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Research ArticleBoltzmannrsquos Six-Moment One-Dimensional Nonlinear SystemEquations with the Maxwell-Auzhan Boundary Conditions
A Sakabekov1 and Y Auzhani2
1Research Laboratory ldquoApplied Modeling Oil and Gas Fieldsrdquo Kazakh-British Technical University 59 Tole Bi StreetAlmaty 050000 Kazakhstan2Al-Farabi Kazakh National University 71 Al-Farabi Avenue Almaty 050040 Kazakhstan
Correspondence should be addressed to A Sakabekov asakabekovkbtukz
Received 2 February 2016 Revised 1 June 2016 Accepted 5 June 2016
Academic Editor Peter G L Leach
Copyright copy 2016 A Sakabekov and Y AuzhaniThis is an open access article distributed under theCreative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited
We prove existence and uniqueness of the solution of the problem with initial and Maxwell-Auzhan boundary conditions fornonstationary nonlinear one-dimensional Boltzmannrsquos six-moment system equations in space of functions continuous in timeand summable in square by a spatial variable In order to obtain a priori estimation of the initial and boundary value problem fornonstationary nonlinear one-dimensional Boltzmannrsquos six-moment system equations we get the integral equality and then use thespherical representation of vector Then we obtain the initial value problem for Riccati equation We have managed to obtain aparticular solution of this equation in an explicit form
1 Introduction
In case of one-atom gas any macroscopic system duringprocess of its evolution to an equilibrium state passes 3 stagesinitial transition period described in terms of a full functionof system distribution the kinetic period by means of a one-partial distribution function and the hydrodynamic periodbymeans of five firstmoments of the distribution function Inkinetic regime the behavior of a rarefied gas in space of timeand velocity is described by Boltzmannrsquos equation It is knownfrom gas dynamics that in most of the encountered problemsthere is no need to use detailed microscopic gas descriptionwith help of the distribution function Therefore it is naturalto look for a less detailed description using macroscopichydrodynamic variables (density hydrodynamic velocitytemperature etc) As these variables are defined in terms ofmoments of the distribution function we are faced with theproblem of analyzing the various moments of Boltzmannrsquosequation
Note that Boltzmannrsquos moment equations are intermedi-ate between Boltzmann (kinetic theory) and hydrodynamiclevels of description of state of the rarefied gas and formclass of nonlinear partial differential equations Existence ofsuch class of equations was noticed by Grad [1 2] in 1949
He obtained the moment system by expanding the particledistribution function in Hermite polynomials near the localMaxwell distribution Grad used Cartesian coordinates ofvelocities and Gradrsquos moment system contained as coeffi-cients such unknown hydrodynamic characteristics like den-sity temperature average speed and so forth Formulation ofboundary conditions for Gradrsquos system is almost impossibleas the characteristic equations for various approximations ofGradrsquos hyperbolic system contain unknown parameters likedensity temperature and average speed However 13- and20-moment Grad equations are widely used in solving manyproblems of the kinetic theory of gases and plasma
Boltzmannrsquos equation is equivalent to an infinite systemof differential equations relative to the moments of theparticle distribution function in the complete system ofeigenfunctions of linearized operator As a rule we limit studyby finitemoment system of equations because solving infinitesystem of equations does not seem to be possible
Finite system ofmoment equations for a specific task witha certain degree of accuracy replaces Boltzmannrsquos equationIt is necessary also roughly to replace the boundary condi-tions for the particle distribution function by a number ofmacroscopic conditions for the moments that is there arisesthe problem of boundary conditions for a finite system of
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2016 Article ID 5834620 8 pageshttpdxdoiorg10115520165834620
2 Journal of Applied Mathematics
equations that approximate the microscopic boundary con-ditions for Boltzmannrsquos equation The question of boundaryconditions for a finite system of moment equations canbe divided into two parts how many conditions must beimposed and how they should be prepared From micro-scopic boundary conditions for Boltzmannrsquos equation therecan be obtained an infinite set of boundary conditionsfor each type of decomposition However the number ofboundary conditions is determined not by the number ofmoment equations that is it is impossible for example totake as many boundary conditions as equations although thenumber of moment equations affects the number of bound-ary conditions In addition the boundary conditions mustbe consistent with the moment equations and the resultingproblemmust be correct Boundary condition problems arisein the following tasks (1) moment boundary conditions inrarefied gas slip-flow problems (2) definition of boundaryconditions on the surfaces of streamlined rarefied gas (3)prediction aerospace aerodynamic characteristics of aircraftat very high speeds and at high altitudes and so forth
In work [3] we have obtained the moment system whichdiffers from Gradrsquos system of equations We used sphericalvelocity coordinates and decomposed the distribution func-tion into a series of eigenfunctions of the linearized collisionoperator [4 5] which is the product of Sonine polynomialsand spherical functions The resulting system of equationswhich correspond to the partial sum of series and whichwe called Boltzmannrsquos moment system of equations is anonlinear hyperbolic system in relation to the moments ofthe particles distribution function
The structure of Boltzmannrsquosmoment systemof equationscorresponds to the structure of Boltzmannrsquos equation namelythe differential part of the resulting system is linear in relationto the moments of the distribution function and nonlinearityis included as moments of collision integral [6]
The linearity of differential part of Boltzmannrsquos momentsystem of equations simplifies the task of formulation of theboundary conditions In work [3] a homogeneous boundarycondition for particles distribution function was approxi-mated and proved the correctness of initial and bound-ary value problem for nonlinear nonstationary Boltzmannrsquosmoment system of equations in three-dimensional spaceIn work [7] the initial and boundary value problem forone-dimensional nonstationary Boltzmannrsquos equation withboundary conditions of Maxwell was approximated by acorresponding problem for Boltzmannrsquos moment system ofequationsThe boundary conditions for Boltzmannrsquosmomentsystem of equations were calledMaxwell-Auzhan conditions
In work [8] a systematic nonperturbative derivation ofa hierarchy of closed systems of moment equations corre-sponding to any classical theory has been presented Thispaper is a fundamental work where closed systems ofmoment equations describe a transition regime Moreoverhydrodynamical model is used to describe charge trans-port in a generic compound semiconductor Compoundsemiconductors have found wide use in the microelectronicindustryThe evolution equation for macroscopic variables isobtained by taking moments of the transport equation [9ndash13]
The study of various problems for Boltzmannrsquos momentsystem of equations is an important and actual task in thetheory of a rarefied gas and other applications of the momentsystem equations The correctness of initial and boundaryproblems for Boltzmannrsquos moment system of equations withMaxwell-Auzhan boundary conditions is being studied forthe first time
2 Existence and Uniqueness of the Solutionsof Initial and Boundary Value Problem forSix-Moment One-Dimensional BoltzmannrsquosSystem of Equations with Maxwell-AuzhanBoundary Conditions
In this section we prove the existence and uniqueness ofsolutions of the initial and boundary value problem forsix-moment one-dimensional Boltzmannrsquos system of equa-tions with Maxwell-Auzhan boundary conditions in space offunctions continuous in time and summable in square byspatial variable Note that the theorem of the existence ofa global solution in time of the initial and boundary valueproblem for 3-dimensional nonlinear Boltzmannrsquos equationwith boundary conditions of Maxwell is proved in work [14]
Wewrite in an expanded form systemof one-dimensionalBoltzmannrsquos moment equations in the kth approximationwhich corresponds to decomposition of the particle distri-bution function by eigenfunctions of the linearized collisionoperator
120597119891119899119897
120597119905+
1
120572
120597
120597119909[119897(radic
2 (119899 + 119897 + 12)
(2119897 minus 1) (2119897 + 1)119891119899119897minus1
minus radic2 (119899 + 1)
(2119897 minus 1) (2119897 + 1)119891119899+1119897minus1
where ⟨1198733119871311989931198973 119897 | 1198991119897111989921198972 119897⟩ are generalized coefficients
of Talmi (11989710119897201198970) are Klebsh-Gordon coefficients 120572 =
1radic119877120579 is the constant 119877 is Boltzmannrsquos constant and 120579 is theideal gas temperature
Journal of Applied Mathematics 3
For generalized coefficients of Talmi there exists a table[15] for each value of quantum number 120585 = 2119899 + 119897 from 0to 6 Moreover there is a program on IBM for calculation ofgeneralized coefficients of Talmi
If in (1) 2119899+ 119897 takes values from 0 to 3 we get Boltzmannrsquosmoment system equations in the third approximation Wewrite it in an expanded form
12059711989100
120597119905+
1
120572
12059711989101
120597119909= 0
12059711989102
120597119905+
1
120572
120597
120597119909(
2
radic311989101
+3
radic511989103
minus2radic2
radic1511989111) = 11986902
12059711989110
120597119905+
1
120572
120597
120597119909(minusradic
2
311989101
+ radic5
311989111) = 0
12059711989101
120597119905+
1
120572
120597
120597119909(11989100
+2
radic311989102
minus radic2
311989110) = 0
12059711989103
120597119905+
1
120572
120597
120597119909
3
radic511989102
= 11986903
12059711989111
120597119905+
1
120572
120597
120597119909(minus
2radic2
radic1511989102
+ radic5
311989110) = 11986911
119909 isin (minus119886 119886) 119905 gt 0
(3)
where 11989100
= 11989100(119905 119909) and 119891
01= 11989101(119905 119909) 119891
