 25, 26] and
references therein). The famous Onsager conjecture raised in [27] is stated
that the critical Hölder regularity of weak solutions keeping the energy is
1/3. A number of works have devoted to the minimal regularity of weak
solutions of the Euler system to preserve the energy(see e.g. [18, 13, 10, 5,
6, 20, 2] and references therein). To be specific, we give a brief survey
here. The energy of weak solutions $v$ of the Euler equations (LABEL:Euler) is
invariant if one of the following conditions is satisfied

  * •

Eyink [18]: $v\in C_{\ast}^{\alpha},\alpha>1/3$;

  * •

Constantin-E-Titi [13] $v\in L^{3}(0,T;B^{\alpha}_{3,\infty}(\mathbb{T}^{3}))$
with $\alpha>1/3$.

  * •

Duchon and Robert: [17]: $\int_{\mathbb{T}^{d}}|v(t,x+\xi)-v(t,x)|^{3}dx\leq
C(t)|\xi|\sigma(|\xi|)$, where $C(t)\in L^{1}(0,T)$ and $\sigma(a)\rightarrow
0$ as $a\rightarrow 0$;

  * •

Cheskidov-Constantin-Friedlander-Shvydkoy [10]: $v\in L^{3}(0,T;\in
B^{1/3}_{3,c(\mathbb{N})});$

  * •

Fjordholm-Wiedemann [20]: $v\in L^{3}(0,T;\underline{B}^{1/3}_{3,VMO});$

  * •

Berselli [5]: $v\in
L^{\frac{1}{\alpha}+\delta}(0,T;C^{\alpha}(\mathbb{T}^{3})$ or $v\in
L^{\frac{1}{\alpha}}(0,T;C_{w}^{\alpha}(\mathbb{T}^{3}))$,
$\alpha\in(1/3,1),\delta>0$;

  * •

Berselli-Georgiadis [6]: $v\in
L^{\frac{1}{\alpha}}(0,T;B^{\beta}_{\frac{2}{1-\alpha},\infty}(\mathbb{T}^{3})),\alpha\in(1/3,1),1/3<\alpha<\beta<1;$
$v\in L^{r}(0,T;W^{1,q}(\mathbb{T}^{3})),r>\frac{5q}{5q-6},q>2.$

The inclusion relations of the aforementioned spaces are that, for
$\alpha>1/3$,

$C_{\ast}^{\alpha}(C_{w}^{\alpha})\subseteq C^{\alpha}\subseteq
B^{\alpha}_{3,\infty}\subseteq
B^{\frac{1}{3}}_{3,c(\mathbb{N})}\subseteq\underline{B}^{\frac{1}{3}}_{3,VMO}\subseteq
B^{\frac{1}{3}}_{3,\infty}.$ (1.5)

Inspired by the recent results [5, 6] and classical critical Onsager
sufficient condition via $B^{1/3}_{3,c(\mathbb{N})}$ in [10], two natural
questions are weather $v\in
L^{\frac{1}{\alpha}}(0,T;B^{\beta}_{\frac{2}{1-\alpha},\infty})$ with
$1/3<\alpha<\beta$ can be replaced by $v\in
L^{\frac{1}{\alpha}}(0,T;B^{\alpha}_{\frac{2}{1-\alpha},c(\mathbb{N})}(\mathbb{T}^{3}))$
and $v\in L^{\frac{5q}{5q-6}}(0,T;W^{1,q}(\mathbb{T}^{3}))$ guarantees the
energy conservation of weak solutions in the Euler equations (LABEL:Euler).
The first result of this paper is formulated as follows.

######  Theorem 1.1.

Let $v$ be a weak solution of the incompressible Euler equations (LABEL:Euler)
on the whole spaces $\mathbb{R}^{d}$ for $d=2,3$. Then for any $0\leq t\leq
T$, the energy equality (1.3) of weak solutions is preserved provided one of
the following condition is satisfied

  1. (1)

$v\in
L^{p}(0,T;B^{\frac{1}{p}}_{\frac{2p}{p-1},c(\mathbb{N})}(\mathbb{R}^{d})),1<p\leq
3;$

  2. (2)

$\nabla v\in L^{p}(0,T;L^{\frac{2dp}{(d+2)(p-1)}}(\mathbb{R}^{d})),1<p\leq 3.$

######  Remark 1.1.

The first part of this theorem is a generalization of classical energy
conservation criterion $v\in L^{3}(0,T;B^{1/3}_{3,c(\mathbb{N})})$ showed by
Cheskidov-Constantin-Friedlander-Shvydkoy for the Euler equations in [10].

######  Remark 1.2.

Actually, by means of interpolation method, the range of $p$ in the first part
of this theorem as well as Theorem 1.4 is $[1,3]$. Indeed, the interpolation
allows us to avoid the finite difference quotients of higher order in the
definition of Besov spaces.

The key point in [5, 6] is to make full use of the kinetic energy in the
Constantin-E-Titi type estimates concerning mollifier kernel on the periodic
domain $\mathbb{T}^{d}$. In the spirit of this and the study of energy
equality for the Navier-Stokes equations in [11], the starting point of this
theorem is the following interpolation inequality

$2^{\frac{1}{3}j}\|\Delta_{j}v\|_{L^{3}(\mathbb{R}^{d})}\leq\|\Delta_{j}v\|_{L^{2}(\mathbb{R}^{d})}^{1-\frac{p}{3}}\Big{[}2^{j\frac{1}{p}}\|\Delta_{j}v\|_{L^{\frac{2p}{p-1}}(\mathbb{R}^{d})}\Big{]}^{\frac{p}{3}},$

which helps us to reduce the first part of this theorem to the classical
Onsager energy conservation sufficient condition $v\in
B^{1/3}_{3,c(\mathbb{N})}$ due to Cheskidov-Constantin-Friedlander-Shvydkoy in
[10]. In addition, it is worth pointing out that there exists an alternative
proof to this part via Littlewood-Paley theory as [10]. Roughly speaking, it
suffices to replace $\omega$ by $v$ and set $k=p,\ell=q$ in
(4.11)-(LABEL:4.13), hence we omit the details here. To the knowledge of the
authors, the statement similar to Theorem 1.1 does not exist in known
literature. Enlightening by the recent sufficient conditions for the energy
equality of the Navier-Stokes equations in [28, 30], we will show the energy
conservation criteria based on the gradient of the veolcity for the weak
solutions of the Euler equations.

For the torus case, we have the following result

######  Theorem 1.2.

Let $v$ be a weak solution to the incompressible Euler equations (LABEL:Euler)
on $\mathbb{T}^{d}$ with $d=2,3$. Then for any $0\leq t\leq T$, the energy
equality (1.3) of weak solutions is preserved provided that one of the
following conditions is satisfied

  1. (1)

$v\in
L^{p}(0,T;B^{\frac{1}{p}}_{\frac{2p}{p-1},c(\mathbb{N})}(\mathbb{T}^{d})),1<p\leq
3;$

  2. (2)

$v\in
L^{p}(0,T;\underline{B}^{\frac{1}{p}}_{\frac{2p}{p-1},VMO}(\mathbb{T}^{d})),1<p\leq
3;$

  3. (3)

$\nabla v\in L^{p}(0,T;L^{\frac{2dp}{(d+2)(p-1)}}(\mathbb{T}^{d})),1<p\leq 3.$

######  Remark 1.3.

The criteria based on the gradient of the velocity here implies that $v\in
L^{\frac{5q}{5q-6}}(0,T;W^{1,q}(\mathbb{T}^{3}))\ \text{with}\ q\geq 9/5$
guarantees the energy conservation. Hence, this improves the recent energy
conservation sufficient condition in [6].

######  Remark 1.4.

According to inclusion relations (1.5), this theorem is an improvement of
corresponding results by Berselli-Georgiadis in [6].

As a byproduct, there holds

######  Corollary 1.3.

Let $\Omega$ be the whole space $\mathbb{R}^{3}$ or the periodic domain
$\mathbb{T}^{3}$. Suppose that $v$ is a Leray-Hopf weak solutions of the 3D
Navier-Stokes on $\Omega$. Then the energy equality, for $t\in[0,T)$,

$\|v(t)\|_{L^{2}(\Omega)}^{2}+2\int_{0}^{t}\|\nabla
v\|_{L^{2}(\Omega)}^{2}ds=\|v_{0}\|_{L^{2}(\Omega)}^{2}.$ (1.6)

is valid if

$v\in L^{p}(0,T;B^{\frac{1}{p}}_{\frac{2p}{p-1},\infty}(\Omega)),1<p\leq 3.$
(1.7)

######  Remark 1.5.

It is worth remarking that the analogous result in this corollary can be found
in [11]. The novel here is the following observation

$\|u\|_{L^{p}(0,T;B^{\frac{1}{p}}_{\frac{2p}{p-1},\infty}(\mathbb{R}^{3}))}\leq
C\|\nabla u\|_{L^{p}(0,T;L^{\frac{6p}{5p-5}}(\mathbb{R}^{3}))},1<p\leq 3,$
(1.8)

from which we know that this corollary immediately yields the energy equality
sufficient condition in terms of gradient of the velocity in space
$L^{p}(0,T;L^{\frac{6p}{5p-5}}(\mathbb{R}^{3}))$ derived in [7, 28, 35, 3, 4]
for the 3D Navier-Stokes equations.

