Stefan Mehedinteanu
email: s.mehedinteanu@gmail.com

What Is Gravity (Graviton) and How Is Embedded in Mater Particles Giving Them Mass (Not the Higgs Field)?

Keywords: Origin of gravity, Planck particle, gravitational charge, virtual black holes, free photons, graviton, color magnetic charges, gluons, photons mass, electrons

It will be shown how the Micro-black-holes particles produced at the horizon entry into a number of 62 61 10   N

by quantum fluctuation as virtual micro-black holes pairs like e+ e-creation, stay at the base of: the origin and evolution of Universe, the Black Holes (BH) nuclei of galaxies, of the free photons creation of near mass-less as by radiation decay that condensate later at Confinement in to the structure of gauge bosons (gluons) . Also, these ex-Micro-black-holes ( BHs  ) by merging release gravitons which deform the space-time, and also decay of all free photons as BH's nuclei of galaxies , or if kipping someone else, as electrons, others leptons (quarks) and uncharged particles (neutrinos). After radiation decay, the heritage of gravitational charge (gravitons) quantized as c M G Planck   inside (or on event horizon) into equilibrium with the inner field like electrostatic Coulomb field as being generated by the few photons remained, finally give the mass. Thus, is explained how the gravity (only few gravitons remaining at Confinement in nucleons) is embedded in matter particles, and in the BH's nuclei of galaxies of gravitational charge only (gravitons), an old waited answer. Also, in this context it results that an equal part of the ex-Micro-black-holes are generated at horizon entry that correspond to Electroweak period during Universe evolution (10 11 GeV; 10 -29 s) .. A proof of the model is done by applying it to light bending due of Earth. It results that the graviton is first unit (indivisible) of Universe. A very stranger result is obtained if we divide the gravitational charge c  to the Compton length for every particle, or in other words the gravitational energy of the particle distributed in their Compton length is just the particle energy or the -rest of mass‖.

numerical examples, not only to reconcile the original idea's that the classical (or perhaps) semi-classical electron has a mass of electromagnetic nature and that the gravitational charge (the inertial mass) is in fact intrinsically-related to the same electromagnetic features. To do this, -I have had to read for many years now, the theories of times past which involved the photon configuration inside of particles --Lloyd Motz [1e]; which he was the first to propose a gravitational charge to a particle and even speculated on bound photon particles in a type of orbital motion; the original idea's brought forth in this paper is how to think of the gravitational charge in terms of the electromagnetic field and we will also study the implications of certain equations under the same investigative field. To make short, the electron is taken in this paper, as a fluctuation of either bound or single photons following toroidal or other topological paths in a dense curved spacetime. To finish off, we will also study what it means to talk about the spin of an electron. I will show that, only few free photons of photons dimension could remain embedded in ex-like-Micro-black-holes which it adjusts its dimensions to electrons, other leptons, noncharged particles (neutrinos, Higgs) in order to equilibrate the gravitational charge Planck M G as it was explained before following electron model [START_REF] Williamson | Is the electron a photon with toroidal topology?[END_REF]. Also we shown how is generated a huge number ( 62 61 10  ) into each Quantum bubble of one Micro-blackholes (virtual micro black holes) as by Quantum fluctuations. In [5b] is interpreted in the topological fluctuations S2 ×S2 bubbles in spacetime foam as virtual black hole loops. It needs electric or magnetic charges to produce a pair of black holes from Ernst solution. However the quantum bubbles form even in absence of any field. The reason for that is that virtual black holes are not classical geometries and hence need not satisfy Einstein's equations. They are rather solutions of the Wheeler-DeWitt equation [START_REF] Faizal | Some Aspects of Virtual Black Holes[END_REF] and hence need not have any electric or magnetic charges as would be required if they satisfied Einstein equations. Virtual black holes lead to the loss of quantum coherence, which is calculated in this paper [START_REF] Faizal | Some Aspects of Virtual Black Holes[END_REF]. Virtual black holes also lead the space-time to have an intrinsic entropy. The soft photons radiated during these virtual micro-black holes inherently decay its being of the same order as the Micro-black-holes, these are later incorporated as gluons into hadrons. In the same context is considered that at horizon entry a large part of the Micro-black-holes by merging become BH's nuclei of galaxies. After Inflation has ended, scalar perturbations begin the re-enter the horizon and interact in such a way as to induce gravitational waves. These gravitational waves are due of gravitons release due of BHs  merging in BHs nuclei of galaxies , and which deform the space-time.

The concept of like-Micro-black-holes particle at the origin of Universe

We consider that the firstly Universe beginning is due of Quantum instability fluctuations when is obtained a lot of Micro-black-holes particles as Planck particles, and which we will consider its as to be the primordial seeds of Universe, in the following we will argue this hypothesis. First of all, I present some of known data. Thus, in [1a] it is therefore assumed, that -potential energy‖ caused by gravitation and -kinetic energy‖ caused by expansion of the Universe are equal to each other (using the relations ct R U  and In fact as already noted [23]

Virtual micro-black holes

The picture of virtual black holes given here also is suggested that macroscopic black holes will evaporate down to the Planck particles ( m 35 10  ) size and then disappear in the sea of virtual black holes [5b]. However, in his paper Hawking says: -I shall be less concerned with real processes like pair creation, which can occur only when there is an external field to provide the energy, than with virtual processes that should occur even in the vacuum or ground state‖. However, in addition to black holes formed by stellar collapse, there might also be much smaller black holes which were formed by density fluctuations in the early universe [5a]. These small black holes, being at a higher temperature, would radiate more than they absorbed. They would therefore presumably decrease in mass. As they got smaller, they would get hotter and so would radiate faster. In quantum gravity, a virtual black hole is a black hole that exists temporarily as a result of a quantum fluctuation of spacetime [5b], [START_REF] Faizal | Some Aspects of Virtual Black Holes[END_REF]. It is an example of quantum foam and is the gravitational analog of the virtual electro-positron pairs found in quantum electrodynamics. At such small scales of time and space, the Heisenberg uncertainty principle allows energy to briefly decay into particles and antiparticles and then annihilate without violating physical conservation laws. Theoretical arguments suggest that virtual black holes should have mass and the lifetime on the order of the Planck particles, but we have changed to a not so small dimension since not respect the general terms of energy and mass of universe. Therefore, we consider also Micro-black-holes pairs, but that occur with a number density of approximately one per Quantum bubble [START_REF] Faizal | Some Aspects of Virtual Black Holes[END_REF] To estimate the horizon entry we use some derivations done in [2a].In Inflation models [START_REF] Mehedinteanu | Numerical Analysis Of Pairs Creation By Schwinger Effect In Nucleons And The Beta-Decay Process Acceleration[END_REF], the scale leaving the horizon at a given epoch is directly related to the number ) ( N of e -folds of slow-roll inflation that occur after the epoch of horizon

 dN k d  
as of eq.( 46) from [START_REF] Mehedinteanu | Numerical Analysis Of Pairs Creation By Schwinger Effect In Nucleons And The Beta-Decay Process Acceleration[END_REF], and therefore ) ( ) ln(

 N k k end  , or, ] [m ke k N end 
where end k is the scale leaving the horizon at the end of slow-roll inflation, or usually ] [

1 1 m k k end    , the correct equation being ] [ 1   m e k k N end . When the wavelength ]) [ ( 1 m k  is large compared to the Hubble length ]) [ ( 1 m H 
, the distance that light can travel in a Hubble time becomes small compared to the wavelength, and hence all motion is very slow and the pattern is essentially frozen in. Since, the FLRW metric of the universe must be of the form

2 2 2 3 2 2 ) ( dt c ds t a ds  
where 2 3 ds is a three-dimensional metric that must be one of (a) flat space, (b) a sphere of constant positive curvature or (c) a hyperbolic space with constant negative curvature, or for small commoving time aHc dt 1 

, we can consider the distance as

end a a ds L   
, so the volume is given by:

] [ 1 ) ( 3 4 s m c a V matter 
A second example is de Sitter space which contains an event horizon. In this case the temperature T is proportional to the Hubble parameter H , i.e. H T  , such a conclusion being used by author in [START_REF] Mehedinteanu | Numerical Analysis Of Pairs Creation By Schwinger Effect In Nucleons And The Beta-Decay Process Acceleration[END_REF] to calculate the evolution of Universe. To estimate the horizon entry we use some derivations done in [START_REF] Mehedinteanu | Numerical Analysis Of Pairs Creation By Schwinger Effect In Nucleons And The Beta-Decay Process Acceleration[END_REF]. During Universe evolution [START_REF] Mehedinteanu | Numerical Analysis Of Pairs Creation By Schwinger Effect In Nucleons And The Beta-Decay Process Acceleration[END_REF], the horizon leave is when

1   leave leave leave H k a , ] [ 10 27 1 1 m H k leave leave      , s c H t leave leave 36 1 10 3 . 3     
at the Electroweak epoch Here the Hubble constant is defined as

3 3 8 2 2 c G a a             2 2 2 8 c p c G a a            
In Newtonian interpretation, the Friedmann equations are equivalent to this pair of equations: a we obtain for outside the object (BH, Universe, planets, stars etc.)

