The Incompressible Navier-Stokes-Fourier Limit from Boltzmann-Fermi-Dirac Equation
aa r X i v : . [ m a t h . A P ] F e b THE INCOMPRESSIBLE NAVIER-STOKES-FOURIER LIMIT FROMBOLTZMANN-FERMI-DIRAC EQUATION
NING JIANG, LINJIE XIONG, AND KAI ZHOU
Abstract.
We study Boltzmann-Fermi-Dirac equation when quantum effects are taken into ac-count in dilute gas dynamics. By employing new estimates on trilinear terms of collision kernels, weprove the global existence of the classical solution to Boltzmann-Fermi-Dirac equation near equilib-rium. Furthermore, the limit from Boltzmann-Fermi-Dirac equation to incompressible Navier-Stokes-Fourier equations is justified rigorously. The corresponding formal analysis was given in the thesis ofZakrevskiy [49]
Keywords.
Boltzmann-Fermi-Dirac equation, Classical solutions, Navier-Stokes-Fourier Limit. Introduction
The Boltzmann-Fermi-Dirac equation.
The evolution of quantum particles obeying Fermi-Dirac statistics can be described by Boltzmann-Fermi-Dirac equation: ( ∂ t F + v · ∇ F = C ( F ) ,F (cid:12)(cid:12) t =0 = F , (1.1)where 0 F ( t, x, v ) t >
0, position x ∈ R , with velocity v ∈ R . The collision integral C ( F ) takes the form ¨ R × S b ( v − v, ω ) h F ′ F ′ (1 − δF )(1 − δF ) − F F (1 − δF ′ )(1 − δF ′ ) i d ω d v . In the expression above, δ = ~ is a constant ( ~ is the Planck constant), F ≡ F ( t, x, v ), F ≡ F ( t, x, v ), F ′ ≡ F ( t, x, v ′ ), F ′ ≡ F ( t, x, v ′ ), and for ω ∈ S , v ′ = v + [( v − v ) · ω ] ω, v ′ = v − [( v − v ) · ω ] ω are velocities after a collision of two particles with velocities v and v before. The particle pairs withthe same mass, during the collision, follow the conservation laws of momentum and kinetic energy: v + v = v ′ + v ′ , | v | + | v | = | v ′ | + | v ′ | . The collision kernel b ( v − v, ω ) is an a.e. positive function defined on R × S , which encodesfeatures of the molecular interaction in kinetic theory. Physically, it is assumed to depend only on themodulus of the relative velocity | v − v | and on the scalar product v − v | v − v | · ω . We assume the collisionkernel takes the factor form: b ( v − v, ω ) = | v − v | γ ˆ b (cos θ ) , cos θ = v − v | v − v | · ω, − < γ . For γ >
0, we call the collision kernel a hard potential; in particular, for γ = 0, we call it a Maxwellcollision kernel and for γ = 1, we call the collision hard sphere collision; and for − < γ < , we callit a soft potential. In Grad angular cutoff, ˆ b (cos θ ) satisfies ˆ π/ ˆ b (cos θ ) sin θdθ < + ∞ . Throughout the paper, we take δ = 1 and the hard sphere collision in (1.1) for convenience, namely,the collision kernel has the following explicit expression b ( v − v, ω ) = | ( v − v ) · ω | . (1.2) February 5, 2021.
Next, we list some basic properties of Boltzmann-Fermi-Dirac equation (1.1). First, the collisionoperator C ( F ) satisfies the conservation laws: ˆ R C ( F ) v | v | d v = 0 . (1.3)Next, as an analog of Boltzmann H -theorem, the following three assertions are equivalent [49]:(1) C ( F ) = 0;(2) The entropy production rate is zero, ˆ R C ( F ) ln 1 − FF d v = 0;(3) F is a Fermi-Dirac distribution, F f,u,θ ( t, x, v ) = 11 + exp( | v − u | θ − f ) . In (3) above, θ = θ ( t, x ) > u = u ( t, x ) ∈ R is the bulk velocity, and f ( t, x ) /θ ( t, x )is the total chemical potential. By the appropriate choice of Galilean frame, the Fermi-Dirac distribu-tion F can be taken with ( f, u, θ ) = (1 , , µ ( v ) = 11 + e | v | − . (1.4)The main goals of the current paper are the global existence of classical solutions to Boltzmann-Fermi-Dirac equation, and the connection between kinetic theory for Fermi-Dirac statistics and macro-scopic fluid equations. In particular, this paper focuses on the incompressible Navier-Stokes scaling,under which the equation (1.1) can be rescaled as ∂ t F ǫ + 1 ǫ v · ∇ F ǫ = 1 ǫ C ( F ǫ ) ,F ǫ (cid:12)(cid:12) t =0 = F ǫ, . (1.5)Here ǫ is the so-called Knudsen number which is the ratio between the mean free path and macroscopiclength scale. It can be derived the incompressible Navier-Stokes equations (for details, see Chapter 3of the thesis of Zakrevskiy [49]) from (1.5) by fluctuating around the global Fermi-Dirac distribution µ with size ǫ : F ǫ = µ + ǫµ (1 − µ ) g ǫ . More specifically, putting the perturbation above into (1.5) yields: ∂ t g ǫ + 1 ǫ v · ∇ x g ǫ + 1 ǫ Lg ǫ = 1 ǫ Q ( g ǫ , g ǫ ) + T ( g ǫ , g ǫ , g ǫ ) ,g ǫ ( t, x, v ) | t =0 = g ǫ, ( x, v ) . (1.6)From now on, we use the notations f = f ( v ) , f = f ( v ) , f ′ = f ( v ′ ) , f ′ = f ( v ′ ) , for any function f , and N = µ ′ µ ′ (1 − µ )(1 − µ ) = µ µ (1 − µ ′ )(1 − µ ′ ) . Then the linear operator L in (1.6) is given by Lg = ν ( v ) g − Kg, (1.7)and K = K − K , where the collision frequency ν ( v ) is ν ( v ) = ¨ R × S | v − v | | cos θ | N µ (1 − µ ) d ω d v , (1.8)the operator K is K g = ¨ R × S | v − v | | cos θ | N µ (1 − µ ) g d ω d v , (1.9) HE INCOMPRESSIBLE NAVIER-STOKES-FOURIER LIMIT FROM BOLTZMANN-FERMI-DIRAC EQUATION 3 and the operator K is K g = ¨ R × S | v − v | | cos θ | N µ (1 − µ ) ( g ′ + g ′ )d ω d v . (1.10)Moreover, in (1.6), the bilinear form Q ( f, g ) is given by Q ( f, g ) = ¨ R × S | v − v | | cos θ | N µ (1 − µ ) (cid:26)(cid:18) − µ ′ − µ ′ (cid:19) f ′ g ′ − (cid:18) − µ − µ (cid:19) f g + (cid:18) µ ′ f ′ + µ ′ f ′ (cid:19) g − (cid:18) f ′ + f ′ (cid:19) µ g + (cid:18) µ ′ f ′ + µ ′ f ′ (cid:19) g − (cid:18) f ′ + f ′ (cid:19) µg (cid:27) d ω d v △ = Q ( f, g ) + Q ( f, g ) + · · · + Q ( f, g ) . (1.11)and the trilinear form T ( f, g, h ) is given by T ( f, g, h ) = ¨ R × S | v − v | | cos θ | N µ (1 − µ ) · (cid:26) µ µf g (cid:18) h ′ + h ′ (cid:19) − µ ′ µ ′ f ′ g ′ (cid:18) h + h (cid:19) + µ µ ′ (cid:18) f ′ g ′ h − f gh ′ (cid:19) + µµ ′ (cid:18) f ′ g ′ h − f gh ′ (cid:19) + µ µ ′ (cid:18) f ′ g ′ h − f gh ′ (cid:19) + µµ ′ (cid:18) f ′ g ′ h − f gh ′ (cid:19) + f g (cid:18) µ ′ h ′ + µ ′ h ′ (cid:19) − f ′ g ′ (cid:18) µ h + µh (cid:19)(cid:27) d ω d v △ = T ( f, g, h ) + T ( f, g, h ) + · · · + T ( f, g, h ) . (1.12)1.2. Well-posedness and hydrodynamic limits.
The Boltzmann-Fermi-Dirac equation, which de-scribes the evolution of rarefied gas with quantum effect, is derived from a modification of the clas-sical Boltzmann equation [15], when the exclusion Pauli principle are taken into account. It is alsooften called Uehling-Uhlenbeck equation or Nordheim equation. However, different from the classi-cal case, the rigorous derivation of the Boltzmann-Fermi-Dirac equation has not been established.Since the heuristic arguments of Nordheim [43], Uehling and Uhlenbeck [47], rigorous derivation ofthe Boltzmann-Fermi-Dirac equation can be found Sphon [46], Erd¨os, Salmhofer and Yau [19] andBenedetto, Pulvirenti, Castella and Esposito [9].For mathematical theory of well-posedness, early results were obtained by Dolbeault [18] and Lions[38]. They studied the global existence of solutions in mild or distributional sense for the whole space R under some assumptions on the collision kernel. Furthermore, Dolbeault [18] obtained that thesolution of Boltzmann-Fermi-Dirac equation converges to the solution of the Boltzmann equation as δ → H -solution.For general initial data, Lu [40] studied the global existence and weak stability of weak solution in T x for very soft potential with a weak angular cutoff. More results are referred to [20, 21, 39, 41].In the context of classical solution near global equilibrium, much less is known for Boltzmann-Fermi-Dirac equation (after we finished the draft of this paper, we were aware of the just posted paper [44] onthis topic). We first review the corresponding results for Boltzmann equation. Ukai [48] obtained thefirst global-in-time smooth solution with cutoff kernel. Later on, Guo [27, 28] developed the so-callednonlinear energy method to get the same type of result for soft potential with γ ≥ −
3. For moreprogress in this direction, we refer to [2, 11, 12, 25, 30, 33].In the other direction, the hydrodynamic limits from kinetic equations to fluid equations has beenvery active in recent decades. One of the important feature of kinetic equations is their connection tothe fluid equations. The smaller the Knudsen number ǫ is, the more the dilute gas behaves like a fluid.Mathematically, the so-called hydrodynamic limits are the process that the Knudsen number goes tozero. Depending on the physically scalings, different fluid equations (incompressible of compressibleNavier-Stokes, Euler, etc.) can be derived from kinetic equations.Bardos and Ukai [8] proved the global existence of classical solution g ǫ to scaled Boltzmann equation(perturbed around the global Maxwellian with size ǫ ) uniformly in 0 < ǫ < NING JIANG, LINJIE XIONG, AND KAI ZHOU the torus for hard cutoff potential, with convergence rate. Recently, Jiang, Xu and Zhao [36] provedagain the same limit for a more general class of collision kernel by using non-isotropic norm developedin the series of work [2, 3, 4, 25]. For the fluid limits of Boltzmann-Fermi-Dirac equation, Zakrevskiy[49] formally derived the compressible Euler and Navier-Stokes limits and incompressible Navier-Stokeslimits. We also mention that, Filbet, Hu and Jin [23] introduced a new scheme for quantum Boltzmannequation to capture the Euler limit by numerical computations.Starting from the solutions to the limiting fluid equations, Caflisch [14] and Nishida [42] proved thecompressible Euler limit from the Boltzmann equation in the context of classical solution by the Hilbertexpansion, and analytic solutions, respectively. Caflisch’s approach was applied to the acoustic limitby Guo, Jang and Jiang [31, 32, 34] by combining with nonlinear energy method. We also mentionsome more results using Hilbert expansions [29, 35].The main concern of the current paper is to justify rigorously the formal derivation of Zakrevskiy[49]. The present paper verifies the incompressible Navier-Stokes limits rigorously in the context ofclassical solution. Precisely, we first prove the uniform in ǫ global existence of classical solution aroundthe equilibrium to the scaled Boltzmann-Fermi-Dirac equation for the hard sphere collision. Moreimportantly, we obtain the uniform energy estimate, using which we rigorously prove the limit fromBoltzmann-Fermi-Dirac equation to Incompressible Navier-Stokes-Fourier equations by taking limit as ǫ → µ (1 − µ ) as the weight, which leads to the gain of newnonlinear terms in (1.6) in comparing with the case of perturbed Boltzmann equation, and brings ussome new difficulties. Notation.
We introduce some notations for the presentation throughout this paper. We write g ∈ L ( µ (1 − µ )d v ) ≡ L v if ´ R | g | µ (1 − µ )d v < + ∞ , and use h· , ·i to denote the inner product inthe Hilbert space L ( µ (1 − µ )d v ), | · | L v the corresponding L norm, and sometimes use h g i to denote ´ R gµ (1 − µ )d v. Similarly, we write g ( x, v ) ∈ H N (cid:0) d x ; L ( µ (1 − µ )d v ) (cid:1) ≡ H Nx L v for integer N > X | α | N ¨ R × R | ∂ αx g | µ (1 − µ )d v d x < + ∞ , for multi-index α = ( α , α , α ) and | α | = P i =1 α i , and use ( · , · ) H Nx L v and ( · , · ) H Nx to denote the innerproduct in the Hilbert space H N (cid:0) d x ; L ( µ (1 − µ )d v ) (cid:1) and L ( µ (1 − µ )d v ) respectively, k · k H Nx L v and k · k H Nx the corresponding norm, sometimes for N = 0. We drop the subscript H Nx L v and H Nx in theinner product and norm. It is also convenient to introduce a weighted inner product as h f, g i ν = h νf, g i for any functions f ( v ) and g ( v ) in L ( µ (1 − µ )d v ), and use | · | ν for the corresponding weighted L norm.Finally, throughout this paper, let N > P as v -projectionin L ( µ (1 − µ )d v ) to the null space N ull ( L ) of L . More specifically, by (2.2), for any g ( t, x, v ) ∈ L ( µ (1 − µ )d v ), there exists constants (in v ) a ( t, x ) ∈ R , b ( t, x ) ∈ R and c ( t, x ) ∈ R such that P g = a ( t, x ) + b ( t, x ) · v + c ( t, x ) | v | , then we have the following macro-micro decomposition g = P g + { I − P } g, (1.13) P g is called the macroscopic part and { I − P } g is called the microscopic part. It’s clear that1 C k ∂ αx P g k ν k ∂ αx ( a, b, c ) k C k ∂ αx P g k , (1.14)for some constant C > E N ( g ) ≡ E N ( g ( t, x, v )) = X | α | N k ∂ αx g ( t, · , · ) k , HE INCOMPRESSIBLE NAVIER-STOKES-FOURIER LIMIT FROM BOLTZMANN-FERMI-DIRAC EQUATION 5 and the microscopic dissipation rate as (cid:0) D micN (cid:1) ( g )( t ) ≡ (cid:0) D micN (cid:1) ( g ( t, x, v )) = X | α | N k ∂ αx { I − P } g ( t, · , · ) k ν , the macroscopic dissipation rate as( D macN ) ( g )( t ) ≡ ( D macN ) ( g ( t, x, v )) = X < | α | N k ∂ αx P g ( t, · , · ) k ν . Clearly, (1.14) implies D macN C E N . (1.15)1.3. Main results.