11= 11989111(119905 119909)
are the moments of the particle distribution function 11986902
=
(1205902minus 1205900)(1198910011989102
minus 1198912
01radic3)2 119869
03= (14)(120590
3+ 31205901minus
41205900)1198910011989103
+ (14radic5)(21205901+ 1205900minus 31205903)1198910111989102 and 119869
11=
(1205901minus 1205900)(1198910011989111
+ (12)radic531198911011989101
minus (radic2radic15)1198910111989102) are
moments of the collision integral where 1205900 1205901 1205902 and 120590
3
are the Fourier coefficients of the cross section expansionby the Legendre polynomials The first third and fourthequations of system (3) correspond to mass conservation lawmomentum conservation law and energy conservation lawrespectively
We study the correctness of initial and boundary valueproblem for six-moment one-dimensional Boltzmannrsquos sys-tem equations
1198601015840 is the transpose matrix 119861 is the positive definite
matrix 1199060(119909) = (119891
0
00(119909) 119891
0
02(119909) 119891
0
10(119909))1015840 and 119908
0(119909) =
(1198910
01(119909) 119891
0
03(119909) 119891
0
11(119909))1015840 are given initial vector-functions
119908+
119906+ are vector moments of falling to the boundary particle
distribution function119908minus 119906minusare vectormoments of reflectingfrom the boundary particle distribution function Equation(4) is vector matrix form of system equations (3)
It is possible to check through direct calculations that
det1198601= det(
0 119860
1198601015840
0) = 0 (9)
and matrix 1198601has three positive and the same number of
negative nonzero eigenvalues From (6)-(7) it follows that thenumber of boundary conditions on the left and right ends ofinterval (minus119886 119886) is equal to the number of positive andnegativeeigenvalues of the matrix 119860
1
Thus system (4) is a symmetric hyperbolic nonlinearpartial differential equations system Let us show that 119869
02
4 Journal of Applied Mathematics
is a sign-nondefined square form It is easy to check that1198910011989102
minus (1198912
01radic3) = (119862119880119880) where 119880 = (119906 119908)
1015840 and
119862 =
((((((((
(
01
20 0 0 0
1
20 0 0 0 0
0 0 0 0 0 0
0 0 0 minus1
radic30 0
0 0 0 0 0 0
0 0 0 0 0 0
))))))))
)
(10)
Eigenvalues of matrix 119862 are minus12 minus1radic3 0 0 0 and 12Therefore 119869
02is a sign-nondefined square form Similarly we
can show that 11986903
and 11986911
are also sign-nondefined squareforms
For problem (4)ndash(7) the following theorem takes place
Theorem 1 If 1198800= (1199060(119909) 119908
0(119909)) isin 119871
2
[minus119886 119886] then problem(4)ndash(7) has a unique solution in domain [minus119886 119886] times [0 119879]
belonging to the space 119862([0 119879] 1198712[minus119886 119886]) moreover
10038171003817100381710038171198712[minus119886119886] minus 1199031(0)) (11)
where 1198621is a constant independent from 119880 and 119879 sim
M(11988001198712[minus119886119886]
minus 1199031(0))minus1
) 1199031(119905) is the particular solution of
Riccati equation (15) in an explicit form
Proof Let 1198800isin 1198712
[minus119886 119886] Let us prove estimation (11) Wemultiply the first equation in system (4) by 119906 and the secondequation by 119908 and perform integration from minus119886 to 119886
where 1198651= ((1 minus 120573)120578120572120573radic120587)119865
Let us use the spherical representation [17] of vector119880(119905 119909) = 119903(119905)120596(119905 119909) where 120596(119905 119909) = (120596
10038171003817100381710038171198712[minus119886119886] minus 1199031(0))
(43)
where 1198622is constant and independent from119898 and 119903
1198981(119905) is a
particular solution of Riccati equation about 119903119898(119905)
Problem (40)-(41) represents a Cauchy problem for theordinary system of differential equations Existence of thesolution of problem (40)-(41) follows from theory of ordi-nary system of differential equations (Picardrsquos existence anduniqueness theorem) [20]Hence the existence of the solutionof problem (34)ndash(37) follows
Thus according to estimation (43) the sequence 119880119898
minus
1199031198981
of approximate solutions of problem (5)ndash(7) is uniformlybounded in function space119862([0 119879] 1198712[minus119886 119886])Moreover thehomogeneous system of equations 120591119864+ (1120572)119860120585with respectto 120591 120585 has only a trivial solution Then it follows from resultsin [21] that 119880
119898minus 1199031198981
rarr 119880minus 1199031is week in 119862([0 119879] 119871
2
[minus119886 119886])
and 119869(119880119898) rarr 119869(119880) is week in 119862([0 119879] 119871
2
[minus119886 119886]) as 119898 rarr
infin Further it can be shown with standard methods that limitelement is a weak solution of problem (5)ndash(7)
The uniqueness of solution of problem (5)ndash(7) is provedby contradiction Let problem (5)ndash(7) have two different solu-tions 119906
1 1199081and 119906
2 1199082 We denote them again by 119906 = 119906
1minus 1199062
Journal of Applied Mathematics 7
and119908 = 1199081minus1199082 Then with respect to new values of 119906 and119908
we obtain the following problem
120597119906
120597119905+ 119860
120597119908
120597119909= 1198691(1199061 1199081) minus 1198691(1199062 1199082)
120597119908
120597119905+ 1198601015840120597119906
120597119909= 1198692(1199061 1199081) minus 1198692(1199062 1199082)
119909 isin (minus119886 119886)
(44)
119906|119905=0
= 0
119908|119905=0
= 0
(45)
(119860119908minus
+ 119861119906minus
)1003816100381610038161003816119909=minus119886 =
1
120573(119860119908+
minus 119861119906+
)1003816100381610038161003816119909=minus119886
(119860119908minus
minus 119861119906minus
)1003816100381610038161003816119909=119886 =
1
120573(119860119908+
+ 119861119906+
)1003816100381610038161003816119909=119886
(46)
We prove that solution of problem (44)ndash(46) is trivialHence the uniqueness of the solution of problem (5)ndash(7)follows
Using the method which was mentioned before we get(see equality (14))
Once again we use spherical representation of 119906 = 119903(119905)1199081(119905
119909) and 119908 = 119903(119905)1199082(119905 119909) where 119903(119905) = 119880(119905 sdot)
1198712[minus119886119886]
Thenconcerning 119903(119905) we obtain new initial value problem
119889119903
119889119905+ 119903119875 (119905) = 119903
2
1198761(119905) (49)
119903 (0) = 0 (50)
where 119875(119905) has the same value as in (15) and
1198761(119905) =
1205902minus 1205900
2int
119886
minus119886
[(120596001
minus 120596002
) 120596021
+ 120596002
(120596021
minus 120596022
) minus1
radic3(120596011
minus 120596012
) (120596011
+ 120596012
)] (120596021
minus 120596022
) 119889119909 +1
4(1205903+ 31205901
minus 41205900) int
119886
minus119886
[(120596001
minus 120596002
) 120596031
+ +120596002
(120596031
minus 120596032
)] (120596031
minus 120596032
) 119889119909 +1
4radic5(21205901+ 1205900minus 31205903)
sdot int
119886
minus119886
[(120596011
minus 120596012
) 120596021
+ 120596012
(120596021
minus 120596022
)]
sdot (120596031
minus 120596032
) 119889119909 + (1205901minus 1205900) int
119886
minus119886
[(120596001
minus 120596002
) 120596111
+ 120596002
(120596111
minus 120596112
) +1
2
8 Journal of Applied Mathematics
sdot radic5
2[(120596101
minus 120596102
) 120596011
+ 120596102
(120596011
minus 120596012
)]
minus radic2
15[(120596011
minus 120596012
) 120596021
+ 120596012
(120596021
minus 120596022
)]
sdot (120596111
minus 120596112
)] 119889119909
(51)
The general solution of (49) is
119903 (119905) = exp(minusint
119905
0
119875 (120591) 119889120591)
sdot [119862 minus int
119905
0
1198761(120591) exp(minusint
120591
0
119875 (120585) 119889120585) 119889120591]
minus1
(52)
Solution of (49) which is satisfying homogeneous condition(50) is trivial that is 119903(119905) = 0 Hence 119880
119862([0119879]1198712[minus119886119886])
= 0
and 1199061= 1199062and 119908
1= 1199082
The theorem is proved
Competing Interests
The authors declare that they have no competing interests
References
[1] H Grad ldquoOn the kinetic theory of rarefied gasesrdquo Communica-tions on Pure andAppliedMathematics vol 2 no 4 pp 331ndash4071949
[2] H Grad ldquoPrinciple of the kinetic theory of gasesrdquo in Thermo-dynamics of Gases vol 12 ofHandbuch der Physik pp 205ndash294Springer Berlin Germany 1958
[3] A Sakabekov Initial-Boundary Value Problems for the Boltz-mannrsquos Moment System Equations Gylym Almaty 2002
[4] C Cercignani Theory and Application of the Boltzmann Equa-tion Instituto di Matematica Milano Italy 1975
[5] M N KoganDynamic of Rarefied Gas NaukaMoscow Russia1967
[6] K Kumar ldquoPolynomial expansions in kinetic theory of gasesrdquoAnnals of Physics vol 37 no 1 pp 113ndash141 1966
[7] A Sakabekov and Y Auzhani ldquoBoundary conditions for theonedimensional nonlinear nonstationary Boltzmannrsquos momentsystem equationsrdquo Journal of Mathematical Physics vol 55Article ID 123507 2014
[8] C D Levermore ldquoMoment closure hierarchies for kinetictheoriesrdquo Journal of Statistical Physics vol 83 no 5-6 pp 1021ndash1065 1996
[9] G Mascali and V Romano ldquoA hydrodynamical model for holesin silicon semiconductors the case of non-parabolic warpedbandsrdquo Mathematical and Computer Modelling vol 53 no 1-2pp 213ndash229 2011
[10] G Mascali and V Romano ldquoA non parabolic hydrodynamicalsubband model for semiconductors based on the maximumentropy principlerdquoMathematical and Computer Modelling vol55 no 3-4 pp 1003ndash1020 2012
[11] V D Camiola and V Romano ldquo2DEG-3DEG charge transportmodel for MOSFET based on the maximum entropy principlerdquo
SIAM Journal on Applied Mathematics vol 73 no 4 pp 1439ndash1459 2013
[12] G Alı G Mascali V Romano and R C Torcasio ldquoA hydrody-namical model for covalent semiconductors with a generalizedenergy dispersion relationrdquo European Journal of Applied Math-ematics vol 25 no 2 pp 255ndash276 2014
[13] V D Camiola and V Romano ldquoHydrodynamical model forcharge transport in graphenerdquo Journal of Statistical Physics vol157 no 6 pp 1114ndash1137 2014
[14] S Mischler ldquoKinetic equations with Maxwell boundary condi-tionsrdquo Annales Scientifiques de lrsquoEcole Normale Superieure vol43 no 5 pp 719ndash760 2010
[15] V G Neudachin and U F SmirnovNucleon Association of EasyKernel Nauka Moscow Russia 1969
[16] MMoshinskyTheHarmonic Oscillator inModern Physics fromAtoms to Quarks 1960
[17] S I Pokhozhaev ldquoOn an approach to nonlinear equationrdquoDoklady Akademii Nauk SSSR vol 247 pp 1327ndash1331 1979
