Quantum Euler-Poisson Systems: Global Existence and Exponential Decay

Transcription

1 Center for Scientific Computation And Mathematical Modeling University of Maryland, College Park Quantum Euler-Poisson Systems: Global Eistence and Eponential Decay Ansgar Jungel and Hailiang Li June 3 CSCAMM Report 3- CSCAMM is part of the College of Computer, Mathematical & Physical Sciences CMPS

2 Quantum Euler-Poisson Systems: Global Eistence and Eponential Decay Ansgar Jüngel Hailiang Li Fachbereich Mathematik und Statistik, Universität Konstanz, Fach D93, Konstanz, Germany; Institut für Mathematik, Universität Wien, 9 Vienna, Austria, and Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University Toyonaka 56-45, Osaka, Japan, lihl@math.sci.osaka-u.ac.jp Abstract A one-dimensional transient quantum Euler-Poisson system for the electron density, the current density and the electrostatic potential in bounded intervals is considered. The equations include the Bohm potential accounting for quantum mechanical effects and are of dispersive type. They are used, for instance, for the modeling of quantum semiconductor devices. The eistence of local-in-time solutions with small initial velocity is proven for general pressure-density functions. If a stability condition related to the subsonic condition for the classical Euler equations is imposed, the local solutions are proven to eist globally in time and tend to the corresponding steady-state solution eponentially fast as the time tends to infinity. Keywords. Quantum Euler-Poisson system, eistence of global-in-time classical solutions, nonlinear fourth-order wave equation, eponential decay rate, long-time behavior of the solutions. Introduction

3 Quantum Euler-Poisson Systems. The Model Equations In 97, Madelung gave a fluid-dynamical description of quantum systems governed by the Schrödinger equation for the wave function ψ: iε t ψ = ε ψ V ψ in Rd,, ψ, = ψ in R d, where d is the space dimension, ε > denotes the scaled Planck constant, and V = V, t is some given potential. Separating the amplitude and phase of ψ = ψ epis/ε, the particle density ρ = ψ and the particle current density j = ρ S for irrotational flow satisfy the so-called Madelung equations [] t ρ + divj =,. j j t j + div ρ φ ε ρ ρ ρ = in R d,,. ρ where the i-th component of the convective term divj j/ρ equals d k= k ji j k. ρ The equations.-. can be interpreted as the pressureless Euler equations including the quantum Bohm potential ε ρ..3 ρ They have been used for the modeling of superfluids like Helium II [6, ]. Recently, Madelung-type equations have been derived to model quantum phenomena in semiconductor devices, like resonant tunneling diodes, starting from the Wigner- Boltzmann equation [6] or from a mied-state Schrödinger-Poisson system [8, 9]. There are several advantages of the fluid-dynamical description of quantum semiconductors. First, kinetic equations, like the Wigner equation, or Schrödinger systems are computationally very epensive, whereas for Euler-type equations efficient numerical algorithms are available [5, 5]. Second, the macroscopic description allows for a coupling of classical and quantum models. Indeed, setting the Planck constant ε in. equal to zero, we obtain the classical pressureless equations, so in both pictures, the same macroscopic variables can be used. Finally, as semiconductor devices are modeled in bounded domains, it is easier to find physically relevant boundary conditions for the macroscopic variables than for the Wigner function or for the wave function. The Madelung-type equations derived by Gardner [6] and Gasser et al. [8] also include a pressure term and a momentum relaation term taking into account interactions of the