Our main results are as follows: the first theorem is about the global existence ofthe Boltzmann-Fermi-Dirac equation (1.6) uniform with respect to the Knudsen number ǫ . Theorem 1.1.
Let < ǫ < . Assume F ǫ, ( x, v ) = µ + ǫµ (1 − µ ) g ǫ, , then there are constants δ > , c > , independent of ǫ , such that if E N ( g ǫ, ) δ , then there exists a unique global solution g ǫ ∈ L ∞ (cid:0) [0 , + ∞ ); H N (d x ; L ( µ (1 − µ )d v )) (cid:1) to the Cauchy problem (1.6) . Furthermore, F ǫ = µ + ǫµ (1 − µ ) g ǫ , and we have the followingglobal energy estimate sup t > E N ( g ǫ )( t ) + c ˆ ∞ ǫ (cid:0) D micN (cid:1) ( g ǫ )( t ) dt + c ˆ ∞ ( D macN ) ( g ǫ )( t ) dt E N ( g ǫ, ) . (1.16)The second theorem is about the limit to the incompressible Navier-Stokes-Fourier equations: E ∂ t u + E u · ∇ x u + ∇ x p = ν ∗ ∆ x u , ∇ x · u = 0 ,C A ∂ t θ + C A u · ∇ x θ = κ ∗ ∆ x θ, (1.17)where ν ∗ = (cid:10) β L ( | v | ) v v (cid:11) , κ ∗ = * α L ( | v | ) (cid:18) | v | − K A (cid:19) v + . One can find the definition of constants E , C A , K A , K g , positive functions α L ( | v | ) and β L ( | v | )appeared here in section 4. Theorem 1.2.
Let < ǫ < , and δ > be as in Theorem 1.1. For any ( ρ , u , θ ) ∈ H N (d x ) with k ( ρ , u , θ ) k H Nx < δ , and g ǫ, ∈ N ull ( L ) ⊥ with k e g ǫ, k H Nx L v < δ , let g ǫ, ( x, v ) = (cid:26) ρ ( x ) + u ( x ) · v + θ ( x ) (cid:18) | v | − K g (cid:19)(cid:27) + e g ǫ, ( x, v ) . Let g ǫ be the family of solutions to the Boltzmann-Fermi-Dirac equation (1.6) constructed in Theorem1.1. Then, g ǫ → u · v + θ (cid:18) | v | − K A (cid:19) , as ǫ → , where the convergence is weak- ⋆ for t , strongly in H N − η (d x ) for any η > , and weakly in L ( µ (1 − µ )d v ) , and (u , θ ) ∈ C (cid:0) [0 , ∞ ); H N − (d x ) (cid:1) ∩ L ∞ (cid:0) [0 , ∞ ); H N (d x ) (cid:1) is the solution of the incompressibleNavier-Stokes-Fourier equation (1.17) with initial data: u (cid:12)(cid:12) t =0 ( x ) = P u ( x ) , θ (cid:12)(cid:12) t =0 = K g θ ( x ) − ρ ( x ) K g + 1 , where P is the Leray projection. Furthermore, the convergence of the moments holds: as ǫ → P h g ǫ , v i → u , in C (cid:0) [0 , ∞ ); H N − − η (d x ) (cid:1) ∩ L ∞ (cid:0) [0 , ∞ ); H N − η (d x ) (cid:1) , (cid:28) g ǫ , C A (cid:18) | v | − K A (cid:19)(cid:29) → θ, in C (cid:0) [0 , ∞ ); H N − − η (d x ) (cid:1) ∩ L ∞ (cid:0) [0 , ∞ ); H N − η (d x ) (cid:1) for any η > . The paper is organized as follows: we give some basic estimates for the operators L , Q and T in thenext section; Then prove the global existence of solution to scaled Boltzmann-Fermi-Dirac equation(1.5) in section 3; In the last section, we prove the incompressible Navier-Stokes-Fourier limit. NING JIANG, LINJIE XIONG, AND KAI ZHOU Some basic estimates
This section is a preparation for energy estimates including microscopic and macroscopic estimatesin next section. The operators L , Q and T defined by (1.7), (1.11) and (1.12) are studied respectively.2.1. Properties of the linear operator L . The most crucial point in energy estimate is that L islocally coercive, namely, there holds h Lg, g i > λ |{ I − P } g | ν , ∀ g ∈ D ( L ) , for some constant λ >
0, where D ( L ) = (cid:8) g ∈ L ( µ (1 − µ )d v ) (cid:12)(cid:12) νg ∈ L ( µ (1 − µ )d v ) (cid:9) is the domain of L . For convenience, we list some results without proof since they are parallel to theresults in perturbed Boltzmann equation [24]. Proposition 2.1.
For the operator L = ν − K , we have the following results: • There are two constants C , C > such that C (1 + | v | ) ν ( v ) C (1 + | v | ) . (2.1) • The operator K : L ( µ (1 − µ )d v ) −→ L ( µ (1 − µ )d v ) is compact. • The linearized collision operator L is symmetric, nonnegative, and its null space is N ull ( L ) = span (cid:8) , v, | v | (cid:9) . (2.2) • There exists a constant λ > such that h Lg , g i > λ |{ I − P } g | ν . (2.3)We denote the global Maxwellian by M ( v ) = e − | v | . and the collision frequency in perturbed Boltzmann equation by ν M = ¨ R × S | v − v | | cos θ | M ( v )d ω d v . (2.1) holds since ν is just replacing M in ν M with N µ (1 − µ ) . The compactness of K is a consequence ofcompactness criterion in [45], and (2.3) is the corollary of the three properties above. For more details,see [24, 27, 49].2.2. Estimates for the nonlinear operators Q and T . In this subsection, we devote to L -typeestimates for the bilinear form Q ( f, g ) and the trilinear form T ( f, g, h ), which plays an important rolein the nonlinear problem. Before stating the estimates, we need a useful Proposition. Recall v ′ and v ′ as in ( ?? ), using the pole coordinates with the direction of v − v as the positive direction of Z -axis,then ω = ( ω , ω , ω ) = (sin θ cos ϕ, sin θ sin ϕ, cos θ ) , θ ∈ [0 , π ] , ϕ ∈ [0 , π ] , thus the rotation through π : ω ω ⊥ = ( − cos θ cos ϕ, − cos θ sin ϕ, sin θ ) ,θ π − θ, ϕ ϕ − π, yields: Proposition 2.2.
Let H be a measurable function with arguments v ′ and v ′ defined in ( ?? ) , then ˆ S | cos θ | H ( v ′ , v ′ )d ω = ˆ S | cos θ | H ( v ′ , v ′ )d ω. (2.4)Now we establish the estimates for Q and T . We always use the fact that M and µ (1 − µ ) arebounded by each other and the relation (2.1). HE INCOMPRESSIBLE NAVIER-STOKES-FOURIER LIMIT FROM BOLTZMANN-FERMI-DIRAC EQUATION 7
Lemma 2.1.
Let f, g, h and e h be smooth functions, then for Q ( f, g ) we have (cid:12)(cid:12)(cid:12)(cid:0) Q ( f, g ) , e h (cid:1)(cid:12)(cid:12)(cid:12) C ˆ R (cid:16) | ν / f | L v | g | L v + | f | L v | ν / g | L v (cid:17) (cid:12)(cid:12)(cid:12) ν / e h (cid:12)(cid:12)(cid:12) L v d x, (2.5) and (cid:13)(cid:13)(cid:13)(cid:10) Q ( f , g ) , e h (cid:11)(cid:13)(cid:13)(cid:13) C sup x,v (cid:12)(cid:12)(cid:12) ( µ (1 − µ )) e h (cid:12)(cid:12)(cid:12) sup x | f | L v k g k ; (2.6) for T ( f, g, h ) we have (cid:12)(cid:12)(cid:12)(cid:0) T ( f, g, h ) , e h (cid:1)(cid:12)(cid:12)(cid:12) C ˆ R (cid:16) | ν / f | L v | g | L v + | f | L v | ν / g | L v (cid:17) | h | L v (cid:12)(cid:12)(cid:12) ν / e h (cid:12)(cid:12)(cid:12) L v d x, (2.7) and (cid:13)(cid:13)(cid:13)D T ( f , g, h ) , e h E(cid:13)(cid:13)(cid:13) C sup x,v (cid:12)(cid:12)(cid:12) ( µ (1 − µ )) e h (cid:12)(cid:12)(cid:12) sup x | f | L v sup x | g | L v k h k . (2.8) where ( f , g ) and ( f , g, h ) are permutations of ( f, g ) and ( f, g, h ) respectively.Proof. For simplicity, we suppress the x -dependence in f ( x, v ) , g ( x, v ) , h ( x, v ) and e h ( x, v ), to prove(2.5), it’s sufficient to establish the same inequality for Q i , i = 1 , , · · · ,
6. Recall that Q ( f, g ) = ¨ R × S | v − v | | cos θ | N µ (1 − µ ) (cid:18) − µ ′ − µ ′ (cid:19) f ′ g ′ d ω d v , and Q ( f, g ) = − ¨ R × S | v − v | | cos θ | N µ (1 − µ ) (cid:18) − µ − µ (cid:19) f g d ω d v , note that N µ (1 − µ ) (cid:18) − µ ′ − µ ′ (cid:19) , N µ (1 − µ ) (cid:18) − µ − µ (cid:19) is bounded by M , thus, similar to Lemma 2.3 in [26], we have for i = 1 , (cid:12)(cid:12)(cid:12)(cid:10) Q i ( f, g ) , e h (cid:11)(cid:12)(cid:12)(cid:12) C (cid:16) | ν / f | L v | g | L v + | f | L v | ν / g | L v (cid:17) | ν / e h | L v . (2.9)In the following, we will deal with the nonlinear terms that are different from those in the case ofperturbed Boltzmann equation. For Q ( f, g ) = ¨ R × S | v − v | | cos θ | N µ (1 − µ ) (cid:18) µ ′ f ′ + µ ′ f ′ (cid:19) f d ω d v we have (cid:12)(cid:12)(cid:12)(cid:10) Q ( f, g ) , e h (cid:11)(cid:12)(cid:12)(cid:12) C ˚ R × R × S | v − v | | cos θ | M M M ′ | f ′ | | f | | e h | d ω d v d v + C ˚ R × R × S | v − v | | cos θ | M M M ′ | f ′ | | f | | e h | d ω d v d v, Due to Proposition 2.2, we need only to estimate the second term on the right hand side above. UsingCauchy-Schwarz inequality to get (cid:12)(cid:12)(cid:12)(cid:10) Q ( f, g ) , e h (cid:11)(cid:12)(cid:12)(cid:12) (cid:18) ˚ R × R × S | v ′ − v ′ | M M M ′ | f ′ | d ω d v ′ d v ′ (cid:19) × (cid:18) ˚ R × R × S | v − v | M M M ′ | f | | e h | d ω d v d v (cid:19) C ˆ S ˆ R v ′ | f ′ | M ′ d v ′ ˆ R v ′ | v ′ − v ′ | M ′ M ′ d v ′ ! d ω | g | L v | ν / e h | L v C | ν / f | L v | g | L v | ν / e h | L v . (2.10)where we have used that | v − v | = | v ′ − v ′ | and | v − v | M C (1 + | v | ). NING JIANG, LINJIE XIONG, AND KAI ZHOU
The estimate for Q is same as Q , we also have (cid:12)(cid:12)(cid:12)(cid:10) Q ( f, g ) , e h (cid:11)(cid:12)(cid:12)(cid:12) C | ν / f | L v | g | L v | ν / e h | L v . (2.11)The estimate for Q and Q need more care. Recall that Q ( f, g ) = ¨ R × S | v − v | | cos θ | N µ (1 − µ ) (cid:18) µ ′ f ′ + µ ′ f ′ (cid:19) g d ω d v By Proposition 2.2, using Cauchy-Schwarz inequality we have (cid:12)(cid:12)(cid:12)(cid:10) Q ( f, g ) , e h (cid:11)(cid:12)(cid:12)(cid:12) C ˚ R × R × S | v − v | | cos θ | M M M ′ | f ′ | | g | | e h | d ω d v d v + C ˚ R × R × S | v − v | | cos θ | M M M ′ | f ′ | | g | | e h | d ω d v d v C ˆ R | g | | e h | M d v (cid:18) ˆ R M d v (cid:19) (cid:18) ¨ R × S | v − v | | cos θ | M M ′ | f ′ | d v (cid:19) , using the notation V k = ( V · ω ) ω, V ⊥ = V − ( V · ω ) ω, (2.12)then the variable changing v − v V and the fact (see [24]) dωdV = 2 dV k dV ⊥ | V k | , (2.13)implies (cid:12)(cid:12)(cid:12)(cid:10) Q ( f, g ) , e h (cid:11)(cid:12)(cid:12)(cid:12) C ˆ R v | g | | e h | M d v × ˆ R V k | V k | " ˆ R V ⊥ M ( v + V ) M ( v + V k )d V ⊥ | f ( v + V k ) | M ( v + V k )d V k | V k | C ˆ R v | g | | e h | M d v ˆ R V k " ˆ R V ⊥ M ( V ⊥ )d V ⊥ | f ( v + V k ) | M ( v + V k )d V k / C | f | L v | g | L v | e h | L v . (2.14)here we have used the fact 12 | v + V | + 12 | v + V k | > | V ⊥ | . Similar to Q , we also have (cid:12)(cid:12)(cid:12)(cid:10) Q ( f, g ) , e h (cid:11)(cid:12)(cid:12)(cid:12) C | f | L v | g | L v | e h | L v . (2.15)Then by summing up (2.9)-(2.11), (2.14)-(2.15) and further integrating over R x , we complete the proofof (2.5).To prove (2.6), we need only to put the weight function ν ( v ) on the function e h when we deal with D Q ( f, g ) , e h E , D Q ( f, g ) , e h E , · · · , D Q ( f, g ) , e h E . We can bound D Q ( f, g ) , e h E by C (cid:20) ˆ R | f ( x, v ) | µ (1 − µ )d v (cid:21) / (cid:20) ˆ R | g ( x, v ) | µ (1 − µ )d v (cid:21) / (cid:20) ˆ R ν ( v ) | e h ( x, v ) | µ (1 − µ )d v (cid:21) / C sup x,v (cid:12)(cid:12)(cid:12) ( µ (1 − µ )) e h ( x, v ) (cid:12)(cid:12)(cid:12) (cid:20) ˆ R | f ( x, v ) | µ (1 − µ )d v (cid:21) / (cid:20) ˆ R | g ( x, v ) | µ (1 − µ )d v (cid:21) / . We conclude that (2.6) holds by integrating over x and taking L and L ∞ norm in x for the last twofactors. HE INCOMPRESSIBLE NAVIER-STOKES-FOURIER LIMIT FROM BOLTZMANN-FERMI-DIRAC EQUATION 9
