Well-posedness of a non-local abstract Cauchy problem with a singular integral
aa r X i v : . [ m a t h - ph ] J a n Well-posedness of a non-local abstract Cauchy problem witha singular integral
Haiyan Jiang ∗ , Tiao Lu † , Xiangjiang Zhu ‡ January 25, 2019
Abstract
A non-local abstract Cauchy problem with a singular integral is studied, which isa closed system of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banach space, the well-posedness of theevolution system is proved under some boundedness and smoothness conditions on thecoefficient functions. Furthermore, an isomorphism is established to extend the resultto a partial integro-differential equation with singular convolution kernel, which is ageneralized form of the stationary Wigner equation. Our investigation considerablyimproves the understanding of the open problem concerning the well-posedness of thestationary Wigner equation with inflow boundary conditions.
Keywords: Partial integro-differential equations, Singular integral, Well-posedness, Wigner equation
In this paper we consider the following initial value problem for the unknown functions c ( t ) and v ( t, x ), d c ( t )d t = h ( t, c ( t ) + Z + ∞−∞ K ( t, − x ′ ) v ( t, x ′ )d x ′ , ( t, x ) ∈ [0 , T ] × R ∂v ( t, x ) ∂t = h ( t, x ) − h ( t, x c ( t )++ Z + ∞−∞ K ( t, x − x ′ ) − K ( t, − x ′ ) x v ( t, x ′ )d x ′ , ( t, x ) ∈ [0 , T ] × R c (0) = c ∈ R , v (0 , x ) = v ( x ) , x ∈ R , ∗ School of Mathematical Sciences, Beijing Institute of Technology, Beijing 100081, China. Email: [email protected] . † CAPT, HEDPS, LMAM, IFSA Collaborative Innovation Center of MoE, & School of MathematicalSciences, Peking University, Beijing 100871, China. Email: [email protected] . ‡ School of Mathematical Sciences, Peking University, Beijing 100871, China. Email: [email protected] . h ( t, x ) and K ( t, x ) are given real-valued functions. It can be rewritten as an abstractCauchy problem dd t (cid:20) c ( t ) v ( t ) (cid:21) = B ( t ) (cid:20) c ( t ) v ( t ) (cid:21) , t ∈ [0 , T ] (cid:20) c (0) v (0) (cid:21) = (cid:20) c v (cid:21) , (1.1)where v ( t, x ) is viewed as a vector-valued function of t , i.e., v ( t ) = v ( t, · ), and B ( t ) is alinear operator, B ( t ) (cid:20) cv (cid:21) = ch ( t,
0) + Z + ∞−∞ K ( t, − x ) v ( x )d xc h ( t, x ) − h ( t, x + Z + ∞−∞ K ( t, x − x ′ ) − K ( t, − x ′ ) x v ( x ′ )d x ′ . (1.2)We will put forward an appropriate Banach space X (See Section 2), on which B ( t ) is abounded linear operator under some boundedness and smoothness assumptions on h and K . Therefore, the well-posedness of the abstract Cauchy problem (1.1) is proved usingthe semigroup theory of linear evolution systems.Meanwhile, the well-posedness result of (1.1) is applied to initial value problem of thefollowing partial integro-differential equation (PIDE), ∂u ( t, x ) ∂t = 1 x Ψ[ K ] u ( t, x ) , t ∈ [0 , T ] ,u (0 , x ) = u ( x ) = c x + v ( x ) , (1.3)where Ψ[ K ] is the convolution operator with kernel K ,Ψ[ K ] u ( t, x ) = Z + ∞−∞ K ( t, x − x ′ ) u ( t, x ′ )d x ′ . The relation of problems (1.1) and (1.3) will be revealed and an isomorphism betweentheir solutions is established (see Section 3). In this way, the well-posedness of (1.3) canalso be obtained applying previous analysis.The PIDE appeared in (1.3) is a generalized form of the stationary Wigner transportequation [10, 13], which is a popular tool in the quantum transport simulation (especiallyin the nano semiconductor simulation). The one-dimensional stationary Wigner equationcan be written as v ∂f ( x, v ) ∂x = Z + ∞−∞ V w ( x, v − v ′ ) f ( x, v ′ )d v ′ , (1.4)where f ( x, v ) is the quasi-probability density function in the phase space ( x, v ), and theWigner potential V w ( x, v ) is related to the potential V ( x ) through V w ( x, v ) = i2 π Z + ∞−∞ e i vy [ V ( x + y/ − V ( x − y/ y. (1.5)Eq. (1.4) with inflow boundary conditions f (0 , v ) = f ( v ) ( v > , f (1 , v ) = f ( v ) ( v <
0) (1.6)2s widely used to obtain the current-voltage curve that is an important characteristic ofsemiconductor devices [4, 5, 7]. However, the well-posedness of this inflow boundary valueproblem is still an open problem which has attracted the attention of many mathematiciansand is only partly solved in [1, 2, 3, 8, 9]. One big issue is whether L ( R ) is a suitablesolution space for (1.4). The well-posedness results previously explored enable us to someextent to investigate the stationary Wigner equation with inflow boundary conditions,which will be illustrated at the end of the paper. In this section, the well-posedness of the Cauchy problem (1.1) is studied. For ease ofunderstanding, we first state some results for the general evolution system (2.1) (see e.g.Refs [11] and [12]). Then we propose a proper solution space for the Cauchy problem (1.1)and verify that the conditions required by these general results are fulfilled if sufficientsmoothness and boundedness of the coefficient functions are assumed. In this way, the well-posedness of (1.1) is proved as an application of the semigroup theory of linear operatorsLet X be a Banach space and A ( t ): D ( A ( t )) ⊂ X → X be a linear operator in X, ∀ t ∈ T . Consider the initial value problem: d u ( t )d t = A ( t ) u ( t ) , ≤ s < t ≤ T,u ( s ) = x ∈ X. (2.1)A solution u of (2.1) is called a classical solution if u ∈ C ([0 , T ]; X ) ∩ C ((0 , T ]; X ).Moreover, let U ( t, s ) be the solution operator, which is a two-parameter family satisfying u ( t ) = U ( t, s ) u ( s ). Then we have the following theorems. Theorem 1.