[18] E Kamke Differentialgleichungen Losungsmethoden und Losu-ngen I Gewohuliche Differentialgleichungen BGTeubnerLeipzig Germany 1977
[19] A Tungatarov and D K Akhmed-Zaki ldquoCauchy problemfor one class of ordinary differential equationsrdquo InternationalJournal of Mathematical Analysis vol 6 no 14 pp 695ndash6992012
[20] M Tenenbaum andH PollardOrdinary Differential EquationsHarper and Row New York NY USA 1963
[21] L Tartar ldquoCompensated compactness and applications topartial differential equationsrdquo in Proceedings of the Non-LinearAnalysis and Mechanics Heriot-Watt Symposium R J KnopsEd vol 4 of Research Notes in Math pp 136ndash212 1979
equations that approximate the microscopic boundary con-ditions for Boltzmannrsquos equation The question of boundaryconditions for a finite system of moment equations canbe divided into two parts how many conditions must beimposed and how they should be prepared From micro-scopic boundary conditions for Boltzmannrsquos equation therecan be obtained an infinite set of boundary conditionsfor each type of decomposition However the number ofboundary conditions is determined not by the number ofmoment equations that is it is impossible for example totake as many boundary conditions as equations although thenumber of moment equations affects the number of bound-ary conditions In addition the boundary conditions mustbe consistent with the moment equations and the resultingproblemmust be correct Boundary condition problems arisein the following tasks (1) moment boundary conditions inrarefied gas slip-flow problems (2) definition of boundaryconditions on the surfaces of streamlined rarefied gas (3)prediction aerospace aerodynamic characteristics of aircraftat very high speeds and at high altitudes and so forth
In work [3] we have obtained the moment system whichdiffers from Gradrsquos system of equations We used sphericalvelocity coordinates and decomposed the distribution func-tion into a series of eigenfunctions of the linearized collisionoperator [4 5] which is the product of Sonine polynomialsand spherical functions The resulting system of equationswhich correspond to the partial sum of series and whichwe called Boltzmannrsquos moment system of equations is anonlinear hyperbolic system in relation to the moments ofthe particles distribution function
The structure of Boltzmannrsquosmoment systemof equationscorresponds to the structure of Boltzmannrsquos equation namelythe differential part of the resulting system is linear in relationto the moments of the distribution function and nonlinearityis included as moments of collision integral [6]
The linearity of differential part of Boltzmannrsquos momentsystem of equations simplifies the task of formulation of theboundary conditions In work [3] a homogeneous boundarycondition for particles distribution function was approxi-mated and proved the correctness of initial and bound-ary value problem for nonlinear nonstationary Boltzmannrsquosmoment system of equations in three-dimensional spaceIn work [7] the initial and boundary value problem forone-dimensional nonstationary Boltzmannrsquos equation withboundary conditions of Maxwell was approximated by acorresponding problem for Boltzmannrsquos moment system ofequationsThe boundary conditions for Boltzmannrsquosmomentsystem of equations were calledMaxwell-Auzhan conditions
In work [8] a systematic nonperturbative derivation ofa hierarchy of closed systems of moment equations corre-sponding to any classical theory has been presented Thispaper is a fundamental work where closed systems ofmoment equations describe a transition regime Moreoverhydrodynamical model is used to describe charge trans-port in a generic compound semiconductor Compoundsemiconductors have found wide use in the microelectronicindustryThe evolution equation for macroscopic variables isobtained by taking moments of the transport equation [9ndash13]
The study of various problems for Boltzmannrsquos momentsystem of equations is an important and actual task in thetheory of a rarefied gas and other applications of the momentsystem equations The correctness of initial and boundaryproblems for Boltzmannrsquos moment system of equations withMaxwell-Auzhan boundary conditions is being studied forthe first time
2 Existence and Uniqueness of the Solutionsof Initial and Boundary Value Problem forSix-Moment One-Dimensional BoltzmannrsquosSystem of Equations with Maxwell-AuzhanBoundary Conditions
In this section we prove the existence and uniqueness ofsolutions of the initial and boundary value problem forsix-moment one-dimensional Boltzmannrsquos system of equa-tions with Maxwell-Auzhan boundary conditions in space offunctions continuous in time and summable in square byspatial variable Note that the theorem of the existence ofa global solution in time of the initial and boundary valueproblem for 3-dimensional nonlinear Boltzmannrsquos equationwith boundary conditions of Maxwell is proved in work [14]
Wewrite in an expanded form systemof one-dimensionalBoltzmannrsquos moment equations in the kth approximationwhich corresponds to decomposition of the particle distri-bution function by eigenfunctions of the linearized collisionoperator
120597119891119899119897
120597119905+
1
120572
120597
120597119909[119897(radic
2 (119899 + 119897 + 12)
(2119897 minus 1) (2119897 + 1)119891119899119897minus1
minus radic2 (119899 + 1)
(2119897 minus 1) (2119897 + 1)119891119899+1119897minus1
where ⟨1198733119871311989931198973 119897 | 1198991119897111989921198972 119897⟩ are generalized coefficients
of Talmi (11989710119897201198970) are Klebsh-Gordon coefficients 120572 =
1radic119877120579 is the constant 119877 is Boltzmannrsquos constant and 120579 is theideal gas temperature
Journal of Applied Mathematics 3
For generalized coefficients of Talmi there exists a table[15] for each value of quantum number 120585 = 2119899 + 119897 from 0to 6 Moreover there is a program on IBM for calculation ofgeneralized coefficients of Talmi
If in (1) 2119899+ 119897 takes values from 0 to 3 we get Boltzmannrsquosmoment system equations in the third approximation Wewrite it in an expanded form
12059711989100
120597119905+
1
120572
12059711989101
120597119909= 0
12059711989102
120597119905+
1
120572
120597
120597119909(
2
radic311989101
+3
radic511989103
minus2radic2
radic1511989111) = 11986902
12059711989110
120597119905+
1
120572
120597
120597119909(minusradic
2
311989101
+ radic5
311989111) = 0
12059711989101
120597119905+
1
120572
120597
120597119909(11989100
+2
radic311989102
minus radic2
311989110) = 0
12059711989103
120597119905+
1
120572
120597
120597119909
3
radic511989102
= 11986903
12059711989111
120597119905+
1
120572
120597
120597119909(minus
2radic2
radic1511989102
+ radic5
311989110) = 11986911
119909 isin (minus119886 119886) 119905 gt 0
(3)
where 11989100
= 11989100(119905 119909) and 119891
01= 11989101(119905 119909) 119891
11= 11989111(119905 119909)
are the moments of the particle distribution function 11986902
=
(1205902minus 1205900)(1198910011989102
minus 1198912
01radic3)2 119869
03= (14)(120590
3+ 31205901minus
41205900)1198910011989103
+ (14radic5)(21205901+ 1205900minus 31205903)1198910111989102 and 119869
11=
(1205901minus 1205900)(1198910011989111
+ (12)radic531198911011989101
minus (radic2radic15)1198910111989102) are
moments of the collision integral where 1205900 1205901 1205902 and 120590
3
are the Fourier coefficients of the cross section expansionby the Legendre polynomials The first third and fourthequations of system (3) correspond to mass conservation lawmomentum conservation law and energy conservation lawrespectively
We study the correctness of initial and boundary valueproblem for six-moment one-dimensional Boltzmannrsquos sys-tem equations
1198601015840 is the transpose matrix 119861 is the positive definite
matrix 1199060(119909) = (119891
0
00(119909) 119891
0
02(119909) 119891
0
10(119909))1015840 and 119908
0(119909) =
(1198910
01(119909) 119891
0
03(119909) 119891
0
11(119909))1015840 are given initial vector-functions
119908+
119906+ are vector moments of falling to the boundary particle
distribution function119908minus 119906minusare vectormoments of reflectingfrom the boundary particle distribution function Equation(4) is vector matrix form of system equations (3)
It is possible to check through direct calculations that
det1198601= det(
0 119860
1198601015840
0) = 0 (9)
and matrix 1198601has three positive and the same number of
negative nonzero eigenvalues From (6)-(7) it follows that thenumber of boundary conditions on the left and right ends ofinterval (minus119886 119886) is equal to the number of positive andnegativeeigenvalues of the matrix 119860
1
Thus system (4) is a symmetric hyperbolic nonlinearpartial differential equations system Let us show that 119869
02
4 Journal of Applied Mathematics
is a sign-nondefined square form It is easy to check that1198910011989102
minus (1198912
01radic3) = (119862119880119880) where 119880 = (119906 119908)
1015840 and
119862 =
((((((((
(
01
20 0 0 0
1
20 0 0 0 0
0 0 0 0 0 0
0 0 0 minus1
radic30 0
0 0 0 0 0 0
0 0 0 0 0 0
))))))))
)
(10)
Eigenvalues of matrix 119862 are minus12 minus1radic3 0 0 0 and 12Therefore 119869
02is a sign-nondefined square form Similarly we
can show that 11986903
and 11986911
are also sign-nondefined squareforms
For problem (4)ndash(7) the following theorem takes place
Theorem 1 If 1198800= (1199060(119909) 119908
0(119909)) isin 119871
2
[minus119886 119886] then problem(4)ndash(7) has a unique solution in domain [minus119886 119886] times [0 119879]
belonging to the space 119862([0 119879] 1198712[minus119886 119886]) moreover
10038171003817100381710038171198712[minus119886119886] minus 1199031(0)) (11)
where 1198621is a constant independent from 119880 and 119879 sim
M(11988001198712[minus119886119886]
minus 1199031(0))minus1
) 1199031(119905) is the particular solution of
Riccati equation (15) in an explicit form
Proof Let 1198800isin 1198712
[minus119886 119886] Let us prove estimation (11) Wemultiply the first equation in system (4) by 119906 and the secondequation by 119908 and perform integration from minus119886 to 119886
where 1198651= ((1 minus 120573)120578120572120573radic120587)119865
Let us use the spherical representation [17] of vector119880(119905 119909) = 119903(119905)120596(119905 119909) where 120596(119905 119909) = (120596
10038171003817100381710038171198712[minus119886119886] minus 1199031(0))