4 A. Jüngel & H.-L. Li 3 electrons with the semiconductor crystal, and are self-consistently coupled to the Poisson equation for the electrostatic potential φ: t ρ + divj =,.4 j j t j + div + P ρ ρ φ ε ρ ρ ρ = j ρ τ,.5 λ φ = ρ C in Ω,,.6 where Ω R d is a bounded domain, τ > is the scaled momentum relaation time constant, λ > the scaled Debye length, and C is the doping concentration modeling the semiconductor device under consideration [, 4]. The pressure is assumed to depend only on the particle density and, like in classical fluid dynamics, often the epression P ρ = T γ ργ, ρ, γ,.7 with a constant T > is employed [6, ]. Isothermal fluids correspond to γ =, isentropic fluids to γ >. Notice that the particle temperature is T ρ = T ρ γ. In this paper we consider general smooth pressure functions. The equations.4-.6 are referred to as the quantum Euler-Poisson system or as the quantum hydrodynamic model. In this paper, we investigate the local and global eistence and long-time behavior of solutions of the following one-dimensional quantum Euler-Poisson system: ρ t + j =,.8 j j t + ρ + P ρ = ρφ + ρ ε ρ j ρ τ,.9 with the following initial and boundary conditions φ = ρ C,. ρ, = ϱ >, j, = j =: ϱ v,. ρ, t = ρ, ρ, t = ρ, ρ, t = ρ, t =,. φ, t =, φ, t = Φ,.3 for, t,,, where ρ, ρ, Φ >, and v is the initial velocity. The eistence and uniqueness of steady-state classical solutions to the quantum Euler- Poisson system for current density j = thermal equilibrium has been studied in [, 7]. The stationary equations for j > have been considered in [4,, 7] for general monotone pressure functions, however, with different boundary conditions, assuming Dirichlet data for the velocity potential S [] or employing nonlinear boundary conditions [4, 7]. Eistence of steady-state solutions to.8. subject to the boundary conditions..3 is proven in [] for the linear pressure function P ρ = ρ and in [4] for

5 4 Quantum Euler-Poisson Systems general pressure functions P ρ also allowing for non-conve or non-monotone pressuredensity relations. So far, to our knowledge, the only known results on the eistence of the time-dependent system.4-.6 have been obtained in [3] for smooth local-in-time solutions on bounded domains and in [7] for small irrotational global-in-time solutions in the whole space assuming strictly conve pressure functions and a constant doping profile. In the present paper, we consider the initial-boundary-value problem IBVP.8.3 for general pressure and non-constant doping profile, and we focus on the local and global eistence of classical solutions ρ, j, φ of the IBVP.8.3 and their timeasymptotic convergence to the stationary solutions ρ, j, φ obtained in [4]. First, we show that there eists a classical local-in-time solution for regular initial data. Second, we prove that if a certain subsonic condition see.5 holds and if the initial data is a perturbation of a stationary solution ρ, j, φ, a classical solution ρ, j, φ eists globally in time and tends to ρ, j, φ eponentially fast as time tends to infinity. In dealing with the IBVP.8.3 we have to overcome the following difficulties. First, since the general pressure P ρ can be non-conve even zero or negative, see Remark.6, the left part of equations.8. may be not hyperbolic any more. Unlike [7], we cannot apply the local eistence theory of quasilinear symmetric hyperbolic systems [3, 5,, 3]. We have to establish a new local eistence theory. Second, the appearance of the nonlinear quantum Bohm potential in.9 requires that the density is strictly positive for regular solutions. This together with the structure of the quantum term causes problems in the local and global eistence proofs.. Main results Before stating our main results we introduce some notation. We denote by L = L, and H k = H k, the Lebesgue space of square integrable functions and the Sobolev space of functions with square integrable weak derivatives of order k, respectively. The norm of L is denoted by =, and the norm of H k is k. The space H k = Hk, is the closure of C, in the norm of Hk. Let T > and let B be a Banach space. Then C k, T ; B C k [, T ]; B, respectively denotes the space of B-valued k-times continuously differentiable functions on, T [, T ], respectively, L, T ; B is the space of B-valued L -functions on, T, and W k,p, T ; B the space of B-valued W k,p -functions on, T. Finally, C always denotes a generic positive constant. It is convenient to make use of the variable transformation ρ = w in.8. which yields the following IBVP for w, j, φ: ww t + j =,.4 j j t + w + P w = w φ + ε w w j w τ,.5 φ = w C,.6