] [ ] [ 2 2 3 4 3 4 3 2 3 2 2 2 2           m n a c G a c GE a c G a GM s H a a g g   ; (1)
, where:

g g E n   ;   g P g merging at n n     ;   3 4 3 a M  ; 2 Mc E  ; BH BH m n M    ; 1   H R .
Or, for inside of these objects:

] [ 3 4 3 4 ] [ 3 4 2 2 2 3 2 2 3 3 2 2        m c l n M G c G s a a G a a C merging at U       (2)
During Universe evolution at Electroweak epoch

     , ] [ 10 20 1 1 1 m H k H EW laeve leae        we found 88 . 15  N
to match the iterations cycle:

N a H R T k m end end B g        1    .
The number of gravitons that has been released following BHs To mention that only this dataset match the model.

Another proof-the light bending by Earth

The number of graviton in case of Earth is 58 26

10 2 . 3 10     U n g ; J R GM U Earth Earth 32 2 10 2 . 3   
The number of gravitons that has been released following

            ; it results 1 _  BH end a
, and from eq. ( 2) ] [ 10 8 . 8

3 1 m r H Sch end      , s c H t end end 11 1 10 9 . 2      , we found 7 . 35  N
to match the iterations cycle:

N a H R T k m end end B g        1    .
At BHs merging, the curvature radius R from eq. ( 2) with the number of merging nucleons during which release gravitons as

merging at n  , Sch end r m H R       ] [ 10 3 . 5 3 1
.

The gravitons release from Earth as a gravitational wave

Therefore, the new horizon leave as gravitational wave (GW) is just when the gravitons escape from the Schwarzschild radius (the Universe is a viewed as an inverse big black hole) :

1   leave leave leave H k a , or ] [ 10 4 . 7 3 1 1 m r H k Schw leave leave        .
Now, the new horizon-entry is when the wave length

N leave end e k k   ; ] [ 10 8 1 4     m k end , ] [ 10 2 . 1 3 1 m k end   
; and the scale arrives factor at The strain at Earth surface Now, based on eq. ( 1) we can derive for the G-wave effect in the deformation (strain) of the space-time between Earth and a detector site by using the gravitational pressure due of gravity charges on the area of Schwarzschild radius Schw r , we have:

end end end H k a  , the Hubble length with Compton length ] [ 16 _ m c m GW g GW C      ; from eq. (1) ] [ 10 5 . 6 6 1 m R H end     ; it results 3 _ 10 3 . 5   GW end a , s c H t end end 02 . 0 1    , we found 12  N to match the iterations cycle: N a H R T k m end end B g        1    .
18 3 _ 4 2 2 10 24 . 1 3 4               GW end Schw g g Schw a c r n G R r   Separately, ] [ 10 86 . 8 2 3 2 m c M G r Earth Schw     
, we have

9 18 2 2 10 1 . 1 10 34 . 1        Earth Schw Earth Schw R r R r 8
, so, in both cases the strain is around

9 10 1 . 1     R r Schw 
, that is near equally with strain as light bending.

The average magnitude of the electric field (negative charge) in the event horizon of a micro-black-hole is like that of the model electron given in [START_REF] Williamson | Is the electron a photon with toroidal topology?[END_REF], and where the inside -trapped‖ photon is similar with the -absorbed‖ photon from thermal energy V in case of BH  particle, or in other words the electron is a decaying BH  particle, see below equation ( 4). In other words the contribution to the space-time deformation is due of , alls correspond to the situation at cosmic microwave background radiation (CMBR) time. Since, at LGO experiment [START_REF] Mehedinteanu | The Quantization of the Gravitational Charges At the Origin of the Gravitational Waves From a Binary Black Hole Merger[END_REF], it was observed also a radiation, is advanced the idea that the all-sky map (pattern) of the CMB represents in fact the radiation emitted during BHs  merging (clustering), that is contrary to the thermal radiation at Recombination due of photon decoupling.

The formation of the Primordial BHs as galaxies nuclei by

] [ 111m R  , s c R c H t end end 7 1 10 3 . 3       , with ] [ 10 2 . 2 7 _ 1 1 m k H BH C laeve leae         we found 5 . 20  N
to match the iterations cycle:

N a H R T k m end end B g        1    . 4 10 7 . 1   T , ] [ 10 26 . 1 7 _ m mgBH C     , Hz k c end 6 1 10    
The number of gravitons that has been released following BHs  merging into primordial galaxies is only , these generate the increase of curvature radius of the object (Universe) to R . To note, that this process it seems to be equivalent with the use of the cosmological constant in CDM  model (but is not necessary to explicit it use-Einstein was right!) .

The strain at LIGO site Now, based on eq. ( 1) we can derive for the G-wave effect in the deformation (strain) of the space-time between BH and LIGO site by using the gravitational pressure due of gravity charges on the area of Schwarzschild radius Schw r of BH, respectively with the values calculated in section 5, we have:

72 3 _ _ 4 2 2 10 6 . 1 3 4               GW mgBH end Schw g g Schw a c r n G R r  
, where J a

           LIGO Schw LIGO Schw R r R r
, so, in both cases at (LIGO laser) the strain is around

36 10 3 . 1     R r Schw 
, which is undetectable.

The Confinement into nucleons

The effect of the potential B is the same as shifting the effective mass

) 1 2 ( 2 4 2 4 2       j c qB c m c m e 
for fermions for each Landau level.

Where Because the magnetic induction of the color magnetic gluons current which is powered by electric field given by a pair of quarks ( 0 H ),

2 0 2 c gluon H H B  
, it has the raw flow consequences squeezing this cromoelectrical flux into a vortex line, followed by forcing an organization into a triangular Abrikosov lattice, see figure 1. From [2a], we have the lower critical field:

          2 15 2 2 0 10 ) log( log 2 2 ) ( Am J c c x B       
, where 01114 



, and when near the axis, for

   116 . 0 x
, when the induction is

1 15 ] [ 10 2 ) ( c H T B     , ] [ 10 6 23 C N cB E   
In the case of a homogeneous potential directed along the z-axis [START_REF] Mehedinteanu | The Quantization of the Gravitational Charges At the Origin of the Gravitational Waves From a Binary Black Hole Merger[END_REF], the Einstein stressenergy tensor is:

   8 2 2 0 33 22 11 00 B c T T T T B       ; 0 0  i T
, where ] [ Also, the interaction energy at interface

B E  is: ] [ 8 2 2 0 J V V B c V vol B vol        , (2.a) 3 8 ) 4 ( 2 C C c C vol V      
, at Compton length equally with the penetration length

   C , that results 3 0 2 ) ( ) (   e C V E   (2.b) With gluons V   as above is obtained ] [ 10 98 . 2 15 T c E B q q   
, where 23 10 9 . 8   q q E with eq. (2.a), that are identically with the above values, indubitable meaning that this force is equilibrated by the gravity charges from quarks tubes, see below. With equation ( 4), see below,

The Quantization of Mass (or Gravitational Charge)

In the classical Abraham-Lorentz theory of the electron, as in references [26, 27, 28] cited in [START_REF] Williamson | Is the electron a photon with toroidal topology?[END_REF], the energy contained in the Coulomb field of a charge e in all space outside its radius R is From [START_REF] Williamson | Is the electron a photon with toroidal topology?[END_REF], our main motivation for the central postulate stated above arises from a consideration of the experimentally well-established (parapositronium) electron-positron annihilation and creation processes as in the here cited reference [36].

       R r R elect R
    e e
Otherwise, in all the Feynman diagrams is considered the particles transformation as been accompanied by  (vortex).

We envisage a quantized solution where, just as is the case for the free photon, we have time varying fields, but where the field distribution is self-confined in space. The mass of any confined photon will be The magnitude of the apparent charge of our model object is based on the length scales estimated in the previous section. We confine an arbitrary photon with wavelength  to a spherical volume

3 2 3 4          V .
The energy density of the electromagnetic field in the volume is

  2 1 0 2 0 2 1 B E W      .
For a propagating photon inside the volume, where space is curved, we take cB E  and

0 0 2     c
as is the case for a free-space photon.

The electric field energy E U and the magnetic field energy B U are then one half of the total confined photon energy

U (i.e. U U U B E 2 1  
). We find for the average energy density of the electric field in the volume

V , V U V U W E E 2 1  
and also To estimate the charge in our model we need to compare the magnitude of the inward directed electric field to that for a point charge at the origin. Making the plausible assumptions that the relevant length scale from where the electric field is effectively inward-directed is the mean radius of energy transport

  4  r
, and that the average electric field of the confined photon, Eq. ( 3), is a good estimate of the field at this radius, we obtain the effective charge, q , by comparing this to the Coulomb field of a point charge at distance r r  14

, where this apparent charge arises from the electric field of the confined photon. Note that q is independent of the energy of the photon (the size of the object) and is a result of the toroidal topology.