Next, we turn our attention to the helicity conservation (1.4) of the Euler
equations. We briefly list the known results in this direction as follows: the
weak solutions $v$ of the 3D Euler equations conserve the helicity if one of
the following conditions is satisfied

  * •

Chae [8]: $\omega\in L^{3}(0,T;B^{\alpha}_{\frac{9}{5},\infty})$ with
$\alpha>\frac{1}{3};$

  * •

Chae [9]: $v\in L^{r_{1}}(0,T;\dot{B}^{\alpha}_{\frac{9}{2},q})$ and
$\omega\in L^{r_{2}}(0,T;\in\dot{B}^{\alpha}_{\frac{9}{5},q})$, with
$\alpha>\frac{1}{3}$, $q\in[2,\infty]$, $r_{1}\in[2,\infty]$,
$r_{2}\in[1,\infty]$ and $\frac{2}{r_{1}}+\frac{1}{r_{2}}=1;$

  * •

Cheskidov-Constantin-Friedlander-Shvydkoy [10]: $v\in
L^{3}(0,T;B^{\frac{2}{3}}_{3,c(\mathbb{N})});$

  * •

De Rosa [16]: $v\in L^{2r}(0,T;W^{\theta,2p})$ and $\omega\in
L^{\kappa}(0,T;W^{\alpha,q})$ with
$\frac{1}{p}+\frac{1}{q}=\frac{1}{r}+\frac{1}{\kappa}$ and $2\theta+\alpha\geq
1$;

  * •

[32]: $v\in L^{k}(0,T;\dot{B}^{\alpha}_{p,c(\mathbb{N})})\ \text{and}\
\omega\in L^{\ell}(0,T;\dot{B}^{\beta}_{q,\infty})$ with
$\frac{2}{k}+\frac{1}{\ell}=1,\frac{2}{p}+\frac{1}{q}=1,2\alpha+\beta\geq 1.$

For the helicity conservation of weak solutions of the Euler equations, our
results read as follows:

######  Theorem 1.4.

Let $v$ be a weak solution of incompressible Euler equations (LABEL:Euler) in
the sense of Definition 2.1 and $\omega\in
C([0,T];L^{\frac{3}{2}}(\mathbb{R}^{3}))$. Then for any $t\in[0,T]$, the
helicity conservation (1.4) holds provided that one of the following
conditions is satisfied

  1. (1)

$v\in
L^{p}(0,T;B^{\frac{2}{p}}_{\frac{2p}{p-1},c(\mathbb{N})}(\mathbb{R}^{3})),2<p\leq
3;$

  2. (2)

$\nabla v\in L^{p}(0,T;L^{\frac{6p}{5p-7}}(\mathbb{R}^{3})),2<p\leq 3;$

  3. (3)

$v\in L^{k}(0,T;{B}^{\alpha}_{\ell,c(\mathbb{N})})\ \text{and}\ \omega\in
L^{p}(0,T;{B}^{\beta}_{q,\infty})$ with
$\frac{\theta}{k}+\frac{1}{p}=1,\frac{\theta}{\ell}+\frac{1}{q}=\frac{\theta}{2},\theta\alpha+\beta\geq
1;$

  4. (4)

$v\in L^{k}(0,T;{B}^{\alpha}_{\ell,\infty})\ \text{and}\ \omega\in
L^{p}(0,T;{B}^{\beta}_{q,c(\mathbb{N})})$ with
$\frac{\theta}{k}+\frac{1}{p}=1,\frac{\theta}{\ell}+\frac{1}{q}=\frac{\theta}{2},\theta\alpha+\beta\geq
1.$

######  Theorem 1.5.

Let $v$ be a weak solution of incompressible Euler equations (LABEL:Euler) in
the sense of Definition 2.1 and $\omega\in
C([0,T];L^{\frac{3}{2}}(\mathbb{T}^{3}))$. Then for any $t\in[0,T]$, the
helicity conservation (1.4) holds provided that one of the following
conditions is satisfied

  1. (1)

$v\in
L^{p}(0,T;B^{\frac{2}{p}}_{\frac{2p}{p-1},c(\mathbb{N})}(\mathbb{T}^{3})),2<p\leq
3;$

  2. (2)

$v\in
L^{p}(0,T;\underline{B}^{\frac{2}{p}}_{\frac{2p}{p-1},VMO}(\mathbb{T}^{3})),2<p\leq
3;$

  3. (3)

$\nabla v\in L^{p}(0,T;L^{\frac{6p}{5p-7}}(\mathbb{T}^{3})),2<p\leq 3;$

  4. (4)

$v\in L^{k}(0,T;{B}^{\alpha}_{\ell,c(\mathbb{N})})\ \text{and}\ \omega\in
L^{p}(0,T;{B}^{\beta}_{q,\infty})$ with
$\frac{\theta}{k}+\frac{1}{p}=1,\frac{\theta}{\ell}+\frac{1}{q}=\frac{\theta}{2},\theta\alpha+\beta\geq
1;$

  5. (5)

$v\in L^{k}(0,T;{B}^{\alpha}_{\ell,\infty})\ \text{and}\ \omega\in
L^{p}(0,T;{B}^{\beta}_{q,c(\mathbb{N})})$ with
$\frac{\theta}{k}+\frac{1}{p}=1,\frac{\theta}{\ell}+\frac{1}{q}=\frac{\theta}{2},\theta\alpha+\beta\geq
1.$

######  Remark 1.6.

These theorems extend the known sufficient class for implying the helicity of
weak solutions in the Euler equations.

There may be potential applications of the strategy of this paper. It is
useful to study physical conserved quantities in other hydrodynamic equations.
A possible candidate is the following EMHD equations in [22, 31]

$\left\\{\begin{aligned} &h_{t}+\nabla\times\Big{[}(\nabla\times h)\times
h\Big{]}=0,\\\ &\mathrm{div}\,h=0.\end{aligned}\right.$ (1.9)

It should be pointed out that the progress of the negative part of Onsager
conjecture can be found in [23, 14, 15].

The rest of the paper is organized as follows. In Section 2, we present some
notations such as the definitions of various spaces and auxiliary lemmas which
will be frequently used throughout this paper. A kind of the Constantin-E-
Titi type estimates concerning mollifier kernel are established to allow us to
use the kinetic energy. Section 3 is devoted to the study of energy
conservation of weak solutions to ideal fluid for both the whole space and the
periodic case. We are concerned with the conservation law of helicity of weak
solutions to invicid flow in Section 4. Eventually, in Section 5, we present
some concluding remarks.

##  2 Notations and some auxiliary lemmas

Sobolev spaces: First, we introduce some notations used in this paper. For
$p\in[1,\,\infty]$, the notation $L^{p}(0,\,T;X)$ stands for the set of
measurable functions on the interval $(0,\,T)$ with values in $X$ and
$\|f(t,\cdot)\|_{X}$ belonging to $L^{p}(0,\,T)$. The classical Sobolev space
$W^{k,p}(\Omega)$ is equipped with the norm
$\|f\|_{W^{k,p}(\Omega)}=\sum\limits_{|\alpha|=0}^{k}\|D^{\alpha}f\|_{L^{p}(\Omega)}$,
where $\Omega$ is either the whole space $\mathbb{R}^{d}$ or the periodic
domain $\mathbb{T}^{d}$ with $d=2,3$.
Besov spaces: We denote $\mathcal{S}$ the Schwartz class of rapidly decreasing
functions, $\mathcal{S}^{\prime}$ the space of tempered distributions and
$\mathcal{S}^{\prime}/\mathcal{P}$ the quotient space of tempered
distributions which modulo polynomials. We use $\mathcal{F}f$ or $\widehat{f}$
to denote the Fourier transform of a tempered distribution $f$. To define
Besov spaces, we need the following dyadic unity partition(see e.g. [1]).
Choose two nonnegative radial functions $\varrho$, $\varphi\in
C^{\infty}(\mathbb{R}^{d})$ supported respectively in the ball
$\mathcal{B}=\\{\xi\in\mathbb{R}^{d}:|\xi|\leq\frac{4}{3}\\}$ and the shell
$\mathcal{C}=\\{\xi\in\mathbb{R}^{d}:\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\\}$
such that

$\varrho(\xi)+\sum_{j\geq
0}\varphi(2^{-j}\xi)=1,\quad\forall\xi\in\mathbb{R}^{d};\qquad\sum_{j\in\mathbb{Z}}\varphi(2^{-j}\xi)=1,\quad\forall\xi\neq
0.$

Write $h=\mathcal{F}^{-1}\varphi$ and $\tilde{h}=\mathcal{F}^{-1}\varrho$,
then nonhomogeneous dyadic blocks $\Delta_{j}$ are defined by

$\Delta_{j}u:=0~{}~{}\text{if}~{}~{}j\leq-2,~{}~{}\Delta_{-1}u:=\varrho(D)u=\int_{\mathbb{R}^{d}}\tilde{h}(y)u(x-y)dy,$
$\text{and}~{}~{}\Delta_{j}u:=\varphi\left(2^{-j}D\right)u=2^{jd}\int_{\mathbb{R}^{d}}h(2^{j}y)u(x-y)dy~{}~{}\text{if}~{}~{}j\geq
0.$

The nonhomogeneous low-frequency cut-off operator $S_{j}$ is defined by

$S_{j}u:=\sum_{k\leq
j-1}\Delta_{k}u=\varrho(2^{-j}D)u=2^{jd}\int_{\mathbb{R}^{d}}\tilde{h}(2^{j}y)u(x-y)dy,~{}j\in\mathbb{N}\cup{0}.$

Analogously, the homogeneous dyadic blocks $\dot{\Delta}_{j}$ and homogeneous
low-frequency cut-off operators $\dot{S}_{j}$ are defined for $\forall
j\in\mathbb{Z}$ by

$\dot{\Delta}_{j}u:=\varphi(2^{-j}D)u=2^{jd}\int_{\mathbb{R}^{d}}h(2^{j}y)u(x-y)dy,~{}j\in\mathbb{Z}$
$\text{ and
}~{}~{}\dot{S}_{j}u:=\varrho(2^{-j}D)u=2^{jd}\int_{\mathbb{R}^{d}}\tilde{h}(2^{j}y)u(x-y)dy,~{}j\in\mathbb{Z}$

Now we introduce the definition of Besov spaces. Let
$(p,r)\in[1,\infty]^{2},s\in\mathbb{R}$, the nonhomogeneous Besov space

$B_{p,r}^{s}:=\left\\{f\in\mathcal{S}^{\prime}\left(\mathbb{R}^{d}\right);\|f\|_{B_{p,r}^{s}}:=\left\|2^{js}\right\|\Delta_{j}f\left\|{}_{L^{p}}\right\|_{\ell^{r}(\mathbb{Z})}<\infty\right\\}$

and the homogeneous space

$\dot{B}_{p,r}^{s}:=\left\\{f\in\mathcal{S}^{\prime}\left(\mathbb{R}^{d}\right)/\mathcal{P}\left(\mathbb{R}^{d}\right);\|f\|_{\dot{B}_{p,r}^{s}}:=\left\|2^{js}\right\|\dot{\Delta}_{j}f\left\|{}_{L^{p}}\right\|_{\ell^{r}(\mathbb{Z})}<\infty\right\\}.$

