Global Existence and Decay of Solutions to the Fokker-Planck-Boltzmann Equation
aa r X i v : . [ m a t h . A P ] J un Global Existence and Decay of Solutions to theFokker-Planck-Boltzmann Equation
Linjie Xiong ∗ , Tao Wang , and
Lusheng Wang
School of Mathematics and StatisticsWuhan University, Wuhan 430072, China
Abstract
The Cauchy problem to the Fokker-Planck-Boltzmann equation under Grad’s angular cut-off assump-tion is investigated. When the initial data is a small perturbation of an equilibrium state, global existenceand optimal temporal decay estimates of classical solutions are established. Our analysis is based on thecoercivity of the Fokker-Planck operator and an elementary weighted energy method.
The Fokker-Planck-Boltzmann equation models the motion of particles in a thermal bath where the bilinearinteraction is one of the main characters [2, 3, 26]. Mathematically, the Fokker-Planck-Boltzmann equationtakes the following form: ∂ t f + ξ · ∇ x f = Q ( f, f ) + ǫ ∇ ξ · ( ξf ) + κ ∆ ξ f, (1.1)where the nonnegative unknown function f = f ( t, x, ξ ) represents the density of particles at position x ∈ R and time t ≥ ξ ∈ R and ǫ, κ are given nonnegative constants. The collision operator Q is abilinear operator which acts only on the velocity variables ξ and is local in ( t, x ) as Q ( f, g )( ξ ) = Z R × S q ( ξ − ξ ∗ , ω ) { f ( ξ ′∗ ) g ( ξ ′ ) − f ( ξ ∗ ) g ( ξ ) } dωdξ ∗ . (1.2)Here ξ , ξ ∗ and ξ ′ , ξ ′∗ are the velocities of a pair of particles before and after collision. we assume these collisionsto be elastic so that ξ ′ = ξ − [( ξ − ξ ∗ ) · ω ] ω, ξ ′∗ = ξ ∗ + [( ξ − ξ ∗ ) · ω ] ω, ω ∈ S . The Boltzmann collision kernel q ( ξ − ξ ∗ , ω ) for a monatomic gas is, on physical grounds, a non-negativefunction which only depends on the relative velocity | ξ − ξ ∗ | and on the angle θ through cos θ = ω · ( ξ − ξ ∗ ) / | ξ − ξ ∗ | . There are two important model cases in physics: • Hard spheres, i.e., particles which collide bounce on each other like billiard balls. In this case q ( | ξ − ξ ∗ | , ω ) = | ( ξ − ξ ∗ ) · ω | = | ξ − ξ ∗ || cos θ | . • Inverse-power law potentials, i.e., particles which interact according to a spherical intermolecular repul-sive potential of the form φ ( r ) = r − ( s − , s ∈ (2 , ∞ ) , then one can show that q ( | ξ − ξ ∗ | , ω ) = | ξ − ξ ∗ | γ B ( θ ) , γ = 1 − s − . ∗ Corresponding author. E-mail: [email protected]; [email protected]; [email protected] B , it is only implicitly defined, locally smooth, and has a non-integrable singularity B ( θ ) = | cos θ | − γ ′ q ( θ ) , γ ′ = 1 + 2 s − , where q ( θ ) is bounded, q ( θ ) = 0 near θ = π/ . We consider the Cauchy problem of (1.1) with prescribed initial data f (0 , x, ξ ) = f ( x, ξ ) . (1.3)Throughout this manuscript, we assume that ǫ = κ > M = (2 π ) − / e −| ξ | / is an equilibrium state of (1.1) and the collision kernels satisfy Grad’s angular cut-off assumption: q ( | ξ − ξ ∗ | , ω ) = | ξ − ξ ∗ | γ B ( θ ) , ≤ B ( θ ) ≤ C | cos θ | , − < γ ≤ . (1.4)Our goal in this paper is to obtain the global existence and optimal temporal decay estimates of classicalsolutions for (1.1) and (1.3) with ǫ = κ > f is near the global Maxwellian M =(2 π ) − / e −| ξ | / . To this end, if we use u to denote the perturbation of f around the Maxwellian M as f = M + M / u, then the Cauchy problem (1.1) and (1.3) can be reformulated as ∂ t u + ξ · ∇ x u = Lu + Γ( u, u ) + ǫL F P u, (1.5) u (0 , x, ξ ) = u ( x, ξ ) = M − / ( f − M ) . (1.6)Here, the linear operator L , the bilinear form Γ( u , u ) and the classical linearized Fokker-Planck operator L F P are, respectively, given by Lu = M − n Q ( M, M / u ) + Q ( M / u, M ) o , Γ( u , u ) = M − Q ( M / u , M / u ) ,L F P u = ∆ ξ u + 14 (6 − | ξ | ) u. It is well known that for the linearized collision operator L , one has Lg ( ξ ) = − ν ( ξ ) g ( ξ ) + Kg ( ξ ) , where the collision frequency is ν ( ξ ) = Z R × S | ξ − ξ ∗ | γ q ( θ ) M ( ξ ∗ ) dωdξ ∗ ∼ (1 + | ξ | ) γ , and the operator K is defined by Ku ( ξ ) = Z R × S | ξ − ξ ∗ | γ q ( θ ) M / ( ξ ∗ ) M / ( ξ ′∗ ) u ( ξ ′ ) dωdξ ∗ + Z R × S | ξ − ξ ∗ | γ q ( θ ) M / ( ξ ∗ ) M / ( ξ ′ ) u ( ξ ′∗ ) dωdξ ∗ − Z R × S | ξ − ξ ∗ | γ q ( θ ) M / ( ξ ∗ ) M / ( ξ ) u ( ξ ∗ ) dωdξ ∗ . Furthermore, the operator L is non-positive, the null space of L is the five dimensional space N = span n M / , ξ j M / ( j = 1 , , , | ξ | M / o , okker-Planck-Boltzmann Equation 3and − L is locally coercive in the sense that there is a positive constant λ such that (see [4], [17], [27]) − Z R uLudξ ≥ λ Z R ν ( ξ ) |{ I − P } u | dξ (1.7)holds for u = u ( ξ ), where I means the identity operator and P denotes its ξ -projection from L ξ ( R ) onto thenull space N . As in [18], for any function u ( t, x, ξ ), we can write P as P u = { a ( t, x ) + b ( t, x ) · ξ + c ( t, x )( | ξ | − } M / ,a = R R M / udξ, b = R R ξM / udξ,c = R R ( | ξ | − M / udξ, so that we have the macro-micro decomposition introduced in [18] u ( t, x, ξ ) = P u ( t, x, ξ ) + { I − P } u ( t, x, ξ ) . (1.8)Here, P u and { I − P } u is called the macroscopic component and the microscopic component of u ( t, x, ξ ),respectively. For later use, one can rewrite P as P u = P u ⊕ P u, P u = a ( t, x ) M / , P u = (cid:8) b ( t, x ) · ξ + c ( t, x )( | ξ | − (cid:9) M / . Notations.
Throughout this paper, C denotes some positive (generally large) constant and λ denotessome positive (generally small) constant, where both C and λ may take different values in different places. A . B means there exists a constant C > A ≤ CB holds uniformly. A ∼ B means A . B and B . A . For the multi-indices α = ( α , α , α ) and β = ( β , β , β ), ∂ αβ = ∂ α x ∂ α x ∂ α x ∂ β ξ ∂ β ξ ∂ β ξ . Similarly, thenotation ∂ α will be used when β = 0, and likewise for ∂ β . The length of α is denoted by | α | = α + α + α . β ≤ α means that β j ≤ α j for each j = 1 , ,
3, and α < β means that β ≤ α and | β | < | α | . For notationalsimplicity, let h· , ·i denote the L inner product in R ξ with the L norm | · | , and let ( · , · ) denote the L innerproduct either in R x × R ξ or in R x with the L norm k · k . Moreover, we define | g | ν = h ν ( ξ ) g, g i , k g k ν = ( ν ( ξ ) g, g ) . For an integer m ≥
0, we use H m to denote the usual Sobolev space. We also define the space Z q = L ( R ξ ; L q ( R x )) for q ≥ k u k Z q = Z R (cid:18)Z R | u ( x, ξ ) | q dx (cid:19) /q dξ ! / , u = u ( x, ξ ) ∈ Z q . For an integrable function g : R → R , its Fourier transform b g = F g is defined by b g ( k ) = F g ( k ) = Z R e − πix · k g ( x ) dx, x · k = X j x j k j . for k ∈ R , where i = √− ∈ C is the imaginary unit. For two complex vectors a, b ∈ C , ( a | b ) = a · b denotesthe dot product over the complex filed, where b is the complex conjugate of b .For q ∈ R , the velocity weight function w q = w q ( ξ ) is always denoted by w q ( ξ ) = h ξ i q − γ (1.9)with h ξ i = (1 + | ξ | ) / . For an integer N and l ≥ N , we define the instant energy functional E q,l ( u )( t ) ≡ X | α | + | β |≤ N k w l −| β | q ∂ αβ u ( t ) k , (1.10) L.-J. Xiong, T. Wang, and L.-S. Wangand the dissipation rate D q,l ( u )( t ) ≡ X ≤| α |≤ N k ∂ αx P u ( t ) k + ǫ X | α |≤ N k{ I − P } ∂ αx u k + X | α | + | β |≤ N k w l −| β | q ∂ αβ { I − P } u ( t ) k ν . (1.11)We remark that our energy functional and dissipation rate which are not necessary to include the temporalderivatives which are different from [36]. The main result of this paper is stated as follows: For the hardpotential case, we have Theorem 1.1.