The rotational energy of a relativistic object is

 L U rot 
, with L the angular momentum, and  the angular frequency. For a photon

  L
, and the total energy of a photon with frequency

 is    photon U
. Thus, the energy of a photon is entirely electromagnetic and contained in its spin. The confined photon in our model has to travel around twice to complete its path of length

   c 2 
. Consequently, the internal rotational frequency of the model is twice the photon frequency

  2  s .
The internal rotational energy is equal to the confined photon energy, and we may write

     s el L U mod
. Our model must then have an intrinsic angular momentum

  2 1   s L  
. We see that this describes an object of half-integer spin. If the spinstatistics theorem applies, our self-confined photon should be a fermions. This is again a direct consequence of the topology of our model; the field vectors must rotate through  720 before coming back to their starting position with the same orientation. In quantum mechanics, the spin angular momentum has a fixed value 2 1  s , therefore we cannot take the intrinsic spin to a classical limit by letting   s and there is no classical correspondence with half-integer spin. In our model this is ensured because, for our topology, we have necessarily one and only one wavelength, and this gives a fixed, length-scale independent of s . If the electron is indeed constituted by a photon, other elementary particles may also be composed of photon states, but in some other configuration [START_REF] Williamson | Is the electron a photon with toroidal topology?[END_REF]. The possibility that muons and tauons may be formed by electron-like states with a different internal curvature has been discussed in the literature [START_REF] Hestenes | The Zitterbewegung Interpretation of Quantum Mechanics[END_REF]. We speculate that the hadrons may be described by composite confined photon states as gluons [START_REF] Mehedinteanu | Numerical Analysis Of Pairs Creation By Schwinger Effect In Nucleons And The Beta-Decay Process Acceleration[END_REF] together with quarks pairs at the origin of an electrical field and of gravity. If we identify a quark with a confined photon state which is not sufficient in itself to complete a closed loop in space, but transforms a photon going in one spatial direction to one travelling in another, it would then only be possible to build closed three-dimensional loops from these elements with qqq and q q combinations. The gravitational theory here proposed [1e] says that the intense interior gravitational field of the electron is just sufficient to compensate for the repulsive forces in an electron of finite size. It shows that if the electronic charge is pictured as being distributed over a region with a radius equal to the classical radius of electron ) (

2 2 mc e
, and the gravitational mass is distributed over a region with a radius of order mc C    , a balance can be achieved between the repulsive force of spin and electric and attractive gravitational forces. We see from the basic equation of the quantized gravitational charge see section 3. , respectively ) gives a positive self energy of 2 mc . Since the two self energies are of the same order of magnitude, they be brought into balance. Consider now a photon and suppose that it maintain its identity in virtue of an interior gravitational filed. I we picture the gravitational mass m of the photon as being distributed over a volume whose radius is of the order of  , the wavelength of the photon, we see that the energy of the photon, in virtue of gravitational mass

2 P M c G   , (6) that



, it results the photons energy

GeV J c Gm 2 ] [ 10 8 . 1 10 2     
There are Higgs particles? At LHC were measured, in fact, two high energy photons, that following my model they resulting from the disrupturing of quarks (like the electrons) as constituents of the two collided protons pp , respectively 

m c m H H C       ; GeV J c H C H 125 ] [ 10 25 . 1 8        
, or in other words the energy of particle as to be distributed in their Compton length is just the energy or the -mass‖. The neutrino, according to this picture, must be represented as consisting of two photons bound together gravitationally and revolving around a common center. To account for the spin of the neutrino, which is a Fermi particle, we need assume that the sum of the orbital angular momenta of photons equals 2  . The two photons must then be revolving such a way that : 1) their own intrinsic spins (each equal to  are antiparallel , and 2) that the electromagnetic field of one cancels that of the other. This can always be achieved. A neutrino consisting of two photons, has, of course, zero rest mass, and therefore this , it results the total number of primordial high energy photons following the -decay‖ of a like-BH  particle and which "fill" the vacuum till are embedded as a Bose Einstein Condensate (B.E.C) in matter (gluons) at symmetries breakings, or when is attained a critical temperature (like in case of superconductors!) as: The Micro-black-holes decay into photons in P t is the Hawking radiation where, the gravitational field is so strong that it causes the spontaneous production of photon pairs (with black body energy distribution) and even of particle pairs. Sure, only few free photons of photons dimension could remain embedded in ex-like-Micro-black-holes which it adjusts its dimensions to electrons, other leptons, noncharged particles (neutrinos, Higgs) in order to equilibrate the gravitational charge P m G as it was explained before following electron model ref. [START_REF] Williamson | Is the electron a photon with toroidal topology?[END_REF] is the gravitational charge defined by L. Motz [1d] as resulting from:

2 2 1 r m G m G F G 
, where it can say that the gravitational charge

1 m G q 
is the source of gravitational field

    G c c V m G P    2  , for c v  , see below.
The gravimagnetic field, in GEM equations, we have:

  B v E q F e   
, or equivalently ) 4 (

g g B v E m F    ; rmv L  ; the electrical field 2 0 4 r q E   ; the gravitelectric field ] [ ] [ 4 1 2         s B ms B v m F g E g g g



; the mass with the charge and the mass density with the charge density, so, the gravimagnetic field is

G c m G F q F E P C C g     .
This cross product term give rice to an angular momentum which is proportional to moment of inertia I and angular speed  , Therefore, it is obtained the quantization condition as:

 2 2 2    r m L P .
Since it was interested in the quantization of fundamental gravitational charge, it is found a value for r and  that must be associated with such charge. To do this, L.Motz considered the Universe to two such charges in gravitational equilibrium and resolving about each other in the first Bohr orbit. The radius of this orbit is just 3 2 P Gm r   , which it was taken as to be r . In this case (unlike the standard Coriolis force), the force is not fictional, but is due to frame dragging induced by the rotating body when Therefore, if we consider the string force due of the Coulomb flux tube as given by the colors quarks q q pairs, see figure 1 Gamma-gamma interactions can either give up mass in the form of special types of decay processes or, vice versa, the energy can come from antiparticle interactions. This is called the parapositronium decay. Such a phase transition is given as . Therefore, into a Planck particle are still embedded an huge number of such tiny particles ~35 10 , in other words it seems that this tiny particle graviton is the fundamental unity (indivisible) of the Universe. Even the graviton have a such infinitely small dimension, them action length (wave length) and the Compton length is much longer.

Cosmological and other consequences of the uniton existence

If the square of quantum charge is, indeed c  , we can account for the stable electrically charged particles such as electrons by balancing the explosive positive electro-static energy with the negative binding energy of the gravitationally charge. Accordingly to this point of view electrons and nucleons are the lowest bound states of two or more unitons that collapsed down to the appropriate dimensions gravitationally and radiated anyway most of their energy in the process. It is clear that a gravitational charge of magnitude c  will contribute negative gravitational binding energy of the order of 2 mc if this charge is distributed over a region whose dimensions are equal to the Compton wave length of the particle. To note again that each uniton of a total 96 10 it produces by radiation

Conclusions

Accounting that the mass of graviton results as been of J a c ), that representing the component of the magnetic field (B ). The further Universe expansion is assured all the time by the gravitons released during the merging, and finally is accelerated when an another fraction of BHs  merging (clustering) and later into greater structures as BH's nuclei of galaxies and, that fraction derives from the same number of the initially BH  . In this way it is entirely confirmed the timeline of Universe. A proof of the model is done by applying it to light bending due of Earth.

  generate the curvature radius of the object R . In other words the contribution to the space-time deformation is due of

  The energy of the graviton becomes at an eventually detector (like LIGO) with the above of particles released as the gravitational wave (like the photons of the light wave) remains equally with the above value curvature radius it results from eq.(1) with the integral (which the starts at the outside of the Schw r



  base of BH's nuclei of galaxies production An important contribution to further Universe expansion is given by the BH's nuclei of galaxies which result from another fraction ( 75 . 0 ) of the initiallyBHsThe number of gravitons that has been released following BHs  merging is only generate the curvature radius of the object (Universe) R .

  The equivalence between the Lorenz force energy which squeezes the electrical field 

  near the values calculated before by both methods.

  a lower limit on its size of the order of the so-called classical electron radius ]

.

  that for the case where the electron and positron annihilate at rest, the decay photon wavelengths  are just the electron Compton wavelength ] We therefore, in the first instance, look for a quantized solution defined by periodic boundary conditions of length one Compton wavelength C  , which is confined to some closed path in 3-D space.

.

  The average magnitude of the electric field inside the model electron is then

  the charge from our model in terms of the elementary charge e

  gravitational charge is distributed over a region of the order of mc  it gives a negative energy of the order of while the electric charge distributed over a region of the order of

  the case of Higgs particle at LHC is argued the decay  diagrams A very stranger result is obtained if we divide the gravitational charge to Compton length of any particle (example Higgs particle)

  cited in [3a]. The energy of graviton Very early in 1960 in a seminal paper L.Motz ref. [1d] has elaborated a new theory of the structure of fundamental particles, which introduces gravitational field into the interior of particles such as electrons to account for their stability. Now,

  r , and that this field is coupled to the gravitational charge 2 m G at the position r (relative to the source of the field ) via the product of the field strength and the charge. L. Motz derived the quantization condition on gravitational charge in a similar manner by noting that moving particle with velocity V  and with gravitational chargeP m Gis coupled not only to the Newtonian gravitational fields of all other particles (the gravielectric field ) in the usual way, but also to the Coriolis field v gravitoelectromagnetism (GEM) equations (see below), due of the inertial Coriolis force, see figure2, (defined as r

  the motion of the particle which is of the order of  to the field   .

2 

 2 a spherical body the Coriolis field is the gravitomagnetic field g B near a rotating body can be derived from the GEM equations. It is exactly half of the Lenseto Mach's principle, produces the centrifugal force c m . -A very general statement of Mach's principle is "Local physical laws are determined by the large-scale structure of the universe".This will be so only if c substituting for r as Bohr orbit when the electron is held in a circular orbit by electrostatic attraction, after some manipulations: ; , introducing in L we have:

Figure 2 ,

 2 Figure 2, Coriolis field H(B)

Fig. 1 .