Moreover, for $s>0$ and $1\leq p,q\leq\infty$, we may write the equivalent
norm below in the nonhomogeneous Besov norm $\left\|f\right\|_{B^{s}_{p,q}}$
of $f\in\mathcal{S}^{{}^{\prime}}$ as

$\left\|f\right\|_{B^{s}_{p,q}}=\left\|f\right\|_{{L^{p}}}+\left\|f\right\|_{\dot{B}^{s}_{p,q}}.$

Motivated by [10], we define ${B}^{\alpha}_{p,c(\mathbb{N})}$ to be the class
of all tempered distributions $f$ for which

$\left\|f\right\|_{{B}^{\alpha}_{p,\infty}}<\infty~{}\text{and}~{}\lim_{j\rightarrow\infty}2^{j\alpha}\left\|{\Delta}_{j}f\right\|_{L^{p}}=0,~{}~{}\text{for
any}~{}1\leq p\leq\infty.$ (2.1)

A function $f$ is said to be in
$L^{p}(0,T;\dot{B}^{\alpha}_{q,c(\mathbb{N})}(\Omega))$ if

$\lim_{z\rightarrow
0}\frac{\left(\int_{0}^{T}\Big{(}\int_{\Omega}|f(x+z,t)-f(x,t)|^{q}dx\Big{)}^{\frac{p}{q}}dt\right)^{\frac{1}{p}}}{|z|^{\alpha}}=0.$

For $0<\alpha<1$, it is worth pointing out that

$\lim_{z\rightarrow
0}\frac{\left(\int_{0}^{T}\Big{(}\int_{\Omega}|f(x+z,t)-f(x,t)|^{p}dx\Big{)}^{\frac{p}{q}}dt\right)^{\frac{1}{p}}}{|z|^{\alpha}}=0\Leftrightarrow
f\in L^{p}(0,T;\dot{B}^{\alpha}_{p,c(\mathbb{N})}(\Omega).$

A function $f$ belongs to the Besov-VMO space
$L^{p}(0,T;\underline{B}^{\alpha}_{q,VMO}(\mathbb{T}^{d}))$ if it satisfies

$\|f\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}<\infty,$

and

$\displaystyle\lim_{\varepsilon\rightarrow
0}\frac{1}{\varepsilon^{\alpha}}\left(\int_{0}^{T}\Big{[}\int_{\mathbb{T}^{d}}\int_{B_{\varepsilon}(x)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-~{}\,\,~{}\,~{}\,|f(x)-f(y)|^{q}dydx\Big{]}^{\frac{p}{q}}dt\right)^{\frac{1}{p}}$
$\displaystyle=$ $\displaystyle\lim_{\varepsilon\rightarrow
0}\frac{1}{\varepsilon^{\alpha}}\left(\int_{0}^{T}\Big{[}\int_{\mathbb{T}^{d}}\int_{B_{\varepsilon}(0)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-~{}\,~{}\,~{}\,~{}\,|f(x)-f(x-y)|^{q}dydx\Big{]}^{\frac{p}{q}}dt\right)^{\frac{1}{p}}=0.$

Mollifier kernel: Let $\eta_{\varepsilon}:\mathbb{R}^{d}\rightarrow\mathbb{R}$
be a standard mollifier.i.e. $\eta(x)=C_{0}e^{-\frac{1}{1-|x|^{2}}}$ for
$|x|<1$ and $\eta(x)=0$ for $|x|\geq 1$, where $C_{0}$ is a constant such that
$\int_{\mathbb{R}^{d}}\eta(x)dx=1$. For $\varepsilon>0$, we define the
rescaled mollifier
$\eta_{\varepsilon}(x)=\frac{1}{\varepsilon^{d}}\eta(\frac{x}{\varepsilon})$
and for any function $f\in L^{1}_{loc}(\mathbb{R}^{d})$, its mollified version
is defined as

$f^{\varepsilon}(x)=(f*\eta_{\varepsilon})(x)=\int_{\mathbb{R}^{d}}f(x-y)\eta_{\varepsilon}(y)dy,\
\ x\in\mathbb{R}^{d}.$

Next, we collect some Lemmas which will be used in the present paper.

######  Lemma 2.1.

(Bernstein inequality [1]) Let $\mathcal{B}$ be a ball of $\mathbb{R}^{d}$,
and $\mathcal{C}$ be a ring of $\mathbb{R}^{d}$. There exists a positive
constant $C$ such that for all integer $k\geq 0$, all $1\leq a\leq
b\leq\infty$ and $u\in L^{a}\left(\mathbb{R}^{d}\right)$, the following
estimates are satisfied:

$\begin{gathered}\sup_{|\alpha|=k}\left\|\partial^{\alpha}u\right\|_{L^{b}\left(\mathbb{R}^{d}\right)}\leq
C^{k+1}\lambda^{k+d\left(\frac{1}{a}-\frac{1}{b}\right)}\|u\|_{L^{a}\left(\mathbb{R}^{d}\right)},\quad\operatorname{supp}\hat{u}\subset\lambda\mathcal{B},\\\
C^{-(k+1)}\lambda^{k}\|u\|_{L^{a}\left(\mathbb{R}^{d}\right)}\leq\sup_{|\alpha|=k}\left\|\partial^{\alpha}u\right\|_{L^{a}\left(\mathbb{R}^{d}\right)}\leq
C^{k+1}\lambda^{k}\|u\|_{L^{a}\left(\mathbb{R}^{d}\right)},\quad\operatorname{supp}\hat{u}\subset\lambda\mathcal{C}.\end{gathered}$

This lemma yields that if $1\leq p_{1}\leq p_{2}\leq\infty,1\leq r_{1}\leq
r_{2}\leq\infty$, there holds

$B^{s}_{p_{1},r_{1}}\subseteq
B^{s-d(\frac{1}{p}_{1}-\frac{1}{p}_{2})}_{p_{2},r_{2}}$ (2.2)

######  Lemma 2.2.

([32, 8, 34]) Let $\Omega$ denote the whole space $\mathbb{R}^{d}$ or the
periodic domain $\mathbb{T}^{d}$. Suppose that $f\in
L^{p}(0,T;{B}^{\alpha}_{q,\infty}(\Omega))$, $g\in
L^{p}(0,T;{B}^{\beta}_{q,c(\mathbb{N})}(\Omega))$ with $\alpha,\beta\in(0,1)$,
$p,q\in[1,\infty]$, then there holds that, for any $k\in\mathbb{N}^{+}$, as
$\varepsilon\rightarrow 0,$

  1. (1)

$\|f^{\varepsilon}-f\|_{L^{p}(0,T;L^{q}(\Omega))}\leq\text{O}(\varepsilon^{\alpha})\|f\|_{L^{p}(0,T;\dot{B}^{\alpha}_{q,\infty}(\Omega))}$;

  2. (2)

$\|\nabla^{k}f^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\Omega))}\leq\text{O}(\varepsilon^{\alpha-k})\|f\|_{L^{p}(0,T;\dot{B}^{\alpha}_{q,\infty}(\Omega))}$;

  3. (3)

$\|g^{\varepsilon}-g\|_{L^{p}(0,T;L^{q}(\Omega))}\leq\text{o}(\varepsilon^{\beta})\|g\|_{L^{p}(0,T;\dot{B}^{\beta}_{q,c(\mathbb{N})}(\Omega))}$;

  4. (4)

$\|\nabla^{k}g^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\Omega))}\leq\text{o}(\varepsilon^{\beta-k})\|g\|_{L^{p}(0,T;\dot{B}^{\beta}_{q,c(\mathbb{N})}(\Omega))}$.

######  Remark 2.1.

The results still hold for $g\in
L^{p}(0,T;\underline{B}^{\beta}_{q,VMO}(\mathbb{T}^{d}))$, whose proof is
proposed in [31, 2].

Next, we will modify the Constantin-E-Titi type commutator estimates to allow
us to take full use of kinetic energy.

######  Lemma 2.3.

Let $\Omega$ denote the whole space $\mathbb{R}^{d}$ or the periodic domain
$\mathbb{T}^{d}$. Assume that $0<\alpha<1$,$0<\theta\leq 2$, $1\leq
r,s,p_{1},p_{2}\leq\infty$, $\frac{1}{r}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$ and
$\frac{1}{s}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$, then as $\varepsilon\rightarrow
0,$ there holds,

$\displaystyle\|(ff)^{\varepsilon}-f^{\varepsilon}f^{\varepsilon}\|_{L^{r}(0,T;L^{s}(\Omega))}\leq
C\text{o}(\varepsilon^{\theta\alpha}),$ (2.3)

if one of the following conditions is satisfied

  1. (1)

$f\in L^{(2-\theta)p_{2}}(0,T;L^{(2-\theta)q_{2}}(\Omega))$ and $f\in
L^{\theta p_{1}}(0,T;B_{\theta q_{1},c(\mathbb{N})}^{\alpha}(\Omega));$

  2. (2)

$f\in L^{(2-\theta)p_{2}}(0,T;L^{(2-\theta)q_{2}}(\Omega))$ and $f\in
L^{\theta p_{1}}(0,T;\underline{B}_{\theta
q_{1},VMO}^{\alpha}(\Omega))$,$q_{2}\geq\frac{q_{1}}{q_{1}-1}$,
$p_{2}\geq\frac{q_{1}}{q_{1}-1}$ .