Let ≤ γ ≤ , l ≥ N ≥ , and q ≥ . Assume that Grad’s angular cut-off (1.4) is satisfiedand that f ( x, ξ ) = M + M / u ( x, ξ ) ≥ . Then we have (i) If there exists a sufficiently small δ > such that E q,l ( u ) ≤ δ and ( q − γ ) ǫ ≤ δ , the Cauchy problem(1.5)-(1.6) admits a unique global solution u which satisfies f ( t, x, ξ ) = M + M / u ( t, x, ξ ) ≥ for every t ≥ ; (ii) If we assume further that γ ≤ l ( q − γ ) and that there exists a sufficiently small positive constant δ > such that E q,l ( u ) + k u k Z ≤ δ and ( q − γ ) ǫ ≤ δ , the unique global solution u ( t, x, ξ ) obtained abovesatisfies the following optimal temporal decay estimates sup t ≥ n (1 + t ) E q,l ( u )( t ) o . δ . For the soft potential case, we have
Theorem 1.2.
Let − < γ < , l ≥ N ≥ , and q ≥ . Assume that Grad’s angular cut-off (1.4) is satisfiedand that f ( x, ξ ) = M + M / u ( x, ξ ) ≥ . Then we have (i) If there exists a sufficiently small δ > such that E q,l ( u ) ≤ δ and ( q − γ ) ǫ ≤ δ , the Cauchy problem(1.5)-(1.6) admits a unique global solution u ( t, x, ξ ) which satisfies f ( t, x, ξ ) = M + M / u ( t, x, ξ ) ≥ for every t ≥ ; (ii) If we assume further that l ≥ N + 1 and γ (1 − l ) ≤ q − γ )( l − for some l > / and that thereexists a sufficiently small δ > such that E q,l ( u ) + kh ξ i − γl / u k Z ≤ δ and ( q − γ ) ǫ ≤ δ , the uniqueglobal solution u ( t, x, ξ ) obtained above satisfy the following optimal temporal decay estimate sup t ≥ n (1 + t ) E q,l − ( u )( t ) o . δ . Remark 1.1.
The analysis here can be used to deal with the case when ǫ = ǫ ( t ) > and similar results canalso be obtained provided that ( q − γ ) ǫ ( t ) ≤ δ i hold for i = 0 , and every t ≥ . This means that for theFokker-Planck-Boltzmann equation (1.1) with ǫ ≡ and κ > , i.e. ∂ t f + ξ · ∇ x f = Q ( f, f ) + κ ∆ ξ f, we can use the scaling used in [23] to transform the above problem into (1.1) with ǫ = κ = κ (1 + 3 κt ) − andsimilar results can also be obtained provided that ( q − γ ) ǫ ( t ) = ( q − γ ) κ (1 + 3 κt ) − ≤ δ i hold for i = 0 , and every t ≥ . It is easy to see that a sufficient condition to guarantee the validity of the above inequalitiesis that κ > is sufficiently small as imposed in [23] and it is worth to pointing out that when γ → − andby taking q = 1 , one can see that the assumptions ( q − γ ) κ (1 + 3 κt ) − ≤ δ i hold even without the smallnessrestriction on κ . In such a sense, our result generalizes the result obtained in [23] even for the hard sphereintermolecular interaction. Remark 1.2.
It is worth to point out that here we use the weight function w l −| β | q to capture the term | ξ || ∂ β u | generated by the ξ -derivatives ∂ β acting on the Fokker-Planck operator in term of the weaker dissipation rate k ∂ β u k ν . okker-Planck-Boltzmann Equation 5 Remark 1.3.
The rates of convergence are optimal under the corresponding assumptions in the sense thatthey coincide with those rates given in (4 . at the level of linearization. There have been a lot of studies on the Fokker-Planck-Boltzmann equation (1.1).
DiPerna and
Lions [5]proved the global existence of the renormalized solutions for the Cauchy problem (1.1) and (1.3).
Hamdache [20] obtained the global existence near the vacuum state in terms of a direct construction. It is shown in[23] that a strong solution of the equation (1.1) for initial data near the global Maxwellian exists globally intime and tends asymptotically to another time-dependent self-similar Maxwellian in the large-time limit forthe hard sphere case (1.4) with γ = 1. Li and Matsumura in [23] first introduced an appropriate scaling totransform (1.1) with ǫ ≡ κ > ǫ = κ → κ (1 + 3 κt ) − and then achieved their goals byemploying the pioneering L energy method based on macro-micro decomposition around a local Maxwelliandeveloped for the Boltzmann equation [24], [25]. For the case − ≤ γ ≤
1, the long time behavior to theCauchy problem of (1.1), (1.3) is studied by constructing the compensating functions to this system, whilethe main goal of this paper is to obtain the global existence of classical solutions for (1.1) and (1.3) and thecorresponding optimal time decay of the solutions under Grad’s angular cut-off assumption for the whole rangeof intermolecular interaction − < γ ≤ − < γ <
1. Our approach is based on the methods in[11, 12] for the Vlasov-Poisson-Boltzmann system. For more information related to the Boltzmann equationand the kinetic theory, the reader can also refer to [4, 3, 13, 30] and references therein.Before concluding this section, we sketch main ideas used in deducing our results. One of the maindifficulties lies in the fact that the dissipation of the linearized Boltzmann operator L for non hard-spherepotentials can not control the full nonlinear dynamics due to the velocity growth effect of | ξ || ∂ β u | generated bythe ξ -derivatives ∂ β acting on the Fokker-Planck operator. A suitable application of a weight function w l −| β | q can indeed yield a satisfactory global existence of classical solution to the Fokker-Planck-Boltzmann equationfor the case − ≤ γ ≤
1, while for the very soft potential case − < γ < −
2, we cannot close our energyestimate by only employing the coercivity of the linearized collision L as for the case of − ≤ γ ≤
1. Still andall, we can combine both the coercivity of L and L F P and divide the integral domain about ξ into two parts:the first part { ξ |h ξ i ≤ R } can be control by the coercivity of L with the smallness of ǫ while the second part { ξ |h ξ i > R } by the coercivity of L F P when we choose R large enough.The time rate of convergence to equilibrium is an important topic in the mathematical theory of the physicalworld. As pointed out in [31], the exist general structures in which the interaction between a conservativepart and a degenerate dissipative part lead to the convergence to equilibrium, where this property was calledhypocoercivity. Here, indeed, we provide a concrete example of hypocoercivity property for the nonlinearFokker-Planck-Boltzmann equation in the framework of perturbation. We employ the methods developingby Duan and Strain [9 , L . We need a more delicate estimate on the time decay of solutionto the corresponding linearized equation in the case of the whole space R based on the weighted energyestimates, a time-frequency analysis method, and the construction of some interactive energy functionals. Wealso mention that Zhang and Li [36] have obtained the similar decay rate for the case − ≤ γ ≤ In this section, we will obtain the macroscopic dissipation rate X ≤| α |≤ N k ∂ αx P u ( t ) k ∼ X | α |≤ N − k ∂ αx ∇ x ( a, b, c )( t ) k . To this end, we shall first apply the macro-micro decomposition (1.8) to the equation (1.5) to discover themacroscopic balance laws satisfied by ( a, b, c ). Multiply (1.1) by the collision invariants 1, ξ and | ξ | to findthe local balance laws ∂ t R R f dξ + ∇ x · R R ξf dξ = 0 ,∂ t R R ξf dξ + ∇ x R R ξ ⊗ ξf dξ + ǫ R R ξf dξ = 0 ,∂ t R R | ξ | f dξ + ∇ x · R R | ξ | ξf dξ + 2 ǫ R R ( | ξ | − ξf dξ = 0 . (2.1)As in [9], define the high-order moment functions A = ( A jm ) × and B = ( B , B , B ) by A jm ( u ) = D ( ξ j ξ m − M / , u E , B j ( u ) = 110 D ( | ξ | − ξ j M / , u E . (2.2)Plugging f = M + M / P u + M / { I − P } u into (2.1), one can deduce the first system of macroscopic equations ∂ t a + ∇ x · b = 0 ,∂ t b + ∇ x ( a + 2 c ) + ∇ x A ( { I − P } u ) + ǫb = 0 ,∂ t c + ∇ x · b + ∇ x · B ( { I − P } u ) + 2 ǫc = 0 . (2.3)To obtain the second system of macroscopic equations, we split u = P u + { I − P } u to decompose the equation(1.5) as ∂ t P u + ξ · ∇ x P u − ǫL F P P u = − ∂ t { I − P } u + R + G, (2.4)with R = − ξ · ∇ x { I − P } u + ǫL F P { I − P } u + L { I − P } u, G = Γ( u, u ) . (2.5)Applying A jm ( · ) and B j ( · ) to both sides of (2.4), and using L F P P u = − b · ξM / − c ( | ξ | − M / and the balance law of mass (2.3) , one has ∂ j b j + 2 ∂ t c + 4 ǫc = − ∂ t A jj ( { I − P } u ) + A jj ( R + G ) ,∂ j b m + ∂ m b j = − ∂ t A jm ( { I − P } u ) + A jm ( R + G ) , j = m,∂ t B j ( { I − P } u ) + ∂ j c = B j ( R + G ) . (2.6)Now we focus on the macroscopic equations (2.3) and (2.6) to estimate the higher order derivatives of themacroscopic coefficients ( a, b, c ) in L norm. For this purpose, we first give a lemma without proofs. Roughlyspeaking, the idea is just based on the fact that the velocity-coordinate projector is bounded uniformly in t and x , and the velocity polynomials and velocity derivatives can be absorbed by the global Maxwellian M which exponentially decays in ξ . Lemma 2.1.