 1 Fig.1. The gluons embedded in Giant-Vortex[START_REF] Mehedinteanu | Numerical Analysis Of Pairs Creation By Schwinger Effect In Nucleons And The Beta-Decay Process Acceleration[END_REF] that could be also the arrangement for the nucleon (only illustration). A spin-orbit nonabelian field is shown.

  due to hair concept of micro-black-holes [5c], that represent the gluons which again together with quarks condensate at Confinement ( V=2.1GeV, ) into nucleons, see more details in [2a]. The binding energy will be sufficient to balance the positive electrostatic energy of the charge e distributed over the classical radius

  -at Reheating), the merging of the primordial Micro-blackholes generated as due of Quantum fluctuation at Electroweak epoch, it conducts to a can generate at Confinement epoch inside the nucleons an electrical field ( E ) that condensate the free photons, also as resulting from the radiating of the Micro-black-holes, but as gluons of near the same number ( 96 10 

  from [1b], a Planck mass particle decays via the Bekenstein radiation within a Planck time s

			42 10  , see below.
	In fact as already noted [23] from [1b], a Micro-black-holes mass particle decays via the
	Bekenstein radiation.
			s
	equations proportional to	t E  	, see [1b].
	This immense energy

The Maxwell equations in a vacuum with a non zero conductivity coefficient, can be shown to lead to a loss of energy ( zshift ) of a photon during its propagation, see [2a]. This is because the dissipating mechanism leads to an extra term in the usual Maxwell'BH  can constitutes a spectrum for blackbody radiation when photon creation takes place has also been proposed by author in [2a] , but now I consider that this radiation is obtained by the decaying of Micro-black-holes ( BH  ) viewed as virtual micro black-holes with hair [5a; [5b]; [5c].

  , initially it was considered per Planck volume [2b] , that means one per quantum is de Sitter space which contains an event horizon. In this case the temperature T is proportional to the Hubble parameter H , i.e. H T  , such a conclusion being used by author in [2a] to calculate the evolution of Universe.

	bubble volume, that means	n	P	3     d H	10 60	[ m	3 	]	; where Hubble constant	d H			20 10 	m	,
	see below..
	A second example

  exit. Indeed, since H -the Hubble length is slowly varying, we

	have	d	ln	k		d	(ln(	aH	))		d	ln	a		a dt a 		Hdt	. From the definition Eq. (38) of [2] this
	gives	ln				(	)


Lorenzo Brandolese
email: brandolese@math.univ-lyon1.fr

Charafeddine Mouzouni
email: mouzouni@math.univ-lyon1.fr

A SHORT PROOF OF THE LARGE TIME ENERGY GROWTH FOR THE BOUSSINESQ SYSTEM

Keywords:

We give a direct proof of the fact that the L p -norms of global solutions of the Boussinesq system in R 3 grow large as t → ∞ for 1 < p < 3 and decay to zero for 3 < p ≤ ∞, providing exact estimates from below and above using a suitable decomposition of the space-time space R + × R 3 . In particular, the kinetic energy blows up as u(t) 2 2 ∼ ct 1/2 for large time. This constrasts with the case of the Navier-Stokes equations.

Introduction

The incompressible Boussinesq system describes the dynamics of an incompressible fluid, taking into account heat exchanges. In addition to the flow, the transport and diffusion of temperature, we have also convection currents created by the vertical force of buoyancy. The Boussinesq approximation consists in neglecting the variations of the density in the continuity equation. Accordingly with this approximation, we also neglect the local heat source due to the viscous dissipation. Taking all the physical constants equal to 1 we can write the Boussinesq system in the following form, (1.1)

           ∂ t θ + u • ∇θ = ∆θ ∂ t u + u • ∇u + ∇p = ∆u + θe 3 ∇ • u = 0 u| t=0 = u 0 , θ| t=0 = θ 0 x ∈ R 3 , t ∈ R + .
Here u : R 3 × R + → R 3 is the velocity field and the scalar field θ : R 3 × R + → R denotes the temperature. The function p : R 3 × R + → R is the Lagrange multiplier related to the constraint of incompressibility, and is closely related to the pressure of the flow. Moreover, e 3 = (0, 0, 1) is a constant unit vertical vector.

In the particular case of incompressible Navier Stokes system (θ ≡ 0), starting with T. Kato [START_REF] Kato | Strong L p -solutions of the Navier-Stokes equation in R m , with applications to weak solutions[END_REF] and M. E. Schonbek [START_REF] Schonbek | L 2 decay for weak solutions of the Navier-Stokes equations[END_REF], many authors studied the decay problem of various L p norms as t → +∞. For example, it is known that if u 0 3 is small enough and if |u 0 (x)|(1 + |x|)dx < ∞, then one has u(t) p ≤ Ct -2+3/(2p) , for t > 0 and 1 ≤ p ≤ ∞ for the solutions of the Navier-Stokes equations. Asymptotic profiles like those constructed in Y. Fujigaki and T. Miyakawa [START_REF] Fujigaki | Asymptotic profiles of nonstationary incompressible Navier-Stokes flows in the whole space[END_REF] prove the optimality of such decay rates. In particular this shows that the kinetic energy vanishes when t → ∞.

The asymptotic behavior of the L p -norms for Boussinesq system was expected to be similar to the particular case of the incompressible Navier Stokes system 1 . But the main result in [START_REF] Brandolese | Large time decay and growth for a viscous Boussinesq system[END_REF] put in evidence that this is true only if the initial temperature has zero mean: indeed, if the data satisfies appropriate size and localization conditions then one has, for t large enough, (1.2) c(1

+ t) -1/2+3/2p ≤ u(t) p ≤ C(1 + t) -1/2+3/2p , for 1 < p ≤ ∞,
with c > 0 if and only if θ 0 = 0. This proves in particular that u(t) L 2 ∼ ct 1/4 if the condition θ 0 = 0 is satisfied. The drawbacks of the approach of [START_REF] Brandolese | Large time decay and growth for a viscous Boussinesq system[END_REF] are that the proof of (1.2) is involved and the conditions on the data too restrictive. For example, the smallness assumptions needed in [START_REF] Brandolese | Large time decay and growth for a viscous Boussinesq system[END_REF] was of the form ess sup R 3 |x| |u 0 (x)|+ ess sup R 3 |x| 3 |θ 0 (x)|+ θ 0 1 < . Moreover, the data were assumed to satisfy additional pointwise estimates. It does not look natural to put such restrictive pointwise conditions for studying the long time behaviour of L p -norms. A similar remark could apply to the conditions proposed in [START_REF] Weng | Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system[END_REF],

where the results of [START_REF] Brandolese | Large time decay and growth for a viscous Boussinesq system[END_REF] are extended to higher-order derivatives, but assuming the data in the Schwartz class.

In this paper we aim to give a simpler and shorter proof for (1.2) by assuming much weaker assumptions on the data and using the natural functional settings, inspired by Kato's classical work [START_REF] Kato | Strong L p -solutions of the Navier-Stokes equation in R m , with applications to weak solutions[END_REF] for the incompressible Navier Stokes system. The subtle part will be the proof of the lower bounds: the classical approach (see [START_REF] Fujigaki | Asymptotic profiles of nonstationary incompressible Navier-Stokes flows in the whole space[END_REF] and the references therein) of writing an asymptotic profile for u putting in evidence the leading term as t → ∞ is not applicable to our situation. Indeed, it turns out that, when θ 0 = 0, the linear and the nonlinear terms behave at the same rates in L p . The main idea will be to split the analysis into different regions of space-time: the size of the linear terms contributing to u will be more important than the size of the nonlinear terms in some of these regions, namely, in {(x, t) : |x| ≥ A √ t} provided A is large enough. A careful linear analysis will finally lead to the lower bound in (1.2).

The use of Duoandikoetxea and Zuazua decompositions for distributions, and of variants of Young-type estimates (useful for estimating convolution integrals outside balls of large radii) are two original features of this paper. The crucial technical step is carried in Proposition 3.3 and the main result is stated in Theorem 4.1.

The Boussinesq equations

The integral formulation of the Boussinesq system (1.1) reads:

(2.1)

             θ(t) = e t∆ θ 0 - t 0 e (t-s)∆ ∇ • (θu)(s) ds u(t) = e t∆ u 0 - t 0 e (t-s)∆ P∇ • (u ⊗ u)(s) ds + t 0 e (t-s)∆ Pθ(s)e 3 ds. ∇ • u 0 = 0
Here P denotes the projector on the space of divergence-free fields, which is also called Leray's projector. We will not discuss here the issue of the equivalence between the system (2.1) and the original system (1.1). One could see [10, Theorems 1.1-1.2] for a discussion of this issue in the particular case of incompressible Navier Stokes system. In all this paper the Boussinesq system will be treated in the integral form above.