###### Proof.

First, we recall the following identity observed by Constantin-E-Titi in [13]
that

$\displaystyle(ff)^{\varepsilon}(x)-f^{\varepsilon}f^{\varepsilon}(x)$ (2.4)
$\displaystyle=$
$\displaystyle\int_{\Omega}\eta_{\varepsilon}(y)[f(x-y)-f(x)][f(x-y)-f(x)]dy-(f-f^{\varepsilon})(f-f^{\varepsilon})(x).$

(1) Using the Hölder inequality and the Minkowski inequality and the
definition of $B_{\theta q_{1},c(\mathbb{N})}^{\alpha}$, we obtain

$\displaystyle\|(ff)^{\varepsilon}-f^{\varepsilon}f^{\varepsilon}\|_{L^{r}(0,T;L^{s}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
C\int_{|y|\leq\varepsilon}\eta_{\varepsilon}(y)\|f(\cdot-y)-f(\cdot)\|^{\theta}_{L^{p_{1}\theta}(0,T;L^{q_{1}\theta}(\Omega))}\|f(\cdot-y)-f(\cdot)\|^{2-\theta}_{L^{p_{2}(2-\theta)}(0,T;L^{q_{2}(2-\theta)}(\Omega))}dy$
$\displaystyle+C\|f-f^{\varepsilon}\|^{\theta}_{L^{p_{1}\theta}(0,T;L^{q_{1}\theta}(\Omega))}\|f-f^{\varepsilon}\|^{2-\theta}_{L^{p_{2}(2-\theta)}(0,T;L^{q_{2}(2-\theta)}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
C\varepsilon^{\theta\alpha}\|f\|^{\theta}_{L^{\theta p_{1}}(0,T;B_{\theta
q_{1},c(\mathbb{N})}^{\alpha}(\Omega))}\|f\|^{2-\theta}_{L^{p_{2}(2-\theta)}(0,T;L^{q_{2}(2-\theta)}(\Omega))}$
$\displaystyle\leq$ $\displaystyle o(\varepsilon^{\theta\alpha}),\ \text{as}\
\varepsilon\to 0,$

where Lemma 2.2 was used.

(2) The Hölder inequality allows us to discover that

$|I|\leq\frac{1}{\varepsilon^{d}}\Big{[}\int_{B_{\varepsilon}(0)}\big{|}\eta\big{(}\frac{y}{\varepsilon}\big{)}[|f(x-y)-f(x)|^{2-\theta}]\big{|}^{\frac{q_{1}}{q_{1}-1}}dy\Big{]}^{1-\frac{1}{q_{1}}}\Big{[}\int_{B_{\varepsilon}(0)}|f(x-y)-f(x)|^{\theta
q_{1}}dy\Big{]}^{\frac{1}{q_{1}}}.$

Using the Hölder inequality once again and Minkowski inequality, we observe
that

$\displaystyle\|I\|_{L^{s}(\Omega)}$ $\displaystyle\leq$ $\displaystyle
C\Big{[}\frac{1}{\varepsilon^{d}}\int_{B_{\varepsilon}(0)}\big{\|}|f(x-y)-f(x)|^{2-\theta}\|_{L^{q_{2}}}^{\frac{q_{1}}{q_{1}-1}}dy\Big{]}^{1-\frac{1}{q_{1}}}\Big{[}\int_{\Omega}\int_{B_{\varepsilon}(0)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-~{}\,~{}\,~{}\,~{}\,|f(x-y)-f(x)|^{\theta
q_{1}}dydx\Big{]}^{\frac{1}{q_{1}}},$

where we require $\frac{1}{s}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$ and
$q_{2}\geq\frac{q_{1}}{q_{1}-1}$.

We deduce from the Hölder inequality, $p_{2}\geq\frac{q_{1}}{q_{1}-1}$ and
Minkowski inequality once again, as $\varepsilon\to 0$, that

$\displaystyle\|I\|_{L^{r}(0,T;L^{s}((\Omega))}$ (2.5) $\displaystyle\leq$
$\displaystyle
C\Big{[}\frac{1}{\varepsilon^{d}}\int_{B_{\varepsilon}(0)}\||f(x-y)-f(x)|^{2-\theta}\|_{L^{p_{2}}(0,T;L^{q_{2}})}^{\frac{q_{1}}{q_{1}-1}}dy\Big{]}^{1-\frac{1}{q_{1}}}$
$\displaystyle\times\Big{\|}\int_{\Omega}\int_{B_{\varepsilon}(0)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-~{}\,~{}\,~{}\,~{}\,|f(x-y)-f(x)|^{\theta
q_{1}}dydx\Big{\|}_{L^{p_{1}}(0,T)}$ $\displaystyle\leq$ $\displaystyle
o(\varepsilon^{\theta\alpha}).$

Thanks to the Hölder inequality and Remark 2.1, we know that, as
$\varepsilon\to 0$,

$\displaystyle\|II\|_{L^{r}(0,T;L^{s}(\Omega))}\leq
C\||f-f^{\varepsilon}|^{\theta}\|_{L^{p_{1}}(0,T;L^{q_{1}}(\Omega))}\||f-f^{\varepsilon}|^{2-\theta}\|_{L^{p_{2}}(0,T;L^{q_{2}}(\Omega))}\leq
o(\varepsilon^{\theta\alpha}).$ (2.6)

Plugging (2.5) and (2.6) into (2.4), we arrive at the desired estimate. The
proof of this lemma is completed.

Then the proof of this lemma is completed. ∎

######  Lemma 2.4.

([33, 32]) Let $\Omega$ denote the whole space $\mathbb{R}^{d}$ or the
periodic domain $\mathbb{T}^{d}$. Let
$p,q,p_{1},q_{1},p_{2},q_{2}\in[1,+\infty)$ with
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}},\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$.
Assume $f\in L^{p_{1}}(0,T;L^{q_{1}}(\Omega))$ and $g\in
L^{p_{2}}(0,T;L^{q_{2}}(\Omega))$, then as $\varepsilon\to 0$, it holds

$\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\Omega))}\rightarrow
0,$ (2.7)

and

$\|(f\times g)^{\varepsilon}-f^{\varepsilon}\times
g^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\Omega))}\rightarrow 0.$ (2.8)

For the convenience of readers, we present the definition of the weak
solutions of the Euler equation (LABEL:Euler).

######  Definition 2.1.

We say that $(v,\Pi)$ is a weak solution of the incompressible homogeneous
Euler equations (LABEL:Euler) if $v\in C_{\text{weak}}([0,T];L^{2}(\Omega))$,
$\Pi\in L^{1}_{loc}([0,T]\times\Omega)$ with initial data $v_{0}\in
L^{2}(\Omega)$ and

  1. (i)

for every test vector field $\varphi\in
C_{0}^{\infty}([0,T]\times\Omega)^{d}$, there holds,

$\displaystyle\int_{\Omega}[v(x,T)\varphi(x,T)-v(x,0)\varphi(x,0)]\,dx$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\Omega}[v(x,t)\partial_{t}\varphi(x,t)+v(x,t)\otimes
v(x,t)\cdot\nabla\varphi(x,t)+\Pi(x,t)\mathrm{div}\,\varphi(x,t)]\,dxdt.$

  2. (ii)

$v$ is weakly divergence free, that is, for every test function $\psi\in
C_{0}^{\infty}([0,T]\times\Omega)$,

$\int_{0}^{T}\int_{\Omega}v(x,t)\cdot\nabla\psi(x,t)\,dxdt=0.$

##  3 Energy conservation of weak solutions for the ideal flow

In this section, we consider the energy conservation of weak solutions for the
Euler equations on the whole space $\mathbb{R}^{d}$ and for the case of the
torus $\mathbb{T}^{d}$, respectively.

###  3.1 Energy conservation for ideal flows on $\mathbb{R}^{d}$

###### Proof of Theorem 1.1.

(1) One can achieve the proof of this part via a modification the proof
presented in [10] for $1<p\leq 3$ (see also the proof of (1) in Theorem 1.4
below). We provided a short proof here $1\leq p\leq 3$. The interpolation
inequality in Lebesgue space means that

$\|\Delta_{j}v\|_{L^{3}(\mathbb{R}^{d})}\leq\|\Delta_{j}v\|_{L^{2}(\mathbb{R}^{d})}^{1-\frac{p}{3}}\|\Delta_{j}v\|_{L^{\frac{2p}{p-1}}(\mathbb{R}^{d})}^{\frac{p}{3}},\
\text{for\ any }\ p\in[1,3],$ (3.1)

which implies that

$\displaystyle 2^{\frac{1}{3}j}\|\Delta_{j}v\|_{L^{3}(\mathbb{R}^{d})}\leq$
$\displaystyle\|\Delta_{j}v\|_{L^{2}(\mathbb{R}^{d})}^{1-\frac{p}{3}}\Big{[}2^{j\frac{1}{p}}\|\Delta_{j}v\|_{L^{\frac{2p}{p-1}}(\mathbb{R}^{d})}\Big{]}^{\frac{p}{3}}$
(3.2) $\displaystyle\leq$ $\displaystyle
C\|v\|_{L^{2}(\mathbb{R}^{d})}^{1-\frac{p}{3}}\|v\|^{\frac{p}{3}}_{B^{\frac{1}{p}}_{\frac{2p}{p-1},\infty}(\mathbb{R}^{d})}.$

As a consequence, we discover that

$\|v\|_{B^{\frac{1}{3}}_{3,\infty}(\mathbb{R}^{d})}\leq
C\|v\|_{L^{2}(\mathbb{R}^{d})}^{1-\frac{p}{3}}\|v\|^{\frac{p}{3}}_{B^{\frac{1}{p}}_{\frac{2p}{p-1},\infty}(\mathbb{R}^{d})},$

which turns out that

$\|v\|_{L^{3}(0,T;B^{\frac{1}{3}}_{3,\infty}(\mathbb{R}^{d}))}\leq
C\|v\|_{L^{\infty}(0,T;L^{2}(\mathbb{R}^{d}))}^{1-\frac{p}{3}}\|v\|^{\frac{p}{3}}_{L^{p}(0,T;B^{\frac{1}{p}}_{\frac{2p}{p-1},\infty}(\mathbb{R}^{d}))}.$

Moreover, it follows from (3.2) that

$\displaystyle 2^{\frac{1}{3}j}\|\Delta_{j}v\|_{L^{3}(\mathbb{R}^{d})}\leq$
$\displaystyle
C\|v\|_{L^{2}(\mathbb{R}^{d})}^{1-\frac{p}{3}}\Big{[}2^{j\frac{1}{p}}\|\Delta_{j}v\|_{L^{\frac{2p}{p-1}}(\mathbb{R}^{d})}\Big{]}^{\frac{p}{3}},$

where $C$ is independent of $j$. This ensures that $v\in
L^{p}(0,T;B^{\frac{1}{p}}_{\frac{2p}{p-1},c(\mathbb{N})}(\mathbb{R}^{d}))$
means $v\in L^{3}(0,T;B^{\frac{1}{3}}_{3,c(\mathbb{N})}(\mathbb{R}^{d}))$.
Thus, the proof of this part is completed.