(43)
where 1198622is constant and independent from119898 and 119903
1198981(119905) is a
particular solution of Riccati equation about 119903119898(119905)
Problem (40)-(41) represents a Cauchy problem for theordinary system of differential equations Existence of thesolution of problem (40)-(41) follows from theory of ordi-nary system of differential equations (Picardrsquos existence anduniqueness theorem) [20]Hence the existence of the solutionof problem (34)ndash(37) follows
Thus according to estimation (43) the sequence 119880119898
minus
1199031198981
of approximate solutions of problem (5)ndash(7) is uniformlybounded in function space119862([0 119879] 1198712[minus119886 119886])Moreover thehomogeneous system of equations 120591119864+ (1120572)119860120585with respectto 120591 120585 has only a trivial solution Then it follows from resultsin [21] that 119880
119898minus 1199031198981
rarr 119880minus 1199031is week in 119862([0 119879] 119871
2
[minus119886 119886])
and 119869(119880119898) rarr 119869(119880) is week in 119862([0 119879] 119871
2
[minus119886 119886]) as 119898 rarr
infin Further it can be shown with standard methods that limitelement is a weak solution of problem (5)ndash(7)
The uniqueness of solution of problem (5)ndash(7) is provedby contradiction Let problem (5)ndash(7) have two different solu-tions 119906
1 1199081and 119906
2 1199082 We denote them again by 119906 = 119906
1minus 1199062
Journal of Applied Mathematics 7
and119908 = 1199081minus1199082 Then with respect to new values of 119906 and119908
we obtain the following problem
120597119906
120597119905+ 119860
120597119908
120597119909= 1198691(1199061 1199081) minus 1198691(1199062 1199082)
120597119908
120597119905+ 1198601015840120597119906
120597119909= 1198692(1199061 1199081) minus 1198692(1199062 1199082)
119909 isin (minus119886 119886)
(44)
119906|119905=0
= 0
119908|119905=0
= 0
(45)
(119860119908minus
+ 119861119906minus
)1003816100381610038161003816119909=minus119886 =
1
120573(119860119908+
minus 119861119906+
)1003816100381610038161003816119909=minus119886
(119860119908minus
minus 119861119906minus
)1003816100381610038161003816119909=119886 =
1
120573(119860119908+
+ 119861119906+
)1003816100381610038161003816119909=119886
(46)
We prove that solution of problem (44)ndash(46) is trivialHence the uniqueness of the solution of problem (5)ndash(7)follows
Using the method which was mentioned before we get(see equality (14))
Once again we use spherical representation of 119906 = 119903(119905)1199081(119905
119909) and 119908 = 119903(119905)1199082(119905 119909) where 119903(119905) = 119880(119905 sdot)
1198712[minus119886119886]
Thenconcerning 119903(119905) we obtain new initial value problem
119889119903
119889119905+ 119903119875 (119905) = 119903
2
1198761(119905) (49)
119903 (0) = 0 (50)
where 119875(119905) has the same value as in (15) and
1198761(119905) =
1205902minus 1205900
2int
119886
minus119886
[(120596001
minus 120596002
) 120596021
+ 120596002
(120596021
minus 120596022
) minus1
radic3(120596011
minus 120596012
) (120596011
+ 120596012
)] (120596021
minus 120596022
) 119889119909 +1
4(1205903+ 31205901
minus 41205900) int
119886
minus119886
[(120596001
minus 120596002
) 120596031
+ +120596002
(120596031
minus 120596032
)] (120596031
minus 120596032
) 119889119909 +1
4radic5(21205901+ 1205900minus 31205903)
sdot int
119886
minus119886
[(120596011
minus 120596012
) 120596021
+ 120596012
(120596021
minus 120596022
)]
sdot (120596031
minus 120596032
) 119889119909 + (1205901minus 1205900) int
119886
minus119886
[(120596001
minus 120596002
) 120596111
+ 120596002
(120596111
minus 120596112
) +1
2
8 Journal of Applied Mathematics
sdot radic5
2[(120596101
minus 120596102
) 120596011
+ 120596102
(120596011
minus 120596012
)]
minus radic2
15[(120596011
minus 120596012
) 120596021
+ 120596012
(120596021
minus 120596022
)]
sdot (120596111
minus 120596112
)] 119889119909
(51)
The general solution of (49) is
119903 (119905) = exp(minusint
119905
0
119875 (120591) 119889120591)
sdot [119862 minus int
119905
0
1198761(120591) exp(minusint
120591
0
119875 (120585) 119889120585) 119889120591]
minus1
(52)
Solution of (49) which is satisfying homogeneous condition(50) is trivial that is 119903(119905) = 0 Hence 119880
119862([0119879]1198712[minus119886119886])
= 0
and 1199061= 1199062and 119908
1= 1199082
The theorem is proved
Competing Interests
The authors declare that they have no competing interests
References
[1] H Grad ldquoOn the kinetic theory of rarefied gasesrdquo Communica-tions on Pure andAppliedMathematics vol 2 no 4 pp 331ndash4071949
[2] H Grad ldquoPrinciple of the kinetic theory of gasesrdquo in Thermo-dynamics of Gases vol 12 ofHandbuch der Physik pp 205ndash294Springer Berlin Germany 1958
[3] A Sakabekov Initial-Boundary Value Problems for the Boltz-mannrsquos Moment System Equations Gylym Almaty 2002
[4] C Cercignani Theory and Application of the Boltzmann Equa-tion Instituto di Matematica Milano Italy 1975
[5] M N KoganDynamic of Rarefied Gas NaukaMoscow Russia1967
[6] K Kumar ldquoPolynomial expansions in kinetic theory of gasesrdquoAnnals of Physics vol 37 no 1 pp 113ndash141 1966
[7] A Sakabekov and Y Auzhani ldquoBoundary conditions for theonedimensional nonlinear nonstationary Boltzmannrsquos momentsystem equationsrdquo Journal of Mathematical Physics vol 55Article ID 123507 2014
[8] C D Levermore ldquoMoment closure hierarchies for kinetictheoriesrdquo Journal of Statistical Physics vol 83 no 5-6 pp 1021ndash1065 1996
[9] G Mascali and V Romano ldquoA hydrodynamical model for holesin silicon semiconductors the case of non-parabolic warpedbandsrdquo Mathematical and Computer Modelling vol 53 no 1-2pp 213ndash229 2011
[10] G Mascali and V Romano ldquoA non parabolic hydrodynamicalsubband model for semiconductors based on the maximumentropy principlerdquoMathematical and Computer Modelling vol55 no 3-4 pp 1003ndash1020 2012
[11] V D Camiola and V Romano ldquo2DEG-3DEG charge transportmodel for MOSFET based on the maximum entropy principlerdquo
SIAM Journal on Applied Mathematics vol 73 no 4 pp 1439ndash1459 2013
[12] G Alı G Mascali V Romano and R C Torcasio ldquoA hydrody-namical model for covalent semiconductors with a generalizedenergy dispersion relationrdquo European Journal of Applied Math-ematics vol 25 no 2 pp 255ndash276 2014
[13] V D Camiola and V Romano ldquoHydrodynamical model forcharge transport in graphenerdquo Journal of Statistical Physics vol157 no 6 pp 1114ndash1137 2014
[14] S Mischler ldquoKinetic equations with Maxwell boundary condi-tionsrdquo Annales Scientifiques de lrsquoEcole Normale Superieure vol43 no 5 pp 719ndash760 2010
[15] V G Neudachin and U F SmirnovNucleon Association of EasyKernel Nauka Moscow Russia 1969
[16] MMoshinskyTheHarmonic Oscillator inModern Physics fromAtoms to Quarks 1960
[17] S I Pokhozhaev ldquoOn an approach to nonlinear equationrdquoDoklady Akademii Nauk SSSR vol 247 pp 1327ndash1331 1979
[18] E Kamke Differentialgleichungen Losungsmethoden und Losu-ngen I Gewohuliche Differentialgleichungen BGTeubnerLeipzig Germany 1977
[19] A Tungatarov and D K Akhmed-Zaki ldquoCauchy problemfor one class of ordinary differential equationsrdquo InternationalJournal of Mathematical Analysis vol 6 no 14 pp 695ndash6992012
[20] M Tenenbaum andH PollardOrdinary Differential EquationsHarper and Row New York NY USA 1963
[21] L Tartar ldquoCompensated compactness and applications topartial differential equationsrdquo in Proceedings of the Non-LinearAnalysis and Mechanics Heriot-Watt Symposium R J KnopsEd vol 4 of Research Notes in Math pp 136ndash212 1979
For generalized coefficients of Talmi there exists a table[15] for each value of quantum number 120585 = 2119899 + 119897 from 0to 6 Moreover there is a program on IBM for calculation ofgeneralized coefficients of Talmi
If in (1) 2119899+ 119897 takes values from 0 to 3 we get Boltzmannrsquosmoment system equations in the third approximation Wewrite it in an expanded form
12059711989100
120597119905+
1
120572
12059711989101
120597119909= 0
12059711989102
120597119905+
1
120572
120597
120597119909(
2
radic311989101
+3
radic511989103
minus2radic2
radic1511989111) = 11986902
12059711989110
120597119905+
1
120572
120597
120597119909(minusradic
2
311989101
+ radic5
311989111) = 0
12059711989101
120597119905+
1
120572
120597
120597119909(11989100
+2
radic311989102
minus radic2
311989110) = 0
12059711989103
120597119905+
1
120572
120597
120597119909
3
radic511989102
= 11986903
12059711989111
120597119905+
1
120572
120597
120597119909(minus
2radic2
radic1511989102
+ radic5
311989110) = 11986911
119909 isin (minus119886 119886) 119905 gt 0
(3)
where 11989100
= 11989100(119905 119909) and 119891
01= 11989101(119905 119909) 119891
11= 11989111(119905 119909)
are the moments of the particle distribution function 11986902
=
(1205902minus 1205900)(1198910011989102
minus 1198912
01radic3)2 119869
03= (14)(120590
3+ 31205901minus
41205900)1198910011989103
+ (14radic5)(21205901+ 1205900minus 31205903)1198910111989102 and 119869
11=
(1205901minus 1205900)(1198910011989111
+ (12)radic531198911011989101
minus (radic2radic15)1198910111989102) are
moments of the collision integral where 1205900 1205901 1205902 and 120590
3
are the Fourier coefficients of the cross section expansionby the Legendre polynomials The first third and fourthequations of system (3) correspond to mass conservation lawmomentum conservation law and energy conservation lawrespectively
We study the correctness of initial and boundary valueproblem for six-moment one-dimensional Boltzmannrsquos sys-tem equations
1198601015840 is the transpose matrix 119861 is the positive definite
matrix 1199060(119909) = (119891
0
00(119909) 119891
0
02(119909) 119891
0
10(119909))1015840 and 119908
0(119909) =
(1198910
01(119909) 119891
0
03(119909) 119891
0
11(119909))1015840 are given initial vector-functions
119908+
119906+ are vector moments of falling to the boundary particle
distribution function119908minus 119906minusare vectormoments of reflectingfrom the boundary particle distribution function Equation(4) is vector matrix form of system equations (3)
It is possible to check through direct calculations that
det1198601= det(
0 119860
1198601015840
0) = 0 (9)
and matrix 1198601has three positive and the same number of
negative nonzero eigenvalues From (6)-(7) it follows that thenumber of boundary conditions on the left and right ends ofinterval (minus119886 119886) is equal to the number of positive andnegativeeigenvalues of the matrix 119860