Let us write the unknown as v = u θ . It is convenient to rewrite the above integral system in the following abstract form:

(2.2) v = a + B(v, v),
where B : E × E → E is a bilinear operator in a suitable Banach space E, and a ∈ E is given in terms of the initial data. Here E = X × Y where X denotes the Banach space of the velocity and Y the Banach space of the temperature. We define X to be the space of all C([0, ∞), L 3 ) divergence-free vector fields u such that u X < ∞, and Y the space of all C([0, ∞), L 1 ) functions such that θ Y < ∞. Here,

u X ≡ sup t>0 u(t) 3 + sup t>0 √ t u(t) ∞ , (2.3) θ Y ≡ sup t>0 θ(t) 1 + sup t>0 t 3/2 θ(t) ∞ . (2.4)
To write (2.1) in the form (2.2), replace the equation of θ inside t 0 e (t-s)∆ Pθ(s)e 3 ds. A simple computation transforms (2.1) into the equivalent system (2.5)

             θ(t) = e t∆ θ 0 - t 0 e (t-s)∆ ∇ • (θu)(s) ds u(t) = e t∆ [u 0 + tPθ 0 e 3 ] - t 0 e (t-s)∆ (t -s)P∇ • (θu)(s) ds e 3 - t 0 e (t-s)∆ P∇ • (u ⊗ u)(s) ds ∇ • u 0 = 0
This system has clearly the form (2.2), with (2.6) a = e t∆ [u 0 + tPθ 0 e 3 ] e t∆ θ 0 and

B(v, ṽ) = -B 1 (u, ũ) + B 2 (u, θ) -B 3 (u, θ) .
Here, v = u θ , ṽ = ũ θ , and the three bilinear operators B 1 : X × X → X, next

B 2 : X × Y → X and B 3 : X × Y → Y , are defined by B 1 (u, ũ) = t 0 e (t-s)∆ P∇ • (u ⊗ ũ)(s) ds, (2.7a) B 2 (u, θ) = t 0 e (t-s)∆ (t -s)P∇ • (uθ)(s) ds e 3 , (2.7b) B 3 (u, θ) = t 0 e (t-s)∆ ∇ • (uθ)(s) ds. (2.7c)
We will make of use the following notations for the standard gaussian

G t (x) = e -|x| 2 /(4t) (4πt) 3/2 .
The starting point of our analysis will be the following theorem:

Theorem 2.1.

(1) There exist two constants > 0 and C > 0 such that if

θ 0 ∈ L 1 (R 3 ), u 0 ∈ L 3 (R 3 ), and
(2.8)

u 0 3 + θ 0 1 < ,
then there exist a unique solution (u, θ) ∈ X × Y of the integral system (2.1) such that

u X + θ Y ≤ C .
(2) If 1 < p < 3 and under the additional conditions u 0 ∈ L p and u 0 3 < p for some 0 < p ≤ , then the solution constructed in the previous item satisfies

(2.9) sup t>0 (1 + t) 1 2 (1-3 p ) u(t) p < ∞. If 3 ≤ p ≤ ∞, then sup t>0 t 1 2 (1- 3
p ) u(t) p < ∞ without any additional assumption on u 0 .

In this functional setting the proof becomes quite standard: the first conclusion of the theorem relies on the following estimates, implying the continuity of the bilinear operator B on the Banach space E.

Lemma 2.2. The following estimates hold for any u, ũ ∈ X and θ, θ ∈ Y :

(2.10)

B 1 (u, ũ) X ≤ C u X ũ X , B 2 (u, θ) X ≤ C u X θ Y , B 3 (u, θ) Y ≤ C u X θ Y .
In particular, for all v, ṽ ∈ E, the following estimate holds:

(2.11) B(v, ṽ) E ≤ C v E ṽ E .
In fact, we will establish a slightly more general version of Lemma 2.2. For this, let 1 < p ≤ 3 and let us define the following Banach space through the norm (2.12)

u Xp = u X + sup t>0 (1 + t) 1 2 (1-3 p ) u(t) p .
Notice that X 3 = X and for 1 < p ≤ 3 the space X p is continuously embedded in X. Next we set E p = X p × Y .

Lemma 2.3. The first two estimates in (2.10) can be generalized as follows:

B 1 (u, ũ) Xp ≤ C p u Xp ũ X , B 2 (u, θ) Xp ≤ C p u Xp θ Y , 1 < p ≤ 3. (2.13)
In particular, we have the estimate

(2.14) B(v, ṽ) Ep ≤ C p v Ep ṽ E , 1 < p ≤ 3.
Proof. The first of estimate (2.10) is due to Kato, see, for example, [START_REF] Meyer | Wavelets, paraproducts, and Navier-Stokes equations[END_REF]. The proof of the second and of the third estimates are very similar and they are left to the reader. The generalisation (2.13) of the above estimates is straightforward. The only thing that are needed to establish all these estimates are the standard Hölder and Young inequalities, and the well known fact that the kernel F (x, t) of the operator e t∆ P∇ satisfies

(2.15) F (x, t) = t -2 F ( x √ t , 1), and
F (•, 1) ∈ L 1 ∩ L ∞
and that the kernel K(x, t) of the operator e t∆ P satisfies

(2.16) K(x, t) = t -3/2 K( x √ t , 1), and
K(•, 1) ∈ 1<p≤∞ L p (R 3 ).
Indeed, we recall the well-known pointwise estimates for these kernels (see, e.g., [START_REF] Miyakawa | On space-time decay properties of nonstationary incompressible Navier-Stokes flows in R n[END_REF])

(2.17) |F (x, 1)| ≤ C(1 + |x|) -4 , and |K(x, 1)| ≤ C(1 + |x|) -3 .
We need also the corresponding linear estimates:

Lemma 2.4. Let u 0 ∈ L 3 (R 3 ) and θ 0 ∈ L 1 .
Let a be defined as in (2.6). Then, for an absolute constant c > 0,

(2.18) a E ≤ c( u 0 3 + θ 0 1 ).
Moreover, if 1 < p ≤ 3 and we have also

u 0 ∈ L p (R 3 ), then a ∈ E p .
The conclusion of Lemma 2.4 follows immediately from the usual properties of the heat kernel, and from the fact that P is a bounded operator in L p , for 1 < p < ∞.

Proof of Theorem 2.1. Estimates (2.11)-(2.18) and the standard fixed point Lemma (see, for example, [START_REF] Lemarié-Rieusset | Recent developments in the Navier-Stokes problem[END_REF]) imply Part 1 of the theorem. The solution is obtained as the limit in E of the sequence (v n ) recursively defined by v 0 = a and

v n+1 = a + B(v n , v n ), n ∈ N.
Take now 1 < p ≤ 3. Lemma 2.4, and estimate (2.14), imply that the sequence of approximating solutions v n remains bounded in E p , provided that u 0 ∈ L p (R 3 ), u 0 3 ≤ p and p is small enough. But the balls {v ∈ E p : v Ep ≤ R} are closed subsets of E (this last claim follows from Fatou's Lemma) and so the solution must belong to E p This establishes Part 2 of the theorem.

Analysis of the solution in the region |x| ≥ A √ t

The basic Young L p -L q convolution estimates read

(3.1) f * g p ≤ f r g q , 1 + 1 p = 1 r + 1 q , 1 ≤ p, q, r ≤ ∞.
Here we are interested in the following variant of (3.1), that provides more information about the behavior of f * g(x) as x → ∞ in the L p sense. Notice that estimate (3.2) below boils down (excepted for the unimportant coefficient 2) to (3.1) in the particular case R = 0.

Proposition 3.1. For R ≥ 0, let B R = {x ∈ R 3 : |x| < R} and B c
R its complementary in R 3 . Let also 1 ≤ p, q, r, r, q ≤ ∞, such that

1 + 1 p = 1 r + 1 q = 1 r + 1 q .
Then,

(3.2) f * g L p (B c R ) ≤ 2 f L r (B c R/2 ) g q + f r g L q (B c R/2 ) . Proof. We decompose f = f 1 B R/2 +f 1 B c R/2 and g = g1 B R/2 +g1 B c R/2
, where 1 A denotes the indicator function of the set A. So f * g is naturally written as the sum of four terms. As the support of (f

1 B R/2 ) * (g1 B R/2 ) is contained in B R , its L p (B c
R )-norm is zero. Applying the triangle inequality, next the classical Young inequality to the three other remaining terms,

f * g L p (B c R ) ≤ f L r (B c R/2 ) g L q (B R/2 ) + g L q (B c R/2 ) + f r g L q (B c R/2
) , which is even slightly stronger than (3.2). Lemma 3.2. There exist positive constants C 0 , η and A 0 such that, for all A ≥ A 0 ,

lim sup t→+∞ e t∆ θ 0 L 1 (B c A √ t ) ≤ C 0 e -ηA 2 θ 0 L 1 , lim sup t→+∞ e t∆ u 0 + t e t∆ Pθ 0 e 3 L 3 (B c A √ t ) ≤ C 0 A -2 u 0 3 + θ 0 1 and lim sup t→+∞ √ t e t∆ u 0 + t e t∆ Pθ 0 e 3 L ∞ (B c A √ t ) ≤ C 0 A -3 u 0 3 + θ 0 1 .
Proof. We claim that there exist two constants C, δ > 0 such that lim sup

t→+∞ e t∆ θ 0 L 1 (B c A √ t ) ≤ Ce -δA 2 θ 0 L 1 , lim sup t→+∞ e t∆ u 0 L 3 (B c A √ t ) ≤ Ce -δA 2 u 0 3 , and lim sup t→+∞ √ t e t∆ u 0 L ∞ (B c A √ t ) ≤ Ce -δA 2 u 0 3 . (3.3)
To see this, apply Proposition 3.1 with f = G t , g = θ 0 , f = G t , g = u 0 , and (p, r, r) = (1, 1, 1), (p, r, r) = (3, 1, 1), (p, r, r) = (∞, 3 2 , 3 2 ) respectively. Then our claim follows by the dominated convergence theorem.