For any | α | ≤ N and ≤ j, m ≤ , it holds that k ∂ αx A jm ( { I − P } u ) , ∂ αx B j ( { I − P } u ) k . min {k ∂ αx { I − P } u k , k ∂ αx { I − P } u k ν } . (2.7)okker-Planck-Boltzmann Equation 7 Moreover, for any | α | ≤ N − and ≤ j, m ≤ , it holds that k ∂ αx A jm ( R ) , ∂ αx B j ( R ) k . X | α |≤| α | +1 k ∂ α x { I − P } u k ν (2.8) and k ∂ αx A jm ( G ) , ∂ αx B j ( G ) k . E q,l ( u )( t ) D q,l ( u )( t ) . (2.9)Next we state the key estimates on the macroscopic dissipation in the following theorem. Theorem 2.1.
There is an interactive energy functional E int ( u )( t ) such that |E int ( u )( t ) | . X | α |≤ N k ∂ αx u ( t ) k (2.10) and ddt E int ( u )( t ) + λ X | α |≤ N − k ∂ αx ∇ x ( a, b, c )( t ) k . X | α |≤ N k ∂ αx { I − P } u k ν + ǫ X | α |≤ N − k ∂ αx ( b, c ) k + E q,l ( u )( t ) D q,l ( u )( t ) , (2.11) where E int ( u )( t ) is the linear combination of the following terms over | α | ≤ N − and ≤ j ≤ : I aα ( u ( t )) = h ∂ αx b, ∇ x ∂ αx a i , I bα,j ( u ( t )) = * X m = j ∂ j ∂ αx A mm ( { I − P } u ) − X m ∂ m ∂ αx A jm ( { I − P } u ) , ∂ αx b j + , I cα,j ( u ( t )) = h ∂ αx B j ( { I − P } u ) , ∂ j ∂ αx c i . Proof.
Step 1.
Estimate on b . For any η >
0, it holds that ddt X | α |≤ N − X j I bα,j ( u ( t )) + 12 X | α |≤ N − k ∂ αx ∇ x b k ≤ Cη X | α |≤ N − k ∂ αx ∇ x ( a, c ) k + Cη X | α |≤ N − ǫ k ∂ αx b k + C η X | α |≤ N k ∂ αx { I − P } u k ν + C η E q,l ( u )( t ) D q,l ( u )( t ) . (2.12)In fact, for fixed j ∈ { , , } , one can deduce from (2.6) that − ∆ x b j − ∂ j ∂ j b j = − ∂ t X m = j ∂ j A mm ( { I − P } u ) − X m ∂ m A jm ( { I − P } u ) + 12 X m = j ∂ j A mm ( R + G ) − X m ∂ m A jm ( R + G ) . (2.13) L.-J. Xiong, T. Wang, and L.-S. WangLet | α | ≤ N −
1. Apply ∂ αx to the elliptic-type equation (2.13), multiply it by ∂ αx b j , and then integrate it over R to find ddt I bα,j ( u ( t )) + k∇ x ∂ αx b j k + k ∂ j ∂ αx b j k = * X m = j ∂ j ∂ αx A mm ( { I − P } u ) − X m ∂ m ∂ αx A jm ( { I − P } u ) , ∂ αx ∂ t b j + + * X m = j ∂ j ∂ αx A mm ( R + G ) − X m ∂ m ∂ αx A jm ( R + G ) , ∂ αx b j + = I b + I b . (2.14)Using (2.3) (the second equation of (2.3)) to replace ∂ t b j , we get I b ≤ η k ∂ αx ∂ t b j k + C η X | β |≤ N (cid:13)(cid:13) ∂ βx A ( { I − P } u ) (cid:13)(cid:13) ≤ η (cid:8) k ∂ αx ∇ x ( a, c ) k + ǫ k ∂ αx b j k (cid:9) + C η X | β |≤ N (cid:13)(cid:13) ∂ βx { I − P } u (cid:13)(cid:13) ν . (2.15)Here we have used (2.1). For I b , integrating by parts implies I b = − X m = j h ∂ αx A mm ( R + G ) , ∂ j ∂ αx b j i + X m h ∂ αx A jm ( R + G ) , ∂ m ∂ αx b j i≤ k∇ x ∂ αx b j k + C X m k ∂ αx A jm ( R, G ) k ≤ k∇ x ∂ αx b j k + C X | β |≤ N (cid:13)(cid:13) ∂ βx { I − P } u (cid:13)(cid:13) ν + C E q,l ( u )( t ) D q,l ( u )( t ) . (2.16)Thus, (2.12) follows by plugging (2.15) and (2.16) into (2.14) and then taking summation over 1 ≤ j ≤ | α | ≤ N − Estimate on c . For any η >
0, it holds that ddt X | α |≤ N − X j I cα,j ( u ( t )) + 12 X | α |≤ N − k∇ x ∂ αx c k ≤ η X | α |≤ N − k ∂ αx ∇ x b k + 12 η X | α |≤ N − ǫ k ∂ αx c k + C η X | α |≤ N k ∂ αx { I − P } u k ν + C η E q,l ( u )( t ) D q,l ( u )( t ) . (2.17)Indeed, applying ∂ αx with | α | ≤ N − , multiplying it by ∂ j ∂ αx c and thenintegrating it over R , we have ddt I cα,j ( u ( t )) + k ∂ j ∂ αx c k = h ∂ αx B j ( { I − P } u ) , ∂ t ∂ j ∂ αx c i + h ∂ αx B j ( R + G ) , ∂ j ∂ αx c i = I c + I c . (2.18)okker-Planck-Boltzmann Equation 9Use (2 . to replace ∂ t c and estimate I c as I c = − h ∂ j ∂ αx B j ( { I − P } u ) , ∂ αx ∂ t c i≤ η k ∂ αx ∂ t c k + C η k ∂ j ∂ αx B j ( { I − P } u ) k ≤ η (cid:8) k ∂ αx ∇ x b k + 4 ǫ ( t ) k ∂ αx c k (cid:9) + C η X | β |≤ N (cid:13)(cid:13) ∂ βx { I − P } u (cid:13)(cid:13) ν . (2.19) I c is bounded by I c ≤ k ∂ j ∂ αx c k + k ∂ αx B j ( R, G ) k ≤ k ∂ j ∂ αx c k + C E q,l ( u )( t ) D q,l ( u )( t ) . (2.20)Thus, (2.17) follows by plugging (2.19) and (2.20) into (2 . ≤ j ≤ | α | ≤ N − Estimate on a . Let | α | ≤ N −
1. Apply ∂ αx to(2.3) , multiply it by ∂ αx ∇ x a and then integrate itover R to discover ∂ t h ∂ αx b, ∂ αx ∇ x a i + k ∂ αx ∇ x a k = − h ∂ αx ∇ x c + ∂ αx ∇ x A ( { I − P } u ) + ǫ∂ αx b, ∂ αx ∇ x a i + h ∂ αx b, ∂ t ∂ αx ∇ x a i = − h ∂ αx ∇ x c + ∂ αx ∇ x A ( { I − P } u ) + ǫ∂ αx b, ∂ αx ∇ x a i + h ∂ αx ∇ x · b, ∂ αx ∇ x · b i≤ k ∂ αx ∇ x a k + ǫ k ∂ αx b k + C k ∂ αx ∇ x ( b, c ) k + C k ∂ αx ∇ x A ( { I − P } u ) k . (2.21)Here we used the conservation of mass (2.3) . Take summation (2.21) over | α | ≤ N − ddt X | α |≤ N − h ∂ αx b, ∇ x ∂ αx a i + 12 X | α |≤ N − k ∂ αx ∇ x a k . X | α |≤ N − k ∂ αx ∇ x ( b, c ) k + X | α |≤ N − ǫ k ∂ αx b k + C X | α |≤ N k ∂ αx { I − P } u k ν . (2.22)Step 4. Combination.