We then turn to study the trilinear form T ( f, g, h ). For T ( f, g, h ), we have | T ( f, g, h ) | = (cid:12)(cid:12)(cid:12)(cid:12) ¨ R × S | v − v | | cos θ | N µ − µ f g (cid:18) h ′ + h ′ (cid:19) d ω d v (cid:12)(cid:12)(cid:12)(cid:12) C ¨ R × S | v − v | | cos θ | M M | f gh ′ | d ω d v + C ¨ R × S | v − v | | cos θ | M M | f gh ′ | d ω d v △ = I + I . By Proposition 2.2, it’s sufficient to estimate I . We use Cauchy-Schwarz inequality to get I (cid:18) ¨ R × S | v − v | | cos θ | M M | f g | d ω d v (cid:19) / × (cid:18) ¨ R × S | v − v | | cos θ | M M | h ′ | d ω d v (cid:19) / . Note that | v − v | M M is bounded, the first factor above is bounded by | f | L v | g | . The second factorneed more careful estimate, note that M M = M ′ M ′ , the variable changing v − v V and (2.12),(2.13) imply the second factor is bounded by (cid:18) ¨ R × R | V k | M / M ( v + V ) | h ( v + V k ) | M ( v + V k ) | V k | d V ⊥ d V k (cid:19) / C ˆ R V k | V k | M ( V k ) | h ( v + V k ) | M ( v + V k )d V k ˆ R V ⊥ M ( V ⊥ )d V ⊥ / C | h | L v , here we have used M / ( v ) M ( v + V ) M ( V ). Then I C | f | L v | h | L v | g | , hence (cid:12)(cid:12)(cid:12)D T ( f, g, h ) , e h E(cid:12)(cid:12)(cid:12) C | f | L v | g | L v | h | L v | e h | L v . (2.16)For T ( f, g, h ), we have | T ( f, g, h ) | = (cid:12)(cid:12)(cid:12)(cid:12) ¨ R × S | v − v | | cos θ | N µ ′ µ ′ µ (1 − µ ) f ′ g ′ (cid:18) h + h (cid:19) d ω d v (cid:12)(cid:12)(cid:12)(cid:12) C ¨ R × S | v − v | | cos θ | M M ′ M ′ | h || f ′ g ′ | d ω d v + C ¨ R × S | v − v | | cos θ | M M ′ M ′ | h || f ′ g ′ | d ω d v △ = I + I . Next we use different methods to estimate D I , e h E and D I , e h E respectively. For D I , e h E , we useCauchy-Schwarz inequality to get (cid:12)(cid:12)(cid:12)D I , e h E(cid:12)(cid:12)(cid:12) C ˚ R × R × S | v − v | | cos θ || f ′ g ′ || h || e h | M M M ′ M ′ d ω d v d v C (cid:18) ˚ R × R × S | f ′ | | g ′ | M ′ M ′ d ω d v ′ d v ′ (cid:19) / × (cid:18) ˚ R × R × S | h | | e h | M M d ω d v d v (cid:19) / C | f | L v | g | L v | h | L v | e h | L v , (2.17)here we have used that | v − v | M M is bounded and d ω d v ′ d v ′ = d ω d v d v . We further estimate I then D I , e h E . Using Cauchy-Schwarz inequality again to get I C (cid:18) ¨ R × S | v − v | | cos θ | M M ′ M ′ | f ′ | d ω d v (cid:19) / × (cid:18) ¨ R × S | v − v | | cos θ | M M ′ M ′ | g ′ | d ω d v (cid:19) / | h | . By Proposition 2.2, we need only to estimate the second factor on the right hand side above, thenusing variable changing v − v V and (2.12), (2.13), it’s bounded by (cid:18) ¨ R × S | v − v | | cos θ | M M | f ′ | M ′ d ω d v (cid:19) / C (cid:18) ¨ R × R | V k | M M ( v + V ) | f ( v + V k ) | M ( v + V k ) | V k | d V ⊥ d V k (cid:19) / C (cid:18) ˆ R | V k | M ( V k ) | f ( v + V k ) | M ( v + V k )d V k ˆ R M ( V ⊥ )d V ⊥ (cid:19) / C | f | L v , where we have used the fact M M ( v + V ) M ( V ). The estimate for the second factor is same asabove. Then we get (cid:12)(cid:12)(cid:12)D I , e h E(cid:12)(cid:12)(cid:12) C | f | L v | g | L v | h | L v | e h | L v , (2.18)hence the estimates (2.17) and (2.18) imply that (cid:12)(cid:12)(cid:12)D T ( f, g, h ) , e h E(cid:12)(cid:12)(cid:12) C | f | L v | g | L v | h | L v | e h | L v . (2.19)For T ( f, g, h ), we have | T ( f, g, h ) | = (cid:12)(cid:12)(cid:12)(cid:12) ¨ R × S | v − v | | cos θ | N µ (1 − µ ) µ µ ′ (cid:18) f ′ g ′ h − f gh ′ (cid:19) d ω d v (cid:12)(cid:12)(cid:12)(cid:12) C ¨ R × S | v − v | | cos θ | M M ′ | f ′ g ′ || h | d ω d v + C ¨ R × S | v − v | | cos θ | M M ′ | f g || h ′ | d ω d v △ = I + I . HE INCOMPRESSIBLE NAVIER-STOKES-FOURIER LIMIT FROM BOLTZMANN-FERMI-DIRAC EQUATION 11
Then for I , we use Cauchy-Schwarz inequality to get (cid:12)(cid:12)(cid:12)D I , e h E(cid:12)(cid:12)(cid:12) C (cid:18) ˚ R × R × S | v − v | | cos θ | M | f ′ | | g ′ | M ′ M ′ d ω d v ′ d v ′ (cid:19) / (cid:18) ˚ R × R × S | v − v | | cos θ | M | h | | e h | M M d ω d v d v (cid:19) / C (cid:18) ˚ R × R × S ( ν ( v ′ ) + ν ( v ′ )) | f ′ | | g ′ | M ′ M ′ d ω d v ′ d v ′ (cid:19) / (cid:18) ˚ R × R × S ν ( v ) | h | | e h | M M d ω d v d v (cid:19) / C (cid:16) | ν / f | L v | g | L v + | f | L v | ν / g | L v (cid:17) | h | L v | ν / e h | L v , (2.20)where we have used | v − v | M C (1 + | v | ) C ( ν ( v ′ ) + ν ( v ′ )). For I , by Proposition 2.2, usingCauchy-Schwarz inequality again to get I C (cid:18) ¨ R × S | f | M d ω d v (cid:19) / (cid:18) ¨ R × S ( | v − v || cos θ | ) M ′ M | h ′ | d ω d v (cid:19) / | g | C | f | L v | g | (cid:18) ¨ R × R | V k | M ( v + V k ) M ( v + V ) | h ( v + V k ) | M ( v + V k ) | V k | d V ⊥ d V k (cid:19) / C | f | L v | h | L v | g | , here we have used M ( v + V k ) M ( v + V ) M ( V ⊥ ). Then (cid:12)(cid:12)(cid:12)D I , e h E(cid:12)(cid:12)(cid:12) C | f | L v | g | L v | h | L v | e h | L v , (2.21)hence by (2.20) and (2.21) we get (cid:12)(cid:12)(cid:12)D T ( f, g, h ) , e h E(cid:12)(cid:12)(cid:12) C (cid:16) | ν / f | L v | g | L v + | f | L v | ν / g | L v (cid:17) | h | L v | ν / e h | L v . (2.22)For T ( f, g, h ), we have | T ( f, g, h ) | = (cid:12)(cid:12)(cid:12)(cid:12) ¨ R × S | v − v | | cos θ | N µ ′ − µ (cid:18) f ′ g ′ h − f gh ′ (cid:19) d ω d v (cid:12)(cid:12)(cid:12)(cid:12) C ¨ R × S | v − v | | cos θ | M M M ′ | f ′ g ′ || h | d ω d v + C ¨ R × S | v − v | | cos θ | M M M ′ | f g || h ′ | d ω d v △ = I + I . Using the same way to treat I and I as to treat I , we get I C | f | L v | g | L v | h | , I C | f | L v | h | L v | g | . Hence (cid:12)(cid:12)(cid:12)D T ( f, g, h ) , e h E(cid:12)(cid:12)(cid:12) C | f | L v | g | L v | h | L v | e h | L v . (2.23)For the rest terms in T ( f, g, h ), we have the following relations: • The estimate for T ( f, g, h ) is similar to the estimate for T ( f, g, h ), we have (cid:12)(cid:12)(cid:12)D T ( f, g, h ) , e h E(cid:12)(cid:12)(cid:12) C (cid:16) | ν / f | L v | g | L v + | f | L v | ν / g | L v (cid:17) | h | L v | ν / e h | L v . (2.24) • The estimate for T ( f, g, h ) is similar to the estimate for T ( f, g, h ), we have (cid:12)(cid:12)(cid:12)D T ( f, g, h ) , e h E(cid:12)(cid:12)(cid:12) C | f | L v | g | L v | h | L v | e h | L v . (2.25) • The estimate for T ( f, g, h ) is similar to the estimate for T ( f, g, h ), we have (cid:12)(cid:12)(cid:12)D T ( f, g, h ) , e h E(cid:12)(cid:12)(cid:12) C | f | L v | g | L v | h | L v | e h | L v . (2.26) • The estimate for T ( f, g, h ) is similar to the estimate for T ( f, g, h ), we have (cid:12)(cid:12)(cid:12)D T ( f, g, h ) , e h E(cid:12)(cid:12)(cid:12) C (cid:16) | ν / f | L v | g | L v + | f | L v | ν / g | L v (cid:17) | h | L v | ν / e h | L v . (2.27)Thus the relations above combining with (2.16), (2.19), (2.22) and (2.23) imply that (cid:12)(cid:12)(cid:12)D T ( f, g, h ) , e h E(cid:12)(cid:12)(cid:12) C (cid:16) | ν / f | L v | g | L v + | f | L v | ν / g | L v (cid:17) | h | L v | ν / e h | L v , (2.28)we get (2.7) by further integrating over x . To prove (2.8), we need only to put the weight function ν ( v ) on the function e h in the estimates for T ( f, g, h ) , · · · , T ( f, g, h ), we can bound D T ( f, g, h ) , e h E by C (cid:20) ˆ R | f ( x, v ) | µ (1 − µ )d v (cid:21) / (cid:20) ˆ R | g ( x, v ) | µ (1 − µ )d v (cid:21) / × (cid:20) ˆ R | h ( x, v ) | µ (1 − µ )d v (cid:21) / (cid:20) ˆ R ν ( v ) | e h ( x, v ) | µ (1 − µ )d v (cid:21) / C sup x,v (cid:16) ( µ (1 − µ )) | e h ( x, v ) | (cid:17) (cid:20) ˆ R | f ( x, v ) | µ (1 − µ )d v (cid:21) / × (cid:20) ˆ R | g ( x, v ) | µ (1 − µ )d v (cid:21) / (cid:20) ˆ R | h ( x, v ) | µ (1 − µ )d v (cid:21) / We conclude that (2.8) holds by integrating over R x and taking L and L ∞ norm in x for the lastthree factors. (cid:3) As a corollary, we want to give higher x -derivative estimates of nonlinear terms. If we apply themacro-micro decomposition as in (1.13) to f and g , and plug them into (2.5) and (2.7), then by theSobolev embedding H (d x ) ֒ → L ∞ (d x ) we have (cid:12)(cid:12)(cid:12)(cid:0) Q ( f , g ) , e h (cid:1)(cid:12)(cid:12)(cid:12) C h k f k H x L v (cid:16) k P g k L x,v + k ν / { I − P } g k L x,v (cid:17) + k ν / { I − P } f k H x L v k g k L x,v i k ν / e h k L x,v , (2.29)and (cid:12)(cid:12)(cid:12)(cid:0) T ( f , g, h ) , e h (cid:1)(cid:12)(cid:12)(cid:12) C h k f k H x L v (cid:16) k P g k L x,v + k ν / { I − P } g k L x,v (cid:17) + k ν / { I − P } f k H x L v k g k L x,v i k h k H x L v k ν / e h k L x,v , (2.30)where ( f , g ) is a permutation of ( f, g ). If in the estimate of (cid:0) T ( f , g, h ) , e h (cid:1) , we apply the Sobolevembedding H (d x ) ֒ → L ∞ (d x ) to f and g , we also have (cid:12)(cid:12)(cid:12)(cid:0) T ( f , g, h ) , e h (cid:1)(cid:12)(cid:12)(cid:12) C (cid:16) k f k H x L v k g k H x L v + k f k H x L v k ν / { I − P } g k H x L v + k ν / { I − P } f k H x L v k g k H x L v (cid:17) k h k L x,v k ν / e h k L x,v , (2.31)3. The existence of global solutions
We use the standard strategy to obtain the global existence of solution to (1.6): construct the localsolutions for a sequence of iterating approximate equations and take limit to get local solution to (1.6);then obtain a uniform energy estimate; finally, the use of the continuation argument. The uniformenergy estimate consists of two parts: one part is to obtain microscopic dissipation rate, the morecomplicated part is to deal with macroscopic dissipation rate.3.1.