(2) We multiply the Euler equations (LABEL:Euler) by $S_{N}(S_{N}v)$ and
utilize integration by parts to get

$\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^{2}}|S_{N}v|^{2}dx=-\int_{\mathbb{R}^{3}}\partial_{j}(v_{j}v_{i})S^{2}_{N}v_{i}\,dx.$

Performing a time integration, we know that

$\frac{1}{2}\|S_{N}v\|_{L^{2}(\mathbb{R}^{3})}^{2}-\frac{1}{2}\|S_{N}v_{0}\|_{L^{2}(\mathbb{R}^{3})}^{2}=-\int_{0}^{t}\int_{\mathbb{R}^{3}}\partial_{j}(v_{j}v_{i})S^{2}_{N}v_{i}\,dxds.$
(3.3)

Our plan is to show that, as $N\rightarrow+\infty$, the right hand side of the
above equation (3.3) converges to 0.

To do this, adapting the incompressibility condition $\operatorname{div}v=0$,
we discover that

$\displaystyle-\int_{0}^{t}\int_{\mathbb{R}^{3}}\partial_{j}(v_{j}S^{2}_{N}v_{i})S^{2}_{N}v_{i}\,dxds=$
$\displaystyle-\frac{1}{2}\int_{0}^{t}\int_{\mathbb{R}^{3}}v_{j}\partial_{j}(S^{2}_{N}v_{i})^{2}\,dxds=0.$

As a consequence, we rewrite the right hand side of (3.3) as

$\begin{split}-\int_{0}^{t}\int_{\mathbb{R}^{3}}\partial_{j}(v_{j}v_{i})S^{2}_{N}v_{i}\,dxds=\int_{0}^{t}\int_{\mathbb{R}^{3}}S_{N}\big{[}v_{j}(\text{I}_{\textnormal{d}}-S^{2}_{N})v_{i}\big{]}S_{N}\partial_{j}v_{i}\,dxds.\end{split}$

Utilizing the Hölder inequality, we find

$\displaystyle\left|\int_{0}^{t}\int_{\mathbb{R}^{3}}S_{N}\big{[}v_{j}(I-S^{2}_{N})v_{i}\big{]}S_{N}\partial_{j}v_{i}\,dxds\right|$
(3.4) $\displaystyle\leq$ $\displaystyle
C\big{\|}S_{N}\big{[}v_{j}(\text{I}_{\textnormal{d}}-S^{2}_{N})v_{i}\big{]}\big{\|}_{L^{\frac{p}{p-1}}(0,T;L^{\frac{2dp}{dp+d-2p+2}}(\mathbb{R}^{3}))}\big{\|}\nabla
S_{N}v\big{\|}_{L^{p}(0,T;L^{\frac{2dp}{(d+2)(p-1)}}(\mathbb{R}^{3}))}$
$\displaystyle\leq$ $\displaystyle
C\big{\|}v_{j}(\text{I}_{\textnormal{d}}-S^{2}_{N})v_{i}\big{\|}_{L^{\frac{p}{p-1}}(0,T;L^{\frac{2dp}{dp+d-2p+2}}(\mathbb{R}^{3}))}\big{\|}S_{N}\partial_{j}v\big{\|}_{L^{p}(0,T;L^{\frac{2dp}{(d+2)(p-1)}}(\mathbb{R}^{3}))}$
$\displaystyle\leq$ $\displaystyle
C\|v\|_{L^{\frac{2p}{p-1}}(0,T;L^{L^{\frac{4dp}{dp+d-2p+2}}}(\mathbb{R}^{3}))}\big{\|}\big{(}\text{I}_{\textnormal{d}}-S^{2}_{N}\big{)}v\big{\|}_{L^{\frac{2p}{p-1}}(0,T;L^{\frac{4dp}{dp+d-2p+2}}(\mathbb{R}^{3}))}\|\nabla
v\|_{L^{p}(0,T;L^{\frac{2dp}{(d+2)(p-1)}}(\mathbb{R}^{3}))}.$

We apply the Gagliardo-Nirenberg inequality to get

$\|v\|_{L^{\frac{4dp}{dp+d-2p+2}}(\mathbb{R}^{3})}\leq
C\|v\|^{\frac{6-2p-pd+3d}{2d+4}}_{L^{2}(\mathbb{R}^{3})}\|\nabla
v\|^{\frac{pd+2p-2-d}{2d+4}}_{L^{\frac{2dp}{(d+2)(p-1)}}(\mathbb{R}^{3})},$

from which follows that

$\|v\|_{L^{\frac{2p}{p-1}}(0,T;L^{\frac{4dp}{dp+d-2p+2}}(\mathbb{R}^{3}))}\leq
C\|v\|^{\frac{6-2p-pd+3d}{2d+4}}_{L^{\infty}(0,T;L^{2}(\mathbb{R}^{3}))}\|\nabla
v\|_{L^{p}(0,T;L^{\frac{2dp}{(d+2)(p-1)}}(\mathbb{R}^{3}))}^{\frac{pd+2p-2-d}{2d+4}}.$
(3.5)

Invoking Littlewood-Paley theory and Dominated Convergence Theorem, we see
that, as $N\rightarrow\infty$,

$\big{\|}\big{(}\text{I}_{\textnormal{d}}-S^{2}_{N}\big{)}v\big{\|}_{L^{\frac{2p}{p-1}}(0,T;L^{\frac{4dp}{dp+d-2p+2}}(\mathbb{R}^{3}))}\rightarrow
0,$

which follows from that, as $N\rightarrow\infty$,

$-\int_{0}^{t}\int_{\mathbb{R}^{3}}\partial_{j}(v_{j}v_{i})S^{2}_{N}v_{i}\,dxds\rightarrow
0.$

Taking $N\rightarrow\infty$ in (3.3), we derive from (3.3) that

$\frac{1}{2}\|v(t)\|_{L^{2}(\mathbb{R}^{3})}^{2}=\frac{1}{2}\|v_{0}\|_{L^{2}(\mathbb{R}^{3})}^{2}.$

Thus we complete the proof.

∎

###  3.2 Energy conservation for invicid flows on torus

Unlike the interpolation and Littlewood-Paley theory utilized in the last sub-
section, the mollifier kernel and Constantin-E-Titi type estimates are applied
to deal with periodic case.

###### Proof of Theorem 1.2.

(1)-(2) Let us begin by mollifying the equations (LABEL:Euler) in spatial
direction to get

$\displaystyle\partial_{t}{v^{\varepsilon}}+\text{div}(v\otimes
v)^{\varepsilon}+\nabla\Pi^{\varepsilon}=0.$ (3.6)

The standard energy estimate means that

$\displaystyle\frac{1}{2}\|v^{\varepsilon}(T)\|^{2}_{L^{2}(\mathbb{T}^{d})}-\frac{1}{2}\|v^{\varepsilon}(0)\|^{2}_{L^{2}(\mathbb{T}^{d})}=-\int_{0}^{T}\int_{\mathbb{T}^{d}}\text{div}(v\otimes
v)^{\varepsilon}v^{\varepsilon}dxdt.$ (3.7)

In view of divergence-free condition and the integration by parts, we
reformulate the right hand side of (3.7) as

$\displaystyle-\int_{0}^{T}\int_{\mathbb{T}^{d}}\text{div}(v\otimes
v)^{\varepsilon}v^{\varepsilon}dxdt$ (3.8) $\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{T}^{d}}\text{div}\big{[}(v\otimes
v)^{\varepsilon}-(v^{\varepsilon}\otimes
v^{\varepsilon})\big{]}v^{\varepsilon}dxdt-\int_{0}^{T}\int_{\mathbb{T}^{d}}\text{div}(v^{\varepsilon}\otimes
v^{\varepsilon})v^{\varepsilon}dxdt$ $\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{T}^{d}}\text{div}\big{[}(v\otimes
v)^{\varepsilon}-(v^{\varepsilon}\otimes
v^{\varepsilon})\big{]}v^{\varepsilon}dxdt$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\big{[}(v\otimes
v)^{\varepsilon}-(v^{\varepsilon}\otimes v^{\varepsilon})\big{]}:\nabla
v^{\varepsilon}dxdt.$

Plugging this into (3.7), we arrive at

$\displaystyle\frac{1}{2}\|v^{\varepsilon}(T)\|^{2}_{L^{2}(\mathbb{T}^{d})}-\frac{1}{2}\|v^{\varepsilon}(0)\|^{2}_{L^{2}(\mathbb{T}^{d})}=\int_{0}^{T}\int_{\mathbb{T}^{d}}\big{[}(v\otimes
v)^{\varepsilon}-(v^{\varepsilon}\otimes v^{\varepsilon})\big{]}:\nabla
v^{\varepsilon}dxdt.$ (3.9)

On the one hand, in virtue of the Hölder inequality, we have

$\displaystyle\Big{|}\int_{0}^{T}\int_{\mathbb{T}^{d}}\big{[}(v\otimes
v)^{\varepsilon}-(v^{\varepsilon}\otimes v^{\varepsilon})\big{]}:\nabla
v^{\varepsilon}dxdt\Big{|}$ $\displaystyle\leq$ $\displaystyle
C\|(v^{\varepsilon}\otimes v^{\varepsilon})-(v\otimes
v)^{\varepsilon}\|_{L^{\frac{p}{p-1}}(0,T;L^{\frac{q}{q-1}}(\mathbb{T}^{d}))}\|\nabla
v^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}.$

We select $p_{2}(2-\theta)=\infty,q_{2}(2-\theta)=2$, $s=\frac{q}{q-1}$ and
$r=\frac{p}{p-1}$ in Lemma 2.3 to derive that $\theta
p_{1}=\frac{p\theta}{p-1}$, $\theta q_{1}=\frac{2q\theta}{q\theta-2}$ and let
$\theta p_{1}=p$ and $\theta q_{1}=q$, we can obtain

$\displaystyle\|(v^{\varepsilon}\otimes v^{\varepsilon})-(v\otimes
v)^{\varepsilon}\|_{L^{\frac{p}{p-1}}(0,T;L^{\frac{q}{q-1}}(\mathbb{T}^{d}))}\leq
o(\varepsilon^{\theta\alpha})\leq o(\varepsilon^{(p-1)\alpha}),$

where we have used the facts that $\theta=\frac{2}{q-2}=p-1$ and
$q=\frac{2p}{p-1}$.