Next, recalling the scaling and decay properties of the kernel K(•, t) of the operator e t∆ P, (2.16)-(2.17), we see that, for 1 < r ≤ ∞:

K(•, t) L r (B c A √ t ) ≤ C(A √ t) -3+3/r .
application of Proposition 3.1 with f = K(•, t), g = θ 0 and (p, r, r) = (p, p, p), by the dominated convergence theorem lim sup

t→+∞ t 1 2 (1-3/p) t e t∆ Pθ 0 e 3 L p (B c A √ t ) ≤ C θ 0 1 A -3+3/p .
Taking here p = 3 and p = ∞ yields the assertion of the Lemma.

Proposition 3.3. Let u 0 ∈ L 3 (R 3 ) be a divergence-free vector field, θ 0 ∈ L 1 (R n ), such that u 0 3 + θ 0 1 < .
If > 0 is small enough ( may need to be smaller than the constant in Theorem 2.1), then there exist two constants A 0 ≥ 1 and κ > 0 such that for all A ≥ A 0 the solution (u, θ) obtained in Theorem 2.1 satisfies

lim sup t→+∞ θ(t) L 1 (B c A √ t ) ≤ κA -1 , lim sup t→+∞ u(t) L 3 (B c A √ t ) ≤ κA -2
, and

lim sup t→+∞ √ t u(t) L ∞ (B c A √ t ) ≤ κA -3 . (3.4)
Proof. Recalling the properties (2.15) and (2.17) of the kernel F (x, t) of the operator e t∆ P∇, we obtain the two estimates:

(3.5a)

F (t -s) r ≤ C(t -s) -2+3/(2r) , and
F (t -s) L r (B c A √ t ) = (t -s) -2+3/(2r) |x|≥A √ t/ √ t-s |F (x, 1)| r dx 1/r
≤ CA -4+3/r t -2+3/(2r) .

(3.5b)

Consider the sequence of approximate solutions (u n , θ n ) (n ≥ 1), where u 1 = e t∆ u 0 + t e t∆ Pθ 0 e 3 , θ 1 = e t∆ θ 0 and (3.6)

u n+1 = u 1 -B 1 (u n , u n ) -B 2 (u n , θ n ), θ n+1 = θ 1 -B 3 (u n , θ n ).
By the usual fixed point argument the sequence, (u n , θ n ) converges to the solution (u, θ) obtained in Theorem 2.1 in the (X × Y )-norm. Moreover, there is an absolute constant C > 0 such that, for all n ≥ 1, sup n≥1

( u n X + θ n Y ) ≤ C ≡ ε,
and so, in particular, u X + θ Y ≤ ε.

We first need to prove that, for all n ≥ 1 and some constant κ n > 0, we have

(3.7a) lim sup t→+∞ θ n (t) L 1 (B c A √ t ) ≤ κ n A -1 , (3.7b) lim sup t→+∞ u n (t) L 3 (B c A √ t ) ≤ κ n A -2 and (3.7c) lim sup t→+∞ √ t u n (t) L ∞ (B c A √ t ) ≤ κ n A -3 .
Let us proceed by induction. For n = 1, (3.7a)-(3.7c) hold true for A 0 big enough because of Lemma 3.2. Assume now (3.7a)-(3.7c) hold at the step n and let us prove their validity at the step n + 1. Let t A ≥ 1 (with t A possibly depending also on n) such that for all

t ≥ t A , (3.8) θ n (t) L 1 (B c A √ t ) ≤ 2κ n A -1 , u n (t) L 3 (B c A √ t ) ≤ 2κ n A -2 and √ t u n (t) L ∞ (B c A √ t ) ≤ 2κ n A -3
We now write

B 3 (u n , θ n )(4t) = t A 0 + 4t t A F (4t -s) * (u n θ n )(s) ds, B 1 (u n , u n )(4t) = t A 0 + 4t t A F (4t -s) * (u n ⊗ u n )(s) ds and B 2 (u n , θ n )(4t) = t A 0 + 4t t A (4t -s)F (4t -s) * (u n θ n )(s) ds
The reason for considering the time 4t instead of t will be clear in (3.14) below. The kernel F (x, t) of the operator e t∆ ∇ has similar properties as the kernel F (x, t) and satisfies

(3.9) F (x, t) = t -2 F ( x √ t , 1), and F (•, 1) ∈ L 1 ∩ L ∞ Moreover, (3.10a) F (t -s) r ≤ C(t -s) -2+3/(2r) , and
F (t -s) L r (B c A √ t ) = (t -s) -2+3/(2r) |x|≥A √ t/ √ t-s | F (x, 1)| r dx 1/r ≤ CA -4+3/r t -2+3/(2r) .
(3.10b)

Only the three integrals 4t t A will play a role as t → ∞. Indeed, for t > t A ,

t A 0 F (4t -s) * (u n ⊗ u n )(s) 3 ≤ C t A 0 F (4t -s) 3/2 u n (s) 2 3 ds ≤ C(4t -t A ) -1 t A u n 2 X ≤ Cε 2 (4t -t A ) -1 t A . Moreover, √ 4t t A 0 F (4t -s) * (u n ⊗ u n )(s) ∞ ≤ C √ t t A 0 F (4t -s) 3 u n (s) 2 3 ds ≤ C √ t(4t -t A ) -3/2 t A u n 2 X ≤ Cε 2 √ t(t -t A ) -3/2 t A .
Taking the the limit as t → +∞ in the above expressions we get (3.11)

lim t→+∞ t A 0 F (4t-s) * (u n ⊗u n )(s) ds 3 = lim t→+∞ √ 4t t A 0 F (4t-s) * (u n ⊗u n )(s) ds ∞ = 0.
In a similar way (using now θ n (s) 1 ≤ ε and, respectively, u n (s) 3 ≤ ε or √ s u n (s) ∞ ≤ ε), we can prove that (3.12)

lim t→+∞ t A 0 (4t-s)F (4t-s) * (u n θ n )(s) ds 3 = lim t→+∞ √ 4t t A 0 (4t-s)F (4t-s) * (u n θ n )(s) ds ∞ = 0.
Furthermore,

t A 0 F (4t -s) * (u n θ n )(s) ds 1 ≤ t A 0 F (4t -s) 1 u n (s) ∞ θ n (s) 1 ds ≤ Cε 2 t 3/4 A 4t 0 (4t -s) -1/2 s -3/4 ds ≤ Cε 2 t 3/4 A t -1/4
Hence, (3.13) lim

t→+∞ t A 0 F (4t -s) * (u n θ n )(s) ds 1 = 0.
Applying Proposition 3.1 with p = 3, r = 3/2 and r = 1, we get from (3.5a)-(3.5b) and our inductive assumption

4t t A F (4t -s) * (u n ⊗ u n )(s) L 3 (B c A √ 4t ) ≤ C 4t t A A -2 t -1 u n (s) 3 u n (s) 3 + (4t -s) -1/2 u n (s) L 3 (B c A 2 √ 4t ) u n (s) ∞ ds ≤ CA -2 4t 0 t -1 ε 2 + κ n ε(4t -s) -1/2 s -1/2 ds ≤ CA -2 ε(ε + κ n ).
(3.14)

In the same way, applying now Proposition 3.1 with p = 3, r = 3 and r = 3,

4t t A (4t -s)F (4t -s) * (u n θ n )(s) L 3 (B c A √ 4t ) ≤ C 4t t A A -2 t -3/2 (4t -s) u n (s) 3 θ n (s) 3/2 + (4t -s) -1/2 u n (s) L 3 (B c A 2 √ 4t ) θ n (s) 3/2 ds ≤ CA -2 4t 1 t -3/2 (4t -s)s -1/2 ε 2 + κ n ε(4t -s) -1/2 s -1/2 ds ≤ CA -2 ε(ε + κ n ). (3.15)
Summarizing, recalling the relation between u n+1 and (u n , θ n ), we proved so far that

A 2 lim sup t→+∞ u n+1 (t) L 3 (B c A √ t ) = A 2 lim sup t→+∞ u n+1 (4t) L 3 (B c A √ 4t ) ≤ C 0 u 0 3 + θ 0 1 + 2Cε(ε + κ n ).
(3.16)

On the other hand, applying once more Proposition 3.1 with p = ∞, r = 3 and r = 6/5, and our inductive assumption, we get from (3.5a)-(3.5b),

√ 4t 4t t A F (4t -s) * (u n ⊗ u n )(s) L ∞ (B c A √ 4t ) ≤ C √ t 4t t A A -3 t -3/2 u n (s) 3 u n (s) 3 + (4t -s) -3/4 u n (s) L ∞ (B c A 2 √ 4t
) u n (s) 6 ds

≤ CA -3 √ t 4t 1 t -3/2 s -1/2 ε 2 + κ n ε(4t -s) -3/4 s -3/4 ds ≤ CA -3 ε(ε + κ n ).
(3.17)

Moreover, applying now Proposition 3.1 with p = ∞, r = ∞ and r = 6,

√ 4t 4t t A (4t -s)F (4t -s) * (u n θ n )(s) L ∞ (B c A √ 4t ) ≤ C √ t 4t t A A -4 t -2 (4t -s) u n (s) ∞ θ n (s) 1 + (4t -s) -3/4 u n (s) L ∞ (B c A 2 √ 4t
) θ n (s) 6/5 ds

