The Wigner–Poisson-Xα system in Wiener algebra

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The Wigner–Poisson-Xα system in Wiener algebra Bin Li a,∗ , Han Yang b a

Department of Math., Jincheng College of Sichuan University, Chengdu 611731, PR China

b

College of Math., Southwest Jiaotong University, Chengdu 611756, PR China

article

abstract

info

Article history: Received 6 September 2016 Received in revised form 26 February 2017 Accepted 27 February 2017 Available online xxxx

In this paper, we concern over a class of the Wigner–Poisson-Xα system introduced and developed by Bao et al. (2003), Mauser (2001) and Stimming (2005), respectively. The model describes the quantum mechanical motion of particles under the influence of the nonlocal Hartree potential and a local power term (exchange potential). The existence and uniqueness of local mild solution to the n-dimensional (n = 1, 3) Cauchy problems are established on the space of some integrable functions whose inverse Fourier transforms are also integrable. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Wigner–Poisson-Xα system Cauchy problem Mild solution Wiener algebra

1. Introduction In this paper, we have a keen interest in establishing existence and uniqueness to mild solution for the Wigner–PoissonXα system, which was introduced and developed by Mauser, Bao and Stimming in [1–3], respectively. The model describes the quantum-mechanical motion of particles under the influence of the Hartree potential and a local power term (exchange potential). We shall restrict our analysis to n-dimensional (n = 1, 3) problems and consider the real-valued Wigner function f (t , x, v), which is a probabilistic quasi-distribution function in the position–velocity (x, v) ∈ (Rn , Rn ) phase space for the considered quantum system at time t ≥ 0. Let ρ(t , x) be the mass (charge) density of the system at time t and position x, that is to say that,

ρ(t , x) =

 Rn

f (t , x, v)dv.

(1.1)

In this situation, its temporal evolution is governed by the Wigner–Poisson-Xα equations, as in [1–3], namely, ft + v · ∇x f + Θh¯ [VH ]f − α Θh¯ [VE ]f = 0,

−∆x VH = ρ(t , x), 1 n

VE = ρ ,

(1.2) (1.3) (1.4)

on the phase space x ∈ Rn , v ∈ Rn . The function f (t , x, v) is also the real-valued Wigner transform [4,5] of wave function ψ(t , x) ∈ L2 , which satisfies Schrödinger–Poisson-Xα equations, see also [1–3] for details. h¯ denotes the Planck’s constant and α ∈ R. The potential function ϕ (ϕ = VH , or VE ) acting on f (t , x, v) is taken into account by the pseudo-differential

∗

Corresponding author. E-mail addresses: [email protected] (B. Li), [email protected] (H. Yang).

http://dx.doi.org/10.1016/j.camwa.2017.02.037 0898-1221/© 2017 Elsevier Ltd. All rights reserved.

2

B. Li, H. Yang / Computers and Mathematics with Applications (

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operator Θh¯ [ϕ]f , as in [1–3], formally defined by

Θh¯ [ϕ]f (t , x, v) =



i

(2π )

n

 Rn

δϕ(x, η)f (t , x, v ′ )ei(v−v )·η dv ′ dη ′

Rn

(1.5)

with the potential difference

δϕ(x, η) =

1 h¯

     h¯ η h¯ η ϕ x+ −ϕ x− . 2

(1.6)

2

Note that the Hartree potential VH = VH (t , x) obeys Poisson equation (1.3), which models the Hartree interaction within the considered particle system in a mean-field description. The exchange potential VE in (1.2) may be seen as a correction of the so-called mean field approximation, see also [6]. It is clear that if α = 0 then (1.1)–(1.6) reduce to the Wigner–Poisson (WP) equation, which has been extensively studied for both bounded and unbounded domains Ω . For instance, existence and uniqueness of solution have been posed on R3x × R3v in [7,8], on a bounded spatial domain in [9–12], and on a discrete lattice in [13,14]. It is worth pointing out that no rigorous results on Cauchy problem of Eqs. (1.1)–(1.6) have been obtained so far. The present paper is devoted to investigating the Cauchy problem for Wigner–Poisson-Xα equation or establishing certain mathematical results on the existence of local mild solutions with following initial condition f (t = 0, x, v) = f0 (x, v),

(x, v) ∈ Rnx × Rnv .