Let X be a Banach space and let A ( t ) be a bounded linear operator on X forevery t ∈ [0 , T ] . If the function t → A ( t ) is continuous in the uniform operator topology,then the initial value problem (2.1) has a unique classical solution u . Theorem 2.
Suppose that the conditions in Theorem 1 are satisfied. Then U ( t, s ) isbounded and continuous in the uniform operator topology for ≤ s < t ≤ T . Moreover, k U ( t, s ) k ≤ exp (cid:18)Z ts k A ( τ ) k d τ (cid:19) . (2.2)Evidently, Theorems 1 and 2 describe the well-posedness of evolution system (2.1)conclusively. Hence, the issue is to define a proper Banach space, say X , for the system(1.1), such that the operator B ( t ) defined by (1.2) satisfies all the conditions referringto A ( t ) in Theorem 1. In other words, B ( t ) is bounded on X for every t ∈ R and iscontinuous, as an operator-valued function of t , in the uniform operator topology.In this paper, we assume that X = R ⊕ L ( R ), where ( c, v ) ∈ X if and only if c ∈ R and v ∈ L ( R ), and the norm of X is naturally defined by k ( c, v ) k X = max( | c | , k v k L ).For convenience of further discussion, we rewrite the problem (1.1) in the following form dd t (cid:20) c ( t ) v ( t ) (cid:21) = (cid:20) B ( t ) B ( t ) B ( t ) B ( t ) (cid:21) (cid:20) c ( t ) v ( t ) (cid:21) , t ∈ [0 , T ] , (cid:20) c (0) v (0) (cid:21) = (cid:20) c v (cid:21) ∈ X, (2.3)3here B ( t ) : R → R , c ch ( t, ,B ( t ) : R → L ( R ) , c c · h ( t, x ) − h ( t, x ,B ( t ) : L ( R ) → R , w ( x ) Z + ∞−∞ K ( t, − x ) w ( x )d x, and B ( t ) : L ( R ) → L ( R ), B ( t ) w ( x ) = Z + ∞−∞ K ( t, x − x ′ ) − K ( t, − x ′ ) x w ( x ′ )d x ′ . (2.4)In order to study the operator B ( t ), we will first prove that for every t ∈ [0 , T ], B i ( t )( i = 1 , , ,
4) is bounded and continuous in the uniform operator topology under somesmoothness and boundedness assumptions on the coefficient functions h ( t, x ) and K ( t, x ).We begin with the boundedness discussion of B i ( t ). Note that B ( t ) : c ch ( t, h ( t,
0) and is thus bounded as long as h ( t,
0) is well defined. (This isgenerally not true in a measurable function since changing the value of a single point willnot alter the function itself.) To achieve this, we assume that h ( t, · ) is continuous at aneighborhood of zero point. Moreover, we assume K ( t, · ) ∈ L ( R ) for every t ∈ [0 , T ]. Bythe Cauchy-Schwarz inequality, we have Z + ∞−∞ K ( t, − x ) w ( x )d x ≤ k e K ( t, · ) k L · k w ( · ) k L = k K ( t, · ) k L · k w ( · ) k L , (2.5)where e K ( t, x ) = K ( t, − x ). Hence B ( t ) is also bounded and k B ( t ) k ≤ k K ( t, · ) k L . Forsimplicity, in what follows we will express B ( t ) in terms of the inner product, B ( t ) : w ( x ) Z + ∞−∞ K ( t, − x ) w ( x )d x = h e K ( t, · ) , w ( · ) i . The boundedness of B ( t ) and B ( t ) is established by the following lemmas. Lemma 1.