≤ CA -3 √ t 4t 1 A -1 t -2 (4t -s)s -1/2 ε 2 + κ n ε(4t -s) -3/4 s -3/4 ds ≤ CA -3 ε(ε + κ n ). (3.18)
These estimates imply

A 3 lim sup t→+∞ √ t u n+1 (t) L ∞ (B c A √ t ) = A 3 lim sup t→+∞ √ 4t u n+1 (4t) L ∞ (B c A √ 4t ) ≤ C 0 u 0 3 + θ 0 1 + 2Cε(ε + κ n ). (3.19)
Moreover, applying Proposition 3.1 with p = 1, r = 1 and r = 1, we get from (3.10a)-(3.10b) and our inductive assumption

4t t A F (4t -s) * (u n θ n )(s) L 1 (B c A √ 4t ) ≤ C 4t t A A -1 t -1/2 u n (s) ∞ θ n (s) 1 + (4t -s) -1/2 θ n (s) L 1 (B c A 2 √ 4t ) u n (s) ∞ ds ≤ CA -1 4t 0 t -1/2 s -1/2 ε 2 + κ n ε(4t -s) -1/2 s -1/2 ds ≤ CA -1 ε(ε + κ n ).
(3.20)

Hence, we get

A lim sup t→+∞ θ n+1 (t) L 1 (B c A √ t ) = A lim sup t→+∞ θ n+1 (4t) L 1 (B c A √ 4t ) ≤ C 0 u 0 3 + θ 0 1 + 2Cε(ε + κ n ).
κ n+1 ≤ C 0 u 0 3 + θ 0 1 + 2Cε 2 + 2Cεκ n ,
where C > 0 is an absolute constant. Our smallness assumption on the initial data ensures 2Cε < 1, so that one gets κ ≡ sup n≥1 κ n < ∞. We thus deduce from (3.7a)-(3.7c) that lim sup

t→+∞ θ(t) L 1 (B c A √ t ) ≤ κA -1 lim sup t→+∞ u(t) L 3 (B c A √ t ) ≤ κA -2 and lim sup t→+∞ √ t u(t) L ∞ (B c A √ t ) ≤ κA -3 . (3.22) 4. Large time growth of L p -norm, 1 < p < 3 4.1.
Statement of the main result. The goal of this section is to establish lower bound estimates for u(t) p that precisely agree with the corresponding upper bounds obtained in Theorem 2.1 (see (2.9)). Namely, we establish the following. We put the additional conditions on the data |x| q min{1, |θ 0 (x)| q } dx < ∞ for some 1 ≤ q < 3/2 such that q ≤ p. When 1 < p < 3, we also assume u 0 ∈ L p (R 3 ). Then there exist two constants c 1 , c 2 > 0 and t 0 > 0 such that for all t ≥ t 0 , (4.1)

c 1 θ 0 t -1 2 (1-3/p) ≤ u(t) p ≤ c 2 t -1 2 (1-3/p) .
In particular, for 3 < p ≤ ∞, u(t) p → 0. If otherwise 1 < p < 3 and θ 0 = 0, then u(t) p → +∞.

The subtle point in establishing the lower bounds is that all the terms contributing to u(t), i.e., the three terms on the right hand side of the equality, (4.2)

u(t) = e t∆ [u 0 + tPθ 0 e 3 ] -B 1 (u, u) -B 2 (u, θ),
individually have the same behavior t -1 2 (1-3/p) as t → +∞ in the L p -norm. The strategy will consist in proving that these three terms cannot compensate each other. To achieve this, we will compute their L p -norm in regions {x : |x| ≥ A √ t}, finding a behavior of the form C j (A) t -1 2 (1-3/p) as t → +∞ for each of them (j = 1, 2, 3). But for large enough A, the constant C 1 (A) corresponding to the first term e t∆ [u 0 + tPθ 0 e 3 ] turns out to be much larger than the corresponding constants C 2 (A) and C 3 (A) of B 1 (u, u) and B 2 (u, θ) respectively. This implies that u(t) can be bounded from below in the L p -norm on {x : |x| ≥ A √ t}, and so in the L p (R n ) norm as announced by (4.1). The result of Theorem 4.1 is in agreement with that of [START_REF] Brandolese | Large time decay and growth for a viscous Boussinesq system[END_REF], where estimates of the form (4.1) were obtained under much more stringent conditions on the data (including pointwise decay assumptions on u 0 , θ 0 and somewhat artificial smallness smallness conditions on θ 0 and u 0 , etc.).

The condition

|x| q min{1, |θ 0 (x)| q } dx < ∞
in Theorem 4.1 could be seen as a weaker integral version of the pointwise estimates in [START_REF] Brandolese | Large time decay and growth for a viscous Boussinesq system[END_REF] on data. We do not know if we can remove it.

L p -analysis of the bilinear terms in

B c A √ t . Lemma 4.2. Let 1 < p ≤ ∞.
Let (u, θ) be the solution constructed in Proposition 3.3 and the second part of Theorem 2.1, under the usual smallness assumptions on u 0 3 and θ 0 1 . If 1 < p < 3, we require the additional condition u 0 ∈ L p (R n ). There is A 0 ≥ 1 and C > 0 such that, for all A ≥ A 0 ,

(4.3) lim sup t→+∞ t 1 2 (1-3 p ) B 1 (u, u)(t) L p (B c A √ t ) + B 2 (u, θ)(t) L p (B c A √ t ) ≤ CA -4+3/p . Proof. Let t A ≥ 1 such that (4.4) u(t) L 3 (B c A √ t ) ≤ 2κA -2 and √ t u(t) L ∞ (B c A √ t ) ≤ 2κA -3 , for all t ≥ t A .
By interpolation, we have for all t ≥ t A ,

∀r ≥ 3, u(t) L r (B c A √ t ) ≤ 2κA -3+ 3 r .
We have, by the application of Proposition 3.1, as t → +∞ (we assume 3 ≤ p < ∞ in the calculations below; if p = ∞ the result remains true with slight changes in the choice of the exponents in Proposition 3.1):

(4t)

1 2 (1-3 p ) B 1 (u, u)(4t) L p (B c A √ 4t ) ≤ Ct 1 2 (1-3 p ) 4t t A A -4+ 3 p t -2+ 3 2p u(s) p u(s) p + (4t -s) -1 2 u(s) L p (B c A 2 √ 4t ) u(s) L ∞ (B c A 2 √ 4t ) ds + o(1) ≤ Ct 1 2 (1-3 p ) 4t 0 A -4+3/p t -2+3/(2p) s 1/2 + (4t -s) -1/2 A -6+3/p s -1+3/(2p) ds + o(1) ≤ CA -4+3/p + o(1). (4.5)
The same holds for 1 < p < 3, (4t)

1 2 (1-3 p ) B 1 (u, u)(4t) L p (B c A √ 4t ) ≤ Ct 1 2 (1-3 p ) 4t t A A -4+ 3 p t -2+ 3 2p u(s) p u(s) p + (4t -s) -1 2 u(s) p u(s) L ∞ (B c A 2 √ 4t ) ds + o(1) ≤ Ct 1 2 (1-3 p ) 4t 0 A -4+3/p t -2+3/(2p) s 1/2 + (4t -s) -1/2 A -3 s -1+3/(2p) ds + o(1) ≤ CA -4+3/p + o(1). (4.6)
Here, and below, the o(1) arise from the contribution of the integral

t A 0 .
Similarly, as t → +∞, (4t)

1 2 (1-3 p ) B 2 (u, θ)(4t) L p (B c A √ 4t ) ≤ Ct 1 2 (1-3 p ) 4t t A A -4+ 3 p t -2+ 3 2p (4t -s) u(s) ∞ θ(s) 1 + (4t -s) -1+ 3 2p u(s) L ∞ (B c A 2 √ 4t ) θ(s) L 1 (B c A 2 √ 4t
) ds + o(1)

≤ Ct 1 2 (1-3 p ) 4t 0 A -4+3/p t -2+3/(2p) (4t -s)s -1/2 + (4t -s) -1+ 3 2p A -4 s -1/2 ds + o(1)
≤ CA -4+3/p + o(1).

(4.7)

Summing the two last estimates and taking the lim sup t→+∞ leads to (4.3).

L p -analysis of the linear term in

B c A √ t . Proposition 4.3. Let f ∈ L 1 (R n ). Then the function V (x) = -x 1 0 f x λ dλ λ n+1 belongs to L 1 loc (R n
) and the following identity holds in the distributional sense:

(4.8) f = f δ + divV,
where δ denotes the Dirac mass at zero. Remark 4.4. Proposition 4.3 should be compared with a result of J. Duoandikoetxea and E. Zuazua [START_REF] Duoandikoetxea | Moments, masses de Dirac et décompositions de fonctions[END_REF], where such decomposition of f was established assuming f ∈ L 1 (R n ) and |x|f ∈ L q (R n ) for some 1 ≤ q < n/(n -1). In this case, the authors observed the validity of the estimate (4.9)

V q ≤ C |x|f q , 1 ≤ q < n/(n -1),
where C is a constant depending only on q and n.

Estimate (4.9) readily follows applying Minkowski integral inequality to the L q -norm in the definition of V . The restriction 1 ≤ q < n/(n -1) is needed for the convergence of the L q -Bochner integral.