1
Thus system (4) is a symmetric hyperbolic nonlinearpartial differential equations system Let us show that 119869
02
4 Journal of Applied Mathematics
is a sign-nondefined square form It is easy to check that1198910011989102
minus (1198912
01radic3) = (119862119880119880) where 119880 = (119906 119908)
1015840 and
119862 =
((((((((
(
01
20 0 0 0
1
20 0 0 0 0
0 0 0 0 0 0
0 0 0 minus1
radic30 0
0 0 0 0 0 0
0 0 0 0 0 0
))))))))
)
(10)
Eigenvalues of matrix 119862 are minus12 minus1radic3 0 0 0 and 12Therefore 119869
02is a sign-nondefined square form Similarly we
can show that 11986903
and 11986911
are also sign-nondefined squareforms
For problem (4)ndash(7) the following theorem takes place
Theorem 1 If 1198800= (1199060(119909) 119908
0(119909)) isin 119871
2
[minus119886 119886] then problem(4)ndash(7) has a unique solution in domain [minus119886 119886] times [0 119879]
belonging to the space 119862([0 119879] 1198712[minus119886 119886]) moreover
10038171003817100381710038171198712[minus119886119886] minus 1199031(0)) (11)
where 1198621is a constant independent from 119880 and 119879 sim
M(11988001198712[minus119886119886]
minus 1199031(0))minus1
) 1199031(119905) is the particular solution of
Riccati equation (15) in an explicit form
Proof Let 1198800isin 1198712
[minus119886 119886] Let us prove estimation (11) Wemultiply the first equation in system (4) by 119906 and the secondequation by 119908 and perform integration from minus119886 to 119886
where 1198651= ((1 minus 120573)120578120572120573radic120587)119865
Let us use the spherical representation [17] of vector119880(119905 119909) = 119903(119905)120596(119905 119909) where 120596(119905 119909) = (120596
10038171003817100381710038171198712[minus119886119886] minus 1199031(0))
(43)
where 1198622is constant and independent from119898 and 119903
1198981(119905) is a
particular solution of Riccati equation about 119903119898(119905)
Problem (40)-(41) represents a Cauchy problem for theordinary system of differential equations Existence of thesolution of problem (40)-(41) follows from theory of ordi-nary system of differential equations (Picardrsquos existence anduniqueness theorem) [20]Hence the existence of the solutionof problem (34)ndash(37) follows
Thus according to estimation (43) the sequence 119880119898
minus
1199031198981
of approximate solutions of problem (5)ndash(7) is uniformlybounded in function space119862([0 119879] 1198712[minus119886 119886])Moreover thehomogeneous system of equations 120591119864+ (1120572)119860120585with respectto 120591 120585 has only a trivial solution Then it follows from resultsin [21] that 119880
119898minus 1199031198981
rarr 119880minus 1199031is week in 119862([0 119879] 119871
2
[minus119886 119886])
and 119869(119880119898) rarr 119869(119880) is week in 119862([0 119879] 119871
2
[minus119886 119886]) as 119898 rarr
infin Further it can be shown with standard methods that limitelement is a weak solution of problem (5)ndash(7)
The uniqueness of solution of problem (5)ndash(7) is provedby contradiction Let problem (5)ndash(7) have two different solu-tions 119906
1 1199081and 119906
2 1199082 We denote them again by 119906 = 119906
1minus 1199062
Journal of Applied Mathematics 7
and119908 = 1199081minus1199082 Then with respect to new values of 119906 and119908
we obtain the following problem
120597119906
120597119905+ 119860
120597119908
120597119909= 1198691(1199061 1199081) minus 1198691(1199062 1199082)
120597119908
120597119905+ 1198601015840120597119906
120597119909= 1198692(1199061 1199081) minus 1198692(1199062 1199082)
119909 isin (minus119886 119886)
(44)
119906|119905=0
= 0
119908|119905=0
= 0
(45)
(119860119908minus
+ 119861119906minus
)1003816100381610038161003816119909=minus119886 =
1
120573(119860119908+
minus 119861119906+
)1003816100381610038161003816119909=minus119886
(119860119908minus
minus 119861119906minus
)1003816100381610038161003816119909=119886 =
1
120573(119860119908+
+ 119861119906+
)1003816100381610038161003816119909=119886
(46)
We prove that solution of problem (44)ndash(46) is trivialHence the uniqueness of the solution of problem (5)ndash(7)follows
Using the method which was mentioned before we get(see equality (14))
Once again we use spherical representation of 119906 = 119903(119905)1199081(119905
119909) and 119908 = 119903(119905)1199082(119905 119909) where 119903(119905) = 119880(119905 sdot)
1198712[minus119886119886]
Thenconcerning 119903(119905) we obtain new initial value problem
119889119903
119889119905+ 119903119875 (119905) = 119903
2
1198761(119905) (49)
119903 (0) = 0 (50)
where 119875(119905) has the same value as in (15) and
1198761(119905) =
1205902minus 1205900
2int
119886
minus119886
[(120596001
minus 120596002
) 120596021
+ 120596002
(120596021
minus 120596022
) minus1
radic3(120596011
minus 120596012
) (120596011
+ 120596012
)] (120596021
minus 120596022
) 119889119909 +1
4(1205903+ 31205901
minus 41205900) int
119886
minus119886
[(120596001
minus 120596002
) 120596031
+ +120596002
(120596031
minus 120596032
)] (120596031
minus 120596032
) 119889119909 +1
4radic5(21205901+ 1205900minus 31205903)
sdot int
119886
minus119886
[(120596011
minus 120596012
) 120596021
+ 120596012
(120596021
minus 120596022
)]
sdot (120596031
minus 120596032
) 119889119909 + (1205901minus 1205900) int
119886
minus119886
[(120596001
minus 120596002
) 120596111
+ 120596002
(120596111
minus 120596112
) +1
2
8 Journal of Applied Mathematics
sdot radic5
2[(120596101
minus 120596102
) 120596011
+ 120596102
(120596011
minus 120596012
)]
minus radic2
15[(120596011
minus 120596012
) 120596021
+ 120596012
(120596021
minus 120596022
)]
sdot (120596111
minus 120596112
)] 119889119909
(51)
The general solution of (49) is
119903 (119905) = exp(minusint
119905
0
119875 (120591) 119889120591)
sdot [119862 minus int
119905
0
1198761(120591) exp(minusint
120591
0
119875 (120585) 119889120585) 119889120591]
minus1
(52)
Solution of (49) which is satisfying homogeneous condition(50) is trivial that is 119903(119905) = 0 Hence 119880
119862([0119879]1198712[minus119886119886])
= 0
and 1199061= 1199062and 119908
1= 1199082
The theorem is proved
Competing Interests
The authors declare that they have no competing interests
References
[1] H Grad ldquoOn the kinetic theory of rarefied gasesrdquo Communica-tions on Pure andAppliedMathematics vol 2 no 4 pp 331ndash4071949
[2] H Grad ldquoPrinciple of the kinetic theory of gasesrdquo in Thermo-dynamics of Gases vol 12 ofHandbuch der Physik pp 205ndash294Springer Berlin Germany 1958
[3] A Sakabekov Initial-Boundary Value Problems for the Boltz-mannrsquos Moment System Equations Gylym Almaty 2002
[4] C Cercignani Theory and Application of the Boltzmann Equa-tion Instituto di Matematica Milano Italy 1975
[5] M N KoganDynamic of Rarefied Gas NaukaMoscow Russia1967
[6] K Kumar ldquoPolynomial expansions in kinetic theory of gasesrdquoAnnals of Physics vol 37 no 1 pp 113ndash141 1966
[7] A Sakabekov and Y Auzhani ldquoBoundary conditions for theonedimensional nonlinear nonstationary Boltzmannrsquos momentsystem equationsrdquo Journal of Mathematical Physics vol 55Article ID 123507 2014
[8] C D Levermore ldquoMoment closure hierarchies for kinetictheoriesrdquo Journal of Statistical Physics vol 83 no 5-6 pp 1021ndash1065 1996
[9] G Mascali and V Romano ldquoA hydrodynamical model for holesin silicon semiconductors the case of non-parabolic warpedbandsrdquo Mathematical and Computer Modelling vol 53 no 1-2pp 213ndash229 2011
[10] G Mascali and V Romano ldquoA non parabolic hydrodynamicalsubband model for semiconductors based on the maximumentropy principlerdquoMathematical and Computer Modelling vol55 no 3-4 pp 1003ndash1020 2012
[11] V D Camiola and V Romano ldquo2DEG-3DEG charge transportmodel for MOSFET based on the maximum entropy principlerdquo
SIAM Journal on Applied Mathematics vol 73 no 4 pp 1439ndash1459 2013
[12] G Alı G Mascali V Romano and R C Torcasio ldquoA hydrody-namical model for covalent semiconductors with a generalizedenergy dispersion relationrdquo European Journal of Applied Math-ematics vol 25 no 2 pp 255ndash276 2014
[13] V D Camiola and V Romano ldquoHydrodynamical model forcharge transport in graphenerdquo Journal of Statistical Physics vol157 no 6 pp 1114ndash1137 2014
[14] S Mischler ldquoKinetic equations with Maxwell boundary condi-tionsrdquo Annales Scientifiques de lrsquoEcole Normale Superieure vol43 no 5 pp 719ndash760 2010
[15] V G Neudachin and U F SmirnovNucleon Association of EasyKernel Nauka Moscow Russia 1969
[16] MMoshinskyTheHarmonic Oscillator inModern Physics fromAtoms to Quarks 1960
[17] S I Pokhozhaev ldquoOn an approach to nonlinear equationrdquoDoklady Akademii Nauk SSSR vol 247 pp 1327ndash1331 1979
[18] E Kamke Differentialgleichungen Losungsmethoden und Losu-ngen I Gewohuliche Differentialgleichungen BGTeubnerLeipzig Germany 1977
[19] A Tungatarov and D K Akhmed-Zaki ldquoCauchy problemfor one class of ordinary differential equationsrdquo InternationalJournal of Mathematical Analysis vol 6 no 14 pp 695ndash6992012
[20] M Tenenbaum andH PollardOrdinary Differential EquationsHarper and Row New York NY USA 1963
[21] L Tartar ldquoCompensated compactness and applications topartial differential equationsrdquo in Proceedings of the Non-LinearAnalysis and Mechanics Heriot-Watt Symposium R J KnopsEd vol 4 of Research Notes in Math pp 136ndash212 1979
10038171003817100381710038171198712[minus119886119886] minus 1199031(0)) (11)
where 1198621is a constant independent from 119880 and 119879 sim
M(11988001198712[minus119886119886]
minus 1199031(0))minus1
) 1199031(119905) is the particular solution of
Riccati equation (15) in an explicit form
Proof Let 1198800isin 1198712
[minus119886 119886] Let us prove estimation (11) Wemultiply the first equation in system (4) by 119906 and the secondequation by 119908 and perform integration from minus119886 to 119886
where 1198651= ((1 minus 120573)120578120572120573radic120587)119865
Let us use the spherical representation [17] of vector119880(119905 119909) = 119903(119905)120596(119905 119909) where 120596(119905 119909) = (120596
10038171003817100381710038171198712[minus119886119886] minus 1199031(0))
(43)
where 1198622is constant and independent from119898 and 119903
1198981(119905) is a
particular solution of Riccati equation about 119903119898(119905)