Suppose w ( x ) is Lipschitz continuous with a Lipschitz constant M and isbounded in terms of k w ( · ) k ∞ ≤ M . Then w ( x ) − w (0) x ∈ L ( R ) and (cid:13)(cid:13)(cid:13)(cid:13) w ( x ) − w (0) x (cid:13)(cid:13)(cid:13)(cid:13) L ≤ q M + 8 M . Proof. (cid:13)(cid:13)(cid:13)(cid:13) w ( x ) − w (0) x (cid:13)(cid:13)(cid:13)(cid:13) L = Z R (cid:12)(cid:12)(cid:12)(cid:12) w ( x ) − w (0) x (cid:12)(cid:12)(cid:12)(cid:12) d x = Z | x |≤ (cid:12)(cid:12)(cid:12)(cid:12) w ( x ) − w (0) x (cid:12)(cid:12)(cid:12)(cid:12) d x + Z | x | > (cid:12)(cid:12)(cid:12)(cid:12) w ( x ) − w (0) x (cid:12)(cid:12)(cid:12)(cid:12) d x ≤ Z | x |≤ | M | d x + Z | x | > (cid:12)(cid:12)(cid:12)(cid:12) M x (cid:12)(cid:12)(cid:12)(cid:12) d x = 2 M + 8 M , B ( t ) then follows by setting w ( · ) = h ( t, · ) in Lemma 1. Toproceed, let D w denote the first-order weak derivative with respect to the variable x ,where w can be a real function of one or two variables, i.e., w = w ( x ) or w = w ( t, x ). Lemma 2.
Suppose K ( t, · ) ∈ H ( R ) = { f ( v ) ∈ L ( R ) : D f ∈ L ( R ) } . Then B ( t ) is abounded linear operator on L ( R ) and the corresponding operator norm is controlled by k B ( t ) k ≤ √ k K ( t, · ) k H . Proof.
Note that B ( t ) u ( t, x ) = ( K ∗ u )( t, x ) − ( K ∗ u )( t, x . Thus, in light of Lemma 1, we will investigate the Lipschitz continuity and boundednessof the function ( K ∗ u )( t, · ). Using the Fubini theorem and the Cauchy-Schwarz inequality,we obtain | ( K ∗ u )( t, x ) − ( K ∗ u )( t, x ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z + ∞−∞ [ K ( t, x − x ′ ) − K ( t, x − x ′ )] u ( t, x ′ )d x ′ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z + ∞−∞ (cid:20)Z x x D K ( t, ¯ x − x ′ )d¯ x (cid:21) u ( t, x ′ )d x ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z + ∞−∞ Z x x (cid:12)(cid:12) D K ( t, ¯ x − x ′ ) (cid:12)(cid:12) · (cid:12)(cid:12) u ( t, x ′ ) (cid:12)(cid:12) d¯ x d x ′ = Z x x (cid:20)Z + ∞−∞ (cid:12)(cid:12) D K ( t, ¯ x − x ′ ) (cid:12)(cid:12) · | u ( t, x ′ ) | d x ′ (cid:21) d¯ x ≤ Z x x (cid:13)(cid:13) D K ( t, ¯ x − · ) (cid:13)(cid:13) L · k u ( t, · ) k L d¯ x = (cid:13)(cid:13) D K ( t, · ) (cid:13)(cid:13) L · k u ( t, · ) k L · | x − x | . On the other hand, | ( K ∗ u )( t, x ) | ≤ Z R | K ( t, x − x ′ ) u ( t, x ′ ) | d x ′ ≤ (cid:18)Z R | K ( t, x − x ′ ) | d x ′ (cid:19) · k u ( t, · ) k L = k K ( t, · ) k L k u ( t, · ) k L . Since K ( t, · ) ∈ H ( R ), replacing w ( · ) by ( K ∗ u )( t, · ) in Lemma 1, we conclude that B ( t ) u ( t, · ) ∈ L ( R ) and k B ( t ) u ( t, · ) k L = (cid:13)(cid:13)(cid:13)(cid:13) ( K ∗ u )( t, x ) − ( K ∗ u )( t, x (cid:13)(cid:13)(cid:13)(cid:13) L ≤ q k D K ( t, · ) k L k u ( t, · ) k L + 8 k K ( t, · ) k L k u ( t, · ) k L ·≤ √ k K ( t, · ) k H k u ( t, · ) k L , which demonstrates that B ( t ) is a bounded linear operator on L ( R ) and the operatornorm is controlled by 2 √ k K ( t, · ) k H . 5ow we consider the continuity of B i ( t ) in the uniform operator topology (we willsimply refer to continuity below). For sake of brevity, the notation w ′ ( t, x ) is used todenote the partial derivative ∂w ( t, x ) /∂t , where w ( t, x ) is any two variable function definedon [0 , T ] × R . From our point of view, w is considered as a vector-valued function withrespect to t and the notation is thus similar to that of the derivative in a real-valuedfunction. In what follows, we assume that(i) For every x ∈ R , h ( · , x ) is absolutely continuous on the interval [0 , T ], i.e., h ( t , x ) − h ( t , x ) = Z t t h ′ ( t, x )d t, ≤ t ≤ t ≤ T. (ii) Moreover, h ′ ( t, · ) is uniformly bounded in L ∞ ( R ), i.e., k h ′ ( t, · ) k ∞ ≤ k , ∀ t ∈ [0 , T ] , and Lipschitz continuous with a uniform Lipschitz constant, i.e., there exists k > | h ′ ( t, x ) − h ′ ( t, x ) | ≤ k | x − x | , ∀ t ∈ [0 , T ] . Obviously, setting x = 0 in (i) gives the continuity of B ( t ). The continuity of B ( t ) canbe derived via (cid:13)(cid:13)(cid:13)(cid:13) h ( t , x ) − h ( t , x − h ( t , x ) − h ( t , x (cid:13)(cid:13)(cid:13)(cid:13) L = (cid:13)(cid:13)(cid:13)(cid:13)Z t t h ′ ( t, x ) − h ′ ( t, x d t (cid:13)(cid:13)(cid:13)(cid:13) L ≤ Z t t (cid:13)(cid:13)(cid:13)(cid:13) h ′ ( t, x ) − h ′ ( t, x (cid:13)(cid:13)(cid:13)(cid:13) L d t ≤ q k + 8 k · | t − t | . The last inequality comes directly from Lemma 1.For the continuity of B ( t ) and B ( t ), we assume(iii) K ( · , x ) is absolutely continuous for t ∈ [0 , T ]. (Thus K ′ ( t, · ) exists almost everywherein [0 , T ].)(iv) K ′ ( t, · ) is uniformly bounded on H ( R ), i.e., there exists a constant m such that k K ′ ( t, · ) k H ≤ m, ∀ t ∈ [0 , T ] . (2.6)Applying (iii) we have, for all 0 ≤ t < t ≤ T ,[ B ( t ) − B ( t )] u ( x )= Z + ∞−∞ (cid:20) K ( t , x − x ′ ) − K ( t , − x ′ ) x − K ( t , x − x ′ ) − K ( t , − x ′ ) x (cid:21) u ( x ′ )d x ′ = Z + ∞−∞ (cid:20)Z t t K ′ ( t, x − x ′ ) − K ′ ( t, − x ′ )d t (cid:21) u ( x ′ ) x d x ′ = Z t t (cid:20)Z + ∞−∞ K ′ ( t, x − x ′ ) − K ′ ( t, − x ′ ) x u ( x ′ )d x ′ (cid:21) d t B ′ ( t ) : L ( R ) → L ( R ), B ′ ( t ) u ( x ) = Z + ∞−∞ K ′ ( t, x − x ′ ) − K ′ ( t, − x ′ ) x u ( x ′ )d x ′ , ∀ u ∈ L ( R ) . We can thus write [ B ( t ) − B ( t )] u ( x ) = Z t t (cid:2) B ′ ( t ) u ( x ) (cid:3) d t. Replacing K ( t, · ) by K ′ ( t, · ) in (2.4) and Lemma 2 and using condition (iv) (see (2.6)) weknow that B ′ ( t ) is also a bounded linear operator on L ( R ) and k B ′ ( t ) k ≤ √ k K ′ ( t, · ) k H ≤ √ m, ∀ t ∈ [0 , T ] . Thus we obtain k [ B ( t ) − B ( t )] k = sup u ∈ L ( R ) (cid:13)(cid:13)(cid:13)R t t [ B ′ ( t ) u ( · )] d t (cid:13)(cid:13)(cid:13) L k u ( · ) k L ≤ sup u ∈ L ( R ) R t t k B ′ ( t ) u ( · ) k L d t k u ( · ) k L ≤ sup u ∈ L ( R ) √ m k u ( · ) k L | t − t |k u ( · ) k L = 2 √ m | t − t | . On the other hand, by the Cauchy-Schwarz inequality, k B ( t ) − B ( t ) k = sup u ∈ L ( R ) (cid:12)(cid:12)(cid:12)D e K ( t , · ) − e K ( t , · ) , u ( · ) E(cid:12)(cid:12)(cid:12) k u ( · ) k L ≤ (cid:13)(cid:13)(cid:13) e K ( t , · ) − e K ( t , · ) (cid:13)(cid:13)(cid:13) L = (cid:13)(cid:13)(cid:13)(cid:13)Z t t e K ′ ( t, · )d t (cid:13)(cid:13)(cid:13)(cid:13) L ≤ Z t t (cid:13)(cid:13)(cid:13) e K ′ ( t, · ) (cid:13)(cid:13)(cid:13) L d t ≤ k K ′ ( t, · ) k H | t − t | ≤ √ m | t − t | . Hence the continuity of B ( t ) and B ( t ) is also proved.Collecting all the previous results on B i ( t ), we can conclude the following theorem forthe Cauchy problem (1.1). Theorem 3.
Suppose that ∀ t ∈ [0 , T ] , K ( t, · ) ∈ H ( R ) and h ( t, · ) ∈ L ∞ ( R ) is Lipschitzcontinuous on R with the minimal Lipschitz constant L ( t ) . Moreover, suppose the as-sumptions (i) - (iv) hold. Then the Cauchy problem (1.1) has a unique classical solution in C ([0 , T ]; X ) and we have the following estimation (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) c ( t ) v ( t ) (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) X ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) c v (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) X · exp (cid:18)Z t (cid:20)q L ( s ) + 8 k h ( s, · ) k L ∞ + 2 √ k K ( s, · ) k H (cid:21) d s (cid:19) . Proof.