Construction of local solutions.
In order to prove the existence of nonlinear problem (1.6),we shall firstly study the following linear Cauchy problem: ∂ t g + 1 ǫ v · ∇ x g + 1 ǫ Lg = 1 ǫ Q ( f, f ) + T ( f, f, f ) ,g ( t, x, v ) | t =0 = g ( x, v ) , (3.1)where f is a given function. For (3.1), we have the following result: HE INCOMPRESSIBLE NAVIER-STOKES-FOURIER LIMIT FROM BOLTZMANN-FERMI-DIRAC EQUATION 13
Lemma 3.1.
For < ǫ < , assume that g ∈ H N (d x ; L ( µ (1 − µ )d v )) , and that for some T > , f satisfies sup t T E N ( f )( t ) + ˆ T (cid:0) D micN (cid:1) ( f )( t ) dt < + ∞ . Then, the Cauchy problem (3.1) admits a unique solution: g ( t, x, v ) ∈ L ∞ (cid:0) [0 , T ]; H N (d x ; L ( µ (1 − µ )d v )) (cid:1) Proof.
We prove the existence of a solution to the Cauchy problem (3.1) by the Hahn-Banach theorem.We define the linear operator T by T g ≡ ∂ t g + 1 ǫ v · ∇ x g + 1 ǫ Lg.
Then, we rewrite (3.1) into the following form: T g = 1 ǫ Q ( f, f ) + T ( f, f, f ) , g (0) = g . For h ∈ C ∞ (cid:0) [0 , T ]; S ( R x,v ) (cid:1) with h ( T ) = 0, we define T ∗ through( g, T ∗ h ) L ([0 ,T ]; H Nx L v ) = ( T g, h ) L ([0 ,T ]; H Nx L v ) , so that T ∗ is the adjoint of the operator T in the Hilbert space L (cid:0) [0 , T ]; H N (d x ; L ( µ (1 − µ )d v )) (cid:1) .We set W = (cid:8) w = T ∗ h (cid:12)(cid:12) h ∈ C ∞ (cid:0) [0 , T ]; S ( R x,v ) (cid:1) with h ( T ) = 0 (cid:9) , which is a dense subspace of L (cid:0) [0 , T ]; H N (d x ; L ( µ (1 − µ )d v )) (cid:1) . We also have that T ∗ h = − ∂ t h − ǫ v · ∇ x h + 1 ǫ Lh.
Then ( h, T ∗ h ) H Nx L v = −
12 dd t k h ( t ) k H Nx L v − ǫ ( v · ∇ x h, h ) H Nx L v + 1 ǫ ( Lh, h ) H Nx L v . Note that the second term above vanishes, and by the estimate (2.3), we have that ˆ Tt ( h, T ∗ h ) H Nx L v dt > k h ( t ) k H Nx L v + λǫ ˆ Tt (cid:0) D micN (cid:1) ( h )( s ) ds. Thus, for all 0 < t < T , k h ( t ) k H Nx L v + 2 λǫ ˆ Tt (cid:0) D micN (cid:1) ( h )( s ) ds kT ∗ h k L ([ t,T ]; H Nx L v ) k h k L ([ t,T ]; H Nx L v ) . This implies that k h ( t ) k L ∞ ([0 ,T ]; H Nx L v ) + 2 λǫ ˆ Tt (cid:0) D micN (cid:1) ( h )( s ) ds kT ∗ h k L ([0 ,T ]; H Nx L v ) k h k L ([0 ,T ]; H Nx L v ) √ T kT ∗ h k L ([0 ,T ]; H Nx L v ) k h k L ∞ ([0 ,T ]; H Nx L v ) . (3.2)This immediately gives k h k L ∞ ([0 ,T ]; H Nx L v ) √ T kT ∗ h k L ([0 ,T ]; H Nx L v ) . (3.3)Furthermore, (3.2) can be written as follows: let Y = k h k L ∞ ([0 ,T ]; H Nx L v ) , λǫ ˆ Tt (cid:0) D micN (cid:1) ( h )( s ) ds √ T kT ∗ h k L ([0 ,T ]; H Nx L v ) Y − Y T kT ∗ h k L ([0 ,T ]; H Nx L v ) . Thus, 1 ǫ ˆ Tt (cid:0) D micN (cid:1) ( h )( s ) ds ! / √ T √ λ kT ∗ h k L ([0 ,T ]; H Nx L v ) . (3.4) Next, we define a functional G on W as follows: G ( w ) = (cid:0) ǫ Q ( f, f ) + T ( f, f, f ) , h (cid:1) L ([0 ,T ]; H Nx L v ) + (cid:0) g , h (0) (cid:1) H Nx L v , note that 0 < ǫ <
1, using (3.19) and (3.26), we have the estimate | G ( w ) | ǫ ( ˆ T (cid:0) E N ( f ) + E N ( f ) (cid:1) (cid:0) E N ( f ) + D micN ( f ) (cid:1) D micN ( h ) dt ) + k g k H Nx L v k h (0) k H Nx L v Cǫ sup t ∈ [0 ,T ] (cid:16) E N ( f )( t ) + E N ( f )( t ) (cid:17) √ T sup t ∈ [0 ,T ] E N ( f )( t ) + ˆ T (cid:0) D micN (cid:1) ( f )( t ) dt ! / × ˆ T (cid:0) D micN (cid:1) ( h ( t )) dt ! / + k g k H Nx L v k h k L ∞ ([0 ,T ]; H Nx L v ) . Finally (3.3) and (3.4) imply that | G ( w ) | C f,g kT ∗ h k L ([0 ,T ]; H Nx L v ) C k w k L ([0 ,T ]; H Nx L v ) , where C f,g = 2 √ T k g k H Nx L v + C √ T sup t ∈ [0 ,T ] (cid:16) E N ( f )( t ) + E N ( f )( t ) (cid:17) ( √ T sup t ∈ [0 ,T ] E N ( f )( t )+ ˆ T (cid:0) D micN (cid:1) ( f )( t ) dt ! / . Thus, G is a continuous linear functional on (cid:0) W ; k · k L ([0 ,T ]; H Nx L v ) (cid:1) . So by the Hahn-BanachTheorem, G can be extended from W to L (cid:0) [0 , T ]; H N (d x ; L ( µ (1 − µ )d v )) (cid:1) . From the Riesz repre-sentation theorem, there exists g ∈ L (cid:0) [0 , T ]; H N (d x ; L ( µ (1 − µ )d v )) (cid:1) , such that for any w ∈ W , G ( w ) = ( g, w ) L ([0 ,T ]; H Nx L v ) . Thus fixing f ∈ L ∞ (cid:0) [0 , T ]; H N (d x ; L ( µ (1 − µ )d v )) (cid:1) , by the definitions of the operator T ∗ and G , wehave for any h ∈ C ∞ (cid:0) [0 , T ]; S ( R x,v ) (cid:1) with h ( T ) = 0, there exists a unique g ∈ L (cid:0) [0 , T ]; H N (d x ; L ( µ (1 − µ )d v )) (cid:1) such that, ( T g, b h ) L ([0 ,T ]; L x L v ) =( T g, h ) L ([0 ,T ]; H Nx L v ) =( g, T ∗ h ) L ([0 ,T ]; H Nx L v ) = (cid:0) ǫ Q ( f, f ) + T ( f, f, f ) , b h (cid:1) L ([0 ,T ]; L x L v ) + (cid:0) g , b h (0) (cid:1) L x L v . (3.5)where b h = Λ Nx h ∈ C ∞ (cid:0) [0 , T ]; S ( R x,v ) (cid:1) with b h ( T ) = 0 , and Λ = (1 − ∆ x ) / , Λ Nx is used for changing the inner product from H N (d x ; L ( µ (1 − µ )d v )) to L (d x ; L ( µ (1 − µ )d v ) . Since Λ Nx is an isomorphism on h : h ∈ C ∞ (cid:0) [0 , T ]; S ( R x,v ) (cid:1) with h ( T ) = 0, g ∈ L (cid:0) [0 , T ]; H N (d x ; L ( µ (1 − µ )d v )) (cid:1) is a solution of the Cauchy problem (3.1). Furthermore, using (3.5), we also havesup t T E N ( g )( t ) + 1 ǫ ˆ T (cid:0) D micN (cid:1) ( g )( t ) dt < e C f,g , HE INCOMPRESSIBLE NAVIER-STOKES-FOURIER LIMIT FROM BOLTZMANN-FERMI-DIRAC EQUATION 15 here e C f,g = C sup t ∈ [0 ,T ] (cid:16) E N ( f )( t ) + E N ( f )( t ) (cid:17) ( T sup t ∈ [0 ,T ] E N ( f )( t ) + ˆ T (cid:0) D micN (cid:1) ( f )( t ) dt ) + E N ( g ) (cid:3) Now we concentrate on the local existence of solutions for fully nonlinear problem. Consider thefollowing iterative scheme: ∂ t g n +1 + 1 ǫ v · ∇ x g n +1 + 1 ǫ Lg n +1 = 1 ǫ Q ( g n , g n ) + T ( g n , g n , g n ) ,g n +1 ( t, x, v ) | t =0 = g ( x, v ) , (3.6)with g = 0. Lemma 3.2.
There exist constants < δ and < T such that for any < ǫ < , if g ∈ H N (d x ; L ( µ (1 − µ )d v )) with E N ( g ) δ , then the iteration problem (3.6) admits a sequence ofsolutions { g n } n > satisfying sup t ∈ [0 ,T ] E N ( g n )( t ) + 1 ǫ ˆ T (cid:0) D micN (cid:1) ( g n )( t ) dt E N ( g ) (3.7) Proof.
It is enough to prove (3.7) by induction, since for the linear Cauchy problem (3.6), given g n satisfying (3.7), the existence of g n +1 is assured by Lemma 3.1. Applying ∂ αx to (3.6), then takinginner product with ∂ αx g n +1 in L ( µ (1 − µ )d v d x ), using (2.3), (3.19) and (2.24), we get12 dd t E N ( g n +1 ) + λǫ (cid:0) D micN (cid:1) ( g n +1 ) Cǫ (cid:0) E N ( g n ) + E N ( g n ) (cid:1) (cid:8) D macN ( g n ) + D micN ( g n ) (cid:9) D micN ( g n +1 ) C η ǫ (cid:0) D micN (cid:1) ( g n +1 ) + 12 η (cid:0) E N ( g n ) + E N ( g n ) (cid:1) (cid:16) ( D macN ) ( g n ) + (cid:0) D micN (cid:1) ( g n ) (cid:17) , in the last step we have used Cauchy-Schwarz inequality with η >
0. Thus let C η = λ , we getdd t E N ( g n +1 ) + λǫ (cid:0) D micN (cid:1) ( g n +1 ) C λ (cid:0) E N ( g n ) + E N ( g n ) (cid:1) (cid:16) ( D macN ) ( g n ) + (cid:0) D micN (cid:1) ( g n ) (cid:17) . Note that we have (3.17), integrating on [0 , T ] with T t ∈ [0 ,T ] E N ( g n +1 )( t ) + 1 ǫ ˆ T (cid:0) D micN (cid:1) ( g n +1 )( t ) dt E N ( g ) + C sup t ∈ [0 ,T ] (cid:0) E N ( g n ) + E N ( g n ) (cid:1) · ( T sup t ∈ [0 ,T ] E N ( g n ( t )) + ˆ T (cid:0) D micN (cid:1) ( g n ( t )) dt ) , we chose δ > C δ (cid:0) δ (cid:1) , to complete the proof. (cid:3) Finally, we prove the convergence of { g n } using the uniform estimate (3.7). Theorem 3.1 (Local existence) . There exist constants < δ and < T such that for any < ǫ < , if g ǫ, ∈ H N (d x ; L ( µ (1 − µ )d v )) with E N ( g ǫ, ) δ , then there is a unique solution g ǫ ( t, x, v ) ∈ L ∞ (cid:0) [0 , T ]; H N (d x ; L ( µ (1 − µ )d v )) (cid:1) to the Boltzmann-Fermi-Dirac equation (3.8) such that sup t ∈ [0 ,T ] E N ( g ǫ )( t ) + 1 ǫ ˆ T (cid:0) D micN (cid:1) ( g ǫ )( t ) dt E N ( g ǫ, ) . (3.8) Moreover, E N ( g ǫ )( t ) : [0 , T ] → R is continuous. If F (0 , x, v ) = µ + µ (1 − µ ) g ( x, v ) , then F ( t, x, v ) = µ + µ (1 − µ ) g ( t, x, v ) . Proof.