(1.7)

In spite of the many existing results for the WP problem, those analysis strategies (e.g. [9,10]) cannot be straightforwardly generalized to Eqs. (1.1)–(1.6) because the operator Θh¯ [VE ]f presents a heavy nonlinearity which brings about a blank wall of analogous analysis for the nonlinear operator Θh¯ [VH ]f . More specifically, the natural choices of the state space for the study of the WP-like equation are the Banach space L2 (Rnx × Rnv , (1 + |v|2 )k dxdv) with k > 2n or L2 ([0, 1] × Rv , (1 + |v|2 )dxdv) for one dimension, see [9,10,15]. In fact, for Θh¯ [VH ]f , one can take advantage of the regularizing property of elliptic Poisson equation (1.3), see Propositions 2.3 and 2.8 in [10]. In contrast, an analogous attempt is unsuccessful for the operator Θh¯ [VE ]f : 1

1

1

if the operator Θh¯ [VE ]f is well-defined in this weighted L2 space, which must need that ∇x (ρ n ), ∇x2 (ρ n ), . . . , ∇xk (ρ n ) are well-defined in L2 space. It is clear that the function ρ given by (1.1) does not satisfy those needs. So, one of our main ideas is to set up the intersection of Lebesgue space and Wiener algebra



1

X = L1 (Rnx ; L1 (Rnv )) ∩ W Rn ; [L1 (Rnv )] n



,

(1.8)

which allows to ‘‘control’’ the nonlinear operators Θh¯ [VH ]f and Θh¯ [VE ]f . W stands for the Wiener algebra (also called Fourier algebra, according to the context) W = {g ∈ S ′ (Rn ) : F −1 (g ) ∈ L1 (Rn )}, endowed with the norm ∥g ∥W = ∥F

F −1 (g ) =

1

(2π )n

 Rnx

−1

(1.9)

g ∥L1 , and the inverse Fourier transform of g is formally defined by

g (x)eix·y dx,

where  g is actuallya distribution. We also write Z

=

1

W (Rn ; [L1 (Rnv )] n ) endowed with the norm ∥f ∥Z

1  −1  F ∥f (t , x, ·) ∥Ln1  in this paper. In our calculations we shall use the following norm:  v  1 L    1   −1 n  ∥f ∥X = ∥f ∥L1x,v + ∥f ∥Z = ∥f ∥L1x,v + F ∥f (t , x, ·) ∥L1   ,

v

=

(1.10)

L1

where L1x,v = L1 (Rnx ; L1 (Rnv )). With the aforementioned notations, we can describe our main results as follows. Theorem 1. For n = 3, if f0 ∈ X is such that R3 |f0 (x − t v, v)|dv ≥ µ > 0 for all t ∈ (0, tmax ), then the Cauchy problem v of (1.1)–(1.6), with the initial condition (1.7), possesses a unique mild solution f ∈ C ([0, tmax ), X ). Moreover, if tmax < ∞ then one has



lim sup ∥f (t )∥X = ∞.

t →tmax

Note that we do not succeed in repeating the analogous strategies in one dimension. Indeed, the solution of Poisson equation (1.3) is

  cn ∗x ρ(t , x), n ≥ 3, VH (t , x) = |x|n−2  c1 |x| ∗x ρ(t , x), n = 1.