6 A. Jüngel & H.-L. Li 5 with the initial and boundary conditions w, j, = w, j = ϱ, ϱ v,.7 w, t = ρ, w, t = ρ, w, t = w, t =,.8 φ, t =, φ, t = Φ,.9 for,, t. This problem is equivalent to.8.3 for classical solutions with positive particle density. We will assume throughout this paper compatibility conditions for the IBVP.4.9 in the sense that the time derivatives of the boundary values and the spatial derivatives of the initial data are compatible at, t =, and, t =, in.4.6. We will prove the following local eistence result for the IBVP.4.9: Theorem. Assume that P C 4, +, C H,. w, j H 6 H 5 such that w > for [, ], and for some α [ + ε, where v C [,] < αw 8 w,. w = min [,] w >. Then, there is a number T determined by 3.69, such that there eists a unique classical solution w, j, φ of the IBVP.4.9 in the time interval [, T ], with < T T, satisfying w αw > in [, ] [, T ] and wt 6 + jt 5 + φt 4 < for t T. Remark. It is well-known that for classical hydrodynamic equations, monotone pressure-density relations are required to guarantee short-time eistence of classical solutions [, 8]. The condition. means that this condition is not necessary to a certain etent when the quantum effects are taken into account. Condition. is needed to prove the positivity of the particle density. A similar condition has been used to prove the eistence of stationary solutions []. This condition allows for arbitrarily large current densities j = w v, for instance, if w is a sufficiently large constant. 3 We are able to show the statements of Theorem. under the slightly more general condition { v C [,] < min αε, α } w 4 w, α,.. Then. is a special case for α > + ε which is equivalent to αε α/.

7 6 Quantum Euler-Poisson Systems 4 The local eistence of the Cauchy problem in R d or T d can be shown in the same framework, see [9]. Theorem. is proven by an iteration method and compactness arguments. More precisely, we construct a sequence of approimate solutions which is uniformly bounded in a certain Sobolev space in a fied maybe small time interval. Compactness arguments then imply that there is a limiting solution which proves to be a local-in-time solution of.4.9. Unlike [7] we cannot apply the theory of quasilinear symmetric hyperbolic systems [3, 5,, 3] to construct local approimate solutions and obtain uniform bounds in Sobolev spaces because the pressure can be non-conve causing the loss of entropy and hyperbolicity of.4.5. The idea of the local eistence result is first to linearize the system.4.6 around its initial state w, j, φ, where φ solves the Dirichlet problem.6 and.9 with w replaced by w, and to consider the equations for the perturbation ψ, η, e = w w, j j, φ φ. The main idea is to write the evolution equation for the perturbed particle density as a semilinear fourth-order wave equation. Then, we construct approimate solutions ψ i, η i, e i i from a fied-point procedure, which are epected to converge to a solution ψ, η, e of the perturbed problem as i. For this, we derive uniform bounds in Sobolev spaces on a uniform time interval and apply standard compactness arguments see Section 3. A further analysis shows that w, j, φ = w +ψ, j +η, φ +e with w > is the epected local in time solution of the original problem.4.9. To etend the local classical solution globally in time, we need to establish uniform estimates. We consider the situation when the initial data is close to the stationary solution w, j, φ of.4.6 with boundary conditions.8.9. The eistence of stationary solutions w, j, φ of the boundary-value problem.4.6 and.8.9 for general pressure functions P ρ was obtained in [4] see Theorem.3 below. Assume that there is a function A H, satisfying A > for,, A = ρ, A = ρ, A = A =.3 such that for a set E [, ], it holds P A j A {, E, >, [, ]\E..4 Then we conclude the eistence of stationary solutions w, j, φ of.4.6 satisfying the boundary conditions.8.9: Theorem.3 [4] Let.,.3.4 hold. For given κ,, assume that min [,] A 4 κε + P A > j..5