Proof. The fact that V ∈ L 1 loc (R n ) follows from an elementary calculation: for R ≥ 0,

(R n ), f - f δ, ϕ = f (x)[ϕ(x) -ϕ(0)] dx = f (x) 1 0 x • ∇ϕ(λx) dλ dx = 1 0 f (x/λ) dλ λ n+1 x • ∇ϕ(x) dx = divV, ϕ .
The decomposition (4.8) can be applied to θ 0 , and allows to obtain the following:

Lemma 4.5. If θ 0 ∈ L 1 (R 3
), if V is associated to θ 0 as in Proposition 4.3 and K(•, t) and F (•, t) denote respectively the kernels of e t∆ P and e t∆ Pdiv, then for all t > 0 the convolution integral F (•, t) * V (x) = F (x -y, t)V (y) dy is well defined for a.e. x ∈ R n and defines a locally integrable function. Moreover, for j = 1, 2, 3, (4.10) e t∆ Pθ 0 e 3 j = θ 0 K j,3 (•, t)

+ 3 h=1 F j,h,3 (•, t) * V h .
Proof. To establish that the convolution integral F (•, t) * V (x) defines a locally integrable function for all t > 0, by (2.15) and (2.17

(•, t) * θ 0 e 3 ] j = ( θ 0 )(K j,3 (•, t)) * δ + div[K j,3 (•, t)) * V ].
The conclusion of the lemma follows.

Lemma 4.6. Let 1 ≤ p ≤ ∞. Assume that θ 0 ∈ L 1 (R n ) and |x| q |θ 0 (x)| q dx < ∞ for some 1 ≤ q < 3/2, with q ≤ p. Let V be related to θ 0 as before. Then, for all A > 0 and all t > 0, we have

(4.11) lim t→∞ t 3 2 (1-1/p) F (•, t) * V p = 0, 1 ≤ p ≤ ∞.
Proof. Recalling the estimate |V | q ≤ |x| q |θ 0 | q , valid for 1 ≤ q < 3/2 (see (4.9)), one just needs to apply Young convolution inequality to deduce that, for 1

+ 1 p = 1 r + 1 q , F (•, t) * V p ≤ C F (t) r ≤ Ct -2+ 3 2 (1+ 1 p -1 q ) .
As 1 ≤ q < 3/2, this implies the assertion of the Lemma.

Lemma 4.7. Let 1 < p ≤ ∞, The kernel K(•, t) of the operator e t∆ P. There exist two positive constants c, A 0 > 0 such that for all A ≥ A 0 ,

t 3 2 (1-1/p) K(t) L p (B c A √ t
) ≥ cA -3+3/p . (4.12)

Proof. We know from [START_REF] Brandolese | Fine properties of self-similar solutions of the Navier-Stokes equations[END_REF] that the kernel K(•, t) can be decomposed as K(x, t) = K(x) + |x| -3 Ψ(x/ √ t), where the components of K(x) are homogeneous function of degree -3 (given by second-order derivatives of the fundamental solution of the Laplacian in R 3 ), and lim x→∞ Ψ(x) = 0 (in fact Ψ decays to zero exponentially fast).

K(t) L p (B c A √ t ) = B c A √ t K(x) + |x| -3 Ψ(x/ √ t) p dx 1/p ≥ t -3 2 (1-1/p) A -3+3/p |x|≥1 |K(x)| p dx 1/p - |x|≥1 |x| -3 Ψ(Ax) p dx 1/p ≥ c t -3 2 (1-1/p) A -3+3/p
where c > 0 if A is large enough, because of the decay of Ψ.

Lemma 4.8. Let 1 < p ≤ ∞ and u 0 ∈ L 3 (R 3 ). Assume also u 0 ∈ L p (R 3 ) if 1 < p < 3. Then lim t→+∞ t 1 2 (1-3/p) e t∆ u 0 p = 0.
Proof. If 1 < p < 3, the conclusion just follows from the inequality e t∆ u 0 p ≤ u 0 p . If 3 ≤ p ≤ ∞, this is well known: one approaches in the L 3 -norm, u 0 by a sequence of functions in L 1 ∩ L 3 , with L 3 norm not exceeding u 0 3 and next applies the usual L 3 -L p and L 1 -L p heat estimates. In fact, let ψ ∈ L 1 ∩ L p such that ψ -u 0 p ≤ ε. Hence,

t 1 2 -3 2p e t∆ u 0 p ≤ t 1 2 -3 2p ε + t -1 ψ L 1 ,
which proves the claimed result.

Proof of Theorem 4.1. By our assumptions, the data satisfy the additional conditions |x| q min{1, |θ 0 (x)| q } dx < ∞ for some 1 ≤ q < 3/2 such that q ≤ p. Moreover, when 1 < p < 3, u 0 ∈ L p (R 3 ). The solution (u, θ), is such that sup t>0 θ(t) 1 < ∞ and θ(t) ∞ → 0. In particular, after some time t q > 0, θ(t) q ≤ 1 for all t ≥ t q . By a time-translation argument we can consider (u(t q ), θ(t q )) as our new initial datum. Notice that θ 0 = θ(t q ) because the spatial mean of the temperature is preserved by the Boussinesq flow. So, without loss of generality, we can work under the seemingly stronger condition |x| q |θ 0 | q (x) dx < ∞. This observation will be useful later on, when we will apply Lemma 4.6, where such stronger condition was needed.

Let us apply Lemma 4. where the notations here are the same as before. Fix A > 0 large enough. Multiplying by t Here the o(1)-term (as t → +∞) includes the contributions of two terms F (t) * V and e t∆ u 0 , that can indeed be neglected accordingly to Lemma 4.6 and Lemma 4.8. Here we applied also Lemma 4.2 and Lemma 4.7. From the above inequalities we deduce that, for t > 0 large enough,

t 1 2 (1-3/p) u(t) L p ≥ c 4 θ 0 A -3+3/p .
The assertion of the theorem is now proved.

Remark 4.9. The result of the present paper seems to be quite specific of the fact that we set the problem in the whole space. In bounded domains exponential decays are expected due to Poincaré's inequality. More interesting is the case of other unbounded domains: in the half-space case with Dirichlet boundary conditions the energy does not grow large anymore, but it decays to zero. See [START_REF] Han | Large time decay properties of solutions to a viscous Boussinesq system in a half space[END_REF]. So the long time behavior of Boussinesq flows in the half-space and in the whole space are completely different. From the physical point of view, this different behavior seems to be related to the energy dissipation that occurs in the boundary layer. Indeed, experimental analyses of energy spectra of viscous flows are available and put in evidence that walls selectively damp out the higher frequencies: the low-frequency content increases and the high-frequency content decreases as the boundary layer is traversed from the freestream to the wall. See, e.g., [START_REF] Hernon | Enhanced energy dissipation rates in laminar boundary layers subjected to elevated levels of freestream turbulence[END_REF]. In the case of the exterior domain problem, the long time behavior of the energy is addressed in [START_REF] Han | Algebraic L 2 decay for weak solutions of a viscous Boussinesq system in exterior domains[END_REF]: therein, the decay of the energy is obtained under additional (non-generic) conditions on the temperature; in the absence of such conditions energy growth is expected, because there is not enough boundary to maintain the dissipation mechanisms, but this has not been rigorously established yet.

  16)-(3.21), we deduce that (3.7b)-(3.7c) hold true at the step n + 1 with a constant

Theorem 4 . 1 .

 41 Let 1 < p ≤ ∞. Let (u, θ) be the solution constructed in Theorem 2.1.

1 0

 1 |x| |f (x)| dλ dx + |x|≥R R/|x| 0 |x| |f (x)| dλ dx ≤ R |f (x)| dx.Moreover, for any test function ϕ ∈ D

( 1 +C |x|≤R 1 0( 1 +λ - 4

 1114 ) it is sufficient to prove that, for all R > 0, the integral |x|≤R (1 + |x -y|) -4 |V (y)| dy converges. Indeed, |x|≤R |x -y|) -4 |V (y)| dy dx ≤ |x -λy|) -4 |y| |θ 0 (y)| dλ dy dx |y| -3 |θ 0 (y)| dy dλ ≤ CR 3 2R/|y| 0 dλ |y| |θ 0 (y)| dy + CR 3 |y|≥2R λ≥2R/|y| λ -4 dλ |y| -3 |θ 0 (y)| dy ≤ CR 4 |θ 0 (y)| dy + C |y|≥2R |θ 0 (y)| dy < ∞ Owing to decomposition (4.8), [K

  [START_REF] Guo | On the suitable weak solutions for the Cauchy problem of the Boussinesq equations[END_REF] and write the velocity field asu(t) = e t∆ [u 0 + tPθ 0 e 3 ] -B 1 (u, u) -B 2 (u, θ) = θ 0 t K(t) + t F (t) * V + e t∆ u 0 -[B 1 (u, u) + B 2 (u, θ)],(4.13)

1 2 1 2)c 2 θ

 112 (1-3/p) , taking the L p (B c A √ t )-norm and applying the triangle inequality, leads tot (1-3/p) u(t) L p (B -B 1 (u, u)(t) + B 2 (u, θ)(t) L p (B c A √ t ) + o(1) ≥ c θ 0 A -3+3/p -CA -4+3/p + o(1)≥ 0 A -3+3/p + o(1).

Acknowledgements

The authors would like to thank the referees for their careful reading and useful suggestions, that have been incorporated in this revised version of the manuscript.

This work was performed within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Universit de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). L. Brandolese was also supported by project ANR-13-BS01-0003 -DYFICOLTI -DYnamique des Fluides, Couches Limites, Tourbillons et Interfaces. 2000