(1.11)

B. Li, H. Yang / Computers and Mathematics with Applications (

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The proof of Theorem 1 is more dependent on the properties of the inverse Fourier transform of |·|n1−2 . However, F −1 (|x|) is not well-defined on R. Fortunately, we note that VH has some characteristics on a bounded domain in one-dimensional, see (4.1)–(4.3), which can help us to overcome the mentioned difficulties. Therefore, the final result of this paper is concerned with the Wigner–Poisson-Xα in one dimension space with x ∈ [0, 1] and v ∈ R. Theorem 2. For n = 1, let f0 ∈ X1 = L1 ([0, 1]; L1 (Rv )) ∩ W (R; L1 (Rv )), then the Cauchy problem of the Wigner–Poisson-Xα equations (1.1)–(1.6), with the following initial boundary conditions f (t = 0, x, v) = f0 (x, v),

f (t , 0, v) = f (t , 1, v),

VH (t , 0) = VH (t , 1) = 0,

admits a unique mild solution f ∈ C ([0, tmax ), X1 ). Moreover, if tmax < ∞, one then has lim sup ∥f (t )∥X1 = ∞.

t →tmax

2. Preliminaries In this section we shall establish the functional analytic preliminaries for studying the nonlinear Wigner–Poisson-Xα equations (1.1)–(1.6). Or more accurately, we shall introduce main properties of the Wiener algebra, which allows to control the nonlinear term Θh¯ [·]f . 1

For a start, we recall some important properties of W (Rn ) and Z = W (R3 ; [L1 (R3v )] 3 ), see also [16–19]. 1

Lemma 1. Wiener algebra space W (Rn ), Z = W (Rn ; [L1 (Rnv )] n ) and X enjoy the following properties:

i. For all n ≥ 1, W (Rn ) is a Banach space, continuously embedded into L∞ (Rn ). Moreover, W (Rn ) is an algebra, in the sense that the mapping (f , g ) → fg is continuous from W 2 (Rn ) to W (Rn ), and

∥fg ∥W ≤ ∥f ∥W ∥g ∥W . ii. If n = 3, then Wiener algebra space Z is a complete metric space with the metric d(f1 , f2 ) = ∥f1 − f2 ∥Z . Moreover, the space X is also complete. Proof. The first assertion follows directly by [16–19]. The space Z is metric space for all n ≥ 1 based on



1

1

a + b ≤ an + bn

n

,

∀a, b ≥ 0.

On the other hand, if n = 3, the completeness of Z follows the completeness of W , L1v and the inequality:

  1 1 1  3 a − b 3  ≤ |a − b| 3 ,

∀a, b ≥ 0.

Finally, if n = 3, then X is also a complete metric space by the completeness of Z and L1x,v .



Remark 1. Indeed, we can also verify that X1 = L1 ([0, 1]; L1 (Rv )) ∩ W (R; L1 (Rv )) is a Banach space. In what follows, we show that the nonlinear operators Θh¯ [VH ]f and Θh¯ [VE ]f are well defined form the space X to itself, respectively. This is one of the key ingredients for our theorems, and is also one of the main difficulties in establishing a mild solution. In fact, the operator Θh¯ [ϕ]w can be rewritten in a more compact form as, see also [9],

Θh¯ [ϕ ] f =

i h¯

−1 Fη→v (δϕ Fv→η (f )) =

i h¯

−1 (Fη→v [δϕ] ∗v f ),

(2.1)

where the symbol ∗v is the partial convolution with respect to the variable v , Fv→η is the Fourier transformation with respect −1 its inverse: to the variable v and Fη→v

Fv→η (f (x, ·))(η) =

 Rn

f (x, v)e−iv·η dv,

−1 Fη→v (g (x, ·))(v) =

1

(2π )n

 Rn

g (x, η)eiv·η dη

for suitable functions f and g. With these definitions at hand, we can state the following result: Lemma 2. For n ≥ 3, let VH satisfy (1.3) and f ∈ X , then there exists C > 0 such that

∥Θh¯ [VH ] f ∥L1x,v ≤ C (∥f ∥L1x,v + ∥f ∥nZ )∥f ∥L1x,v

(2.2)

  1 n ∥Θh¯ [VH ] f ∥Z ≤ C ∥f ∥L1 + ∥f ∥Z ∥f ∥Z .