On the other hand, in light of Lemma (2.2), we have

$\|\nabla v^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}\leq
o(\varepsilon^{\alpha-1}).$

Hence,

$\displaystyle\Big{|}\int_{0}^{T}\int_{\mathbb{T}^{d}}\big{[}(v\otimes
v)^{\varepsilon}-(v^{\varepsilon}\otimes v^{\varepsilon})\big{]}:\nabla
v^{\varepsilon}dxdt\Big{|}\leq o(\varepsilon^{(p-1)\alpha+\alpha-1})\leq
o(\varepsilon^{p\alpha-1}).$

Then letting $\alpha=\frac{1}{p},q=\frac{2p}{p-1}$ and passing to the limit of
$\varepsilon$ in (3.9), we get the energy conservation.

(3) We deduce from the Hölder inequality that

$\displaystyle\Big{|}\int_{0}^{T}\int_{\mathbb{T}^{d}}\big{[}(v\otimes
v)^{\varepsilon}-(v^{\varepsilon}\otimes v^{\varepsilon})\big{]}:\nabla
v^{\varepsilon}dxdt\Big{|}$ $\displaystyle\leq$ $\displaystyle
C\|(v^{\varepsilon}\otimes v^{\varepsilon})-(v\otimes
v)^{\varepsilon}\|_{L^{\frac{p}{p-1}}(0,T;L^{\frac{2dp}{dp+d-2p+2}}(\mathbb{T}^{d}))}\|\nabla
v^{\varepsilon}\|_{L^{p}(0,T;L^{\frac{2dp}{(d+2)(p-1)}}(\mathbb{T}^{d}))}$

From an analogue of inequality (3.5) on bounded domain and $v\in
L^{\infty}(0,T;L^{2}(\mathbb{T}^{d}))$, we derive from $\nabla v\in
L^{p}(0,T;L^{\frac{2dp}{(d+2)(p-1)}}(\mathbb{R}^{d}))$ that $v\in
L^{\frac{2p}{p-1}}(0,T;L^{\frac{4dp}{dp+d-2p+2}}(\mathbb{T}^{d}))$. Employing
Lemma 2.4, we find, as $\varepsilon\rightarrow 0,$

$\int_{0}^{T}\int_{\mathbb{T}^{d}}\big{[}(v\otimes
v)^{\varepsilon}-(v^{\varepsilon}\otimes v^{\varepsilon})\big{]}:\nabla
v^{\varepsilon}dxdt\rightarrow 0.$

The proof of this theorem is completed. ∎

##  4 Helicity conservation of weak solutions for the ideal flow

In this section, we are devoted to the proof of Theorem 1.4 and 1.5, involving
the minimum regularity of the weak solutions to guarantee the helicity
conservation for the incompressible Euler equations.

###  4.1 Helicity conservation for ideal flows on $\mathbb{R}^{3}$

###### Proof of Theorem 1.4.

(1) For any $p\in[1,3]$, by virtue of

$2^{\frac{2}{3}j}\|\Delta_{j}u\|_{L^{3}(\mathbb{R}^{3})}\leq
C\|\Delta_{j}u\|_{L^{2}(\mathbb{R}^{3})}^{1-\frac{p}{3}}\Big{[}2^{j\frac{2}{p}}\|\Delta_{j}u\|_{L^{\frac{2p}{p-1}}(\mathbb{R}^{3})}\Big{]}^{\frac{p}{3}}$

and the helicity conservation result in [10], one can argue almost exactly as
the proof in the Theorem 1.1 to complete the proof this part. We omit the
detail here. As compensation, we present the proof via the Littlewood-Paley
theory for $2<p\leq 3$.

Combining the vorticity equations (1.2) and some straightforward calculations,
we conclude by integration by parts that

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}^{3}}(S_{N}v\cdot S_{N}\omega)dx$
(4.1) $\displaystyle=$ $\displaystyle\int_{\mathbb{R}^{3}}S_{N}v_{t}\cdot
S_{N}\omega+S_{N}\omega_{t}\cdot S_{N}vdx$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}-\left[S_{N}(\omega\times v)+\nabla
S_{N}(\Pi+\frac{1}{2}|v|^{2})\right]\cdot
S_{N}\omega+\left(\text{curl}S_{N}\left(v\times\omega\right)\right)\cdot
S_{N}vdx$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}-S_{N}\left(\omega\times v\right)\cdot
S_{N}\omega+S_{N}\left(v\times\omega\right)\cdot S_{N}\omega dx.$

Using the fact $(S_{N}v\times S_{N}\omega)\cdot
S_{N}\omega=S_{N}v\cdot(S_{N}\omega\times S_{N}\omega)=0$, we reformulate
(4.1) as

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}^{3}}(S_{N}v\cdot S_{N}\omega)dx=$
$\displaystyle\int_{\mathbb{R}^{3}}-2\Big{[}S_{N}(\omega\times
v)-S_{N}\omega\times S_{N}v\Big{]}\cdot S_{N}\omega dx.$ (4.2)

To control the right hand side of (4.2), we recall the identity

$\nabla(A\cdot B)=A\cdot\nabla B+B\cdot\nabla
A+A\times\text{curl}B+B\times\text{curl}A$

which together with $\mathrm{div}\,\omega=0$ yields that

$\text{div}(v\otimes v)=\frac{1}{2}\nabla|v|^{2}+\omega\times v,$

which gives

$\omega\times v=\text{div}(v\otimes v)-\frac{1}{2}\nabla|v|^{2}.$

As a consequence, we deduce that

$S_{N}(\omega\times v)=S_{N}(v\cdot\nabla v)-\frac{1}{2}\nabla
S_{N}(|v|^{2})=\text{div}S_{N}(v\otimes v)-\frac{1}{2}\nabla S_{N}(|v|^{2})$

and

$S_{N}\omega\times S_{N}v=S_{N}v\cdot\nabla
S_{N}v-\frac{1}{2}\nabla|S_{N}v|^{2}=\text{div}(S_{N}v\otimes
S_{N}v)-\frac{1}{2}\nabla|S_{N}v|^{2}.$

Plugging the latter two equations into (4.2), we use integration by parts to
obtain that

$\displaystyle\frac{d}{dt}\int_{\Omega}S_{N}v\cdot S_{N}\omega dx$ (4.3)
$\displaystyle=$ $\displaystyle-2\int_{\Omega}\text{div}\big{(}S_{N}(v\otimes
v)-(S_{N}v\otimes S_{N}v)\big{)}\cdot
S_{N}\omega+\frac{1}{2}\nabla\left(|S_{N}v|^{2}-S_{N}(|v|^{2})\right)\cdot
S_{N}\omega dx$ $\displaystyle=$ $\displaystyle
2\int_{\Omega}\big{(}S_{N}(v\otimes v)-S_{N}v\otimes S_{N}v\big{)}:\nabla
S_{N}\omega dx.$

Making use of the Hölder inequality, we have

$\displaystyle\Big{|}\int_{\Omega}\big{(}S_{N}(v\otimes v)-S_{N}v\otimes
S_{N}v\big{)}:\nabla S_{N}\omega dx\Big{|}$ $\displaystyle\leq$ $\displaystyle
C\|S_{N}(v\otimes v)-S_{N}v\otimes
S_{N}v\|_{L^{\frac{q}{q-1}}(\mathbb{R}^{3})}\|\nabla
S_{N}\omega\|_{L^{q}(\mathbb{R}^{3})}.$

Thanks to the Constantin-E-Titi identity in [13], we observe that

$\displaystyle S_{N}(v\otimes v)-S_{N}v\otimes S_{N}v$ $\displaystyle=$
$\displaystyle
2^{3N}\int_{\mathbb{R}^{3}}\tilde{h}(2^{N}y)[v_{i}(x-y)-v_{i}(x)][v_{j}(x-y)-v_{j}(x)]dy-(v_{i}-S_{N}v_{i})(v_{j}-S_{N}v_{j}),$

where we used
$2^{3N}\int_{\mathbb{R}^{3}}\tilde{h}(2^{N}y)dy=\mathcal{F}(\tilde{h}(\cdot))|_{\xi=0}=1.$

For $0\leq\theta\leq 2$, we deduce from the Minkowski inequality and $v\in
L^{\infty}(0,T;L^{2}(\mathbb{R}^{3}))$ that,

$\displaystyle\|S_{N}(v_{i}v_{j})-S_{N}v_{i}S_{N}v_{j}\|_{L^{\frac{q}{q-1}}(\mathbb{R}^{3})}$
$\displaystyle\leq$ $\displaystyle
2^{3N}\int_{\mathbb{R}^{3}}|\tilde{h}(2^{N}y)|\|v(x-y)-v(x)\|^{\theta}_{L^{\frac{2q\theta}{q\theta-2}}(\mathbb{R}^{3})}\|v(x-y)-v\|^{2-\theta}_{L^{2}(\mathbb{R}^{3})}dy$
$\displaystyle+\|v_{i}-S_{N}v_{i}\|^{\theta}_{L^{\frac{2q\theta}{q\theta-2}}(\mathbb{R}^{3})}\|v_{j}-S_{N}v_{j}\|^{2-\theta}_{L^{2}(\mathbb{R}^{3})}$
$\displaystyle\leq$ $\displaystyle
2^{3N}\int_{\mathbb{R}^{3}}|\tilde{h}(2^{N}y)|\|v(x-y)-v(x)\|^{\theta}_{L^{\frac{2q\theta}{q\theta-2}}(\mathbb{R}^{3})}dy+\|v_{i}-S_{N}v_{i}\|^{\theta}_{L^{\frac{2q\theta}{q\theta-2}}(\mathbb{R}^{3})}$
$\displaystyle=$ $\displaystyle I+II.$

On the one hand, applying the Bernstein inequality in Lemma 2.1, we observe
that

$\displaystyle\|v(x-y)-v(x)\|_{L^{\frac{2q\theta}{q\theta-2}}(\mathbb{R}^{3})}\leq
C\Big{[}\sum_{j<N}2^{j}|y|\|{\Delta}_{j}v\|_{L^{\frac{2q\theta}{q\theta-2}}(\mathbb{R}^{3})}+\sum_{j\geq
N}\|{\Delta}_{j}v\|_{L^{\frac{2q\theta}{q\theta-2}}(\mathbb{R}^{3})}\Big{]}.$
(4.4)

Before going further, we write

$\displaystyle\Gamma(j)=\left\\{\begin{aligned}
&2^{j\alpha},~{}~{}~{}~{}~{}~{}~{}~{}\text{if}~{}j\leq 0,\\\
&2^{-(1-\alpha)j},~{}~{}\text{if}~{}j>0,\end{aligned}\right.$

and
$d_{j}=2^{j\alpha}\|{\Delta}_{j}v\|_{L^{\frac{2q\theta}{q\theta-2}}(\mathbb{R}^{3})}$.