8 A. Jüngel & H.-L. Li 7 Then there is a unique solution w, j, φ of the stationary version of the boundary-value problem.4.6 and.8.9 such that A w A + A w 3 + φ Cδ, provided δ := A + A C is small enough. Here, A = min [,] A, A = min [,] 4 κε + P A ja >,.6 and C > is a constant depending on j, τ and A. Let ρ = w. Then ρ, j, φ is a solution of the stationary version of the boundaryvalue problem.8. and..3 satisfying A ρ A + A ρ 3 + φ C δ, and C > is a constant depending on j, τ and A. Remark.4 When E = the assumption.4 corresponds eactly to the subsonic condition for classical fluids [, 8]. We recall that a classical fluid is in the subsonic state if the velocity is smaller than the sound speed P ρ. Only for subsonic fluids, we can epect to have eistence of classical solutions [, 8]. Therefore, in order to get eistence of classical solutions of the quantum hydrodynamic equations, we epect that a condition corresponding to the classical subsonic condition is needed. It turns out that.4 is such a condition. Notice that the condition.5 can allow for non-empty sets E when quantum effects are involved. The condition.4 can be replaced by 4 κε + E min E p A j A >, κ,,.7 in order to obtain the eistence and uniqueness of classical solutions. Here, E denotes the volume of the subset E. We recall that in the steady state, the current density j is a constant. If j =, we obtain the thermal equilibrium state. The condition.4 is satisfied if j > is sufficiently small. Thus, Theorem.3 means that we can show the eistence of solutions close to the thermal equilibrium state. In the following, we use the abbreviation ψ = w w, η = j j..8 In view of the uniform a-priori estimates of Section, we are able to etend the local classical solution globally in time and prove its eponential convergence to the stationary solution w, j, φ :

9 8 Quantum Euler-Poisson Systems Theorem.5 Assume that.,.3.5 hold. Let w, j, φ be the stationary solution of the boundary-value problem.4.6 and.8.9 given by Theorem.3 for sufficiently small δ. Assume that the initial datum w, j H 6 H 5 satisfies. and w > in [, ]. Then there is a number m > such that if ψ 6 + η 5 = w w 6 + j j 5 m, the classical solution w, j, φ of the IBVP.4.9 eists globally in time and satisfies w w t 6 + j j t 5 + φ φ t 4 C ψ 6 + η 5e Λ t.9 for all t. Here, C > and Λ > are constants independent of the time variable t. Remark.6 Theorems..5 also apply for non-monotone or even negative pressure functions. These functions are related to quantum mechanical phenomena in which the motion of the particles is affected by their attractive interaction [6]. A typical eample is the focusing nonlinear Schrödinger equation. In fact, this equation is formally equivalent to the quantum Euler-Poisson system with infinite relaation time and with negative pressure. Using Theorems..5 and the variable transformation ρ = w, we also obtain the local and global eistence of classical solutions of the original IBVP.8.3 and can establish their large-time behavior: Theorem.7 Let. hold. Assume that ϱ, j H 6 H 5 such that ϱ > in [, ] and { } α v C [,] < min αε, ϱ 4, α,, ϱ where ϱ = min ϱ. [,] Then there is a number T > such that there eists a classical solution ρ, j, φ of the IBVP.8.3 in t [, T ] satisfying ρ > in [, ] [, T ] and ρt 6 + jt 5 + φt 4 <, t T..3 Furthermore, assume that.3.5 hold and let ρ, j, φ be the stationary solution of the boundary-value problem.8. and..3 given by Theorem.3 with sufficiently small δ. Then, there is a number m > such that if ϱ ρ 6 + η 5 m, the solution ρ, j, φ of the IBVP.4.9 eists globally in time and satisfies ρ ρ t 6 + j j t 5 + φ φ t 4 C ψ 6 + η 5e Λ t, for all t, where C > and Λ > are constants independent of t and the pair ψ, η is defined in.8.