For simplicity we denote v ( t ) = v ( t, x ) and d v ( t ) / d t = ∂v ( t, x ) /∂t . From thediscussion on boundedness (see (2.5), Lemma 1 and Lemma 2), we know that k B ( t ) k ≤ | h ( t, | , k B ( t ) k ≤ q L ( t ) + 8 k h ( t, · ) k L ∞ , k B ( t ) k ≤ k K ( t, · ) k L , k B ( t ) k ≤ √ k K ( t, · ) k H , k · k denotes the corresponding operator norm, respectively. Substituting it into(2.3) we obtain (note that | h ( t, | ≤ k h ( t, · ) k L ∞ and K ( t, · ) k L ≤ k K ( t, · ) k H ) (cid:13)(cid:13)(cid:13)(cid:13) B ( t ) (cid:20) c ( t ) v ( t ) (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) X = max ( | B ( t ) c ( t ) + B ( t ) v ( t ) | , k B ( t ) c ( t ) + B ( t ) v ( t ) k L ) ≤ max( k B ( t ) k + k B ( t ) k , k B ( t ) k + k B ( t ) k ) · max( | c ( t ) | , k v ( t ) k L )= (cid:20)q L ( t ) + 8 k h ( t, · ) k L ∞ + 2 √ k K ( t, · ) k H (cid:21) · (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) c ( t ) v ( t ) (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) X . Thus we see that B ( t ) is bounded on X for every t ∈ [0 , T ] and k B ( t ) k ≤ q L ( t ) + 8 k h ( t, · ) k L ∞ + 2 √ k K ( t, · ) k H . Similarly, given the assumptions (i)-(iv) we can also verify the continuity of B ( t ) (in theuniform operator topology). Hence applying Theorems 1 and 2 the proof is completedimmediately. In this section, we will reveal the relation between the Cauchy problem (1.1) and thePIDE (1.3) and apply the previous results to the latter problem. Throughout this section, h ( t, x ) is determined by h ( t, x ) = Z + ∞−∞ K ( t, x − x ′ ) x ′ d x ′ . (3.1)For simplicity, we will also use the notation Ψ[ K ] u = K ∗ u and h = K ∗ (1 /x ) in thefollowing. To improve understanding, we state the core theorem first. Theorem 4.
Assume ∀ t ∈ [0 , T ] , K ( t, · ) ∈ L ( R ) is Lipschitz continuous. If the PIDE (1.3) has a solution u in terms of u ( t, x ) = c ( t ) /x + v ( t, x ) , where v ( t, · ) ∈ L ( R ) for all t ∈ [0 , T ] , then ( c ( t ) , v ( t, x )) is also a solution of the Cauchy problem (1.1) . Conversely,if ( c ( t ) , v ( t, x )) is the solution of (1.1) and v ( t, · ) ∈ L ( R ) , ∀ t ∈ [0 , T ] , then u ( t, x ) = c ( t ) /x + v ( t, x ) is a solution of the PIDE (1.3) . To begin with, we define the space L ( R ) = n f ( x ) = cx + v ( x ) : c ∈ R , v ( · ) ∈ L ( R ) o , which can be viewed as an one-dimensional extension of L ( R ) since 1 /x / ∈ L ( R ). Wecan thus equate L ( R ) with X through an isomorphism: σ : L ( R ) → X, f ( x ) = cx + v ( x ) ( c, v ( · )) . Accordingly, L ( R ) is a Banach space with the norm k f k L ( R ) = max( | c | , k v k L ).In the following, we will study the PIDE (1.3) in the space L ( R ). Namely, we lookfor solutions in form of u ( t, · ) ∈ L ( R ), i.e., u ( t, x ) = c ( t ) x + v ( t, x ) , v ( t, · ) ∈ L ( R ) , K ] u = K ∗ u is in the sense of the Cauchy principal integration, K ∗ u := lim ǫ → + "Z /ǫǫ K ( t, x − x ′ ) u ( x ′ )d x ′ + Z − ǫ − /ǫ K ( t, x − x ′ ) u ( x ′ )d x ′ = Z + ∞ (cid:2) K ( t, x − x ′ ) u ( x ′ ) − K ( t, x + x ′ ) u ( − x ′ ) (cid:3) d x ′ . Particularly, we have h = K ∗ x = Z + ∞ K ( t, x − x ′ ) − K ( t, x + x ′ ) x ′ d x ′ . (3.2)We have the following lemmas concerning the properties of such integrals. Lemma 3.
Suppose K ( t, · ) ∈ L ( R ) is Lipschitz continuous with a Lipschitz constant k ( t ) . Then h ( t, · ) = K ( t, · ) ∗ (1 /x ) is well defined and k h ( t, · ) k L ∞ ≤ k ( t ) + k K ( t, · ) k L ) . Proof.