Problem (40)-(41) represents a Cauchy problem for theordinary system of differential equations Existence of thesolution of problem (40)-(41) follows from theory of ordi-nary system of differential equations (Picardrsquos existence anduniqueness theorem) [20]Hence the existence of the solutionof problem (34)ndash(37) follows
Thus according to estimation (43) the sequence 119880119898
minus
1199031198981
of approximate solutions of problem (5)ndash(7) is uniformlybounded in function space119862([0 119879] 1198712[minus119886 119886])Moreover thehomogeneous system of equations 120591119864+ (1120572)119860120585with respectto 120591 120585 has only a trivial solution Then it follows from resultsin [21] that 119880
119898minus 1199031198981
rarr 119880minus 1199031is week in 119862([0 119879] 119871
2
[minus119886 119886])
and 119869(119880119898) rarr 119869(119880) is week in 119862([0 119879] 119871
2
[minus119886 119886]) as 119898 rarr
infin Further it can be shown with standard methods that limitelement is a weak solution of problem (5)ndash(7)
The uniqueness of solution of problem (5)ndash(7) is provedby contradiction Let problem (5)ndash(7) have two different solu-tions 119906
1 1199081and 119906
2 1199082 We denote them again by 119906 = 119906
1minus 1199062
Journal of Applied Mathematics 7
and119908 = 1199081minus1199082 Then with respect to new values of 119906 and119908
we obtain the following problem
120597119906
120597119905+ 119860
120597119908
120597119909= 1198691(1199061 1199081) minus 1198691(1199062 1199082)
120597119908
120597119905+ 1198601015840120597119906
120597119909= 1198692(1199061 1199081) minus 1198692(1199062 1199082)
119909 isin (minus119886 119886)
(44)
119906|119905=0
= 0
119908|119905=0
= 0
(45)
(119860119908minus
+ 119861119906minus
)1003816100381610038161003816119909=minus119886 =
1
120573(119860119908+
minus 119861119906+
)1003816100381610038161003816119909=minus119886
(119860119908minus
minus 119861119906minus
)1003816100381610038161003816119909=119886 =
1
120573(119860119908+
+ 119861119906+
)1003816100381610038161003816119909=119886
(46)
We prove that solution of problem (44)ndash(46) is trivialHence the uniqueness of the solution of problem (5)ndash(7)follows
Using the method which was mentioned before we get(see equality (14))
Once again we use spherical representation of 119906 = 119903(119905)1199081(119905
119909) and 119908 = 119903(119905)1199082(119905 119909) where 119903(119905) = 119880(119905 sdot)
1198712[minus119886119886]
Thenconcerning 119903(119905) we obtain new initial value problem
119889119903
119889119905+ 119903119875 (119905) = 119903
2
1198761(119905) (49)
119903 (0) = 0 (50)
where 119875(119905) has the same value as in (15) and
1198761(119905) =
1205902minus 1205900
2int
119886
minus119886
[(120596001
minus 120596002
) 120596021
+ 120596002
(120596021
minus 120596022
) minus1
radic3(120596011
minus 120596012
) (120596011
+ 120596012
)] (120596021
minus 120596022
) 119889119909 +1
4(1205903+ 31205901
minus 41205900) int
119886
minus119886
[(120596001
minus 120596002
) 120596031
+ +120596002
(120596031
minus 120596032
)] (120596031
minus 120596032
) 119889119909 +1
4radic5(21205901+ 1205900minus 31205903)
sdot int
119886
minus119886
[(120596011
minus 120596012
) 120596021
+ 120596012
(120596021
minus 120596022
)]
sdot (120596031
minus 120596032
) 119889119909 + (1205901minus 1205900) int
119886
minus119886
[(120596001
minus 120596002
) 120596111
+ 120596002
(120596111
minus 120596112
) +1
2
8 Journal of Applied Mathematics
sdot radic5
2[(120596101
minus 120596102
) 120596011
+ 120596102
(120596011
minus 120596012
)]
minus radic2
15[(120596011
minus 120596012
) 120596021
+ 120596012
(120596021
minus 120596022
)]
sdot (120596111
minus 120596112
)] 119889119909
(51)
The general solution of (49) is
119903 (119905) = exp(minusint
119905
0
119875 (120591) 119889120591)
sdot [119862 minus int
119905
0
1198761(120591) exp(minusint
120591
0
119875 (120585) 119889120585) 119889120591]
minus1
(52)
Solution of (49) which is satisfying homogeneous condition(50) is trivial that is 119903(119905) = 0 Hence 119880
119862([0119879]1198712[minus119886119886])
= 0
and 1199061= 1199062and 119908
1= 1199082
The theorem is proved
Competing Interests
The authors declare that they have no competing interests
References
[1] H Grad ldquoOn the kinetic theory of rarefied gasesrdquo Communica-tions on Pure andAppliedMathematics vol 2 no 4 pp 331ndash4071949
[2] H Grad ldquoPrinciple of the kinetic theory of gasesrdquo in Thermo-dynamics of Gases vol 12 ofHandbuch der Physik pp 205ndash294Springer Berlin Germany 1958
[3] A Sakabekov Initial-Boundary Value Problems for the Boltz-mannrsquos Moment System Equations Gylym Almaty 2002
[4] C Cercignani Theory and Application of the Boltzmann Equa-tion Instituto di Matematica Milano Italy 1975
[5] M N KoganDynamic of Rarefied Gas NaukaMoscow Russia1967
[6] K Kumar ldquoPolynomial expansions in kinetic theory of gasesrdquoAnnals of Physics vol 37 no 1 pp 113ndash141 1966
[7] A Sakabekov and Y Auzhani ldquoBoundary conditions for theonedimensional nonlinear nonstationary Boltzmannrsquos momentsystem equationsrdquo Journal of Mathematical Physics vol 55Article ID 123507 2014
[8] C D Levermore ldquoMoment closure hierarchies for kinetictheoriesrdquo Journal of Statistical Physics vol 83 no 5-6 pp 1021ndash1065 1996
[9] G Mascali and V Romano ldquoA hydrodynamical model for holesin silicon semiconductors the case of non-parabolic warpedbandsrdquo Mathematical and Computer Modelling vol 53 no 1-2pp 213ndash229 2011
[10] G Mascali and V Romano ldquoA non parabolic hydrodynamicalsubband model for semiconductors based on the maximumentropy principlerdquoMathematical and Computer Modelling vol55 no 3-4 pp 1003ndash1020 2012
[11] V D Camiola and V Romano ldquo2DEG-3DEG charge transportmodel for MOSFET based on the maximum entropy principlerdquo
SIAM Journal on Applied Mathematics vol 73 no 4 pp 1439ndash1459 2013
[12] G Alı G Mascali V Romano and R C Torcasio ldquoA hydrody-namical model for covalent semiconductors with a generalizedenergy dispersion relationrdquo European Journal of Applied Math-ematics vol 25 no 2 pp 255ndash276 2014
[13] V D Camiola and V Romano ldquoHydrodynamical model forcharge transport in graphenerdquo Journal of Statistical Physics vol157 no 6 pp 1114ndash1137 2014
[14] S Mischler ldquoKinetic equations with Maxwell boundary condi-tionsrdquo Annales Scientifiques de lrsquoEcole Normale Superieure vol43 no 5 pp 719ndash760 2010
[15] V G Neudachin and U F SmirnovNucleon Association of EasyKernel Nauka Moscow Russia 1969
[16] MMoshinskyTheHarmonic Oscillator inModern Physics fromAtoms to Quarks 1960
[17] S I Pokhozhaev ldquoOn an approach to nonlinear equationrdquoDoklady Akademii Nauk SSSR vol 247 pp 1327ndash1331 1979
[18] E Kamke Differentialgleichungen Losungsmethoden und Losu-ngen I Gewohuliche Differentialgleichungen BGTeubnerLeipzig Germany 1977
[19] A Tungatarov and D K Akhmed-Zaki ldquoCauchy problemfor one class of ordinary differential equationsrdquo InternationalJournal of Mathematical Analysis vol 6 no 14 pp 695ndash6992012
[20] M Tenenbaum andH PollardOrdinary Differential EquationsHarper and Row New York NY USA 1963
[21] L Tartar ldquoCompensated compactness and applications topartial differential equationsrdquo in Proceedings of the Non-LinearAnalysis and Mechanics Heriot-Watt Symposium R J KnopsEd vol 4 of Research Notes in Math pp 136ndash212 1979
10038171003817100381710038171198712[minus119886119886] minus 1199031(0))
(43)
where 1198622is constant and independent from119898 and 119903
1198981(119905) is a
particular solution of Riccati equation about 119903119898(119905)
Problem (40)-(41) represents a Cauchy problem for theordinary system of differential equations Existence of thesolution of problem (40)-(41) follows from theory of ordi-nary system of differential equations (Picardrsquos existence anduniqueness theorem) [20]Hence the existence of the solutionof problem (34)ndash(37) follows
Thus according to estimation (43) the sequence 119880119898
minus
1199031198981
of approximate solutions of problem (5)ndash(7) is uniformlybounded in function space119862([0 119879] 1198712[minus119886 119886])Moreover thehomogeneous system of equations 120591119864+ (1120572)119860120585with respectto 120591 120585 has only a trivial solution Then it follows from resultsin [21] that 119880
119898minus 1199031198981
rarr 119880minus 1199031is week in 119862([0 119879] 119871
2
[minus119886 119886])
and 119869(119880119898) rarr 119869(119880) is week in 119862([0 119879] 119871
2
[minus119886 119886]) as 119898 rarr
infin Further it can be shown with standard methods that limitelement is a weak solution of problem (5)ndash(7)
The uniqueness of solution of problem (5)ndash(7) is provedby contradiction Let problem (5)ndash(7) have two different solu-tions 119906
1 1199081and 119906
2 1199082 We denote them again by 119906 = 119906
1minus 1199062
Journal of Applied Mathematics 7
and119908 = 1199081minus1199082 Then with respect to new values of 119906 and119908
we obtain the following problem
120597119906
120597119905+ 119860
120597119908
120597119909= 1198691(1199061 1199081) minus 1198691(1199062 1199082)
120597119908
120597119905+ 1198601015840120597119906
120597119909= 1198692(1199061 1199081) minus 1198692(1199062 1199082)
119909 isin (minus119886 119886)
(44)
119906|119905=0
= 0
119908|119905=0
= 0
(45)
(119860119908minus
+ 119861119906minus
)1003816100381610038161003816119909=minus119886 =
1
120573(119860119908+
minus 119861119906+
)1003816100381610038161003816119909=minus119886
(119860119908minus
minus 119861119906minus
)1003816100381610038161003816119909=119886 =
1
120573(119860119908+
+ 119861119906+
)1003816100381610038161003816119909=119886
(46)
We prove that solution of problem (44)ndash(46) is trivialHence the uniqueness of the solution of problem (5)ndash(7)follows
Using the method which was mentioned before we get(see equality (14))
Once again we use spherical representation of 119906 = 119903(119905)1199081(119905
119909) and 119908 = 119903(119905)1199082(119905 119909) where 119903(119905) = 119880(119905 sdot)
1198712[minus119886119886]
Thenconcerning 119903(119905) we obtain new initial value problem
119889119903
119889119905+ 119903119875 (119905) = 119903
2
1198761(119905) (49)
119903 (0) = 0 (50)
where 119875(119905) has the same value as in (15) and
1198761(119905) =
1205902minus 1205900
2int
119886
minus119886
[(120596001
minus 120596002
) 120596021