We have finished the estimates of a, b, c . With them in hand, let us multiply (2 . .
17) by a constant
M >
M > b and c . By fixing M >
0, one can choose η > .
12) and (2 .
17) are absorbed by the full dissipation of b and c . Hence, we have proved(2.11). Cauchy’s inequality and (2.7) yield |E int ( u )( t ) | . X | α |≤ N − (cid:8) k ∂ αx ∇ x ( a, b, c )( t ) k + k ∂ αx { I − P } u k + k ∂ αx b k (cid:9) , which implies (2.10). Therefore one has finished the proof of Theorem 2.1. In this section, we shall devote ourselves to obtaining the existence of classical solutions to (1.5) globallyin time. For this purpose, we first collect some estimates for the linearized Fokker-Planck operator L F P andthe collision operators L and Γ.For the linearized Fokker-Planck operator L F P , we have the following two results. The first one is concernedwith the dissipative property of the linearized Fokker-Planck operator L F P without weight0 L.-J. Xiong, T. Wang, and L.-S. Wang
Lemma 3.1. ([1], [7]) L F P is a linear self-adjoint operator with respect to the duality induced by the L ξ -scalarproduct. Furthermore, there exists a constant λ F P > such that − ( u, L F P u ) ≥ λ F P k{ I − P } u k . (3.1)For the dissipative property of the linearized Fokker-Planck operator L F P with the weight w l −| β | q , we have Lemma 3.2.
It holds that for any l ≥ , (cid:16) L F P ∂ αβ u, w l −| β | ) q ∂ αβ u (cid:17) ≤ − λ F P (cid:13)(cid:13)(cid:13) { I − P } ( w l −| β | q ∂ αβ u ) (cid:13)(cid:13)(cid:13) + C ( q − γ ) (cid:13)(cid:13)(cid:13) w l −| β | q ∂ αβ u (cid:13)(cid:13)(cid:13) ν . (3.2) Proof.
Integrating by parts yields (cid:16) L F P ∂ αβ u, w l −| β | ) ∂ αβ u (cid:17) − (cid:16) L F P ( w l −| β | q ∂ αβ u ) , w l −| β | q ∂ αβ u (cid:17) = − (cid:16) ∇ ξ · (cid:16) ∂ αβ u ∇ ξ w l −| β | q (cid:17) + ∇ ξ w l −| β | q · ∇ ξ ∂ αβ u, w l −| β | q ∂ αβ u (cid:17) = (cid:16) ∇ ξ w l −| β | q ∂ αβ u, ∇ ξ ( w l −| β | q ∂ αβ u ) (cid:17) − (cid:16) ∇ ξ w l −| β | q · ∇ ξ ∂ αβ u, w l −| β | q ∂ αβ u (cid:17) = (cid:16) ∇ ξ w l −| β | q ∂ αβ u, ∇ ξ w l −| β | q ∂ αβ u (cid:17) ≤ C ( q − γ ) (cid:13)(cid:13)(cid:13)(cid:0) χ | ξ | >R + χ | ξ |≤ R (cid:1) h ξ i − w l −| β | q ∂ αβ u (cid:13)(cid:13)(cid:13) (3.3)for each R >
0. Here, we have used the fact that ∇ ξ w l −| β | q = ( q − | β | )(1 − γ ) w l −| β | q ξ | ξ | . We estimate the terms on the right hand side of (3.3). First, (cid:13)(cid:13)(cid:13) χ | ξ | >R h ξ i − w l −| β | q ∂ αβ u (cid:13)(cid:13)(cid:13) ≤ R − (cid:13)(cid:13)(cid:13) { I − P } (cid:16) w l −| β | q ∂ αβ u (cid:17)(cid:13)(cid:13)(cid:13) + C (cid:13)(cid:13)(cid:13) P (cid:16) w l −| β | q ∂ αβ u (cid:17)(cid:13)(cid:13)(cid:13) ≤ R − (cid:13)(cid:13)(cid:13) { I − P } (cid:16) w l −| β | q ∂ αβ u (cid:17)(cid:13)(cid:13)(cid:13) + C (cid:13)(cid:13)(cid:13) w l −| β | q ∂ αβ u (cid:13)(cid:13)(cid:13) ν . (3.4)If ξ is bounded, then h ξ i − ∼ ν ( ξ ) which implies (cid:13)(cid:13)(cid:13) χ | ξ |≤ R h ξ i − w l −| β | q ∂ αβ u (cid:13)(cid:13)(cid:13) . (cid:13)(cid:13)(cid:13) w l −| β | q ∂ αβ u (cid:13)(cid:13)(cid:13) ν . (3.5)Plugging (3.4) and (3.5) into (3.3), and noticing that − (cid:16) L F P ( w l −| β | q ∂ αβ u ) , w l −| β | q ∂ αβ u (cid:17) ≥ λ F P (cid:13)(cid:13)(cid:13) { I − P } ( w l −| β | q ∂ αβ u ) (cid:13)(cid:13)(cid:13) from (3.1), one can prove (3.2) by choosing R > L and the nonlinearcollision operator Γ, we haveokker-Planck-Boltzmann Equation 11 Lemma 3.3. ([19], [17]) Consider the inverse power law with − < γ ≤ . If η > and m ≥ , then thereare C η , C > , such that − (cid:0) h ξ i m ∂ β Lg, ∂ β g (cid:1) ≥ kh ξ i m ∂ β g k ν − η X | β |≤| β | kh ξ i m ∂ β g k ν − C η k g k ν , (3.6) (cid:12)(cid:12)(cid:10) h ξ i m ∂ β Γ( f , f ) , ∂ β h (cid:11)(cid:12)(cid:12) . X i,j X β + β ≤ β |h ξ i m ∂ β f i | |h ξ i m ∂ β f j | ν |h ξ i m ∂ β h | ν . (3.7) Lemma 3.4.
It holds that for any l ≥ , (cid:0) ∂ αx Γ( u, u ) , w lq ∂ αx u (cid:1) . E q,l ( u ) / ( t ) D q,l ( u )( t ) , (3.8)( ∂ αβ Γ( u, u ) , w l −| β | ) q ∂ αβ { I − P } u ) . E q,l ( u ) / ( t ) D q,l ( u )( t ) . (3.9)Next, as the first step, we shall obtain the dissipation rate ǫ X | α |≤ N k{ I − P } ∂ αx u k . To this end, we consider the non-weighted energy estimates on the solution u of (1.5)-(1.6). Taking ∂ αx of theequation (1.5) yields 12 ddt k ∂ αx u k − ( L∂ αx u, ∂ αx u ) − ǫ ( L F P ∂ αx u, ∂ αx u ) = ( ∂ αx Γ( u, u ) , ∂ αx u ) . (3.10)Applying (1.7), (3.1) and (3.8) with l = 0 to (3.10), we thus get the following lemma. Lemma 3.5.
It holds that for each t > , ddt X | α |≤ N k ∂ αx u k + λ X | α |≤ N k ∂ αx { I − P } u k ν + λ F P ǫ X | α |≤ N k{ I − P } ∂ αx u k . E q,l ( u ) / ( t ) D q,l ( u )( t ) . (3.11)For the second step, we consider the weighted energy estimates on u to get the dissipation rate X | α | + | β |≤ N k w l −| β | q ∂ αβ { I − P } u ( t ) k ν . Lemma 3.6.