According to the Lipschitz continuity, | h ( t, x ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z + ∞ K ( t, x − x ′ ) − K ( t, x + x ′ ) x ′ d x ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18)Z + Z + ∞ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) K ( t, x − x ′ ) − K ( t, x + x ′ ) x ′ (cid:12)(cid:12)(cid:12)(cid:12) d x ′ ≤ Z | k ( t ) | d x ′ + Z + ∞ (cid:12)(cid:12)(cid:12)(cid:12) K ( t, x − x ′ ) x ′ (cid:12)(cid:12)(cid:12)(cid:12) d x ′ + Z + ∞ (cid:12)(cid:12)(cid:12)(cid:12) K ( t, x + x ′ ) x ′ (cid:12)(cid:12)(cid:12)(cid:12) d x ′ . Using the Cauchy-Schwarz inequality, we obtain Z + ∞ (cid:12)(cid:12)(cid:12)(cid:12) K ( t, x − x ′ ) x ′ (cid:12)(cid:12)(cid:12)(cid:12) d x ′ ≤ (cid:18)Z + ∞ | K ( t, x − x ′ ) | d x ′ (cid:19) · Z + ∞ (cid:12)(cid:12)(cid:12)(cid:12) x ′ (cid:12)(cid:12)(cid:12)(cid:12) d x ′ ! ≤ k K ( t, · ) k L . Similarly, Z + ∞ (cid:12)(cid:12)(cid:12)(cid:12) K ( t, x + x ′ ) x ′ (cid:12)(cid:12)(cid:12)(cid:12) d x ′ ≤ k K ( t, · ) k L . Therefore the improper integral (3.2) converges and | h ( t, · ) | ≤ k ( t ) + k K ( t, · ) k ) . . Lemma 4.
Suppose D K ( t, · ) ∈ L ( R ) ( D denotes the first-order weak derivative withrespect to the variable x as in the previous section) is Lipschitz continuous with a Lipschitzconstant k ( t ) . Then h ( t, · ) = K ( t, · ) ∗ (1 /x ) is Lipschitz continuous with the Lipschitzconstant k ( t ) + k D K ( t, · ) k ) . roof. Let V K ( t, x, x ′ ) = K ( t, x − x ′ ) − K ( t, x + x ′ ) and D V K ( t, x, x ′ ), D K ( t, x − x ′ ) bethe first-order weak derivative with respect to the (second) variable x . Since D K ( t, · ) ∈ L ( R ) ⊆ L ( R ), we have D V K ( t, · , x ′ ) ∈ L ( R ) for any fixed x ′ and thus V K ( t, · , x ′ ) isabsolutely continuous, i.e., for ∀ x < x , V K ( t, x , x ′ ) − V K ( t, x , x ′ ) = Z x x D V K ( t, ¯ x, x ′ )d¯ x. Therefore, | h ( t, x ) − h ( t, x ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z + ∞ V K ( t, x , x ′ ) − V K ( t, x , x ′ ) x ′ d x ′ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z + ∞ (cid:20)Z x x D V K ( t, ¯ x, x ′ )d¯ x (cid:21) x ′ d x ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z x x (cid:20)Z + ∞ (cid:12)(cid:12) D V K ( t, ¯ x, x ′ ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) x ′ (cid:12)(cid:12)(cid:12)(cid:12) d x ′ (cid:21) d¯ x = Z x x (cid:20)Z + ∞ (cid:12)(cid:12)(cid:12)(cid:12) D K ( t, x − x ′ ) − D K ( t, x + x ′ ) x ′ (cid:12)(cid:12)(cid:12)(cid:12) d x ′ (cid:21) d¯ x = Z x x (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) D K ∗ x (cid:19) ( t, ¯ x ) (cid:12)(cid:12)(cid:12)(cid:12) d¯ x. Finally, replacing K by D K in Lemma 3, we obtain | h ( t, x ) − h ( t, x ) | ≤ Z x x (cid:2) k ( t ) + k D K ( t, · ) k ) (cid:3) d¯ x = 2( k ( t ) + k D K ( t, · ) k ) | x − x | . Proof of Theorem 4.
Assume u ( t, · ) ∈ L ( R ), t ∈ [0 , T ], is a solution of the PIDE (1.3).Substituting it into (1.3), we obtaind c ( t )d t + x ∂v ( t, x ) ∂t = c ( t ) (cid:18) K ( t, x ) ∗ x (cid:19) + ( K ∗ v )( t, x ) . (3.3)According to Lemma 3, h ( t, x ) = K ( t, x ) ∗ (1 /x ) is well defined on [0 , T ] × R and usingYoung’s inequality, k ( K ∗ v )( t, · ) k L ∞ ≤ k K ( t, · ) k L k v ( t, · ) k L . Thus K ∗ v is also well defined. Let x = 0 in both sides of (3.3) and note that h = K ∗ (1 /x ).We have d c ( t )d t = c ( t ) h ( t,
0) + Z + ∞−∞ K ( t, − x ) v ( x )d x. (3.4)In order to describe the evolution of v ( t, x ), we substitute (3.4) into (3.3) and obtain ∂v ( t, x ) ∂t = 1 x (cid:20) c ( t ) h ( t, x ) + ( K ∗ v )( t, x ) − d c ( t )d t (cid:21) = 1 x (cid:20) c ( t ) h ( t, x ) + ( K ∗ v )( t, x ) − c ( t ) h ( t, − Z + ∞−∞ K ( t, − x ) v ( x )d x (cid:21) = c ( t ) h ( t, x ) − h ( t, x + Z + ∞−∞ K ( t, x − x ′ ) − K ( t, − x ′ ) x v ( t, x ′ )d x ′ . (3.5)10pparently, the evolution system composed of (3.4) and (3.5) is exactly the Cauchy prob-lem (1.1). The inverse proposition can also be verified by simply reversing the proofabove.Theorem 3 reveals that the Cauchy problem (1.1) and the PIDE (1.3) are equivalentif h is defined by (3.1). Hence, the well-posedness of the PIDE (1.3) can be investigatedas an application of the previous section (see Theorem 3). In order to correspond to theassumptions in Theorem 3, we shall propose some extra conditions on K .(v) Both ∂K ( t, · ) /∂t and D ( ∂K ( t, · ) /∂t ) are uniformly Lipschitz continuous and uni-formly bounded in L ( R ) for t ∈ [0 , T ]. Namely, there exists a constant M independent of t , such that for every t ∈ [0 , T ], the corresponding Lipschitz constants and L norms areall less than M . Theorem 5.