10 A. Jüngel & H.-L. Li 9 This paper is arranged as follows. Section is concerned with uniform a-priori estimates of local in time solutions. We reformulate the original problem in Section. as a nonlinear fourth-order wave equation and establish the a-priori estimates for local solutions in Section.. The a-priori estimates and the local eistence result of Section 3 imply the global eistence. In order to prove the local eistence result, we first give a result on the eistence of solutions of an abstract fourth-order wave equation Section 3.. This wave equation allows us to construct a sequence of approimate solutions converging to a local solution of the problem under consideration Section 3.. Proof of Theorem.5 In this section, we establish uniform a-priori estimates for local classical solutions of.4.6. This yields, together with the usual continuity argument, the eistence of globalin-time solutions and proves Theorem.5. For notational simplicity, we set τ =.. Reformulation of the original problem Let w, j, φ be the steady-state solution of the boundary-value problem.4.6 and.8.9. For any T >, assume that w, j, φ is a solution to the IBVP.4.9 in [, T ]. Differentiating.4 with respect to t and.5 with respect to and adding the resulting equations leads to a nonlinear fourth-order wave equation for w: w tt + w t + w w t + w w φ w where we have used the identity [ w w w [P w + j w ] ] + 4 ε w 4 ε w w =,. [ ] = w w w.. w Similarly, the steady-state solution of.4.5 satisfies w w φ [ ] P w w + j + 4 ε w 4 ε w =..3 w Introduce the perturbations of the steady-state ψ = w w, η = j j, e = φ φ..4 Then, using.4,..3, and.6, the evolution equations for ψ, η, e read as follows: η t + η = g, t,.5

11 Quantum Euler-Poisson Systems ψ tt + ψ t + 4 ε ψ + w + 3w ψ + φ + ψ ψ [ ] [ ] j w η P j 4 ψ = g, t + g, t,.6 e = w + ψψ,.7 with the following initial-boundary values ψ, = ψ, η, = η,,,.8 η ψ t, = θ =:,,,.9 w + ψ ψ, t = ψ, t = ψ, t = ψ, t =, t,. e, t = e, t =, t,. and the definitions [ j + η ] g, t = w + ψ j + P w + ψ P + ε w + ψ + ψ w + ψ ε + w + ψψφ + w + ψ e,. g, t = ε w + ψ ψ ε w + ψ ψt ψ 4w 4w + ψw w + ψ φ + e ψ w e,.3 g, t = [P w + ψ + j + η ] w + ψ w + ψ [ ] P j 4 ψ [ ] j w η Notice that we can write.4 equivalently as w [ ] P w w + j..4 w + ψψ t + η =,.5 which allows us to estimate the derivatives of η in terms of ψ t.. The a-priori estimates We assume that for given T >, there is a classical solution ψ, η, e of the IBVP.5. satisfying the regularity condition ψ, η, e XT := C [, T ]; H 6 C [, T ]; H 5 C [, T ]; H 4.

12 A. Jüngel & H.-L. Li We also use the definition δ T := ma t T ψt 6 + ηt 5..6 It is easy to verify that if δ T is sufficiently small, there are constants w, w +, j, and j + such that < w w + ψ w +, j j + η j +. In the following we assume that δ T is sufficiently small such that the above estimates hold. Lemma. It holds for ψ, η, e XT and, t,, T, e, t + et 4 C ψt, e t, t + e t, t + e t t 4 C ψ t t,.7 ηt C η ep{ c t} + C ψ t, ψ, ψ,.8 η, t C η ep{ c t} + C ψ t, ψ, ψ,.9 η t t C η ep{ c t} + C ψ t, ψ, ψ,. ψ, ψ C ψ tt, ψ t, ψ, ψ, ψ, ψ t,. provided that δ T + δ is small enough see Theorem.3 for the definition of δ. Here, c, C > are constants independent of time t. The notation f, g,... means f + g +. Proof: The estimates.7 follow directly from the formula e = G, yw y + ψy, tψy, tdy, and Hölder s inequality. Here, G, y denotes the Green s function G, y = { y, < y, y, > y.. To prove.8., it is sufficient to prove.8. In fact, from.5 follows η η d + η η d C η d + C ψ t d, which gives.9 if.8 is proved. In order to see that also. follows from.8, we proceed as follows. We conclude from the boundary condition. that there eists t such that ψ t, t =,