There is a positive constant δ such that if sup ≤ t ≤ T E q,l ( u )( t ) ≤ δ (3.12) and ( q − γ ) ǫ ≤ δ , then ddt E q,l ( u )( t ) + λ D q,l ( u )( t ) ≤ . (3.13) Proof.
Step 1.
Weight estimate on zero-order of { I − P } u : ddt k w lq ( ξ ) { I − P } u ( t ) k + 12 k w lq { I − P } u k ν + ǫλ F P k{ I − P } ( w lq { I − P } u )( t ) k . k{ I − P } u k ν + k∇ x u k ν + E q,l ( u ) / ( t ) D q,l ( u )( t ) . (3.14)2 L.-J. Xiong, T. Wang, and L.-S. WangIn fact, apply { I − P } to (1.5) and then use L F P P u = P L F P u to find ∂ t { I − P } u + ξ · ∇ x { I − P } u − L { I − P } u =Γ( u, u ) + ǫL F P { I − P } u + P ξ · ∇ x u − ξ · ∇ x P u. (3.15)Multiply (3.15) by w lq { I − P } u and integrate it over R × R to have12 ddt k w lq { I − P } u k − ( w lq L { I − P } u, { I − P } u )=( w lq Γ( u, u ) , { I − P } u ) + ǫ ( L F P { I − P } u, w lq { I − P } u )+ ( P ξ · ∇ x u − ξ · ∇ x P u, w lq { I − P } u ) . (3.16)Cauchy’s inequality yields that the third term on the right-hand side of (3.16) is bounded by18 k w lq { I − P } u k ν + C k∇ x u k ν . Plugging (3.6), (3.9) and (3.2) into (3.16), we can prove (3.14) when ( q − γ ) ǫ is suitably small. Step 2.
Weighted estimate on pure space-derivative of u : ddt X ≤| α |≤ N k w lq ∂ αx u k + 12 X ≤| α |≤ N k w lq ∂ αx u k ν + λ F P ǫ X ≤| α |≤ N k{ I − P } ( w lq ∂ αx u ) k . X ≤| α |≤ N k ∂ αx u k ν + E q,l ( u ) / ( t ) D q,l ( u )( t ) . (3.17)In fact, let 1 ≤ | α | ≤ N . Taking ∂ αx of (1.5), multiplying it by w lq ( ξ ) ∂ αx u , and then integrating it over R × R ,one has 12 ddt k w lq ∂ αx u k − ( w lq L∂ αx u, ∂ αx u )=( ∂ αx Γ( u, u ) , w lq ∂ αx u ) + ǫ ( L F P ∂ αx u, w lq ∂ αx u ) . (3.18)Hence, (3.17) follows from plugging the estimates (3.6), (3.8) and (3.2) into (3.18) and then taking summationover 1 ≤ | α | ≤ N . Step 3.
Weighted estimate on mixed space-velocity-derivative of u : ddt N X m =1 C m X | β | = m | α | + | β |≤ N k w l −| β | q ∂ αβ { I − P } u k + λ X | β |≥ | α | + | β |≤ N n k w l −| β | q ∂ αβ { I − P } u k ν + ǫ k{ I − P } ( w l −| β | q ∂ αβ { I − P } u ) k o . X ≤| α |≤ N k ∂ αx u k ν + E q,l ( u ) / ( t ) D q,l ( u )( t ) . (3.19)okker-Planck-Boltzmann Equation 13Indeed, let | β | = m > | α | + | β | ≤ N . For notational simplicity, we denote that u ≡ { I − P } u . Apply ∂ αβ to (3.15), and multiply it by w l −| β | ) q ∂ αβ u and then integrate over R × R to find12 ddt (cid:13)(cid:13)(cid:13) w l −| β | q ∂ αβ u (cid:13)(cid:13)(cid:13) − (cid:16) w l −| β | ) q ∂ αβ Lu , ∂ αβ u (cid:17) = (cid:16) ∂ αβ Γ( u, u ) , w l −| β | ) q ∂ αβ u (cid:17) + ǫ (cid:16) ∂ αβ L F P u , w l −| β | ) q ∂ αβ u (cid:17) − (cid:16) ∂ αβ ( ξ · ∇ x u ) , w l −| β | ) q ∂ αβ u (cid:17) + (cid:16) ∂ αβ ( P ξ · ∇ x u − ξ · ∇ x P u ) , w l −| β | ) q ∂ αβ u (cid:17) . (3.20)Noting that w l −| β | q ≤ w l −| β | q whenever | β | ≤ | β | , we obtain from (3.6) that − (cid:16) w l −| β | ) q ∂ αβ Lu , ∂ αβ u (cid:17) ≥ (cid:13)(cid:13)(cid:13) w l −| β | q ∂ αβ u (cid:13)(cid:13)(cid:13) ν − C η k ∂ αx u k ν − η X | β |≤| β | (cid:13)(cid:13)(cid:13) w l −| β | q ∂ αβ u (cid:13)(cid:13)(cid:13) ν . (3.21)We estimate the terms on the right hand side of (3.20). Recall that w = h ξ i q − γ , which implies that h ξ i . ν ( ξ ) w − ( ξ ) , whenever q ≥
1. Hence we have( ∂ αβ L F P u , w l −| β | ) q ∂ αβ u ) − ( L F P ∂ αβ u , w l −| β | ) q ∂ αβ u )= − X <β ≤ β C β β ( ∂ β | ξ | ∂ αβ − β u , w l −| β | ) q ∂ αβ u ) ≤ C X <β ≤ β ( h ξ i| ∂ αβ − β u | , w l −| β | ) q | ∂ αβ u | ) ≤ C X <β ≤ β ( ν ( ξ ) w l −| β − β | | ∂ αβ − β u | , w l −| β | q | ∂ αβ u | ) ≤ η (cid:13)(cid:13)(cid:13) w l −| β | q ∂ αβ u (cid:13)(cid:13)(cid:13) ν + C η X | β |
0, (3.19) follows by plugging the estimates (3.21), (3.8), (3.22),(3.2), (3.23) and (3.24) into (3 . {| β | = m, | α | + | β | ≤ N } for each given 1 ≤ m ≤ N and taking proper linear combination of those N − C m > ≤ m ≤ N ). Step 4.
Combination.
First, let us multiply (3.11) by a constant M > X | α |≤ N k ∂ αx ( b, c ) k ≤ X | α |≤ N k{ I − P } ∂ αx u k . Thus, one can take M > E int ( u )( t ) + 12 M X | α |≤ N k ∂ αx u k ∼ X | α |≤ N k ∂ αx u k . In the further linear combination(3 .
14) + (3 .
17) + (3 .
19) + M × [ M × (3 .
11) + (2 . , one can take M > ddt E q,l ( u )( t ) + λ D q,l ( u )( t ) . h E q,l ( u ) / ( t ) + E q,l ( u )( t ) i D q,l ( u )( t ) . (3.25)Therefore, (3.13) follows under the a priori assumption (3.12). Proof of Theorem 1.1(i) and Theorem 1.2(i):
Fix N , l as stated in Theorem 1.1 or Theorem 1.2.The local existence and uniqueness of the solution u ( t, x, ξ ) to the Cauchy problem (1 . − (1 .
6) can beproved in terms of the energy functional E q,l ( u )( t ) given by (1 . , ,
23] with a little modification. Now we have obtained the unform-in-time estimate (3 .
13) over0 ≤ t ≤ T with 0 < T ≤ ∞ . By the standard continuity argument, the global existence follows provided theinitial energy functional E ( u ) is sufficiently small. In this subsection, we devote ourselves to obtaining the time decay rate of the global solution u to theFokker-Planck-Boltzmann equation (1 . .
6) in the hard potential case (0 ≤ γ ≤ ∂ t u + ξ · ∇ x u = Lu + ǫL F P u + G,u (0 , x, ξ ) = u ( x, ξ ) , (4.1)where u ( x, ξ ) and G = G ( t, x, ξ ) with P G = 0 are given. Formally, the solution u to the Cauchy problem(4 .
1) can be written as the mild form u ( t ) = e tB u + Z t e ( t − s ) B h ( s ) ds, where e tB denotes the solution operator to the Cauchy problem of (4.1) with G ≡
0. We first show that theoperator e tB has the proposed algebraic decay properties as time tends to infinity. The idea of the proofs is tomake energy estimates for pointwise time t and frequency variable k , which corresponds to the spatial variable x .okker-Planck-Boltzmann Equation 15 Lemma 4.1.