We prove that { g n } is a Cauchy sequence in L ∞ (cid:0) [0 , T ]; L ( µ (1 − µ )d v d x ) (cid:1) . Setting w n = g n +1 − g n , we deduce from (3.6) ∂ t w n + 1 ǫ v · ∇ x w n + 1 ǫ Lw n = (cid:18) ǫ Q ( g n , g n ) + T ( g n , g n , g n ) (cid:19) − (cid:18) ǫ Q ( g n − , g n − ) + T ( g n − , g n − , g n − ) (cid:19) ,w n | t =0 = 0 . Since for any h ∈ L ( µ (1 − µ )), (cid:28)(cid:18) ǫ Q ( g n , g n ) + T ( g n , g n , g n ) (cid:19) − (cid:18) ǫ Q ( g n − , g n − ) + T ( g n − , g n − , g n − ) (cid:19) , P h (cid:29) = 0 (cid:18) ǫ Q ( g n , g n ) + T ( g n , g n , g n ) (cid:19) − (cid:18) ǫ Q ( g n − , g n − ) + T ( g n − , g n − , g n − ) (cid:19) = 1 ǫ (cid:2) Q ( ǫw n − , ǫg n ) + T ( ǫw n − , ǫg n , ǫg n ) (cid:3) + 1 ǫ (cid:2) Q ( ǫg n − , ǫw n − ) + T ( ǫg n − , ǫw n − , ǫg n ) (cid:3) + T ( g n − , g n − , w n − ) , then we have (cid:18)(cid:18) ǫ Q ( g n , g n ) + T ( g n , g n , g n ) (cid:19) − (cid:18) ǫ Q ( g n − , g n − ) + T ( g n − , g n − , g n − ) (cid:19) , w n (cid:19) = 1 ǫ (cid:16) Q ( ǫw n − , ǫg n ) + T ( ǫw n − , ǫg n , ǫg n ) , { I − P } w n (cid:17) + 1 ǫ (cid:0) Q ( ǫg n − , ǫw n − ) + T ( ǫg n − , ǫw n − , ǫg n ) , { I − P } w n (cid:1) + (cid:16) T ( g n − , g n − , w n − ) , { I − P } w n (cid:17) . We need to estimate the three terms on the right hand side above. Note that 0 < ǫ <
1, by (2.29),(2.30) and the Cauchy-Schwarz inequality with η >
0, the first term is bounded by Cǫ (cid:8) E ( w n − ) D micN ( g n ) + E N ( g n ) (cid:0) E ( w n − ) + D mic ( w n − ) (cid:1)(cid:9) (1 + ǫ E N ( g n )) D mic ( w n ) C η n E ( w n − ) (cid:0) D micN (cid:1) ( g n ) + E N ( g n ) (cid:16) E ( w n − ) + (cid:0) D mic (cid:1) ( w n − ) (cid:17)o (cid:0) E N ( g n ) (cid:1) + ηǫ (cid:0) D mic (cid:1) ( w n ) , (3.9)the second term is bounded by Cǫ (cid:8) E ( w n − ) D micN ( g n − ) + E N ( g n − ) (cid:0) E ( w n − ) + D mic ( w n − ) (cid:1)(cid:9) (1 + ǫ E N ( g n )) D mic ( w n ) C η n E ( w n − ) (cid:0) D micN (cid:1) ( g n − ) + E N ( g n − ) (cid:16) E ( w n − ) + (cid:0) D mic (cid:1) ( w n − ) (cid:17)o (cid:0) E N ( g n ) (cid:1) + ηǫ (cid:0) D mic (cid:1) ( w n ) , (3.10)and the third term of is bounded by C (cid:8) E N ( g n − ) + D micN ( g n − ) (cid:9) E N ( g n − ) E ( w n − ) D mic ( w n ) C η n E N ( g n − ) + (cid:0) D micN (cid:1) ( g n − ) o E N ( g n − ) E ( w n − ) + η (cid:0) D mic (cid:1) ( w n ) . (3.11)Thus we have the estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) (cid:18) ǫ Q ( g n , g n ) + T ( g n , g n , g n ) (cid:19) − (cid:18) ǫ Q ( g n − , g n − ) + T ( g n − , g n − , g n − ) (cid:19) , w n (cid:17) L x,v (cid:12)(cid:12)(cid:12)(cid:12) C η (cid:26) E ( w n − ) (cid:0) E N ( g n ) (cid:1) h E N ( g n ) + (cid:0) D micN (cid:1) ( g n ) i + E ( w n − ) (cid:0) E N ( g n ) + E N ( g n − ) (cid:1) h E N ( g n − ) + (cid:0) D micN (cid:1) ( g n − ) i + (cid:0) D micN (cid:1) ( w n − ) (cid:0) E N ( g n ) (cid:1) (cid:2) E N ( g n ) + E N ( g n − ) (cid:3) (cid:27) + 3 ηǫ (cid:0) D mic (cid:1) ( w n ) . (3.12) HE INCOMPRESSIBLE NAVIER-STOKES-FOURIER LIMIT FROM BOLTZMANN-FERMI-DIRAC EQUATION 17
Let 3 η = λ , we get12 dd t k w n k + λ ǫ (cid:0) D micN (cid:1) ( w n ) C η (cid:26) E ( w n − ) (cid:0) E N ( g n ) (cid:1) h E N ( g n ) + (cid:0) D micN (cid:1) ( g n ) i + E ( w n − ) (cid:0) E N ( g n ) + E N ( g n − ) (cid:1) h E N ( g n − ) + (cid:0) D micN (cid:1) ( g n − ) i + (cid:0) D micN (cid:1) ( w n − ) (cid:0) E N ( g n ) (cid:1) (cid:2) E N ( g n ) + E N ( g n − ) (cid:3) (cid:27) , it follows that k w n k L ∞ ([0 ,T ]; L x L v ) + 1 ǫ ˆ T (cid:0) D micN (cid:1) ( w n ) dt C sup t ∈ [0 ,T ] E ( w n − ) sup t ∈ [0 ,T ] (cid:0) E N ( g n ) + E N ( g n − ) (cid:1) × " T sup t ∈ [0 ,T ] E N ( g n ) + ˆ T (cid:0) D micN (cid:1) ( g n ) dt + T sup t ∈ [0 ,T ] E N ( g n − ) + ˆ T (cid:0) D micN (cid:1) ( g n − ) dt + C sup t ∈ [0 ,T ] (cid:0) E N ( g n ) (cid:1) sup t ∈ [0 ,T ] (cid:2) E N ( g n ) + E N ( g n − ) (cid:3) ˆ T (cid:0) D micN (cid:1) ( w n − ) dt. By using (3.7) with 0 < δ < < ǫ < k w n k L ∞ ( [0 ,T ]; L x,v ) + 1 ǫ ˆ T (cid:0) D micN (cid:1) ( w n ) dt ( k w n − k L ∞ ( [0 ,T ]; L x,v ) + 1 ǫ ˆ T (cid:0) D micN (cid:1) ( w n − ) dt ) . Thus we have proved that { g n } is a Cauchy sequence in L ∞ (cid:0) [0 , T ]; L ( µ (1 − µ )d v d x ) (cid:1) .The interpolation combining with the estimate (3.7) implies that { g n } is a Cauchy sequence in L ∞ (cid:0) [0 , T ]; H N − η (d x ; L ( µ (1 − µ )d v )) (cid:1) for any η > L ∞ (cid:0) [0 , T ]; H N (d x ; L ( µ (1 − µ )d v )) (cid:1) .Finally the estimate (3.8) follows from weak lower semicontinuity.Finally we prove the uniqueness of the local solutions. Let g and g be two local solutions of (3.8).Then g = g − g ∈ L ∞ (cid:0) [0 , T ]; H N (d x ; L ( µ (1 − µ )d v )) (cid:1) and it satisfies ∂ t g + 1 ǫ v · ∇ g + 1 ǫ Lg = 1 ǫ Q ( g, g ) + T ( g, g , g ) + 1 ǫ Q ( g , g ) + T ( g , g, g ) + T ( g , g , g ) ,g ( t, x, v ) | t =0 = 0 , a similar procedure proves that g ≡
0, which implies the uniqueness of the solutions. (cid:3)
In the rest subsections, we devote ourselves to the global energy estimate. The global existence ofCauchy problem (3.8) is obtained by a standard continuation argument of local solutions.3.2.
Microscopic energy estimate.
Firstly, we study the estimate on the microscopic part { I − P } g in the function space H N (cid:0) d x ; L ( µ (1 − µ )d v ) (cid:1) . For notational simplification, we drop the sub-index ε of the local solution g ε to the Cauchy problem (3.8), and also drop g in the notations E N , D macN , D micN .Actually, we shall establish Lemma 3.3.
Let g ∈ L ∞ (cid:0) [0 , T ]; H N (d x ; L ( µ (1 − µ )d v )) (cid:1) be a solution of the equation (3.8) con-structed in Theorem 3.1. Then, there exists a constant C > independent of ǫ such that the followingestimate holds: dd t E N + 1 ǫ (cid:0) D micN (cid:1) C (cid:26) ǫ (cid:0) E N + E N (cid:1) (cid:0) D micN (cid:1) + (cid:0) E N + E N (cid:1) ( D macN ) (cid:27) . (3.13) Proof.
Applying ∂ αx to (3.8) and taking inner product with ∂ αx g in L (d x ; L ( µ (1 − µ )d v )) for | α | N ,we get 12 dd t X | α | N ( ∂ αx g, ∂ αx g ) + 1 ǫ X | α | N (cid:16) L∂ αx g, ∂ αx g (cid:17) = 1 ǫ X | α | N (cid:16) ∂ αx ( Q ( ǫg, ǫg ) + T ( ǫg, ǫg, ǫg )) , ∂ αx g (cid:17) . (3.14)where the inner product including v · ∇ g vanishes by integration by parts. By the coercivity of L (2.3)we have (cid:16) L∂ αx g, ∂ αx g (cid:17) > λ k{ I − P } ∂ αx g k ν . (3.15)Then we treat the right hand side of (3.2). Fixing α >
0, since (cid:16) ∂ αx ( Q ( ǫg, ǫg ) + T ( ǫg, ǫg, ǫg )) , ∂ αx P g (cid:17) = 0 , we have (cid:16) ∂ αx ( Q ( ǫg, ǫg ) + T ( ǫg, ǫg, ǫg )) , ∂ αx g (cid:17) = X α + α = α C α α (cid:16) Q ( ǫ∂ α x g, ǫ∂ α x g ) , ∂ αx { I − P } g (cid:17) + X α + α + α = α C α α α (cid:16) T ( ǫ∂ α x g, ǫ∂ α x g, ǫ∂ α x g ) , ∂ αx { I − P } g (cid:17) . (3.16)Without loss of generality, let’s assume | α | N/
2, then | α | >
0. From (2.29), let ( f , g ) be ( f, g ),then replacing ( f, g, e h ) by ( ǫ∂ α x g, ǫ∂ α x g, ∂ αx { I − P } g ) we deduce (cid:12)(cid:12)(cid:12)(cid:16) Q ( ǫ∂ α x g, ǫ∂ α x g ) , ∂ αx { I − P } g (cid:17)(cid:12)(cid:12)(cid:12) Cǫ E N (cid:8) D macN + D micN (cid:9) D micN (3.17)For the second term on the right hand side of (3.16), we establish the estimate (cid:12)(cid:12)(cid:12)(cid:16) T ( ǫ∂ α x g, ǫ∂ α x g, ǫ∂ α x g ) , ∂ αx { I − P } g (cid:17)(cid:12)(cid:12)(cid:12) Cǫ E N (cid:8) D macN + D micN (cid:9) D micN ( g ) . (3.18)for the following three cases:(A) Two of | α | , | α | , | α | are equal to 0.(a) | α | = 0 and | α | = 0, then | α | >
0. Using (2.30) with ( f , g ) = ( f, g ), we get (3.18).(b) | α | = 0 and | α | = 0, then | α | >
0. Using (2.30) with ( f , g ) = ( g, f ), we get (3.18).(c) | α | = 0 and | α | = 0, then | α | >
0. Using (2.31) with ( f , g ) = ( f, g ), we get (3.18).(B) Only one of | α | , | α | , | α | is equal to 0.(a) | α | = 0. Using (2.31) with ( f , g ) = ( f, g ) if | α | | α | , or using (2.30) with ( f , g ) = ( f, g ) if | α | > | α | , to get (3.18).(b) | α | = 0. Using (2.31) with ( f , g ) = ( f, g ) if | α | | α | , or using (2.30) with ( f , g ) = ( g, f ) if | α | > | α | , to get (3.18).(c) | α | = 0. Using (2.30) with ( f , g ) = ( f, g ) if | α | | α | , or using (2.30) with ( f , g ) = ( g, f ) if | α | > | α | , to get (3.18).(C) All of | α | , | α | , | α | are greater than 0. We claim that at least two of the | α | , | α | , | α | are lessthan or equal to | α | /
2. This case is similar to case (A).Thus, plugging (3.17) and (3.18) into (3.16) yields for 0 < | α | N (cid:12)(cid:12)(cid:12)(cid:16) ∂ αx ( Q ( ǫg, ǫg ) + T ( ǫg, ǫg, ǫg )) , ∂ αx g (cid:17)(cid:12)(cid:12)(cid:12) Cǫ (cid:16) E N + ǫ E N (cid:17) (cid:8) D macN + D micN (cid:9) D micN ( g ) . (3.19)Next, for | α | = 0, similar to (3.16), we have (cid:16) Q ( ǫg, ǫg ) + T ( ǫg, ǫg, ǫg ) , g (cid:17) = (cid:16) Q ( ǫg, ǫg ) + T ( ǫg, ǫg, ǫg ) , { I − P } g (cid:17) . (3.20)Splitting g = P g + { I − P } g , to further decompose (cid:16) Q ( ǫg, ǫg ) , { I − P } g (cid:17) into (cid:16) Q ( ǫ P g, ǫ P g ) , { I − P } g (cid:17) + (cid:16) Q ( ǫ { I − P } g, ǫ P g ) , { I − P } g (cid:17) + (cid:16) Q ( ǫg, ǫ { I − P } g ) , { I − P } g (cid:17) . (3.21)Recall the estimate (2.29), replacing f by ǫ { I − P } g and g by ǫg there we get (cid:12)(cid:12)(cid:12)(cid:16) Q ( ǫ { I − P } g, ǫ P g ) , { I − P } g (cid:17) + (cid:16) Q ( ǫg, ǫ { I − P } g ) , { I − P } g (cid:17)(cid:12)(cid:12)(cid:12) Cǫ E N (cid:8) D macN + D micN (cid:9) D micN . (3.22) HE INCOMPRESSIBLE NAVIER-STOKES-FOURIER LIMIT FROM BOLTZMANN-FERMI-DIRAC EQUATION 19
On the other hand, putting P g = a ( t, x ) + b ( t, x ) · v + c ( t, x ) | v | , into the first term in (3.21) and using the H¨older’s inequality to get (cid:12)(cid:12)(cid:12)(cid:16) Q ( ǫ P g, ǫ P g ) , { I − P } g (cid:17)(cid:12)(cid:12)(cid:12) Cǫ ˆ R | P g | L v | ν / P g | L v | ν / { I − P } g | L v dx Cǫ (cid:18) ˆ R | P g | L v dx (cid:19) / (cid:18) ˆ R | ν / P g | L v dx (cid:19) / k ν / { I − P } g k L x L v Cǫ k P g k H x L v k∇ x P g k L x L v k ν / { I − P } g k L x L v Cǫ E N D macN D micN , (3.23)where the third step is due to (3.16), in detail, (cid:18) ˆ R | ν / P g | L v dx (cid:19) / C (cid:13)(cid:13) a + | b | + c (cid:13)(cid:13) C (cid:0) k a ( t, · ) k L (d x ) + k b ( t, · ) k L (d x ) + k c ( t, · ) k L (d x ) (cid:1) × (cid:0) k a ( t, · ) k L (d x ) + k b ( t, · ) k L (d x ) + k c ( t, · ) k L (d x ) (cid:1) C (cid:0) k∇ x a ( t, · ) k L (d x ) + k∇ x b ( t, · ) k L (d x ) + k∇ x c ( t, · ) k L (d x ) (cid:1) × (cid:0) k a ( t, · ) k H (d x ) + k b ( t, · ) k H (d x ) + k c ( t, · ) k H (d x ) (cid:1) C k∇ x P g k L x,v k P g k H x ( L v ) . Plugging (3.22) and (3.23) into (3.21) we get (cid:16) Q ( ǫg, ǫg ) , { I − P } g (cid:17) Cǫ E N (cid:8) D macN + D micN (cid:9) D micN . (3.24)Repeating the process of (3.21)-(3.24) to get (cid:16) T ( ǫg, ǫg, ǫg ) , { I − P } g (cid:17) Cǫ E N (cid:8) D macN + D micN (cid:9) D micN . (3.25)Hence combining (3.20) with (3.24) and (3.25) we obtain (cid:12)(cid:12)(cid:12)(cid:16) Q ( ǫg, ǫg ) + T ( ǫg, ǫg, ǫg ) , g (cid:17)(cid:12)(cid:12)(cid:12) Cǫ (cid:0) E N + ǫ E N (cid:1) (cid:8) D macN + D micN (cid:9) D micN . (3.26)We plug (3.15), (3.19) and (3.26) into (3.2) and use Cauchy-Schwarz inequality with η > t E N + λǫ (cid:0) D micN (cid:1) Cǫ (cid:0) E N + ǫ E N (cid:1) (cid:8) D macN + D micN (cid:9) D micN Cǫ (cid:0) E N + E N (cid:1) (cid:0) D micN (cid:1) + ηǫ (cid:0) D micN (cid:1) + Cη ( D macN ) (cid:0) E N + E N (cid:1) . (3.27)By taking η = λ we complete the proof. (cid:3) Macroscopic energy estimate.