13 Quantum Euler-Poisson Systems and that there are t, 3 t and 4 t satisfying t t 3 t and t 4 t 3 t such that ψ t, t = ψ 3 t, t = ψ 4 t, t =. Thus, by Poincaré s and Hölder s inequality, we obtain ψ d C ψ d = ψ d = ψ d,.3 ψ y, tdy d 3 t ψ y, tdy d 4 t Then, using.5,.7,.5, and.3.4, we can estimate { ηt j + η } d C w + ψ j + P w + ψ P d C C + C [η + ψ φ + e ]d + C [η + ψ + ψ t + ψ + ψ + ψ ]d [η + ψ + ψ t + ψ ]d. ψ d,.4 ψ d..5 [w + ψψ + w + ψ ψ ]d Hence, the estimate. follows as soon as.8 is shown. We now prove.8. Multiplying.5 by η, integrating over, and integrating by parts gives, in view of the boundary conditions., d η d + η d dt [ η j + η j ] + η + ψψφ + w + ψ e d + + C η w j + η w + ψ j + P w + ψ P d ηw + ψψ + η ψ w + ψ d =I + I + I + I 3..6 The integrals I, I, I, and I 3 are estimated as follows. I η j + η j j + η j + η j + ηη ηw 3 d

14 A. Jüngel & H.-L. Li 3 Cδ + η d + C I I C I 3 η d + C ψ t η + ψψ t d η d + C ψ + e d ψ t d,.7 η d + C η d + C ψ d,.8 [ψ t + ψ ]d,.9 [ψ t + ψ + ψ ]d,.3 provided that δ T + δ is small enough. In the above estimates we have used.5,.7,.3 and.4. Substituting.7.3 into.6 yields d η d + c η d C dt [ψ t + ψ + ψ ]d,.3 where c, 4 3 Cδ ] is a constant and δ is chosen so small that Cδ < 4 3. Applying Gronwall s inequality to.3 gives.8. Finally, we prove.. By.6 and.8, it holds ψ d C ψ tt + ψ t + ψ + ψ + ψ + ψ td + Cδ T + δ ψ d..3 The combination of.3 and.5 leads to. provided that δ T + δ is small enough such that Cδ T + δ <. We prove now uniform estimates in Sobolev spaces for ψ, ψ t and ψ tt. Lemma. It holds for ψ, η, e XT and < t < T, ψt 4 + ψ t t + ψ tt t + et + P A j,\e A ψ + ψtd C ψ 4 + η 3 ep{ β 3 t},.33 provided that δ T + δ is small enough. Here, C, β 3 > are constants independent of t. Proof: Step : differential inequality for ψ and ψ t in L. We multiply.6 by ψ, integrate the resulting equation over, and integrate by parts, taking into account the boundary conditions.: d dt [ ] ψ + ψψ t d ψ t d + w + 3w ψ + φ + ψ ψ d

15 4 Quantum Euler-Poisson Systems = + [ 4 ε ψ + g ψd + P w j g ψd w 4 ] ψ d + ψ w j η d =I 4 + I 5 + I 6 + I We estimate the integrals I 4,..., I 7 term by term. From.3 follows I 4 = 4 ε [ 4 ε ψ + P A j A b + A ψ d min [,] P w P A j ] ψ d P A j 4 + j A ψd,\e where A is given by.6 and A E ψ d P A j A b = 4 κε. P w P A j ψd,\e 4 + j A P A j A ψ d ψ d ψ d + Cδ ψd,.35 Notice that A > by assumption.5. Elementary computations, employing.5 and.6, lead to j η = j [w + ψψ t + w + ψ ψ t ] j j 4 w + ψψ t + η..36 With this identity, Cauchy s inequality, integration by parts,.8 and.3, we have where In view of I 5 Cδ T + δ Cδ T + δ + C ep{ c t} ψ + ψ t + η d + j ψ ψ t d w ψt + ψ + ψd + a ψt d + 4 b η d, a = 4j / min [,] w4 b = ψ d 6j κε min [,] g, t C ψ + ψ + ψ t + ψ + e,.38