(2.3)

and

x,v

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B. Li, H. Yang / Computers and Mathematics with Applications (

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Proof. Using (2.1), properties of the Fourier transform, the Young’s inequality for convolution with respect to the v variable and the Hölder inequality with respect to the x variable, we have −1 (δ VH ) ∗v f ∥L1x,v ∥Θh¯ [VH ]f ∥L1x,v ≤ C ∥Fη→v     −1 ≤ C  ∥Fη→v (δ VH ) ∥L1v ∥f ∥L1v  1 Lx  −1  ≤ C Fη→v (δ VH )L∞ (L1 ) ∥f ∥L1x,v x v   1  −1  ≤ C  2 F ρ(·)  1 ∥f ∥L1x,v . |·| L   1  n p Next, using |x|β |x|≤1 ∈ L with p ∈ [1, β ) and |x1|β |x|>1 ∈ Lq with q ∈ ( βn , ∞], we obtain

∥Θh¯ [VH ]f ∥L1x,v ≤ C (∥F −1 ρ∥L∞ + ∥F −1 ρ∥L1 )∥f ∥L1x,v ≤ C (∥ρ∥L1 + ∥F −1 ρ∥L1 )∥f ∥L1x,v ≤ C (∥f ∥L1x,v + ∥F −1 ρ∥L1 )∥f ∥L1x,v . 1

1

1

1

In fact, by [20,21], ρ ≥ 0. One has ρ = ρ n · ρ n · · · ρ n and ρ n ≤ { we get 1

1



1

Rnv

|f (x, v, t )|dv} n . By properties of the Fourier transform,

1

∥F −1 ρ∥L1 = ∥F −1 [ρ n · ρ n · · · ρ n ]∥L1 1

≤ ∥F −1 [ρ n ]∥nL1    1n   n  −1  ≤ F |f (x, v, t )|dv  = ∥f ∥nZ . n  1 Rv L

Moreover, we get

∥Θh¯ [VH ]f ∥L1x,v ≤ C (∥f ∥L1x,v + ∥f ∥nZ )∥f ∥L1x,v . On the other hand, we observe that the same types of estimates are also true for the Z norms. Indeed, using Lemma 1 and Eq. (2.1), we have

  1   n   ∥Θh¯ [VH ]f ∥Z ≤ C  ∥Θ [VH ]f ∥L1  v W   1 1   −1 n n   ≤ C  ∥Fη→v (δ VH ) ∥L1 ∥f ∥L1  v v W  −1  1n   ≤ C Fη→v (δ VH ) L∞ (L1 ) ∥f ∥Z x

≤ C (∥F

−1

ρ∥

v

1

+ ∥F −1 ρ∥L1 ) n ∥f ∥Z

L∞

1

≤ C (∥ρ∥L1 + ∥f ∥nZ ) n ∥f ∥Z , and the second assertion is proved.



Next, we present some results for the nonlinear operator Θh¯ [VE ]f . Lemma 3. For n ≥ 3, let f ∈ X , then nonlinear operator Θh¯ [VE ]f maps X into itself and satisfies 1+ 1n

∥Θh¯ [VE ]f ∥L1x,v ≤ C ∥f ∥Z ∥f ∥L1x,v ,

∥Θh¯ [VE ]f ∥Z ≤ C ∥f ∥Z

.

Proof. Similar to the proof of Lemma 3, using (2.1) and Lemma 1, we have

  1  ∥Θh¯ [VE ]f ∥L1x,v ≤ C F −1 ρ n  1 ∥f ∥L1x,v ≤ C ∥f ∥Z ∥f ∥L1x,v . L

In the last line, by a similar strategy as in the proof of Lemma 2, we get

  1   n   ∥Θh¯ [VE ]f ∥Z ≤  ∥Θh¯ [VE ]f ∥L1  v

W

 −1 1 (δ VE )Ln∞ (L1 ) ∥f ∥Z ≤ C Fη→v x

v

(2.4)

B. Li, H. Yang / Computers and Mathematics with Applications (

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1

1

≤ C ∥F −1 ρ n ∥Ln1 ∥f ∥Z 1+ 1n

≤ C ∥f ∥Z

.