Then we can rewrite (4.4) as

$\displaystyle\|v(x-y)-v(x)\|_{L^{\frac{2q\theta}{q\theta-2}}(\mathbb{R}^{3})}$
$\displaystyle\leq$ $\displaystyle
C\Big{(}2^{N(1-\alpha)}|y|\sum_{j<N}2^{-(N-j)(1-\alpha)}2^{j\alpha}\|{\Delta}_{j}v\|_{L^{\frac{2q\theta}{q\theta-2}}(\mathbb{R}^{3})}+2^{-\alpha
N}\sum_{j\geq
N}2^{(N-j)\alpha}2^{j\alpha}\|{\Delta}_{j}v\|_{L^{\frac{2q\theta}{q\theta-2}}(\mathbb{R}^{3})}\Big{)}$
$\displaystyle\leq$ $\displaystyle C(2^{N}|y|+1)2^{-\alpha
N}\left(\Gamma\ast{d}_{j}\right)(N),$

which turns out that

$\displaystyle\|v(x-y)-v(x)\|^{\theta}_{L^{\frac{2q\theta}{q\theta-2}}(\mathbb{R}^{3})}\leq$
$\displaystyle C(2^{N}|y|+1)^{\theta}2^{-\alpha\theta
N}\left(\Gamma\ast{d}_{j}\right)^{\theta}(N).$

Some straightforward calculations yield that

$\sup_{N}2^{3N}\int_{\mathbb{R}^{3}}|\tilde{h}(2^{N}y)|(2^{N}|y|+1)^{\theta}dy<\infty,$

which means that

$I\leq C2^{-\alpha\theta N}\left(\Gamma\ast{d}_{j}\right)^{\theta}(N).$

On the other hand, notice that

$|II|\leq\|v-{S}_{N}v\|^{\theta}_{L^{\frac{2q\theta}{q\theta-2}}(\mathbb{R}^{3})}\leq\Big{(}\sum_{j\geq
N}\|{\Delta}_{j}v\|_{L^{\frac{2q\theta}{q\theta-2}}(\mathbb{R}^{3})}\Big{)}^{\theta}\leq
C2^{-\alpha\theta N}\left(\Gamma\ast{d}_{j}\right)^{\theta}(N),$

where we used $N>0.$

Therefore, we end up with,

$\displaystyle\|S_{N}(v_{i}v_{j})-S_{N}v_{i}S_{N}v_{j}\|_{L^{\frac{q}{q-1}}(\mathbb{R}^{3})}\leq
C2^{-\theta\alpha N}\left(\Gamma\ast{d}_{j}\right)^{\theta}(N),$ (4.5)

Owing to the Bernstein inequality, we conclude that

$\|\nabla S_{N}\omega\|_{L^{q}(\mathbb{R}^{3})}\leq\sum_{j\leq
N}2^{2j}\|{\Delta}_{j}v\|_{L^{q}(\mathbb{R}^{3})}\leq
2^{N(2-\alpha)}\left(\Gamma\ast{d}_{j}\right)(N),$ (4.6)

Combining (4.5) and (4.6), we infer that

$\displaystyle\Big{|}\int_{\mathbb{R}^{3}}\big{(}S_{N}(v\otimes
v)^{\varepsilon}-S_{N}v\otimes S_{N}v\big{)}:\nabla S_{N}\omega dx\Big{|}$
(4.7) $\displaystyle\leq$ $\displaystyle
C2^{(2-\alpha-\theta\alpha)N}\left(\Gamma\ast{d}_{j}\right)^{\theta}(N)\left(\Gamma\ast{d}_{j}\right)(N).$

To proceed further, we pick $\alpha$ such that $0<\alpha<1$ and
$2-\alpha-\theta\alpha=0$. Hence, we have $\Gamma\in l^{1}(\mathbb{Z})$.
Letting $\alpha=\frac{2}{p}$ and $q=\frac{2p}{p-1}$, it follows from
(LABEL:4.7) that

$\displaystyle\int_{0}^{T}\Big{|}\int_{\mathbb{R}^{3}}\big{(}S_{N}(v\otimes
v)-S_{N}v\otimes S_{N}v\big{)}:\nabla S_{N}\omega dx\Big{|}dt$
$\displaystyle\leq$ $\displaystyle
C\int_{0}^{T}\Big{|}\left[\Gamma\ast{d}_{j}\right]^{\theta}(N)\left[\Gamma\ast{d}_{j}\right](N)\Big{|}dt$
$\displaystyle\leq$ $\displaystyle
C\|v\|^{\theta}_{L^{\frac{p\theta}{p-1}}(0,T;B^{\alpha}_{\frac{2q\theta}{q\theta-2},\infty}(\mathbb{R}^{3}))}\|v\|_{L^{p}(0,T;B^{\alpha}_{q,\infty}(\mathbb{R}^{3}))}$
$\displaystyle\leq$ $\displaystyle
C\|v\|^{\theta+1}_{L^{p}(0,T;B^{\alpha}_{q,\infty}(\mathbb{R}^{3}))}<\infty,$

where we used the facts that $\frac{p\theta}{p-1}=p$ and
$\frac{2q\theta}{q\theta-2}=q.$

Invoking the the dominated convergence theorem, we get

$\displaystyle\int_{0}^{T}\Big{|}\int_{\mathbb{R}^{3}}\big{(}S_{N}(v\otimes
v)-S_{N}v\otimes S_{N}v\big{)}:\nabla S_{N}\omega dx\Big{|}dt$ (4.8)
$\displaystyle\leq$ $\displaystyle
C\int_{0}^{T}\Big{|}\left[\Gamma\ast{d}_{j}\right]^{\theta}(N)\left[\Gamma\ast{d}_{j}\right](N)\Big{|}dt\rightarrow
0,~{}\text{as}\ N\rightarrow+\infty.$

By integrating (LABEL:4.3) in time over $(0,T)$, we find

$\displaystyle\int_{\mathbb{R}^{3}}S_{N}v(x,T)\cdot
S_{N}\omega(x,T)dx-\int_{\mathbb{R}^{3}}S_{N}v(x,0)\cdot S_{N}\omega(x,0)dx$
(4.9) $\displaystyle=$ $\displaystyle
2\int_{0}^{T}\int_{\mathbb{R}^{3}}\big{(}S_{N}v\otimes S_{N}v-S_{N}(v\otimes
v)\big{)}:\nabla S_{N}\omega dxdt.$

With (LABEL:4.8) in hand, taking $N\rightarrow+\infty$ in (4.9), we finish the
proof of this part.

(2) We deduce from the Gagliardo-Nirenberg inequality that

$\|v\|_{L^{\frac{p}{p-2}}(0,T;L^{\frac{3p}{7-2p}}(\mathbb{R}^{3}))}\leq
C\|v\|^{\frac{21-7p}{7}}_{L^{\infty}(0,T;L^{2}(\mathbb{R}^{3}))}\|\nabla
v\|^{\frac{7p-14}{7}}_{L^{p}(0,T;L^{\frac{6p}{5p-7}}(\mathbb{R}^{3}))}.$
(4.10)

In addition, there holds

$\displaystyle\Big{|}\int_{0}^{T}\int_{\Omega}\Big{(}S_{N}(\omega\times
v)-S_{N}\omega\times S_{N}v\Big{)}\cdot S_{N}\omega dxdt\Big{|}$
$\displaystyle\leq$ $\displaystyle\|S_{N}(\omega\times v)-S_{N}\omega\times
S_{N}v\|_{L^{\frac{p}{p-1}}(0,T;L^{\frac{6p}{p+7}}(\mathbb{R}^{3}))}\|\nabla
v\|_{L^{p}(0,T;L^{\frac{6p}{5p-7}}(\mathbb{R}^{3}))}.$

It is enough to apply Lemma 2.4 to finish the proof.

(3)-(4) Before going further, we set

$\Gamma_{2}(j)=\left\\{\begin{aligned}
&2^{j\beta},~{}~{}~{}~{}~{}~{}~{}~{}\text{if}~{}j\leq 0,\\\
&2^{-(1-\beta)j},~{}~{}\text{if}~{}j>0,\end{aligned}\right.$

and
${\tilde{d}}_{j}=2^{j\beta}\|{\Delta}_{j}\omega\|_{L^{\frac{2q\theta}{q\theta-2}}(\mathbb{R}^{3})}$.