Let the convolution kernel K satisfy (iii) - (v) . Moreover, let K ( t, · ) ∈ H ( R ) and K ( t, · ) , D K ( t, · ) be Lipschitz continuous and k ( t ) , k ( t ) are the minimal Lips-chitz constants, respectively. Then the PIDE (1.3) has a unique classical solution u ∈ C ([0 , T ]; L ( R )) and k u ( t, · ) k L ( R )) ≤ k u k L ( R )) · exp (cid:18)Z t √ k ( s ) + 2 k ( s ) + 3 k K ( s, · ) k H ) d s (cid:19) . Proof.
Applying Lemmas 3 and 4, we know that k h ( t, · ) k L ∞ ≤ k ( t ) + k K ( t, · ) k L ) , ∀ t ∈ [0 , T ] , (3.6)and h ( t, · ) is Lipschitz continuous with the minimal Lipschitz constant L ( t ) ≤ k ( t ) + k D K ( t, · ) k L ) , ∀ t ∈ [0 , T ] . (3.7)In addition, replacing K ( t, x ) by K ′ ( t, x ) in Lemmas 3 and 4 and applying (v), weknow that [ K ′ ∗ (1 /x )]( t, · ) is well defined, uniformly bounded in L ∞ ( R ) and uniformlyLipschitz continuous for t ∈ [0 , T ]. Meanwhile, since K ( · , x ) is absolutely continuous, forany 0 ≤ t < t ≤ T and x ∈ R , we have h ( t , x ) − h ( t , x )= Z + ∞ (cid:20) K ( t , x + x ′ ) − K ( t , x − x ′ ) x ′ − K ( t , x + x ′ ) − K ( t , x − x ′ ) x ′ (cid:21) d x ′ = Z + ∞ (cid:20)Z t t K ′ ( t, x + x ′ ) − K ′ ( t, x − x ′ )d t (cid:21) x ′ d x ′ = Z t t (cid:20)Z + ∞ K ′ ( t, x + x ′ ) − K ′ ( t, x − x ′ ) x ′ d x ′ (cid:21) d t = Z t t (cid:20) K ′ ( t, x ) ∗ x (cid:21) d t. Therefore, h ( · , x ) is absolutely continuous and h ′ ( t, x ) = K ′ ( t, x ) ∗ x . Conditions (i) and (ii) then follows according to the previous conclusions on K ′ ∗ (1 /x ).11ow, all the conditions referring to h and K in Theorems 3 and 4 are fulfilled. Hencethe PIDE (1.3) has a unique classical solution u ∈ C ([0 , T ]; L ( R )) and using (3.6), (3.7)and the estimation in Theorem 3 we obtain k u ( t, · ) k L ( R )) ≤ k u k L ( R )) · exp (cid:18)Z t (cid:20)q L ( s ) + 8 k h ( s, · ) k L ∞ + 2 √ k K ( s, · ) k H (cid:21) d s (cid:19) ≤ k u k L ( R )) · exp (cid:18)Z t √ k ( s ) + 2 k ( s ) + 3 k K ( s, · ) k H ) d s (cid:19) . The last inequality holds due to q L ( s ) + 8 k h ( s, · ) k L ∞ + 2 √ k K ( s, · ) k H =2 √ (cid:16)p ( k ( s ) + k D K ( s, · ) k L ) + (2 k ( s ) + 2 k K ( s, · ) k L ) + k K ( s, · ) k H (cid:17) ≤ √ (cid:0) k ( s ) + k D K ( s, · ) k L + 2 k ( s ) + 2 k K ( s, · ) k L + k K ( s, · ) k H (cid:1) ≤ √ k ( s ) + 2 k ( s ) + 3 k K ( s, · ) k H ) . We will conclude this section by an illustrative example regarding the stationaryWigner equation (1.4). First we look at the initial value problem v ∂f ( x, v ) ∂x = Z + ∞−∞ V w ( x, v − v ′ ) f ( x, v ′ )d v ′ ,f (0 , v ) = f ( v ) . (3.8)Evidently, this is a special case of the PIDE (1.3) (simply replacing the variables ( t, x ) by( x, v ) in the discussion above). We put f ( v ) = v exp( − v /
2) and V ( x ) = exp( − x / V w can be expressed analytically, V w ( x, v ) = p /π exp( − v ) sin(2 vx ) . One can thus easily see that V w satisfies the conditions required by Theorem 5. Hence ourtheory indicates that the solution has the following form, f ( x, v ) = c ( x ) v + f ( x, v ) , f ( x, · ) ∈ L ( R ) . This can be further verified by a numerical experiment (see Figure 1). It seems our theoryhandles the initial value problem (3.8) perfectly.The inflow boundary value problem (1.6) can also be investigated through our theoryabsorbing the idea of parity decomposition. One significant aspect of the evolution opera-tor Ψ[ V w ] /v (Ψ[ V w ] is the convolution operator as defined in (1.3)) is the parity-preservingproperty (see e.g. [2, 3, 9]), which states that an odd/even function f ( v ) (see (3.8))will remain odd/even while propagating along the x -axis. This leads us to the paritydecomposition f ( x ) = f o ( x ) + f e ( x ) , ∀ f ( x, · ) ∈ L ( R ) , (3.9)where f o / f e is the odd/even part and we suppress the dependence on variable v at times.Denote L o ( R )/ L e ( R ) the set of odd/even functions in L ( R ). According to our theory andthe parity-preserving property, the operator Ψ[ V w ] /v is a bounded operator on both L o ( R )12 v -4-3-2-101234 f ( x , v ) Direct discretizationDecoupled discretization -3 -2 -1 