Now we study the macroscopic part P g , where g is a solutionof the equation (3.8). The key idea is to show the macroscopic part of the solution, a , b , c , are boundedby the microscopic part { I − P } g . Therefore, we plug the decomposition g = P g + { I − P } g into theBoltzmann-Fermi-Dirac equation (3.8) to get the time evolution of the macroscopic part P g : (cid:26) ∂ t + 1 ǫ v · ∇ x (cid:27) ( P g ) = − (cid:26) ∂ t + 1 ǫ v · ∇ x + 1 ǫ L (cid:27) { I − P } g + 1 ǫ Q ( g, g ) + T ( g, g, g ) . (3.28)In order to study the time evolution of P g precisely, we write P g = a ( t, x ) + b ( t, x ) · v + c ( t, x ) | v | , (3.29)and set the thirteen moments (cid:8) e k ( v ) (cid:9) k =1 = (cid:8) , v i , v i v j , v i | v | (cid:9) , i, j = 1 , , . (3.30)Then for fixed t, x , put the expansion (3.29) into (3.28) to get the following macroscopic equations: ∂ t a = − ∂ t r (0) + 1 ǫ m (0) + 1 ǫ ℓ (0) + h (0) (3.31) ∂ t b i + 1 ǫ ∂ i a = − ∂ t r (1) i + 1 ǫ m (1) i + 1 ǫ ℓ (1) i + h (1) i (3.32) ∂ t c + 1 ǫ ∂ i b i = − ∂ t r (2) i + 1 ǫ m (2) i + 1 ǫ ℓ (2) i + h (2) i (3.33)1 ǫ ∂ i b j + 1 ǫ ∂ j b i = − ∂ t r (2) ij + 1 ǫ m (2) ij + 1 ǫ ℓ (2) ij + h (2) ij , i = j (3.34)1 ǫ ∂ i c = − ∂ t r (3) i + 1 ǫ m (3) i + 1 ǫ ℓ (3) i + h (3) i , (3.35)by collecting the coefficients on both sides with respect to the moments in (3.30), where r (0) , r (1) i , · · · , h (2) ij , h (3) i are coefficients of { I − P } g , − v · ∇ x { I − P } g , − L { I − P } g and ǫ Q ( g, g ) + T ( g, g, g ) with respect tothe moments in (3.30) respectively.On the other hand, we introduce some constants: p = ˆ R µ d v, p = ˆ R | v | µ d v,p = ˆ R µ (1 − µ )d v, p = ˆ R | v | µ (1 − µ )d v,p = ˆ R | v | µ (1 − µ )d v. (3.36)By taking inner product with 1 , v, | v | in L ( µ (1 − µ )d v ) respectively, equation (3.28) yields the localconservation laws: ∂ t a − p p p − ( p ) ǫ ∇ · (cid:10) | v | v, { I − P } g (cid:11) = 0 ,∂ t b i + 1 ǫ ∂ i (cid:18) a + p p c (cid:19) + 3 p ǫ ∇ · h vv i , { I − P } g i = 0 ,∂ t c + 13 ǫ ∇ · b + p p p − ( p ) ǫ ∇ · (cid:10) | v | v, { I − P } g (cid:11) = 0 . (3.37)Next, we claim that the coefficients of the linear terms { I − P } g , − v · ∇ x { I − P } g , − L { I − P } g and the nonlinear term ǫ Q ( g, g ) + T ( g, g, g ) involved in the right-hand side of macroscopic system(3.31)-(3.35) can be bounded by the microscopic dissipation rate. Lemma 3.4.
It holds that X | α | N − X ij (cid:13)(cid:13)(cid:13) ∂ αx h r (0) , r (1) i , r (2) i , r (2) ij , r (3) i i(cid:13)(cid:13)(cid:13) C X | α | N − k ∂ αx { I − P } g k , (3.38) X | α | N − X ij (cid:13)(cid:13)(cid:13) ∂ αx h m (0) , m (1) i , m (2) i , m (2) ij , m (3) i i(cid:13)(cid:13)(cid:13) C X | α | N k ∂ αx { I − P } g k , (3.39) X | α | N − X ij (cid:13)(cid:13)(cid:13) ∂ αx h ℓ (0) , ℓ (1) i , ℓ (2) i , ℓ (2) ij , ℓ (3) i i(cid:13)(cid:13)(cid:13) C X | α | N − k ∂ αx { I − P } g k , (3.40) and X | α | N X ij (cid:13)(cid:13)(cid:13) ∂ αx h h (0) , h (1) i , h (2) i , h (2) ij , h (3) i i(cid:13)(cid:13)(cid:13) Cǫ (cid:0) E N + E N (cid:1) (cid:0) D macN + D micN (cid:1) . (3.41) Proof.
Let { e e i ( v ) } i =1 be its corresponding orthonormal basis such that for some constants ξ ij e e i ( v ) = X j =1 ξ ij e i ( v ) . For the linear term, the coefficients m (0) , m (1) i , m (2) i , m (2) ij , m (3) i of the projection of − v · ∇ x { I − P } g take the form − X i,n =1 ξ ij ξ in ˆ R { v · ∇ x }{ I − P } g · e n ( v ) µ (1 − µ )d v. HE INCOMPRESSIBLE NAVIER-STOKES-FOURIER LIMIT FROM BOLTZMANN-FERMI-DIRAC EQUATION 21
Same is true after we take ∂ αx . Let | α | N −
1, (3.39) holds for that (cid:13)(cid:13)(cid:13)(cid:13) ˆ R v · ∇ x { I − P } ∂ αx g · e n ( v ) µ (1 − µ )d v (cid:13)(cid:13)(cid:13)(cid:13) ˆ R | e n ( v ) | | v | µ (1 − µ )d v ¨ R × R |{ I − P }∇ ∂ αx g | µ (1 − µ )d v d x C X | α | N k ∂ αx { I − P } g k , Note that ν ( v ) C (1 + | v | ) and K is bounded from L ( µ (1 − µ )d v ) to itself, then the same process asabove gives the estimates (3.38) and (3.40) for the coefficients of the rest linear terms.For the coefficients h (0) , h (1) i , h (2) i , h (2) ij , h (3) i , by (2.6) and (2.8), we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:28) ∂ αx (cid:20) ǫ Q ( g, g ) + T ( g, g, g ) (cid:21) , e n (cid:29)(cid:13)(cid:13)(cid:13)(cid:13) Cǫ (1 + ǫ E N ) E N (cid:0) D macN + D micN (cid:1) , thus (3.41) holds. (cid:3) Similarly, those terms containing the microscopic part { I − P } g in the conservation laws (3.37) canbe also bounded by the microscopic dissipation rate. Lemma 3.5.
It holds that X | α | N − (cid:13)(cid:13) ∂ αx ∇ x · (cid:2)(cid:10) | v | v, { I − P } g (cid:11) , h v ⊗ v, { I − P } g i (cid:3)(cid:13)(cid:13) C X < | α | N k ∂ αx { I − P } g k . (3.42)Before we establish the crucial macroscopic energy estimate, we shall introduce the notation: G α,i ( t ) = D ∂ i ∂ αx a, ∂ αx b i E , G α,i ( t ) = D ∂ αx r (1) i , ∂ i ∂ αx a E , G α,i ( t ) = D ∂ αx r (3) i , ∂ i ∂ αx c E ,G α,i ( t ) = − X j = i D ∂ αx r (2) j , ∂ i ∂ αx b i E + X j = i D ∂ αx r (2) ji , ∂ j ∂ αx b i E + 2 D ∂ αx r (2) i , ∂ i ∂ αx b i E , (3.43)and G α ( t ) = X i =1 (cid:2) G α,i ( t ) + G α,i ( t ) + G α,i ( t ) + G α,i ( t ) (cid:3) . (3.44) Lemma 3.6.
Let g be a solution of the scaled Boltzmann-Fermi-Dirac equation (3.32) . Then thereexists a positive constant e C independent of ǫ such that the following estimate holds: ǫ dd t X | α | N − G α ( t ) + X | α | N − k∇ x ∂ αx ( a, b, c ) k e C (cid:26) ǫ (cid:0) D micN (cid:1) + (cid:0) E N + E N (cid:1) (cid:16) ( D macN ) + (cid:0) D micN (cid:1) (cid:17)(cid:27) (3.45) for any t ∈ [0 , T ] .Proof. We fix a constant η ∈ (0 ,
1) to be determined later and let | α | N −
1. Note that from(3.31)-(3.35), we deduce b = ( b , b , b ) satisfies the following elliptic-type equation: − ∆ x b j − ∂ j ∂ j b j = X i = j ∂ j (cid:20) − ǫ∂ t r (2) i + m (2) i + 1 ǫ ℓ (2) i + ǫh (2) i (cid:21) − X i = j ∂ i (cid:20) − ǫ∂ t r (2) ij + m (2) ij + 1 ǫ ℓ (2) ij + ǫh (2) ij (cid:21) − ∂ j (cid:20) − ǫ∂ t r (2) j + m (2) j + 1 ǫ ℓ (2) j + ǫh (2) j (cid:21) . (3.46)Thus we start with estimates on b . Estimates on b . Applying ∂ αx to (3.46), then taking inner product with ∂ αx b j in L (d x ), k∇ x ∂ αx b j k + k ∂ j ∂ αx b j k = ǫR r,b − X i = j (cid:18) ∂ αx (cid:20) ǫ ℓ (2) i + m (2) i + ǫh (2) i (cid:21) , ∂ j ∂ αx b j (cid:19) + X i = j (cid:18) ∂ αx (cid:20) ǫ ℓ (2) ij + m (2) ij + ǫh (2) ij (cid:21) , ∂ i ∂ αx b j (cid:19) + 2 (cid:18) ∂ αx (cid:20) ǫ ℓ (2) j + m (2) j + ǫh (2) j (cid:21) , ∂ j ∂ αx b j (cid:19) . (3.47)Integrating by parts ǫR r,b = − ǫ ∂ t X i = j ∂ j ∂ αx r (2) i − X i = j ∂ i ∂ αx r (2) ij − ∂ j ∂ αx r (2) j , ∂ αx b j = − ǫ ddt X i = j ∂ j ∂ αx r (2) i − X i = j ∂ i ∂ αx r (2) ij − ∂ j ∂ αx r (2) j , ∂ αx b j + ǫ X i = j ∂ j ∂ αx r (2) i − X i = j ∂ i ∂ αx r (2) ij − ∂ j ∂ αx r (2) j , ∂ αx ∂ t b j . Note the first term on the right hand side above is just − ǫ dd t G α,j ( t ), and from the conservation law(3.37), the second term is bounded by ηǫ k ∂ αx ∂ t b j k + 14 η (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i = j ∂ j ∂ αx r (2) i − X i = j ∂ i ∂ αx r (2) ij − ∂ j ∂ αx r (2) j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) η (cid:13)(cid:13)(cid:13)(cid:13) ∂ αx ∂ j ( a + p p c ) + ∂ αx ∇ x · ( vv i , { I − P } g ) (cid:13)(cid:13)(cid:13)(cid:13) + Cη X i = j (cid:13)(cid:13)(cid:13) ∂ j ∂ αx r (2) i (cid:13)(cid:13)(cid:13) + X i = j (cid:13)(cid:13)(cid:13) ∂ i ∂ αx r (2) ij (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ∂ j ∂ αx r (2) j (cid:13)(cid:13)(cid:13) Cη h k ∂ j ∂ αx ( a, c ) k + k ∂ αx ∇ x · ( vv i , { I − P } g ) k i + Cη X ij (cid:13)(cid:13)(cid:13) ∇ x ∂ αx h r (2) i , r (2) ij , r (2) j i(cid:13)(cid:13)(cid:13) . The sum of the rest terms on the right hand side of (3.47) is bounded by η k∇ x ∂ αx b j k + Cη X i = j (cid:13)(cid:13)(cid:13)(cid:13) ∂ αx (cid:20) ǫ ℓ (2) i + m (2) i + ǫh (2) i (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) + X i = j (cid:13)(cid:13)(cid:13)(cid:13) ∂ αx (cid:20) ǫ ℓ (2) ij + m (2) ij + ǫh (2) ij (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13) ∂ αx (cid:20) ǫ ℓ (2) ij + m (2) ij + ǫh (2) ij (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) η k∇ x ∂ αx b j k + Cη X ij (cid:13)(cid:13)(cid:13)(cid:13) ∂ αx (cid:20) ǫ ℓ (2) i , ǫ ℓ (2) ij , m (2) i , m (2) ij , ǫh (2) i , ǫh (2) ij (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) . Putting all estimates into (3.47), by Lemmas 3.4 and 3.5 we get ǫ ddt X j =1 G α,j ( t ) + k∇ x ∂ αx b k Cη n k∇ x ∂ αx ( a, b, c ) k + k ∂ αx { I − P } g k o + Cη (cid:26) ǫ (cid:0) D micN (cid:1) + (cid:0) E N + E N (cid:1) (cid:16) ( D macN ) + (cid:0) D micN (cid:1) (cid:17)(cid:27) . (3.48) Estimates on c.