This concludes the proof of Lemma 3.



3. The proof of Theorem 1 In this section we shall use a contractive fixed point map to establish a local mild solution of n-dimensional (n = 3) Wigner–Poisson-Xα equations (1.1)–(1.6) with initial condition (1.7). In fact, the concept of mild solution of the (nonlinear) Wigner–Poisson-Xα equations (1.1)–(1.6) is then introduced as a solution of an equivalent integral equation involving the fundamental solution of the free-streaming operator: f (t , x, v) = U (t )[f0 ] −

t



U (t − s)[(Θh¯ [VE ] f )(s)]ds + α

t



U (t − s)[(Θh¯ [VH ] f )(s)]ds,

(3.1)

0

0

with U (t )[h(x, v)] = h(x − t v, v),

∀(x, v) ∈ R3 × R3 ,

(3.2)

see [22] for details. Lemmas 2–3 motivate the definition of the following space YT = {f ∈ C ((0, T ]; X )},

(3.3)

endowed with the norm

∥f ∥YT = sup ∥f (t )∥L1x,v + sup ∥f (t )∥Z ,

(3.4)

t ∈[0,T ]

t ∈[0,T ]

for every fixed T > 0. We also define the following closed, bounded subset of YT , YTK = f ∈ YT : ∥f ∥YT ≤ K , K > max{2C1 (∥f0 ∥X + ∥f0 ∥3X ), ∥f0 ∥X } ,





(3.5)

where C1 is given in (3.7). We introduce a mapping

Ψ [f (t )] = U (t )[f0 ] −

t



U (t − s)[(Θh¯ [VH ]f )(s)]ds + α

t



U (t − s)[Θh¯ [VE ]f (s)]ds.

(3.6)

0

0

Indeed, we can show that Ψ is a strict contraction on YTK , for T sufficiently small, which yields the local-in-time solution of Eqs. (3.1)–(3.2). Proof of Theorem 1. We first notice that Ψ is well-defined. Indeed, from Lemmas 2 and 3 and [22] we can get

∥U (t )[f0 ]∥L∞ (L1x,v ) ≤ ∥f0 ∥L1x,v , t   t    U (t − s)[Θh¯ [VH ]f ]ds  ∞ 1 ≤ C ∥Θh¯ [VH ]f ∥L1t (L1x,v ) ,  0 L (Lx,v )  t  t    U (t − s)[Θh¯ [VE ]f ]ds ≤ C ∥Θh¯ [VE ]f ∥L1 (L1x,v ) ,   t

1 L∞ t (Lx,v )

0

where

∥w∥Lr (Lpx,v ) = t

   

T

∥w(t , x, v)∥ 0

r p dt Lx,v

   sup ∥w(t , x, v)∥Lpx,v , t ∈[0,T ]

 1r

,

1 ≤ r < ∞;

r = ∞.

On the other hand, the same type of estimates are also true for the Z norm. In fact, from (3.2) and (3.6) we can get

 R3v

|Ψ [f ](t , x, v)|dv

 13

1

≤ C (κ1 + κ2 + κ3 ) 3

with

κ1 (t , x) :=



κ2 (t , x) :=

 

R3v t 0

|f0 (x − t v, v)|dv,

R3v

|(Θh¯ [VE ]f )(s, x − (t − s)v, v)|dv ds,

6

B. Li, H. Yang / Computers and Mathematics with Applications (

κ3 (t , x) :=

 t R3v

0

)

–

|(Θh¯ [VH ] f ) (s, x − (t − s)v, v)| dv ds.