We mimic the derivation of (4.6) to get

$\|\nabla S_{N}\omega\|_{L^{q}(\mathbb{R}^{3})}\leq\sum_{j\leq
N}2^{j}\|{\Delta}_{j}\omega\|_{L^{q}(\mathbb{R}^{3})}\leq
2^{N(1-\beta)}\left(\Gamma_{2}\ast{\tilde{d}}_{j}\right)(N),$ (4.11)

A combination of (4.5) and (4.11), we arrive at

$\displaystyle\Big{|}\int_{\mathbb{R}^{3}}\big{(}S_{N}(v\otimes
v)^{\varepsilon}-S_{N}v\otimes S_{N}v\big{)}:\nabla S_{N}\omega dx\Big{|}$
(4.12) $\displaystyle\leq$ $\displaystyle
C2^{(1-\beta-\theta\alpha)N}\left(\Gamma\ast{d}_{j}\right)^{\theta}(N)\left(\Gamma_{2}\ast{\tilde{d}}_{j}\right)(N).$

We conclude by the Hölder inequality that

$\displaystyle\int_{0}^{T}\Big{|}\int_{\mathbb{R}^{3}}\big{(}S_{N}(v\otimes
v)^{\varepsilon}-S_{N}v\otimes S_{N}v\big{)}:\nabla S_{N}\omega dx\Big{|}dt$
(4.13) $\displaystyle\leq$ $\displaystyle
C\int_{0}^{T}\left(\Gamma\ast{d}_{j}\right)^{\theta}(N)\left(\Gamma_{2}\ast\tilde{d}_{j}\right)(N)dt$
$\displaystyle\leq$ $\displaystyle
C\|v\|^{\theta}_{L^{\frac{p\theta}{p-1}}(0,T;B^{\alpha}_{\frac{2q\theta}{q\theta-2},\infty})}\|\omega\|_{L^{p}(0,T;B^{\beta}_{q,\infty})}<\infty,$

where we require

$\frac{p\theta}{p-1}=k,\frac{2q\theta}{q\theta-2}=\ell,1-\beta-\theta\alpha=0.$

By means of the dominated convergence theorem, we infer that, as
$N\rightarrow+\infty$,

$\displaystyle\int_{0}^{T}\Big{|}\int_{\Omega}\big{(}S_{N}(v\otimes
v)^{\varepsilon}-S_{N}v\otimes S_{N}v\big{)}:\nabla S_{N}\omega dx\Big{|}dt$
$\displaystyle\leq$ $\displaystyle
C\int_{0}^{T}\left(\Gamma\ast{d}_{j}\right)^{\theta}(N)\left(\Gamma_{2}\ast\tilde{d}_{j}\right)(N)dt\rightarrow
0.$

The proof of this part is completed. ∎

###  4.2 Helicity conservation for ideal flows on $\mathbb{T}^{3}$

###### Proof of Theorem 1.5.

(1)-(2) Arguing in the same manner as in the derivation of (LABEL:4.3), we
find

$\displaystyle\frac{d}{dt}\int_{\mathbb{T}^{3}}v^{\varepsilon}\cdot\omega^{\varepsilon}dx=$
$\displaystyle 2\int_{\mathbb{T}^{3}}\big{[}(v\otimes
v)^{\varepsilon}-v^{\varepsilon}\otimes
v^{\varepsilon}\big{]}:\nabla\omega^{\varepsilon}dx,$

which together with integration with respect to time leads to

$\displaystyle\int_{\mathbb{T}^{3}}v^{\varepsilon}(x,T)\cdot\omega^{\varepsilon}(x,T)dx-\int_{\mathbb{T}^{3}}v^{\varepsilon}(x,0)\cdot\omega^{\varepsilon}(x,0)dx$
(4.14) $\displaystyle=$ $\displaystyle
2\int_{0}^{T}\int_{\mathbb{T}^{3}}\big{(}v^{\varepsilon}\otimes
v^{\varepsilon}-(v\otimes
v)^{\varepsilon}\big{)}:\nabla\omega^{\varepsilon}dxdt.$

Making full use of the Hölder inequality, we discover that

$\displaystyle\Big{|}\int_{0}^{T}\int_{\mathbb{T}^{3}}\big{(}(v\otimes
v)^{\varepsilon}-(v^{\varepsilon}\otimes
v^{\varepsilon})\big{)}:\nabla\omega^{\varepsilon}dxdt\Big{|}$ (4.15)
$\displaystyle\leq$ $\displaystyle C\|(v^{\varepsilon}\otimes
v^{\varepsilon})-(v\otimes
v)^{\varepsilon}\|_{L^{\frac{p}{p-1}}(0,T;L^{\frac{q}{q-1}}(\mathbb{T}^{3}))}\|\nabla\omega^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{3}))}.$

Picking $p_{2}(2-\theta)=\infty,q_{2}(2-\theta)=2$, $s=\frac{q}{q-1}$ and
$r=\frac{p}{p-1}$ in Lemma 2.3 to derive that $\theta
p_{1}=\frac{p\theta}{p-1}$, $\theta q_{1}=\frac{2q\theta}{q\theta-2}$ and
letting $\theta p_{1}=p$ and $\theta q_{1}=q$, we can obtain

$\|(v^{\varepsilon}\otimes v^{\varepsilon})-(v\otimes
v)^{\varepsilon}\|_{L^{\frac{p}{p-1}}(0,T;L^{\frac{q}{q-1}}(\mathbb{T}^{3}))}\leq
o(\varepsilon^{\theta\alpha}),$

where we have used the facts that $\theta=\frac{2}{q-2}=p-1$ and
$q=\frac{2p}{p-1}$.

By virtue of Lemma 2.2, we know that

$\|\nabla\omega^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{3}))}\leq
o(\varepsilon^{\alpha-2}),\ \text{as}\ \varepsilon\to 0.$

Putting the above estimate together, we observe that

$\displaystyle\Big{|}\int_{0}^{T}\int_{\mathbb{T}^{3}}\big{(}(v\otimes
v)^{\varepsilon}-(v^{\varepsilon}\otimes v^{\varepsilon})\big{)}:\nabla
v^{\varepsilon}dxdt\Big{|}\leq o(\varepsilon^{\theta\alpha+\alpha-2}).$

Letting $\varepsilon\rightarrow 0$ in (4.14), we obtain the energy
conservation.

(3) It follows from the Hölder inequality that

$\displaystyle\Big{|}\int_{0}^{T}\int_{\mathbb{T}^{3}}\big{(}(v\times\omega)^{\varepsilon}-(v^{\varepsilon}\times\omega^{\varepsilon})\big{)}:\omega^{\varepsilon}dxdt\Big{|}$
(4.16) $\displaystyle\leq$ $\displaystyle
C\|(v\times\omega)^{\varepsilon}-(v^{\varepsilon}\times\omega^{\varepsilon})\|_{L^{\frac{p}{p-1}}(0,T;L^{\frac{6p}{p+7}}(\mathbb{T}^{3}))}\|\omega\|_{L^{p}(0,T;L^{\frac{6p}{5p-7}}(\mathbb{T}^{3}))}.$

and

$\|v\|_{L^{\frac{p}{p-2}}(0,T;L^{\frac{3p}{7-2p}}(\mathbb{T}^{3}))}\leq
C\|v\|^{\frac{21-7p}{7}}_{L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))}\|\nabla
v\|^{\frac{7p-14}{7}}_{L^{p}(0,T;L^{\frac{6p}{5p-7}}(\mathbb{T}^{3}))}+C\|v\|_{L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))}.$

The Hölder inequality ensures that

$\|\omega\times
v\|_{L^{\frac{p}{p-1}}(0,T;L^{\frac{6p}{p+7}}(\mathbb{T}^{3}))}\leq
C\|v\|_{L^{\frac{p}{p-2}}(0,T;L^{\frac{3p}{7-2p}}(\mathbb{T}^{3}))}\|\nabla
v\|_{L^{p}(0,T;L^{\frac{6p}{5p-7}}(\mathbb{T}^{3}))}.$

We can apply Lemma 2.4 to complete the proof of this part.

(4) Choosing $p_{2}(2-\theta)=\infty,q_{2}(2-\theta)=2$, $q=\frac{q}{q-1}$ and
$p=\frac{p}{p-1}$ in Lemma 2.3, we get $\theta p_{1}=\frac{p\theta}{p-1}$ and
$\theta q_{1}=\frac{2q\theta}{q\theta-2}$ and

$\|(v^{\varepsilon}\otimes v^{\varepsilon})-(v\otimes
v)^{\varepsilon}\|_{L^{\frac{p}{p-1}}(0,T;L^{\frac{q}{q-1}}(\mathbb{T}^{3}))}\leq
o(\varepsilon^{\theta\alpha}),$ (4.17)

where $\theta p_{1}=k$ and $\theta q_{1}=\ell$ were used.

It follows from Lemma 2.2 that

$\|\nabla\omega^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{3}))}\leq
O(\varepsilon^{\beta-1}).$ (4.18)

Inserting (4.17) and (4.18) into (4.16), we arrive at

$\displaystyle\Big{|}\int_{0}^{T}\int_{\mathbb{T}^{3}}\big{(}(v\otimes
v)^{\varepsilon}-(v^{\varepsilon}\otimes v^{\varepsilon})\big{)}:\nabla
v^{\varepsilon}dxdt\Big{|}\leq o(\varepsilon^{\theta\alpha+\beta-1}).$

This ensures that the right-hand side of (LABEL:4.12) vanishes as
$\varepsilon\rightarrow 0$. The proof of this part is finished. (5) Exchanging
$o$ and $O$ in (4.17)-(4.18), we can prove this theorem. ∎

##  5 Conclusion

A celebrated result in the study of Onsager conjecture is that $v\in
L^{3}(0,T;B^{1/3}_{3,c(\mathbb{N})}(\mathbb{R}^{3}))$ guarantees the energy
conservation of the Euler equations on the whole space, which is shown by
Cheskidov-Constantin-Friedlander-Shvydkoy in [10]. Inspired by recent works
[6, 5] of Berselli and Berselli-Georgiadis, we extend it to a more general
case $v\in L^{p}(0,T;B^{\frac{1}{p}}_{\frac{2p}{p-1},c(\mathbb{N})})$ for
$1\leq p\leq 3.$ We provide two different methods to show it. The first is
interpolation technique and the second one is Littlewood-Paley theory as [10]
for the case $1<p\leq 3.$ It seems that the first one is better than the
second one. As a byproduct, a observation is that the general case immediately
implies energy conservation sufficient condition in terms of gradient in space
$L^{p}(0,T;L^{\frac{6p}{5p-5}}(\mathbb{R}^{3}))$ derived in [7, 28, 35, 3, 4]
for the Navier-Stokes equations. In addition, parallel to energy, the helicity
conservation of weak solutions in inviscid flow is also considered. It seems
that our argument can be applied to other fluid equations.

It is shown that the space $B^{1/3}_{3,c(\mathbb{N})}$ for the energy
conservation are sharp in [10]. It is an interesting problem to construct an
example to illustrate the sharpness of space
$B^{\frac{1}{p}}_{\frac{2p}{p-1},c(\mathbb{N})}$ for $1\leq p\leq 3.$

## Acknowledgement

Wang was partially supported by the National Natural Science Foundation of
China under grant (No. 11971446 and No. 12071113) and sponsored by Natural
Science Foundation of Henan (No. 232300421077). Wei was partially supported by
the National Natural Science Foundation of China under grant (No. 11601423 and
No. 12271433). Wu was partially supported by the National Natural Science
Foundation of China under grant No. 11771423. Ye was partially sponsored by
National Natural Science Foundation of China (No. 11701145) and Natural
Science Foundation of Henan (No. 232300420111).

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