For each i = 1 , ,
3, it follows from (3.35) that k ∂ i ∂ αx c k = ǫ ( ∂ i ∂ αx c, ∂ i ∂ αx c )= − ǫ (cid:16) ∂ t ∂ αx r (3) i , ∂ i ∂ αx c (cid:17) + (cid:18) ∂ αx (cid:20) ǫ ℓ (3) i + m (3) i + ǫh (3) i (cid:21) , ∂ i ∂ αx c (cid:19) = − ǫ ddt (cid:16) ∂ αx r (3) i , ∂ i ∂ αx c (cid:17) + ǫ (cid:16) ∂ αx r (3) i , ∂ i ∂ αx ∂ t c (cid:17) + (cid:18) ∂ αx (cid:20) ǫ ℓ (3) i + m (3) i + ǫh (3) i (cid:21) , ∂ i ∂ αx c (cid:19) . (3.49) HE INCOMPRESSIBLE NAVIER-STOKES-FOURIER LIMIT FROM BOLTZMANN-FERMI-DIRAC EQUATION 23
The first term on the right hand side above is just − ǫ dd t G α,i ( t ). From the conservation law (3.37),the second term is bounded by − ǫ (cid:16) ∂ i ∂ αx r (3) i , ∂ αx ∂ t c (cid:17) ηǫ k ∂ αx ∂ t c k + 14 η (cid:13)(cid:13)(cid:13) ∂ i ∂ αx r (3) i (cid:13)(cid:13)(cid:13) η (cid:13)(cid:13)(cid:13)(cid:13) ∂ αx ∇ · b + p p p − ( p ) ∂ αx ∇ · (cid:0) | v | v, { I − P } g (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) + 14 η (cid:13)(cid:13)(cid:13) ∂ i ∂ αx r (3) i (cid:13)(cid:13)(cid:13) Cη k ∂ αx ∇ x · b k + Cη (cid:13)(cid:13) ∂ αx ∇ · (cid:0) | v | v, { I − P } g (cid:1)(cid:13)(cid:13) + 14 η (cid:13)(cid:13)(cid:13) ∂ i ∂ αx r (3) i (cid:13)(cid:13)(cid:13) . The third term is bounded by η k ∂ i ∂ αx c k + 14 η (cid:13)(cid:13)(cid:13)(cid:13) ∂ αx (cid:20) ǫ ℓ (3) i + m (3) i + ǫh (3) i (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) . Note Lemmas 3.4 and 3.5, plugging estimates on the three terms into (3.49), then taking summationfor i over i ∈ { , , } yields ǫ ddt G α ( t ) + k∇ x ∂ αx c k Cη n k∇ x ∂ αx ( b, c ) k + k ∂ αx ∇ x { I − P } g k o + Cη (cid:26) ǫ (cid:0) D micN (cid:1) + (cid:0) E N + E N (cid:1) (cid:16) ( D macN ) + (cid:0) D micN (cid:1) (cid:17)(cid:27) . (3.50) Estimates on a . For each i = 1 , ,
3, it follows from (3.32) that k ∂ i ∂ αx a k = (cid:18) − ǫ∂ αx ∂ t b i − ǫ∂ αx ∂ t r (1) i + ∂ αx (cid:20) ǫ ℓ (1) i + m (1) i + ǫh (1) i (cid:21) , ∂ i ∂ αx a (cid:19) = − ǫ ddt h ( ∂ αx b i , ∂ i ∂ αx a ) + (cid:16) ∂ αx r (1) i , ∂ i ∂ αx a (cid:17)i + ǫ ( ∂ αx b i , ∂ i ∂ αx ∂ t a ) + ǫ (cid:16) ∂ αx r (1) i , ∂ i ∂ αx ∂ t a (cid:17) + (cid:18) ∂ αx (cid:20) ǫ ℓ (1) i + m (1) i + ǫh (1) i (cid:21) , ∂ i ∂ αx a (cid:19) . (3.51)Similarly as before, we need to estimate the four terms on the right-hand side of (3.51). The first termis − ǫ dd t (cid:2) G α ( t ) + G α ( t ) (cid:3) . From the conservation law (3.37), ǫ ( ∂ αx ∂ i b i , ∂ αx ∂ t a ) + ǫ (cid:16) ∂ αx ∂ i r (1) i , ∂ αx ∂ t a (cid:17) = (cid:18) ∂ αx ∂ i b i , p p p − ( p ) ∂ αx ∇ x · (cid:0) | v | v, { I − P } g (cid:1)(cid:19) + (cid:18) ∂ αx ∂ i r (1) i , p p p − ( p ) ∂ αx ∇ x · (cid:0) | v | v, { I − P } g (cid:1)(cid:19) η (cid:18) k ∂ αx ∂ i b i k + (cid:13)(cid:13)(cid:13) ∂ αx ∂ i r (1) i (cid:13)(cid:13)(cid:13) (cid:19) + Cη (cid:13)(cid:13) ∂ αx ∇ x · (cid:0) | v | v, { I − P } g (cid:1)(cid:13)(cid:13) . The final term is bounded by η k ∂ i ∂ αx a k + Cη (cid:13)(cid:13)(cid:13)(cid:13) ∂ αx (cid:20) ǫ ℓ (1) i , m (1) i , ǫh (1) i (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) . Collecting all the estimates above, by Lemmas 3.4 and 3.5 we get ǫ ddt (cid:2) G α ( t ) + G α ( t ) (cid:3) + k∇ x ∂ αx a k η k∇ x ∂ αx ( a, b ) k + Cη k ∂ αx ∇ x { I − P } g k + Cη (cid:26) ǫ (cid:0) D micN (cid:1) + (cid:0) E N + E N (cid:1) (cid:16) ( D macN ) + (cid:0) D micN (cid:1) (cid:17)(cid:27) . (3.52) Finally we add up the inequalities (3.48), (3.50) and (3.52) and take summation for α over | α | N − ǫ ddt X | α | N − X i =1 G iα ( t ) + X | α | N − k∇ x ∂ αx ( a, b, c ) k Cη X | α | N − k∇ x ∂ αx ( a, b, c ) k + Cη (cid:26) ǫ (cid:0) D micN (cid:1) + (cid:0) E N + E N (cid:1) (cid:16) ( D macN ) + (cid:0) D micN (cid:1) (cid:17)(cid:27) (3.53)We complete the proof by choosing η ∈ (0 , Cη = . (cid:3) Uniform estimate and the global existence.
We derive the uniform energy estimate by themicroscopic estimate (3.1) and the macroscopic estimate (3.45). For constant d > E N,d ( t ) ≡ d E N ( t ) + ǫ X | α | N − G α ( t ) , multiplying (3.1) by d , and then adding it to (3.45), since ( D macN ) ( g ) is equivalent with P | α | N − k∇ x ∂ αx ( a, b, c ) k ,we havedd t E N,d + (cid:26) dǫ (cid:0) D micN (cid:1) + ( D macN ) (cid:27) Cd (cid:26) ǫ (cid:0) E N + E N (cid:1) (cid:0) D micN (cid:1) + (cid:0) E N + E N (cid:1) ( D macN ) (cid:27) + e C (cid:26) ǫ (cid:0) D micN (cid:1) + (cid:0) E N + E N (cid:1) (cid:16) ( D macN ) + (cid:0) D micN (cid:1) (cid:17)(cid:27) , which is dd t E N,d + ( d − e Cǫ (cid:0) D micN (cid:1) + ( D macN ) ) Cd + e Cǫ (cid:0) E N + E N + E N (cid:1) (cid:0) D micN (cid:1) + (cid:16) e C + Cd (cid:17) (cid:0) E N + E N (cid:1) ( D macN ) . (3.54)We point out that for some constant c , c >
0, there holds c E N ( g )( t ) E N,d ( g )( t ) c E N ( g )( t ) . (3.55)Indeed, this is due to X | α | N − | G α ( t ) | C X | α | N − X i =1 G iα ( t ) C X | α | N − k ∂ αx { I − P } g k + C X | α | N − k ∂ αx P g k C E N ( g )( t ) . (3.56)Choosing d > d − e C >
0, then the relation (3.55) holds. Hence we prove thefollowing theorem
Theorem 3.2. If g is a solution of the scaled Boltzmann-Fermi-Dirac equation (3.32) , then thereexists a constant c > independent of ǫ such that if E N,d , then dd t E N,d + 1 ǫ (cid:0) D micN (cid:1) + ( D macN ) c E N,d (cid:26) ǫ (cid:0) D micN (cid:1) + ( D macN ) (cid:27) . (3.57)Now, with the above preparations, we are ready to prove Theorem 1.1 by the usual continuationarguments. Proof of Theorem 1.1.
Recall the constants c , c , c in (3.57) and (3.55), and δ appeared in The-orem 3.1, we can define M = min (cid:26) c δ c , c , c c (cid:27) . (3.58)Let the initial data g ,ǫ satisfies E N (0) = k g ,ǫ k H Nx L v M. HE INCOMPRESSIBLE NAVIER-STOKES-FOURIER LIMIT FROM BOLTZMANN-FERMI-DIRAC EQUATION 25
Then Theorem 3.1 shows that for some constant
T >
0, there exists a solution g ∈ L ∞ (cid:0) [0 , T ]; H Nx L v (cid:1) satisfying E N ( t ) M for 0 < t < T . We further define T ∗ = sup (cid:26) t ∈ R + (cid:12)(cid:12) E N ( t ) M c c (cid:27) > . Note that E N,d ( t ) c E N ( t ) c M < , ∀ t T. The global energy estimate (3.57) implies thatdd t E N,d + (1 − c c M ) (cid:26) ǫ (cid:0) D micN (cid:1) + ( D macN ) (cid:27) , from the choice of M such that 1 − c c M > . Thus E N,d ( T ) + 12 ˆ T (cid:26) ǫ (cid:0) D micN (cid:1) + ( D macN ) (cid:27) dt E N,d (0) , which implies E N ( T ) c c M . Thus T ∗ = + ∞ , and we finish the proof of Theorem 1.1. (cid:3) Limit to Incompressible Navier-Stokes-Fourier Equations
The limit from the global energy estimate.
First of all, we shall introduce some constants: E = h i , E = D | v | E ,E = D | v | E , E = D | v v | E ,C A = *(cid:18) | v | − K A (cid:19) v + − µ , K A = E + 2 E E . Based on Theorem 1.1, there exists a δ >
0, such that for any given initial data g ǫ, ( x, v ) = (cid:26) ρ ( x ) + u ( x ) · v + θ ( x ) (cid:18) | v | − K g (cid:19)(cid:27) + e g ǫ, ( x, v ) , satisfying k ( ρ , u , θ ) k H Nx δ C E ,E ,E , where C E ,E ,E > e g ǫ, ∈ N ull ( L ) ⊥ , k e g ǫ, k H Nx L v δ , the scaled Boltzmann-Fermi-Dirac equation (1.6) admits a global solution g ǫ , where K g = K A − C > ǫ such that the global energy estimate (1.16)holds, thus sup t > E N ( t ) = sup t > X | α | N ˆ R x,v | ∂ αx g ǫ ( t ) | µ (1 − µ )d v d x C, (4.1)and ˆ ∞ (cid:0) D micN (cid:1) ( t )d t = X | α | N ˆ ∞ ˆ R x,v | ∂ αx { I − P } g ǫ ( t ) | νµ (1 − µ )d v d x d t Cǫ , (4.2)and ˆ ∞ ( D macN ) ( t )d t = X < | α | N ˆ ∞ ˆ R x,v | ∂ αx P g ǫ ( t ) | νµ (1 − µ )d v d x d t ≤ C. (4.3)We deduce from the energy bound (4.1) that there exists a g in the functional space L ∞ (cid:0) [0 , + ∞ ); H N (cid:0) d x ; L ( µ (1 − µ )d v ) (cid:1)(cid:1) , such that g ǫ → g as ǫ → , (4.4)where the convergence is weak- ⋆ for t, strongly in H N − η (d x ) for any η > , and weakly in L ( µ (1 − µ )d v ).From the energy dissipation bound (4.2) we have { I − P } g ǫ → , in L (cid:0) [0 , + ∞ ); H N (cid:0) d x ; L ( µ (1 − µ )d v ) (cid:1)(cid:1) as ǫ → . (4.5)By combining the convergence (4.4) and (4.5) we have { I − P } g = 0 . Thus, there exists ( ρ, u , θ ) ∈ L ∞ (cid:0) [0 , + ∞ ); H N (d x ) (cid:1) , such that g ( t, x, v ) = ρ ( t, x ) + u( t, x ) · v + θ ( t, x ) (cid:18) | v | − K g (cid:19) . The limiting equations.