Note that if a ≥ µ > 0, b ≥ 0, c ≥ 0 then one has a+b+c

1

(a + b + c ) n = Hence, if



R3v

(a + b + c )

n−1 n

≤ µ−

n−1 n

(a + b + c ),

n ≥ 1.

|f0 (x − t v, v)|dv ≥ µ > 0 we have

∥Ψ [f ]∥Z ≤ C ∥κ1 + κ2 + κ3 ∥W ≤ C (∥κ1 ∥W + ∥κ2 ∥W + ∥κ3 ∥W ). On the other hand, by using Lemmas 1–3 and the properties of Fourier transform, we have

    −1   ∥κ1 ∥W ≤ C  | f ( x − t v, v)| d v F  x→z 3 0 1 Rv Lz     −1  |exp(it v · z )| Fx→z |f0 |dv  ≤ C  R3v

L1z

   31  3    −1 ≤ C Fx→z |f0 |dv  3 1  Rv Lz

f0 3Z

≤ C∥ ∥ and

  t    −1  ds F [ V ] f )( s , x − ( t − s )v, v)| d v |( Θ h¯ E 1  x→ z 3 0 Rv Lz    t     |exp(i(t − s)v · z )| Fx−→1z |Θh¯ [VE ]f |dv  ≤C   ds

∥κ2 ∥W ≤ C

R3v

0

L1z

  t   −1  F  ds ≤C [ V ] f | d v | Θ  x→z 3 h¯ E 1 0 Rv Lz  t 1+ 31 ≤C ∥f (s)∥Z ds. 0

Using the analogous strategies, we can also get

  t   −1  F  ds [ V ] f )( s , x − ( t − s )v, v)| d v |( Θ h H ¯  x→ z 3 1 0 Rv L  z  t     |exp(i(t − s)v · z )| Fx−→1z |Θh¯ [VH ]f |dv  ≤C   ds

∥κ3 ∥W ≤ C

R3v

0

t



L1z



∥f (s)∥Z ∥f (s)∥Z + ∥f (s)∥

≤C 0

1 3 L1x,v



ds.

Hence, for all 0 ≤ t ≤ T , we have

   t  1 ∥f (s)∥L1x,v ∩Z ds ∥Ψ [f ]∥YT ≤ C ∥f0 ∥X + ∥f0 ∥3X + ∥f ∥YT + ∥f ∥Y3T + ∥f ∥3YT 0



≤ C1 (∥f0 ∥X + ∥f0 ∥3X ) + C2 K 2 + K

1+ 31



+ K4 T.

(3.7) 1

Since C1 (∥f0 ∥X + ∥f0 ∥3X ) < 12 K , for sufficiently small T satisfying C2 (K + K 3 + K 3 )T < 12 then it is clear that Ψ maps YTK onto itself. To prove contractivity we shall estimate ∥Ψ [f1 ] − Ψ [f2 ]∥YT for all f1 , f2 ∈ YTK . By (1.11), we have VH [ f 1 − f 2 ] =

cn

|x|n−2

∗x (ρ1 − ρ2 ).

B. Li, H. Yang / Computers and Mathematics with Applications (

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Using t



Ψ [f 1 ] − Ψ [f 2 ] =

U (t − s)([Θh¯ [VH [f1 ]]f1 ] − [Θh¯ [VH [f2 ]]f2 ])(s)ds 0 t



U (t − s)([Θh¯ [VE [f1 ]]f1 ] − [Θh¯ [VE [f2 ]]f2 ])(s)ds

+ 0

=: I1 + I2 with

 t     ∥I1 ∥L∞ (L1x,v ) ≤  U ( t − s )([ Θ [ V [ f ]] f ] − [ Θ [ V [ f ]] f ])( s ) ds h¯ H 1 1 h¯ H 2 2  ∞ 1 t 0 L (Lx,v )  t   t t         ≤  U (t − s)[Θh¯ [VH [f1 ]](f1 − f2 )](s)ds +  U (t − s)[Θh¯ [VH [f1 − f2 ]]f2 ](s)ds  1 L∞ t (Lx,v )