+ 120596002
(120596021
minus 120596022
) minus1
radic3(120596011
minus 120596012
) (120596011
+ 120596012
)] (120596021
minus 120596022
) 119889119909 +1
4(1205903+ 31205901
minus 41205900) int
119886
minus119886
[(120596001
minus 120596002
) 120596031
+ +120596002
(120596031
minus 120596032
)] (120596031
minus 120596032
) 119889119909 +1
4radic5(21205901+ 1205900minus 31205903)
sdot int
119886
minus119886
[(120596011
minus 120596012
) 120596021
+ 120596012
(120596021
minus 120596022
)]
sdot (120596031
minus 120596032
) 119889119909 + (1205901minus 1205900) int
119886
minus119886
[(120596001
minus 120596002
) 120596111
+ 120596002
(120596111
minus 120596112
) +1
2
8 Journal of Applied Mathematics
sdot radic5
2[(120596101
minus 120596102
) 120596011
+ 120596102
(120596011
minus 120596012
)]
minus radic2
15[(120596011
minus 120596012
) 120596021
+ 120596012
(120596021
minus 120596022
)]
sdot (120596111
minus 120596112
)] 119889119909
(51)
The general solution of (49) is
119903 (119905) = exp(minusint
119905
0
119875 (120591) 119889120591)
sdot [119862 minus int
119905
0
1198761(120591) exp(minusint
120591
0
119875 (120585) 119889120585) 119889120591]
minus1
(52)
Solution of (49) which is satisfying homogeneous condition(50) is trivial that is 119903(119905) = 0 Hence 119880
119862([0119879]1198712[minus119886119886])
= 0
and 1199061= 1199062and 119908
1= 1199082
The theorem is proved
Competing Interests
The authors declare that they have no competing interests
References
[1] H Grad ldquoOn the kinetic theory of rarefied gasesrdquo Communica-tions on Pure andAppliedMathematics vol 2 no 4 pp 331ndash4071949
[2] H Grad ldquoPrinciple of the kinetic theory of gasesrdquo in Thermo-dynamics of Gases vol 12 ofHandbuch der Physik pp 205ndash294Springer Berlin Germany 1958
[3] A Sakabekov Initial-Boundary Value Problems for the Boltz-mannrsquos Moment System Equations Gylym Almaty 2002
[4] C Cercignani Theory and Application of the Boltzmann Equa-tion Instituto di Matematica Milano Italy 1975
[5] M N KoganDynamic of Rarefied Gas NaukaMoscow Russia1967
[6] K Kumar ldquoPolynomial expansions in kinetic theory of gasesrdquoAnnals of Physics vol 37 no 1 pp 113ndash141 1966
[7] A Sakabekov and Y Auzhani ldquoBoundary conditions for theonedimensional nonlinear nonstationary Boltzmannrsquos momentsystem equationsrdquo Journal of Mathematical Physics vol 55Article ID 123507 2014
[8] C D Levermore ldquoMoment closure hierarchies for kinetictheoriesrdquo Journal of Statistical Physics vol 83 no 5-6 pp 1021ndash1065 1996
[9] G Mascali and V Romano ldquoA hydrodynamical model for holesin silicon semiconductors the case of non-parabolic warpedbandsrdquo Mathematical and Computer Modelling vol 53 no 1-2pp 213ndash229 2011
[10] G Mascali and V Romano ldquoA non parabolic hydrodynamicalsubband model for semiconductors based on the maximumentropy principlerdquoMathematical and Computer Modelling vol55 no 3-4 pp 1003ndash1020 2012
[11] V D Camiola and V Romano ldquo2DEG-3DEG charge transportmodel for MOSFET based on the maximum entropy principlerdquo
SIAM Journal on Applied Mathematics vol 73 no 4 pp 1439ndash1459 2013
[12] G Alı G Mascali V Romano and R C Torcasio ldquoA hydrody-namical model for covalent semiconductors with a generalizedenergy dispersion relationrdquo European Journal of Applied Math-ematics vol 25 no 2 pp 255ndash276 2014
[13] V D Camiola and V Romano ldquoHydrodynamical model forcharge transport in graphenerdquo Journal of Statistical Physics vol157 no 6 pp 1114ndash1137 2014
[14] S Mischler ldquoKinetic equations with Maxwell boundary condi-tionsrdquo Annales Scientifiques de lrsquoEcole Normale Superieure vol43 no 5 pp 719ndash760 2010
[15] V G Neudachin and U F SmirnovNucleon Association of EasyKernel Nauka Moscow Russia 1969
[16] MMoshinskyTheHarmonic Oscillator inModern Physics fromAtoms to Quarks 1960
[17] S I Pokhozhaev ldquoOn an approach to nonlinear equationrdquoDoklady Akademii Nauk SSSR vol 247 pp 1327ndash1331 1979
[18] E Kamke Differentialgleichungen Losungsmethoden und Losu-ngen I Gewohuliche Differentialgleichungen BGTeubnerLeipzig Germany 1977
[19] A Tungatarov and D K Akhmed-Zaki ldquoCauchy problemfor one class of ordinary differential equationsrdquo InternationalJournal of Mathematical Analysis vol 6 no 14 pp 695ndash6992012
[20] M Tenenbaum andH PollardOrdinary Differential EquationsHarper and Row New York NY USA 1963
[21] L Tartar ldquoCompensated compactness and applications topartial differential equationsrdquo in Proceedings of the Non-LinearAnalysis and Mechanics Heriot-Watt Symposium R J KnopsEd vol 4 of Research Notes in Math pp 136ndash212 1979
10038171003817100381710038171198712[minus119886119886] minus 1199031(0))
(43)
where 1198622is constant and independent from119898 and 119903
1198981(119905) is a
particular solution of Riccati equation about 119903119898(119905)
Problem (40)-(41) represents a Cauchy problem for theordinary system of differential equations Existence of thesolution of problem (40)-(41) follows from theory of ordi-nary system of differential equations (Picardrsquos existence anduniqueness theorem) [20]Hence the existence of the solutionof problem (34)ndash(37) follows
Thus according to estimation (43) the sequence 119880119898
minus
1199031198981
of approximate solutions of problem (5)ndash(7) is uniformlybounded in function space119862([0 119879] 1198712[minus119886 119886])Moreover thehomogeneous system of equations 120591119864+ (1120572)119860120585with respectto 120591 120585 has only a trivial solution Then it follows from resultsin [21] that 119880
119898minus 1199031198981
rarr 119880minus 1199031is week in 119862([0 119879] 119871
2
[minus119886 119886])
and 119869(119880119898) rarr 119869(119880) is week in 119862([0 119879] 119871
2
[minus119886 119886]) as 119898 rarr
infin Further it can be shown with standard methods that limitelement is a weak solution of problem (5)ndash(7)
The uniqueness of solution of problem (5)ndash(7) is provedby contradiction Let problem (5)ndash(7) have two different solu-tions 119906
1 1199081and 119906
2 1199082 We denote them again by 119906 = 119906
1minus 1199062
Journal of Applied Mathematics 7
and119908 = 1199081minus1199082 Then with respect to new values of 119906 and119908
we obtain the following problem
120597119906
120597119905+ 119860
120597119908
120597119909= 1198691(1199061 1199081) minus 1198691(1199062 1199082)
120597119908
120597119905+ 1198601015840120597119906
120597119909= 1198692(1199061 1199081) minus 1198692(1199062 1199082)
119909 isin (minus119886 119886)
(44)
119906|119905=0
= 0
119908|119905=0
= 0
(45)
(119860119908minus
+ 119861119906minus
)1003816100381610038161003816119909=minus119886 =
1
120573(119860119908+
minus 119861119906+
)1003816100381610038161003816119909=minus119886
(119860119908minus
minus 119861119906minus
)1003816100381610038161003816119909=119886 =
1
120573(119860119908+
+ 119861119906+
)1003816100381610038161003816119909=119886
(46)
We prove that solution of problem (44)ndash(46) is trivialHence the uniqueness of the solution of problem (5)ndash(7)follows
Using the method which was mentioned before we get(see equality (14))
Once again we use spherical representation of 119906 = 119903(119905)1199081(119905
119909) and 119908 = 119903(119905)1199082(119905 119909) where 119903(119905) = 119880(119905 sdot)
1198712[minus119886119886]
Thenconcerning 119903(119905) we obtain new initial value problem
119889119903
119889119905+ 119903119875 (119905) = 119903
2
1198761(119905) (49)
119903 (0) = 0 (50)
where 119875(119905) has the same value as in (15) and
1198761(119905) =
1205902minus 1205900
2int
119886
minus119886
[(120596001
minus 120596002
) 120596021
+ 120596002
(120596021
minus 120596022
) minus1
radic3(120596011
minus 120596012
) (120596011
+ 120596012
)] (120596021
minus 120596022
) 119889119909 +1
4(1205903+ 31205901
minus 41205900) int
119886
minus119886
[(120596001
minus 120596002
) 120596031
+ +120596002
(120596031
minus 120596032
)] (120596031
minus 120596032
) 119889119909 +1
4radic5(21205901+ 1205900minus 31205903)
sdot int
119886
minus119886
[(120596011
minus 120596012
) 120596021
+ 120596012
(120596021
minus 120596022
)]
sdot (120596031
minus 120596032
) 119889119909 + (1205901minus 1205900) int
119886
minus119886
[(120596001
minus 120596002
) 120596111
+ 120596002
(120596111
minus 120596112
) +1
2
8 Journal of Applied Mathematics
sdot radic5
2[(120596101
minus 120596102
) 120596011
+ 120596102
(120596011
minus 120596012
)]
minus radic2
15[(120596011
minus 120596012
) 120596021
+ 120596012
(120596021
minus 120596022
)]
sdot (120596111
minus 120596112
)] 119889119909
(51)
The general solution of (49) is
119903 (119905) = exp(minusint
119905
0
119875 (120591) 119889120591)
sdot [119862 minus int
119905
0
1198761(120591) exp(minusint
120591
0
119875 (120585) 119889120585) 119889120591]
minus1
(52)
Solution of (49) which is satisfying homogeneous condition(50) is trivial that is 119903(119905) = 0 Hence 119880
119862([0119879]1198712[minus119886119886])
= 0
and 1199061= 1199062and 119908
1= 1199082
The theorem is proved
Competing Interests
The authors declare that they have no competing interests
References
[1] H Grad ldquoOn the kinetic theory of rarefied gasesrdquo Communica-tions on Pure andAppliedMathematics vol 2 no 4 pp 331ndash4071949
[2] H Grad ldquoPrinciple of the kinetic theory of gasesrdquo in Thermo-dynamics of Gases vol 12 ofHandbuch der Physik pp 205ndash294Springer Berlin Germany 1958
[3] A Sakabekov Initial-Boundary Value Problems for the Boltz-mannrsquos Moment System Equations Gylym Almaty 2002
[4] C Cercignani Theory and Application of the Boltzmann Equa-tion Instituto di Matematica Milano Italy 1975
[5] M N KoganDynamic of Rarefied Gas NaukaMoscow Russia1967
[6] K Kumar ldquoPolynomial expansions in kinetic theory of gasesrdquoAnnals of Physics vol 37 no 1 pp 113ndash141 1966
[7] A Sakabekov and Y Auzhani ldquoBoundary conditions for theonedimensional nonlinear nonstationary Boltzmannrsquos momentsystem equationsrdquo Journal of Mathematical Physics vol 55Article ID 123507 2014
[8] C D Levermore ldquoMoment closure hierarchies for kinetictheoriesrdquo Journal of Statistical Physics vol 83 no 5-6 pp 1021ndash1065 1996
[9] G Mascali and V Romano ldquoA hydrodynamical model for holesin silicon semiconductors the case of non-parabolic warpedbandsrdquo Mathematical and Computer Modelling vol 53 no 1-2pp 213ndash229 2011