There is
M > such that the free energy functional E free ( b u )( t, k ) , defined by E free ( b u )( t, k ) = M X j X m = j ik j | k | A mm ( { I − P } b u ) − X m ik m | k | A jm ( { I − P } b u ) | − b b j ! + M X j (cid:18) B j { I − P } b u | ik j b a | k | (cid:19) + X j (cid:18)b b j | ik j | k | b a (cid:19) (4.2) satisfies Re E free ( b u )( t, k ) . | b u | (4.3) and ∂ t Re E free ( b u ( t, k )) + λ | k | | k | (cid:16) | b a | + | b b | + | b c | (cid:17) . ǫ (cid:16) | b b | + | b c | (cid:17) + |{ I − P } b u | ν + | ν − / b G | (4.4) for any t ≥ and k ∈ R .Proof. Estimate on b b . We claim that for 0 < η <
1, it holds that ∂ t Re X j X m = j ik j A mm ( { I − P } b u ) − X m ik m A jm ( { I − P } b u ) | b b j + (1 − η ) | k | | b b | ≤ η | k | (cid:0) | b a | + | b c | (cid:1) + ǫ | b b j | + C η (1 + | k | ) (cid:16) |{ I − P } b u | ν + | ν − / b G | (cid:17) . (4.5)In fact, the Fourier transform of (2 .
13) gives ∂ t X m = j ik j A mm ( { I − P } b u ) − X m ik m A jm ( { I − P } b u ) + | k | b b j + k j b b j = 12 X m = j ik j A mm ( ˆ R + b G ) − X m ik m A jm ( ˆ R + b G ) , where R = − ξ · ∇ x { I − P } u + ǫL F P { I − P } u + L { I − P } u. We then take the complex inner product with b b j to find ∂ t X m = j ik j A mm ( { I − P } b u ) − X m ik m A jm ( { I − P } b u ) | b b j + (cid:0) | k | + k j (cid:1) | b b j | = X m = j ik j A mm ( ˆ R + b G ) − X m ik m A jm ( ˆ R + b G ) | b b j + X m = j ik j A mm ( { I − P } b u ) − X m ik m A jm ( { I − P } b u ) | ∂ t b b j = I + I . (4.6)6 L.-J. Xiong, T. Wang, and L.-S. WangNote that ˆ R = − iξ · k { I − P } b u + ǫL F P { I − P } b u + L { I − P } b u, which implies | A jm ( ˆ R ) | . (1 + | k | ) |{ I − P } b u | ν . Thus, I is bounded by I ≤ η | k | | b b j | + C η X j,m (cid:16) | A jm ( ˆ R ) | + | A jm ( b G ) | (cid:17) ≤ η | k | | b b j | + C η (1 + | k | ) (cid:16) |{ I − P } b u | ν + | ν − / b G | (cid:17) . (4.7)For I , using the Fourier transform of (2 . ∂ t b b j + ik j ( b a + 2 b c ) + X m ik m A jm ( { I − P } b u ) + ǫ b b j = 0 (4.8)to replace ∂ t b b j , we have I ≤ η | k | (cid:0) | b a | + | b c | (cid:1) + ǫ | b b j | + C η (1 + | k | ) X jm | A j,m { I − P } b u | ≤ η | k | (cid:0) | b a | + | b c | (cid:1) + ǫ | b b j | + C η (1 + | k | ) |{ I − P } b u | ν . (4.9)Therefore, one can take the real part of (4 .
6) and plug the estimates (4.7) and (4.9) into it to discover (4.6).
Estimate on b c . For any 0 < η <
1, we have ∂ t Re X j ( B j ( { I − P } b u ) | ik j b c ) + (1 − η ) | k | | b c | ≤ η | k | || b b j | + ǫ | b c | + C η (1 + | k | ) (cid:16) |{ I − P } b u | ν + | ν − / b G | (cid:17) . (4.10)In fact, multiply the Fourier transform of (2 . ∂ t B j ( { I − P } b u ) + ik j b c = B j ( ˆ R + b G )by − ik j b c to give ∂ t ( B j ( { I − P } b u ) | ik j b c ) + | k j | | b c | = (cid:16) B j ( ˆ R + b G ) | ik j b c (cid:17) + ( B j ( { I − P } b u ) | ik j ∂ t b c )= I + I .I is bounded by I ≤ η | k j | | b c j | + C η X j (cid:16) | B j ( ˆ R ) | + | B j ( b G ) | (cid:17) ≤ η | k j | | b c j | + C η (1 + | k | ) (cid:16) |{ I − P } b u | ν + | ν − / b G | (cid:17) . (4.11)For I , using the Fourier transform of (2 . ∂ t b c + 13 ik · b b + 53 X j ik j B j ( { I − P } b u ) + 2 ǫ b c = 0to replace ∂ t b c , one has I ≤ η | k | || b b j | + ǫ | b c | + C η (1 + | k | ) X j | B j ( { I − P } b u ) | ≤ η | k | || b b j | + ǫ | b c | + C η (1 + | k | ) |{ I − P } b u | ν . (4.12)okker-Planck-Boltzmann Equation 17Hence, (4.10) follows by taking the real part and applying the estimates of (4.11) and (4.12), and then takingthe summation over 1 ≤ j ≤ Estimate on b a . We claim that it holds for any 0 ≤ η < ∂ t Re X j ( b j | ik j b a ) + (1 − η ) | k | | b a | ≤ | k | | b b | + C η (cid:16) | k | | b c | + | k | |{ I − P } b u | ν + ǫ | b b | (cid:17) . (4.13)In fact, using (4 . ik j b a , and then taking the summation over1 ≤ j ≤
3, one has ∂ t X j (cid:16)b b j | ik j b a (cid:17) + | k | | b a | = X j ( − ik j b c | ik j b a ) − X j,m ( ik m A jm ( { I − P } b u ) | ik j b a )+ X j (cid:16) − ǫ b b j | ik j b a (cid:17) + X j (cid:16)b b j | ik j ∂ t b a (cid:17) . (4.14)The first there terms on the right-hand side of (4 .
14) are bounded by η | k | | b a | + C η (cid:16) | k | | b c | + | k | |{ I − P } b u | ν + ǫ | b b | (cid:17) , while for the last term, it holds that X j (cid:16)b b j | ik j ∂ t b a (cid:17) = X j (cid:16)b b j | ik j ( − ik · b b ) (cid:17) = | k · b b | ≤ | k | | b b | . Here we used the Fourier transform of (2 . : ∂ t b a + ik · b b = 0 . Then, one can deduce (4.13) by putting the above estimates into (4 .
14) and taking the real part.Therefore, (4 .
4) follows from the proper linear combination of (4.5), (4.10) and (4.13) by taking
M > < η < |E free ( b u ) | ( t, k ) . (cid:16) | b a | + | b b | + | b c | (cid:17) + X j,m (cid:0) | A jm ( { I − P } b u ) | + | B j ( { I − P } b u ) | (cid:1) . | P b u | + |{ I − P } b u | . | b u | . This completes the proof of lemma 4 . Lemma 4.2. κ > exists such that E ( b u )( t, k ) , which is defined by E ( b u ) = | b u | + κ Re E free ( b u ) , (4.15) satisfies that E ( b u ) ∼ | b u | (4.16) and E ( b u )( t, k ) ≤ E ( b u )(0 , k ) e − λ | k | | k | t + C Z t e − λ | k | | k | ( t − s ) | ν − / b G ( s, k ) | ds (4.17) for any t ≥ and k ∈ R . Proof.
We first claim that for any t ≥ k ∈ R , it holds that ∂ t | b u | + κ (cid:8) |{ I − P } b u | ν + ǫ |{ I − P } b u | (cid:9) . | ν − / b G | . (4.18)In fact, the Fourier transform of (4 .