0 1 2 3 v -4-3-2-101234 f(x,v)f (x,v)c(x)/v Figure 1: Numerical solutions of the IVP (3.8) at x = 1 using finite difference methods.The left picture displays the numerical result of f ( x, v ) using a direct velocity discretizationin (3.8) combined with a RK4 (fourth order Runge-Kutta) method for the configurationpropagation. Note that the zero velocity has been avoided (and should be avoided) inthe direct discretization to ensure stability. Although the initial data is smooth enough,one can see that the singularity appears at the zero velocity while the solution propagatesalong the x -axis. On the right is the numerical solution of system (1.1) with h = V w ∗ (1 /v ),also using RK4 for the configuration propagation (one advantage here is that there is noneed to avoid the zero velocity anymore). In this picture, however, the smooth part of thesolution f ∈ L ( R ) referred by the solid line and the singular part c ( x ) /v referred by thedashed line can be further decoupled. One can see that the solution f ( x, v ) approximates c ( x ) /v near the zero velocity, while approximates f in the region | v | >
1. The numericalsolution of (1.1), referred by the “decoupled discretization”, is also compared to the directdiscretization method of (3.8) in the left picture.13nd L e ( R ). Moreover, since the singular part of f ( x ) is always odd, we have L e ( R ) equalsto L e ( R ), the set of all even functions in L ( R ). Therefore, we can define the solutionoperators U o ( t, s ) and U e ( t, s ), bounded on L o ( R ) and L e ( R ) (see Theorem 2) respectively,such that f o ( t ) = U o ( t, s ) f o ( s ) , f e ( t ) = U e ( t, s ) f e ( s ) . (3.10)To treat the inflow boundary value problem, for any real function w ( v ), we define w + ( v )the restriction of w ( v ) on the half line R + and w − ( v ) for the other half R − . We can thuswrite the inflow boundary conditions (1.6) into f + (0) = f +0 , f − (1) = f − , which, when substituting the decomposition (3.9), can be further interpreted by f + o (0) + f + e (0) = f +0 , − f + o (1) + f + e (1) = f − . (3.11)Meanwhile, since any odd/even function can be identified with its restriction on the halfline R + , the solution operator U o ( t, s )/ U e ( t, s ) induces a corresponding solution operator U + o ( t, s )/ U + e ( t, s ) which, due to the fact proved on the former one, is also bounded in L o ( R + )/ L e ( R + ), where L o ( R + )/ L e ( R + ) is the L / L counterpart on the half line R + .Similar to (3.10), we have f + o (1) = U + o (1 , f + o (0) , f + e (1) = U + e (1 , f + e (0) . (3.12)Putting (3.11) and (3.12) together we obtain a closed system, for which the initial datacan be solved formally, f + o (0) = (cid:2) U + o (1 ,
0) + U + e (1 , (cid:3) − (cid:2) U + e (1 , f +0 − f − (cid:3) ,f + e (0) = (cid:2) U + o (1 ,
0) + U + e (1 , (cid:3) − (cid:2) U + o (1 , f +0 + f − (cid:3) . However, this is not the final answer to the inflow boundary value problem since we haveno clue whether [ U + o (1 ,
0) + U + e (1 , − is a bounded operator on a certain space (a similarresult can be found in [3]). We prove the well-posedness of the abstract Cauchy problem (1.1) and extend ourresults to the study of the partial integro-differential Eq. (1.3), which is a generalizedform of the stationary Wigner equation. Our theory reveals the equivalence of these twoseemingly unrelated problems. Also, the results on the initial value problem (1.3) shedlights on the inflow boundary value problem of the stationary Wigner equation, althougha throughly solution of this problem definitely requires a further investigation. Anotherinteresting question comes from the observation that if c ( t ) ≡ t ∈ [0 , T ] (see Theorem4), then u ( t, x ) is purely a solution of (1.3) in L ( R ). However, whether L ( R ) is a propersolution space for the PIDE (1.3) (or particularly the stationary Wigner equation (1.4)with initial or inflow boundary conditions) is not in the scope of this paper. These topicswill be treated in future study. Acknowledgments
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