If we write A ( v ) = (cid:18) | v | − K A (cid:19) v, B ( v ) = v ⊗ v − | v | I, then from [49], there exist unique A ′ ( v ) = (cid:16) A ′ i ( v ) (cid:17) , B ′ ( v ) = (cid:16) B ′ ij ( v ) (cid:17) , i, j = 1 , ,
3, such that L ( A ′ i ) = A ′ i , L ( B ′ ij ) = B ij . Moreover, A ′ ( v ) = − α L ( | v | ) A ( v ) , B ′ ( v ) = − β L ( | v | ) B ( v ) . for some positive functions α L ( | v | ) , β L ( | v | ). HE INCOMPRESSIBLE NAVIER-STOKES-FOURIER LIMIT FROM BOLTZMANN-FERMI-DIRAC EQUATION 27
Now we define the fluid variables as follows: ρ ǫ = 1 K A E − E / (cid:28) g ǫ , (cid:20) (cid:18) E E K g − (cid:19) | v | (cid:21)(cid:29) , u ǫ = 1 E h g ǫ , v i ,θ ǫ = 1 K A E − E / (cid:28) g ǫ , (cid:18) E E | v | − (cid:19)(cid:29) . Then it follows from (4.4) that ( ρ ǫ , u ǫ , θ ǫ ) → ( ρ, u , θ ) as ǫ → , (4.6)where the convergence is weak- ⋆ for t, strongly in H N − η (d x ) for any η > L v by1 + (cid:16) E E K g − (cid:17) | v | K A E − E / , vE , E E | v | − K A E − E / ∂ t ρ ǫ + 2 K g ǫ ∇ x · u ǫ + E E K g − K A E − E / ∇ x · (cid:28) A ′ , ǫ Lg ǫ (cid:29) = 0 ,∂ t u ǫ + 1 ǫ ∇ x ( ρ ǫ + θ ǫ ) + 1 E ∇ x · (cid:28) B ′ , ǫ Lg ǫ (cid:29) = 0 ,∂ t θ ǫ + 23 ǫ ∇ x · u ǫ + 2 E / E K A E − E / ∇ x · (cid:28) A ′ , ǫ Lg ǫ (cid:29) = 0 . (4.7)Here we use the properties of A, B, A ′ and B ′ , the self-adjointness of L , to obtain the following calcu-lations: h v · ∇ x g ǫ , v i = ∇ x · h B, g ǫ i + ∇ x (cid:28) | v | g ǫ , (cid:29) = ∇ x · h LB ′ , g ǫ i + E ∇ x ( ρ ǫ + θ ǫ )= ∇ x · h B ′ , Lg ǫ i + E ∇ x ( ρ ǫ + θ ǫ ) , and (cid:28) v · ∇ x g ǫ , K A E − E / (cid:18) E E | v | − (cid:19)(cid:29) = 1 K A E − E / ∇ x · (cid:28) g ǫ , E E (cid:18) | v | − K A (cid:19) v + (cid:18) E E K A − (cid:19) v (cid:29) = 2 E / E K A E − E / ∇ x · h A, g ǫ i + 23 ∇ x · u ǫ = 2 E / E K A E − E / ∇ x · h A ′ , Lg ǫ i + 23 ∇ x · u ǫ . Incompressibility and Boussinesq relation.
From the first equation of (4.7),2 K g ∇ x · u ǫ = − ǫ∂ t ρ ǫ − E E K g − K A E − E / ∇ x · h LA ′ , { I − P } g ǫ i . From the global energy bound (4.1) and the global energy dissipation (4.2), it is easy to deduce ∇ x · u ǫ → ǫ → . (4.8)By combining with the convergence (4.6), we have ∇ x · u = 0 . (4.9)From the second equation of (4.7), ∇ x ( ρ ǫ + θ ǫ ) = − ǫ∂ t u ǫ − E ∇ x · ( LB ′ , { I − P } g ǫ ) . From the global energy dissipation (4.2), it follows that ∇ x ( ρ ǫ + θ ǫ ) → ǫ → , (4.10) which gives the Boussinesq relation ∇ x ( ρ + θ ) = 0 . (4.11) Convergence of K g θ ǫ − ρ ǫ K g + 1 . The third equation times K g and then minus the first equation in(4.7) gives ∂ t (cid:18) K g θ ǫ − ρ ǫ K g + 1 (cid:19) + 1 K A ( K A E − E / ∇ x · (cid:28) A ′ , ǫ Lg ǫ (cid:29) = 0 . (4.12)From the global energy estimate (4.1), we have that (cid:13)(cid:13)(cid:13)(cid:13) K g θ ǫ − ρ ǫ K g + 1 ( t ) (cid:13)(cid:13)(cid:13)(cid:13) H N (d x ) C for almost every t ∈ [0 , + ∞ ) , Then there exists a e θ ∈ L ∞ (cid:0) [0 , + ∞ ); H N (d x ) (cid:1) , so that K g θ ǫ − ρ ǫ K g + 1 ( t ) → e θ ( t ) in H N − η (d x ) , for any η > ǫ →
0. Furthermore, using the equation (4.12), we can show the equi-continuity in t .Indeed, for any [ t , t ] ⊂ [0 , ∞ ) , and any test function χ ( x ) and | α | N − ˆ R (cid:20) ∂ αx (cid:18) K g θ ǫ − ρ ǫ K g + 1 ( t ) (cid:19) − ∂ αx (cid:18) K g θ ǫ − ρ ǫ K g + 1 (cid:19) ( t ) (cid:21) χ ( x )d x = − K A ( K A E − E / ˆ t t ˆ R (cid:28) A ′ , ǫ L { I − P }∇ x ∂ αx g ǫ (cid:29) χ ( x )d x d t Cǫ (cid:18) ˆ t t (cid:0) D micN (cid:1) ( g ǫ ) ( t )d t (cid:19) / . Thus the energy dissipation estimate (4.2) implies the equi-continuity in t. From the Arzel`a-AscoliTheorem, e θ ∈ C (cid:0) [0 , ∞ ); H N − − η (d x ) (cid:1) ∩ L ∞ (cid:0) [0 , ∞ ); H N − η (d x ) (cid:1) , and K g θ ǫ − ρ ǫ K g + 1 → e θ in C (cid:0) [0 , ∞ ); H N − − η (d x ) (cid:1) ∩ L ∞ (cid:0) [0 , ∞ ); H N − η (d x ) (cid:1) (4.13)as ε → η >
0. Note that e θ = K g θ − ρK g +1 and θ = K g θ − ρK g +1 + K g +1 ( ρ + θ ) , and the relation (4.11),we get e θ = θ and ρ + θ = 0 Convergence of P u ε . Taking the Leray projection operator P on the second equation of (4.7)gives ∂ t P u ǫ + 1 E P∇ x · (cid:28) B ′ , ǫ Lg ǫ (cid:29) = 0 . Similar arguments as above deduce that there exists a divergence free e u ∈ L ∞ (cid:0) [0 , ∞ ); H N (d x ) (cid:1) such that P u ǫ → e u in C (cid:0) [0 , ∞ ); H N − − η (d x ) (cid:1) ∩ L ∞ (cid:0) [0 , ∞ ); H N − η (d x ) (cid:1) (4.14)as ǫ → η > . Note that e u = P u and (4.9), we have e u = u.Similar to the standard calculations in [6], the local conservation laws can be rewritten as ∂ t ρ ǫ + K g ǫ ∇ x · u ǫ + K A (cid:16) K g E E − (cid:17) ∇ x · (u ǫ θ ǫ ) = κ ∇ x · [ ∇ x θ ǫ ] + ∇ x · R (1) ǫ,θ ∂ t u ǫ + ǫ ∇ x ( ρ ǫ + θ ǫ ) + ∇ x · (cid:16) u ǫ ⊗ u ǫ − | u ǫ | I (cid:17) = ν ∗ E Σ(u ǫ ) + ∇ x · R ǫ, u ∂ t θ ǫ +
23 1 ǫ ∇ x · u ǫ + K A E E ∇ x · (u ǫ θ ǫ ) = κ ∇ x · [ ∇ x θ ǫ ] + ∇ x · R (2) ǫ,θ (4.15)where κ = E E K g − K A E − E / * α L ( | v | ) (cid:18) | v | − K A (cid:19) v + ,κ = 2 E / E K A E − E / * α L ( | v | ) (cid:18) | v | − K A (cid:19) v + , Σ( u ) = ∇ u + ( ∇ u ) T −
23 ( ∇ · u ) I, HE INCOMPRESSIBLE NAVIER-STOKES-FOURIER LIMIT FROM BOLTZMANN-FERMI-DIRAC EQUATION 29 and R ǫ, u , R (1) ǫ,θ , R (2) ǫ,θ have of the form R + S (cid:16) ǫ D ζ ( v ) , ∂ t g ǫ − T ( g ǫ , g ǫ , g ǫ ) E + D ζ ( v ) , v · ∇ x { I − P } g ǫ E − D ζ ( v ) , Q ( { I − P } g ǫ , { I − P } g ǫ ) E − D ζ ( v ) , Q ( { I − P } g ǫ , P g ǫ ) E − D ζ ( v ) , Q ( P g ǫ , { I − P } g ǫ ) E(cid:17) . (4.16)For R (1) ǫ,θ , we take ζ ( v ) = A ′ , R = − K A (cid:16) K g E E − (cid:17) u ǫ ( ρ ǫ + θ ǫ ) and S = E E K g − K A E − E / ; For R (1) ǫ,θ , wetake ζ ( v ) = A ′ , R = − K A E E u ǫ ( ρ ǫ + θ ǫ ) and S = E / E K A E − E / and for R ǫ, u , we take ζ ( v ) = B ′ , R = 0and S = E . The equations of θ and u . Decompose u ǫ = P u ǫ + Q u ǫ , where Q = ∇ x ∆ − x ∇ x · is a gradi-ent. Denote e θ ǫ = K g θ ǫ − ρ ǫ K g + 1 . Then from (4.15), the following equation is satisfied in the sense ofdistributions: ∂ t e θ ǫ + ∇ x · (cid:16) P u ǫ e θ ǫ (cid:17) − E K A E κ ∆ x e θ ǫ = 1 C A ∇ x · e R ǫ,θ , where e R ǫ,θ = 1 C A R ǫ,θ − (cid:18) K A + 1 (cid:19) P u ǫ ( ρ ǫ + θ ǫ ) − (cid:18) K A + 1 (cid:19) Q u ǫ ( ρ ǫ + θ ǫ ) − Q u ǫ e θ ǫ + 3 E K A E κ ∇ x ( ρ ǫ + θ ǫ ) . (4.17)For any T >
0, let ϕ ( t, x ) be a text function satisfying ϕ ( t, x ) ∈ C (cid:0) [0 , T ] , C ∞ c (cid:0) R x (cid:1)(cid:1) with ϕ (0 , x ) = 1and ϕ ( t, x ) = 0 for t > T ′ , where T ′ < T. Note (4.16) and use the global bounds (4.1), (4.2), and (4.3). It is easy to show that ˆ T ˆ R ∇ x · R ǫ ( t, x ) ϕ ( t, x )d x d t → , as ǫ → , (4.18)where R ǫ = R ǫ, u or R ǫ,θ . For other terms in (4.17) noting the convergence (4.8) and (4.10), togetherwith (4.18), we have ˆ T ˆ R ∇ x · e R ǫ,θ ( t, x ) ϕ ( t, x )d x d t → ǫ → . (4.19)From the convergence (4.13) and (4.14), as ǫ → ˆ T ˆ R ∂ t e θ ǫ ( t, x ) ϕ ( t, x )d x d t → − ˆ R (cid:18) K g θ − ρ K g + 1 (cid:19) ( x )d x − ˆ T ˆ R θ ( t, x ) ∂ t ϕ ( t, x )d x d t, ˆ T ˆ R ∆ x e θ ǫ ϕ ( t, x )d x d t → ˆ T ˆ R θ ( t, x )∆ x ϕ ( t, x )d x d t, and ˆ T ˆ R ∇ x · (cid:16) P u ǫ e θ ǫ (cid:17) ϕ ( t, x )d x d t → − ˆ T ˆ R u( t, x ) θ ( t, x ) · ∇ x ϕ ( t, x )d x d t. By taking the Leray projection P on the second equation of (4.15), we have the following equation: ∂ t P u ǫ + P∇ x · ( P u ǫ ⊗ P u ǫ ) − ν ∗ E ∆ x P u ǫ = P∇ x · e R ǫ, u , where e R ǫ, u = 1 E R ǫ, u − P · ( P u ǫ ⊗ Q u ǫ + Q u ǫ ⊗ P u ǫ + Q u ǫ ⊗ Q u ǫ ) . Similar as above we can take the vector-valued test function ψ ( t, x ) with ∇ x · ψ = 0 , and prove thatas ǫ → ˆ T ˆ R (cid:18) ∂ t P u ǫ + P∇ x · ( P u ǫ ⊗ P u ǫ ) − ν ∗ E ∆ x P u ǫ (cid:19) · ψ ( t, x )d x d t → − ˆ R P u ( x ) · ψ (0 , x )d x − ˆ T ˆ R (cid:18) u · ∂ t ψ + u ⊗ u : ∇ x ψ − ν ∗ E u · ∆ x ψ (cid:19) d x d t, and ˆ T ˆ R P∇ x · e R ǫ, u ( t, x ) φ ( t, x )d x d t → ǫ → . By collecting all above convergence results, we have shown that(u , θ ) ∈ C (cid:0) [0 , ∞ ); H N − (d x ) (cid:1) ∩ L ∞ (cid:0) [0 , ∞ ); H N (d x ) (cid:1) satisfies the following incompressible Navier-Stokes equations E ∂ t u + E u · ∇ x u + ∇ x p = ν ∗ ∆ x u , ∇ x · u = 0 ,C A ∂ t θ + C A u · ∇ x θ = κ ∗ ∆ x θ, with initial data: u(0 , x ) = P u ( x ) , θ (0 , x ) = K g θ ( x ) − ρ ( x ) K g + 1 , HE INCOMPRESSIBLE NAVIER-STOKES-FOURIER LIMIT FROM BOLTZMANN-FERMI-DIRAC EQUATION 31
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