1) gives ∂ t b u + iξ · k b u = L b u + ǫL F P b u + b G. (4.19)Further, taking the complex inner product with b u and taking the real part yield12 ∂ t | b u | − Re Z R ( L b u | b u ) dξ = ǫ Re Z R ( L F P b u | b u ) dξ + Re Z R (cid:16) b G | b u (cid:17) dξ. (4.20)For the second term on the left hand side of (4.20), we have from (1.7) that − Re Z R ( L b u | b u ) dξ ≥ λ |{ I − P } b u | ν . For the two terms on the right-hand side of (4.20), we have ǫ Re Z R ( L F P b u | b u ) dξ ≤ − ǫλ F P |{ I − P } b u | and Re Z R (cid:16) b G | b u (cid:17) dξ =Re Z R (cid:16) b G |{ I − P } b u (cid:17) dξ ≤ λ |{ I − P } b u | ν + C | ν − / b G | . Here we used P h = 0. Plugging the above estimates into (4.20) yields (4.18). Note that | b b | + | b c | . |{ I − P } b u | .By taking κ > ∂ t E ( b u )( t, k ) + λ | k | | k | | P b u | + λ |{ I − P } b u | ν . | ν − / b G | . (4.21)(4.3) implies (4.16) by further taking κ > ≤ γ ≤
1. Thus, we have E ( b u )( t, k ) . | b u | . | P b u | + |{ I − P } b u | ν . (4.22)Pplug (4.22) into (4.21) to find ∂ t E ( b u )( t, k ) + λ | k | | k | E ( b u )( t, k ) . | ν − / b G | (4.23)which by the Gronwall’s inequality, implies (4.17). This completes the proof of Lemma 4.2.Now, to prove , let h = 0 so that u ( t ) = e tB u is the solution to the Cauchy problem (4 .
1) and hencesatisfies the estimate (4.17) with h = 0: E ( c u )( t, k ) ≤ E ( c u )(0 , k ) e − λ | k | | k | t . (4.24)Write k α = k α k α k α . Paseval’s identity and (4 .
16) yield k ∂ αx u k . Z R | k α || c u ( t, k ) | dk . Z R | k α |E ( c u )( t, k ) dk. (4.25)okker-Planck-Boltzmann Equation 19Then, from (4.24) and (4.16), one has k ∂ αx u k . Z R | k α | e − λ | k | | k | t | c u | dk. (4.26)As in [22], one can further estimate (4.26) as k ∂ αx u k . Z | k |≤ | k α | e − λ | k | | k | t | c u | dk + Z | k |≥ | k α | e − λ | k | | k | t | c u | dk . Z | k |≤ | k α | e − λ | k | | k | t dk k u k Z + e − λ t k ∂ αx u k . (1 + t ) − −| α | (cid:0) k u k Z + k ∂ αx u k (cid:1) . (4.27)Here, we used the Hausdorff-Young inequalitysup | k |≤ | c u ( k, ξ ) | . Z R | u | ( x, ξ ) dx. Next, let u = 0 so that u ( t ) = Z t e ( t − s ) B G ( s ) ds is the solution of the Cauchy problem (4 .
1) with u = 0. Then, similar to (4.25) and (4.27), one has (cid:13)(cid:13)(cid:13)(cid:13) ∂ αx Z t e ( t − τ ) B G ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) . Z t Z R | k α | e − | k | | k | ( t − s ) | ν − / b G ( s ) | dkds . Z t (1 + t − s ) − −| α | (cid:16) k ν − / G ( s ) k Z + k ν − / ∂ αx G ( s ) k (cid:17) ds. (4.28)Recall that the solution u to the Cauchy problem (1 . .
6) can be formally written as u ( t ) = e tB u + Z t e ( t − s ) B Γ( u, u )( s ) ds. Thus, (4 .
27) and (4 .
28) yield k u k . (1 + t ) − (cid:0) k u k Z + k u k (cid:1) + Z t (1 + t − s ) − (cid:16) k ν − / Γ( u, u )( s ) k Z + k ν − / Γ( u, u )( s ) k (cid:17) ds. (4.29)In the following, we shall estimate the terms on the right hand side of (4.29). For this, we first note that for0 ≤ γ ≤ | ν − / Γ( u, u ) | . | ν / u | | u | , k ν − / Γ( u, u ) k Z . k ν / u kk u k , (4.30)which are proved in [14] and [33], respectively. Thus, one can discover from (4.30) that if γ ≤ l ( q − γ ), thenit holds that k ν − / Γ( u, u )( s ) k Z + k ν − / Γ( u, u )( s ) k . k ν / u k (cid:18) k u k + sup x | u | (cid:19) . E q,l ( u )( t ) . (4.31)0 L.-J. Xiong, T. Wang, and L.-S. WangFor t ≥
0, define a temporal function by X q,l ( u )( t ) = sup ≤ s ≤ t (1 + s ) E q,l ( u )( s ) . (4.32)Hence, it follows from (4.29) and (4.31) that k u k . (1 + t ) − (cid:0) k u k Z + k u k (cid:1) + Z t (1 + t − s ) − (1 + s ) − X q,l ( u ) ( s ) ds . (1 + t ) − (cid:0) k u k Z + k u k + X q,l ( u ) ( t ) (cid:1) . (4.33)Here we used X N,l ( t ) is nondecreasing in t and Z t (1 + t − s ) − (1 + s ) − ds . (1 + t ) − . By comparing (1.10) and (1.11), it holds that D q,l ( u )( t ) + k ( a, b, c )( t ) k ≥ κ E q,l ( u )( t ) . Then it follows from (3 .
13) that ddt E q,l ( u )( t ) + κ E q,l ( u )( t ) . k ( a, b, c )( t ) k . k u ( t ) k . (4.34)Due to the Gronwall inequality, (4.34) together with (4.33) imply E q,l ( u )( t ) . E q,l ( u ) e − κt + Z t e − κ ( t − s ) k u ( s ) k ds . (1 + t ) − (cid:0) k u k Z + E q,l ( u ) + X q,l ( u ) ( t ) (cid:1) , which implies X q,l ( u )( t ) . k u k Z + E q,l ( u ) + X q,l ( u ) ( t ) . This proves the decay rate stated in our Theorem for the hard potential case, i.e., 0 ≤ γ ≤ In this subsection, we shall obtain the time decay of the solution u to the Cauchy problem (1.5)-(1.6) inthe soft potential case ( − < γ < e tB , which is stated as follows. Lemma 4.3.
Define µ = µ ( ξ ) = h ξ i − γ . Let − < γ < , l ≥ and l > . If (cid:13)(cid:13) µ l + l u (cid:13)(cid:13) Z + (cid:13)(cid:13) µ l + l u (cid:13)(cid:13) < ∞ , then the evolution operator e tB satisfies (cid:13)(cid:13) µ l e tB u (cid:13)(cid:13) . (1 + t ) − (cid:16)(cid:13)(cid:13) µ l + l u (cid:13)(cid:13) Z + (cid:13)(cid:13) µ l + l u (cid:13)(cid:13)(cid:17) (4.35) for each t ≥ . okker-Planck-Boltzmann Equation 21 Proof.
Let G = 0 so that u ( t ) = e tB u is the solution to the Cauchy problem (4 . { I − P } to (4.19)with G = 0 to find ∂ t { I − P } c u + iξ · k { I − P } c u = L { I − P } c u + ǫL F P { I − P } c u + P iξ · k c u − iξ · k P c u . By further taking the complex inner product of the above equation with µ l { I − P } c u and integrating it over R ξ , we have ∂ t (cid:12)(cid:12) µ l { I − P } c u (cid:12)(cid:12) + κ (cid:12)(cid:12) µ l { I − P } c u (cid:12)(cid:12) ν . |{ I − P } c u | ν + Re Z R (cid:0) P iξ · k c u − iξ · k P c u | µ l { I − P } c u (cid:1) dξ, (4.36)whenever ( q − γ ) ǫ is small enough. Here, we used (3.6) and (3.2). For the second term on the right hand sideof (4.36), it holds that for each | k | ≤ Z R (cid:0) P iξ · k c u − iξ · k P c u | µ l { I − P } c u (cid:1) dξ . |{ I − P } c u | ν + | k | (cid:16) | P c u | + |{ I − P } c u | ν (cid:17) . |{ I − P } c u | ν + | k | | k | | P c u | . Thus, using µ = h ξ i − γ , we get ∂ t (cid:12)(cid:12) µ l { I − P } c u (cid:12)(cid:12) χ | k |≤ + κ (cid:12)(cid:12) µ l − { I − P } c u (cid:12)(cid:12) χ | k |≤ . |{ I − P } c u | ν + | k | | k | | P c u | . (4.37)To obtain the velocity-weighted estimate for the pointwise time-frequency variables over | k | ≥
1, we directlytake the complex inner product of (4.19) with G = 0 with µ l c u and integrate in over R ξ to discover ∂ t (cid:12)(cid:12) µ l c u (cid:12)(cid:12) + κ (cid:12)(cid:12) µ l − c u (cid:12)(cid:12) . | c u | ν , (4.38)whenever ( q − γ ) ǫ is small enough. Note that | k | | k | χ | k |≥ ≥ . It follows that ∂ t (cid:12)(cid:12) µ l c u (cid:12)(cid:12) χ | k |≥ + κ (cid:12)(cid:12) µ l − c u (cid:12)(cid:12) χ | k |≥ . |{ I − P } c u | ν + | k | | k | | P c u | . (4.39)Therefore, for κ > h ≡ ∂ t E ( c u )( t, k ) + λ | k | | k | | P c u | + λ |{ I − P } c u | ν ≤ l ≥ ∂ t E l ( c u ) + κD l ( c u ) ≤ E l ( c u ) and D l ( c u ) are given by E l ( c u ) = E ( c u ) + κ (cid:16)(cid:12)(cid:12) µ l { I − P } c u (cid:12)(cid:12) χ | k |≤ + (cid:12)(cid:12) µ l c u (cid:12)(cid:12) χ | k |≥ (cid:17) ,D l ( c u ) = (cid:12)(cid:12) µ l − { I − P } c u (cid:12)(cid:12) + | k | | k | | P c u | . .