0

∥I2 ∥L∞ (L1x,v ) t

0

≤ C ∥Θh¯ [VH [f1 ]](f1 − f2 )∥L1 (L1x,v ) + C ∥Θh¯ [VH [f1 − f2 ]]f2 ∥L1 (L1x,v ) , t  t  t    ≤  U (t − s)(Θh¯ [VE [f1 ]]f1 − [Θh¯ [VE [f2 ]]f2 ])(s)ds ∞ 1 0 L (Lx,v )  t t   t        +  U (t − s)[Θh¯ [VE [f1 − f2 ]]f2 ](s)ds ≤  U (t − s)[Θh¯ [VE [f1 ]](f1 − f2 )](s)ds  1 L∞ t (Lx,v )

0

0

1 L∞ t (Lx,v )

1 L∞ t (Lx,v )

≤ C ∥Θh¯ [VE [f1 ]](f1 − f2 )∥L1 (L1x,v ) + C ∥(Θh¯ [VE [f1 ]] − Θh¯ [VE [f1 ]])f2 ∥L1 (L1x,v ) , t

t

we can get t



∥Θh¯ [VH [f1 ]](f1 − f2 )∥L1x,v + ∥Θh¯ [VH [f1 − f2 ]]f2 ∥L1x,v ds  t +C ∥Θh¯ [VE [f1 ]](f1 − f2 )∥L1x,v + ∥Θh¯ [VE [f1 − f2 ]]f2 ∥L1x,v ds 0  t  t ≤C (∥f1 ∥L1x,v + ∥f1 ∥3Z )∥f1 − f2 ∥L1x,v ds + C (∥f1 − f2 ∥L1x,v + ∥f1 − f2 ∥3Z )∥f2 ∥L1x,v ds 0 0  t +C ∥Θh¯ [VE [f1 − f2 ]]f2 ∥L1x,v ds.

∥Ψ [f1 ] − Ψ [f2 ]∥L1x,v ≤ C

0

0

On the other hand, if a, b ≥ 0, then one has

 1  1 1  3 a − b 3  ≤ |a − b| 3 . Indeed, without loss of generality, let a ≥ b, we get



1

1

a3 − b3

3 1

 1  1 1 1 = a − b − 3a 3 b 3 a 3 − b 3 ≤ a − b. 1

1

It is clear that a 3 − b 3 ≤ (a − b) 3 . Moreover, we have

∥Θh¯ [VE [f1 − f2 ]]f2 ∥L1x,v

  1  31 3  ≤ C ρ1 − ρ2  ∥f2 ∥L1x,v W   1  3 ≤ C (ρ1 − ρ2 )  ∥f2 ∥L1x,v W   1   3   ≤ C  ∥(f1 − f2 ) ∥L1  ∥f2 ∥L1x,v v

W

≤ C ∥f1 − f2 ∥Z ∥f2 ∥L1x,v . Combining these estimates, then one has

∥Ψ [f1 ] − Ψ [f2 ]∥L1x,v ≤ C (1 + K + K + K ) 2

3

t

 0

∥f1 − f2 ∥L1x,v ∩Z ds.

In the last line, by similarly strategy, we get

  t 1 2 ∥f1 − f2 ∥L1x,v ∩Z ds. ∥Ψ [f1 ] − Ψ [f2 ]∥Z ≤ C 1 + K + K 3 + K − 3 0

8

B. Li, H. Yang / Computers and Mathematics with Applications (

)

–

Moreover,

∥Ψ [f1 ] − Ψ [f2 ]∥YT ≤ CK

t

 0

∥f1 − f2 ∥L1x,v ∩Z ds ≤ CK T ∥f1 − f2 ∥YT .