[10] G Mascali and V Romano ldquoA non parabolic hydrodynamicalsubband model for semiconductors based on the maximumentropy principlerdquoMathematical and Computer Modelling vol55 no 3-4 pp 1003ndash1020 2012
[11] V D Camiola and V Romano ldquo2DEG-3DEG charge transportmodel for MOSFET based on the maximum entropy principlerdquo
SIAM Journal on Applied Mathematics vol 73 no 4 pp 1439ndash1459 2013
[12] G Alı G Mascali V Romano and R C Torcasio ldquoA hydrody-namical model for covalent semiconductors with a generalizedenergy dispersion relationrdquo European Journal of Applied Math-ematics vol 25 no 2 pp 255ndash276 2014
[13] V D Camiola and V Romano ldquoHydrodynamical model forcharge transport in graphenerdquo Journal of Statistical Physics vol157 no 6 pp 1114ndash1137 2014
[14] S Mischler ldquoKinetic equations with Maxwell boundary condi-tionsrdquo Annales Scientifiques de lrsquoEcole Normale Superieure vol43 no 5 pp 719ndash760 2010
[15] V G Neudachin and U F SmirnovNucleon Association of EasyKernel Nauka Moscow Russia 1969
[16] MMoshinskyTheHarmonic Oscillator inModern Physics fromAtoms to Quarks 1960
[17] S I Pokhozhaev ldquoOn an approach to nonlinear equationrdquoDoklady Akademii Nauk SSSR vol 247 pp 1327ndash1331 1979
[18] E Kamke Differentialgleichungen Losungsmethoden und Losu-ngen I Gewohuliche Differentialgleichungen BGTeubnerLeipzig Germany 1977
[19] A Tungatarov and D K Akhmed-Zaki ldquoCauchy problemfor one class of ordinary differential equationsrdquo InternationalJournal of Mathematical Analysis vol 6 no 14 pp 695ndash6992012
[20] M Tenenbaum andH PollardOrdinary Differential EquationsHarper and Row New York NY USA 1963
[21] L Tartar ldquoCompensated compactness and applications topartial differential equationsrdquo in Proceedings of the Non-LinearAnalysis and Mechanics Heriot-Watt Symposium R J KnopsEd vol 4 of Research Notes in Math pp 136ndash212 1979
Once again we use spherical representation of 119906 = 119903(119905)1199081(119905
119909) and 119908 = 119903(119905)1199082(119905 119909) where 119903(119905) = 119880(119905 sdot)
1198712[minus119886119886]
Thenconcerning 119903(119905) we obtain new initial value problem
119889119903
119889119905+ 119903119875 (119905) = 119903
2
1198761(119905) (49)
119903 (0) = 0 (50)
where 119875(119905) has the same value as in (15) and
1198761(119905) =
1205902minus 1205900
2int
119886
minus119886
[(120596001
minus 120596002
) 120596021
+ 120596002
(120596021
minus 120596022
) minus1
radic3(120596011
minus 120596012
) (120596011
+ 120596012
)] (120596021
minus 120596022
) 119889119909 +1
4(1205903+ 31205901
minus 41205900) int
119886
minus119886
[(120596001
minus 120596002
) 120596031
+ +120596002
(120596031
minus 120596032
)] (120596031
minus 120596032
) 119889119909 +1
4radic5(21205901+ 1205900minus 31205903)
sdot int
119886
minus119886
[(120596011
minus 120596012
) 120596021
+ 120596012
(120596021
minus 120596022
)]
sdot (120596031
minus 120596032
) 119889119909 + (1205901minus 1205900) int
119886
minus119886
[(120596001
minus 120596002
) 120596111
+ 120596002
(120596111
minus 120596112
) +1
2
8 Journal of Applied Mathematics
sdot radic5
2[(120596101
minus 120596102
) 120596011
+ 120596102
(120596011
minus 120596012
)]
minus radic2
15[(120596011
minus 120596012
) 120596021
+ 120596012
(120596021
minus 120596022
)]
sdot (120596111
minus 120596112
)] 119889119909
(51)
The general solution of (49) is
119903 (119905) = exp(minusint
119905
0
119875 (120591) 119889120591)
sdot [119862 minus int
119905
0
1198761(120591) exp(minusint
120591
0
119875 (120585) 119889120585) 119889120591]
minus1
(52)
Solution of (49) which is satisfying homogeneous condition(50) is trivial that is 119903(119905) = 0 Hence 119880
119862([0119879]1198712[minus119886119886])
= 0
and 1199061= 1199062and 119908
1= 1199082
The theorem is proved
Competing Interests
The authors declare that they have no competing interests
References
[1] H Grad ldquoOn the kinetic theory of rarefied gasesrdquo Communica-tions on Pure andAppliedMathematics vol 2 no 4 pp 331ndash4071949
[2] H Grad ldquoPrinciple of the kinetic theory of gasesrdquo in Thermo-dynamics of Gases vol 12 ofHandbuch der Physik pp 205ndash294Springer Berlin Germany 1958
[3] A Sakabekov Initial-Boundary Value Problems for the Boltz-mannrsquos Moment System Equations Gylym Almaty 2002
[4] C Cercignani Theory and Application of the Boltzmann Equa-tion Instituto di Matematica Milano Italy 1975
[5] M N KoganDynamic of Rarefied Gas NaukaMoscow Russia1967
[6] K Kumar ldquoPolynomial expansions in kinetic theory of gasesrdquoAnnals of Physics vol 37 no 1 pp 113ndash141 1966
[7] A Sakabekov and Y Auzhani ldquoBoundary conditions for theonedimensional nonlinear nonstationary Boltzmannrsquos momentsystem equationsrdquo Journal of Mathematical Physics vol 55Article ID 123507 2014
[8] C D Levermore ldquoMoment closure hierarchies for kinetictheoriesrdquo Journal of Statistical Physics vol 83 no 5-6 pp 1021ndash1065 1996
[9] G Mascali and V Romano ldquoA hydrodynamical model for holesin silicon semiconductors the case of non-parabolic warpedbandsrdquo Mathematical and Computer Modelling vol 53 no 1-2pp 213ndash229 2011
[10] G Mascali and V Romano ldquoA non parabolic hydrodynamicalsubband model for semiconductors based on the maximumentropy principlerdquoMathematical and Computer Modelling vol55 no 3-4 pp 1003ndash1020 2012
[11] V D Camiola and V Romano ldquo2DEG-3DEG charge transportmodel for MOSFET based on the maximum entropy principlerdquo
SIAM Journal on Applied Mathematics vol 73 no 4 pp 1439ndash1459 2013
[12] G Alı G Mascali V Romano and R C Torcasio ldquoA hydrody-namical model for covalent semiconductors with a generalizedenergy dispersion relationrdquo European Journal of Applied Math-ematics vol 25 no 2 pp 255ndash276 2014
[13] V D Camiola and V Romano ldquoHydrodynamical model forcharge transport in graphenerdquo Journal of Statistical Physics vol157 no 6 pp 1114ndash1137 2014
[14] S Mischler ldquoKinetic equations with Maxwell boundary condi-tionsrdquo Annales Scientifiques de lrsquoEcole Normale Superieure vol43 no 5 pp 719ndash760 2010
[15] V G Neudachin and U F SmirnovNucleon Association of EasyKernel Nauka Moscow Russia 1969
[16] MMoshinskyTheHarmonic Oscillator inModern Physics fromAtoms to Quarks 1960
[17] S I Pokhozhaev ldquoOn an approach to nonlinear equationrdquoDoklady Akademii Nauk SSSR vol 247 pp 1327ndash1331 1979
[18] E Kamke Differentialgleichungen Losungsmethoden und Losu-ngen I Gewohuliche Differentialgleichungen BGTeubnerLeipzig Germany 1977
[19] A Tungatarov and D K Akhmed-Zaki ldquoCauchy problemfor one class of ordinary differential equationsrdquo InternationalJournal of Mathematical Analysis vol 6 no 14 pp 695ndash6992012
[20] M Tenenbaum andH PollardOrdinary Differential EquationsHarper and Row New York NY USA 1963
[21] L Tartar ldquoCompensated compactness and applications topartial differential equationsrdquo in Proceedings of the Non-LinearAnalysis and Mechanics Heriot-Watt Symposium R J KnopsEd vol 4 of Research Notes in Math pp 136ndash212 1979
Solution of (49) which is satisfying homogeneous condition(50) is trivial that is 119903(119905) = 0 Hence 119880
119862([0119879]1198712[minus119886119886])
= 0
and 1199061= 1199062and 119908
1= 1199082
The theorem is proved
Competing Interests
The authors declare that they have no competing interests
References
[1] H Grad ldquoOn the kinetic theory of rarefied gasesrdquo Communica-tions on Pure andAppliedMathematics vol 2 no 4 pp 331ndash4071949
[2] H Grad ldquoPrinciple of the kinetic theory of gasesrdquo in Thermo-dynamics of Gases vol 12 ofHandbuch der Physik pp 205ndash294Springer Berlin Germany 1958
[3] A Sakabekov Initial-Boundary Value Problems for the Boltz-mannrsquos Moment System Equations Gylym Almaty 2002
[4] C Cercignani Theory and Application of the Boltzmann Equa-tion Instituto di Matematica Milano Italy 1975
[5] M N KoganDynamic of Rarefied Gas NaukaMoscow Russia1967
[6] K Kumar ldquoPolynomial expansions in kinetic theory of gasesrdquoAnnals of Physics vol 37 no 1 pp 113ndash141 1966
[7] A Sakabekov and Y Auzhani ldquoBoundary conditions for theonedimensional nonlinear nonstationary Boltzmannrsquos momentsystem equationsrdquo Journal of Mathematical Physics vol 55Article ID 123507 2014
[8] C D Levermore ldquoMoment closure hierarchies for kinetictheoriesrdquo Journal of Statistical Physics vol 83 no 5-6 pp 1021ndash1065 1996
[9] G Mascali and V Romano ldquoA hydrodynamical model for holesin silicon semiconductors the case of non-parabolic warpedbandsrdquo Mathematical and Computer Modelling vol 53 no 1-2pp 213ndash229 2011
[10] G Mascali and V Romano ldquoA non parabolic hydrodynamicalsubband model for semiconductors based on the maximumentropy principlerdquoMathematical and Computer Modelling vol55 no 3-4 pp 1003ndash1020 2012
[11] V D Camiola and V Romano ldquo2DEG-3DEG charge transportmodel for MOSFET based on the maximum entropy principlerdquo
SIAM Journal on Applied Mathematics vol 73 no 4 pp 1439ndash1459 2013
[12] G Alı G Mascali V Romano and R C Torcasio ldquoA hydrody-namical model for covalent semiconductors with a generalizedenergy dispersion relationrdquo European Journal of Applied Math-ematics vol 25 no 2 pp 255ndash276 2014
[13] V D Camiola and V Romano ldquoHydrodynamical model forcharge transport in graphenerdquo Journal of Statistical Physics vol157 no 6 pp 1114ndash1137 2014
[14] S Mischler ldquoKinetic equations with Maxwell boundary condi-tionsrdquo Annales Scientifiques de lrsquoEcole Normale Superieure vol43 no 5 pp 719ndash760 2010
[15] V G Neudachin and U F SmirnovNucleon Association of EasyKernel Nauka Moscow Russia 1969
[16] MMoshinskyTheHarmonic Oscillator inModern Physics fromAtoms to Quarks 1960
[17] S I Pokhozhaev ldquoOn an approach to nonlinear equationrdquoDoklady Akademii Nauk SSSR vol 247 pp 1327ndash1331 1979
[18] E Kamke Differentialgleichungen Losungsmethoden und Losu-ngen I Gewohuliche Differentialgleichungen BGTeubnerLeipzig Germany 1977
[19] A Tungatarov and D K Akhmed-Zaki ldquoCauchy problemfor one class of ordinary differential equationsrdquo InternationalJournal of Mathematical Analysis vol 6 no 14 pp 695ndash6992012
[20] M Tenenbaum andH PollardOrdinary Differential EquationsHarper and Row New York NY USA 1963
[21] L Tartar ldquoCompensated compactness and applications topartial differential equationsrdquo in Proceedings of the Non-LinearAnalysis and Mechanics Heriot-Watt Symposium R J KnopsEd vol 4 of Research Notes in Math pp 136ndash212 1979