16) and the fact P c u decays exponentially in ξ , it is clear that E l ( c u ) ∼ | P c u | + | µ l { I − P } c u | ∼ (cid:12)(cid:12) µ l c u (cid:12)(cid:12) . (4.42)Set ρ ( k ) = | k | | k | . Let 0 < η ≤ J > .
41) by [1 + ηρ ( k ) t ] J , we have from (4.42) that ∂ t (cid:8) [1 + ηρ ( k ) t ] J E l ( c u ) (cid:9) + κ [1 + ηρ ( k ) t ] J D l ( c u ) ≤ J [1 + ηρ ( k ) t ] J − ηρ ( k ) E l ( c u ) ≤ CJ [1 + ηρ ( k ) t ] J − ηρ ( k ) | P c u | + CJ [1 + ηρ ( k ) t ] J − ηρ ( k ) | µ l { I − P } c u | ≤ ηC [1 + ηρ ( k ) t ] J D l ( c u ) + CJ [1 + ηρ ( k ) t ] J − ηρ ( k ) | µ l { I − P } c u | . (4.43)In what follows, we estimate the second term on the right hand side of (4.43). To this end, let p > (cid:12)(cid:12) µ l { I − P } c u (cid:12)(cid:12) ≤ (cid:12)(cid:12) µ l { I − P } c u χ µ ( ξ ) ≤ [1+ ηρ ( k ) t ] (cid:12)(cid:12) + (cid:12)(cid:12) µ l { I − P } c u χ µ ( ξ ) > [1+ ǫρ ( k ) t ] (cid:12)(cid:12) ≤ [1 + ηρ ( k ) t ] (cid:12)(cid:12) µ l − { I − P } c u (cid:12)(cid:12) + [1 + ηρ ( k ) t ] − p − J +1 (cid:12)(cid:12) µ l + p + J − { I − P } c u (cid:12)(cid:12) ≤ [1 + ηρ ( k ) t ] D l ( c u ) + C [1 + ηρ ( k ) t ] − p − J +1 E l + p + J − ( c u ) . (4.44)Here, we used the splitting 1 = χ µ ( ξ ) ≤ [1+ ηρ ( k ) t ] + χ µ ( ξ ) > [1+ ηρ ( k ) t ] and (4.42). Plugging (4.44) into (4.43) and noting that E l + p + J − ( c u ) ≤ E l + p + J − ( c u ) from (4.41) due to l + p + J − ≥
0, one has ∂ t (cid:8) [1 + ηρ ( k ) t ] J E l ( c u ) (cid:9) + κ [1 + ηρ ( k ) t ] J D l ( c u ) ≤ ǫC [1 + ηρ ( k ) t ] J D l ( c u ) + C [1 + ηρ ( k ) t ] − p ηρ ( k ) E l + p + J − ( c u ) , which implies ∂ t (cid:8) [1 + ηρ ( k ) t ] J E l ( c u ) (cid:9) + λ [1 + ηρ ( k ) t ] J D l ( c u ) . [1 + ηρ ( k ) t ] − p ηρ ( k ) E l + p + J − ( c u ) , whenever η > Z t [1 + ηρ ( k ) s ] − p ηρ ( k ) ds ≤ Z ∞ [1 + s ] − p ds < ∞ for p >
1, and noting p + J − >
0, we have[1 + ηρ ( k ) t ] J E l ( c u ) . E l ( c u ) + E l + p + J − ( c u ) . E l + p + J − ( c u ) . Now, for any given l > , we choose J > and p > p + J − l to get E l ( c u ) . [1 + ηρ ( k ) t ] − J E l + l ( c u ) . (4.45)okker-Planck-Boltzmann Equation 23Since J > , (4.42), (4.45) and Hausdorff-Young inequality yield (cid:13)(cid:13) µ l u (cid:13)(cid:13) . Z R (cid:12)(cid:12) µ l c u (cid:12)(cid:12) dk . Z R E l ( c u ) dk . sup k E l + l ( c u ) Z | k |≤ [1 + ηρ ( k ) t ] − J dk + (1 + t ) − J Z | k |≥ E l + l ( c u ) dk . (1 + t ) − (cid:16)(cid:13)(cid:13) µ l + l u (cid:13)(cid:13) Z + (cid:13)(cid:13) µ l + l u (cid:13)(cid:13) (cid:17) . This completes the proof.Recall that the solution u to the Cauchy problem (1 . .
6) can be formally written as u ( t ) = e tB u + Z t e ( t − s ) B Γ( u, u )( s ) ds. Thus, one has k u k . (1 + t ) − (cid:0) k µ l u k Z + k µ l u k (cid:1) + Z t (1 + t − s ) − (cid:16) k µ l Γ( u, u )( s ) k Z + k µ l Γ( u, u )( s ) k (cid:17) ds, from Lemma 4.3. To estimate the time integral term on the right hand side of the above inequality, we notethat k µ l Γ( u, u )( t ) k Z + k µ l Γ( u, u )( t ) k . X | α | + | β |≤ N k ∂ αβ u k X | α |≤ N kh ξ i max { , γ (1 − l ) } ∂ αx u k , which is proved in [12]. Then we have k µ l Γ( u, u )( t ) k Z + k µ l Γ( u, u )( t ) k . E q,l − ( u )( t ) , whenever γ (1 − l ) ≤ ( q − γ )( l − k u k . (1 + t ) − (cid:0) k µ l u k Z + k µ l u k + X q,l − ( u ) ( t ) (cid:1) . (4.46)Let 0 < η < /
2. Notice that (3 .
13) also holds when l is replaced by l − l ≥ N + 1,sup ≤ t ≤ T E q,l ( s ) ≤ δ and ( q − γ ) ǫ ≤ δ . Thus, it holds that ddt E q,l − ( u )( t ) + λ D q,l − ( u )( t ) ≤ . Multiplying the above inequality by (1 + t ) / η gives ddt n (1 + t ) + η E q,l − ( u )( t ) o + λ (1 + t ) + η D q,l − ( u )( t ) . (1 + t ) + η E q,l − ( u )( t ) . (4.47)Similarly, from (3 .
13) with l replaced by l − and further multiplying it by (1 + t ) / η , one has ddt n (1 + t ) + η E q,l − ( u )( t ) o + λ (1 + t ) + η D q,l − ( u )( t ) . (1 + t ) − + η E q,l − ( u )( t ) . E q,l − ( u )( t ) . (4.48)Note from (1 . .
11) that E q,l ′ − ( u )( t ) . D q,l ′ ( u )( t ) + k P u k l ′ . Then, from taking integration over [0 , t ] of (4.47), (4.48) and (3.13) and further takingthe appropriate linear combination, we have(1 + t ) + η E q,l − ( u )( t ) . E q,l ( u ) + Z t (1 + s ) + η k P u ( s ) k ds. Thus, applying the estimate (4 .
46) to the second term on the right hand side of the above inequality andnoticing Z t (1 + s ) + η (1 + s ) − ds . (1 + t ) η , we have (1 + t ) + η E q,l − ( u )( t ) . E q,l ( u ) + (1 + t ) η (cid:8) k µ l u k Z + k µ l u k + X q,l − ( u ) ( t ) (cid:9) , which implies sup ≤ s ≤ t (1 + s ) E q,l − ( u )( s ) . E q,l ( u ) + k µ l u k Z + k µ l u k + X q,l − ( u ) ( t ) . This proves the decay rate stated in our Theorem for the soft potential case, i.e., − < γ < Acknowledgments
This work was supported by “the Fundamental Research Funds for the Central Universities”. This workwas completed when Tao Wang was visiting the Mathematical Institute at the University of Oxford under thesupport of the China Scholarship Council 201206270022. He would like to thank Professor Gui-Qiang Chenand his group for their kind hospitality.
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