By appropriately choosing K and T it is clear that Ψ is a contractive map from YTK onto itself, thus it has a unique fixed point (i.e. there exists a unique mild solution in the sense of (3.1)–(3.2) for sufficiently small T > 0 only depending on f0 ). Let us denote by tmax the maximal existence time of the mild solution. By the general results contained in [23], the initial value problem (1.1)–(1.7) admits a unique mild solution f ∈ X , which satisfies (3.1)–(3.2), for all t < tmax . Moreover, if tmax < ∞, then lim ∥f ∥X = ∞.

t →tmax

This concludes the proof of theorem.



4. The proof of Theorem 2 This section is devoted to the study of the one-dimensional Wigner–Poisson-Xα system on Ω = [0, 1] × Rv . In order to proceed in the analysis of this problem, we need a solution VH of the Poisson problem (1.3) whose extension with value zero out of [0,1] by the definition of (1.5). Such solution grants us that the operator Θh¯ [ϕ]f (1.5) is well-defined from X1 into itself, see also [9,10,15]. Indeed, the solution of Poisson equation VH (t , 0) = VH (t , 1) = 0,

VH′′ = −ρ,

(4.1)

can be easily calculated by VH (t , x) = −

 x 0

VH′ (t , x) = −

y

ρ(t , z )dzdy + x 0

ρ(t , z )dz + 0



1

0

y



0

x



1



ρ(t , z )dzdy,

(4.2)

0

y



ρ(t , z )dzdy.

(4.3)

0

VH and VH , its extension with value zero outside [0,1], are essentially bounded and similarly for VH′ and for its zero extension VH′ . Moreover, we can get

∥VH ∥L∞ (R) + ∥VH′ ∥L∞ (R) ≤ C ∥ρ∥L1 (I ) ≤ C ∥f ∥L1 (Ω ) , where I = [0, 1], see also [9,24].

(4.4)

Lemma 4. For n = 1, let VH satisfy (4.1)–(4.3) and f ∈ X1 , then there exists C > 0 such that

∥Θh¯ [VH ] f ∥L1x,v ≤ C ∥f ∥2L1

x,v (Ω )

,

∥Θh¯ [VH ] f ∥W (L1 ) ≤ C ∥f ∥W (L1 ) ∥f ∥L1x,v (Ω ) .

(4.5)

Proof. By Lemma 3 and (2.1), we can obtain

∥Θh¯ [VH ]f ∥L1x,v ≤ C ∥F −1 [VH ]∥L1 ∥f ∥L1x,v . Furthermore, using (4.4) and Hölder inequality, we have

∥Θh¯ [VH ]f ∥L1x,v (Ω ) ≤ C ∥VH ∥L∞ (I ) ∥f ∥L1x,v (Ω ) ≤ C ∥ρ∥L1 (I ) ∥f ∥L1x,v (Ω ) ≤ C ∥f ∥2L1

x,v (Ω )

.

Similarly, then one has

∥Θh¯ [VH ]f ∥W (L1 ) ≤ C ∥F −1 [VH ]∥L1 ∥f ∥W (L1 ) ≤ C ∥VH ∥L∞ (I ) ∥f ∥W (L1 ) ≤ C ∥ρ∥L1 (I ) ∥f ∥W (L1 ) ≤ C ∥f ∥W (L1 ) ∥f ∥L1x,v (Ω ) .  Lemma 5. For n = 1, let f ∈ X1 , then nonlinear operator Θh¯ [VE ]f maps X1 into itself and satisfies

∥Θh¯ [VE ]f ∥L1x,v ≤ C ∥f ∥W (L1 ) ∥f ∥L1x,v

(4.6)

∥Θh¯ [VE ]f ∥W (L1 ) ≤ C ∥f ∥2W (L1 ) .

(4.7)

and

Proof. It is similar to the proof of Lemma 3.



Proof of Theorem 2. Using Lemmas 4–5, the assertion is proved by repeating the analogous strategies as in the proof of Theorem 1. 

B. Li, H. Yang / Computers and Mathematics with Applications (

)

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9

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