16 A. Jüngel & H.-L. Li 5 Cauchy s inequality,.7 and.3, we infer I 6 Cδ T + δ [ψ + ψ + ψ t ]d. By.6,.5,.8 and.3, we obtain, after a tedious calculation, that g, t Cδ T + δ ψ + ψ + ψ + ψ t + η..4 From the above estimate,.9 and Cauchy s inequality follows I 7 Cδ T + δ [ψ + ψ + ψ + ψ t ]d + C ep{ c t} Substituting the estimates for I 4,..., I 7 into.34, we conclude d [ ] dt ψ + ψ t ψ d + a ψt d + + A b Cδ T + δ w + 3w ψ + φ + ψ ψ dd ψd + P A j,\e A ψd ψ + ψ + ψ + ψ t d + C ep{ c t} η d. η d..4 Multiply now.6 by ψ t, integrate the resulting equation over, and integrate by parts, noticing ψ t, t = ψ t, t = : d [ ψt + w + 3 dt w ψ + φ + 4 ] ψ ψ d + d [ ] dt 4 ε ψ + P j 4 ψ d + ψt d = ψ j t η d + g ψ t d + g ψ t d =I 8 + I 9 + I..4 Employing.36, integration by parts and.8, we estimate I 8 Cδ T + δ j ψ t ψ t d + Cδ T + δ ψ + ψ t + η d ψ t + ψ + ψ d + C ep{ c t} η d. In view of.38,.4,.7,.8, and., the integrals I 9 and I can be bounded as follows: I 9 Cδ T + δ ψ + ψ + ψ t d,

17 6 Quantum Euler-Poisson Systems I Cδ T + δ ψ + ψ + ψ + ψ t d + C ep{ c t} Substitution of the above three estimates into.4 yields d [ ψt + w + 3 dt w ψ + φ + 4 ] ψ ψ d + d [ ] dt 4 ε ψ + P j 4 ψ d + ψt d Cδ T + δ ψ + ψ + ψ + ψ t d + C ep{ c t} η d. η d..43 We add.4 and.43, multiplied by + a here we recall that a is denoted by.37, to obtain d [ ] dt ψ + ψ t ψ + + a ψt d + d + a [w + 3 dt w ψ + φ + 4 ] ψ ψ d + d [ ] + a dt 4 ε ψ + P j 4 ψ d + [ w + 3w ψ + φ + ψ ψ + + a ψ ] t d + A b ψd + P A j,\e A ψd Cδ T + δ ψ + ψ + ψ + ψ t d + C ep{ c t} η d..44 Applying Grownwall Lemma to.44, we can estimate the H -norm of ψ and L -norm of ψ t in terms of the initial data and ψ. However, the differential inequality for ψ and ψ t is enough for the following considerations. Step : differential inequality for ψ tt in L. The starting point of the following estimates is.6, differentiated with respect to t: ψ ttt + ψ tt + 4 ε ψ t + w + 3w ψ + φ + 3 ψ ψ t [ ] j w η t P j 4 ψ t = g t, t + g t, t..45 This equation holds pointwise almost everywhere in,, T since ψ C [, T ]; H 6 H 3, T ; L see the proof of Theorem.. We multiply.45 first by ψ t, integrate the resulting equation over, and integrate by parts, using the boundary conditions ψ t, t = ψ t, t = ψ t, t = ψ t, t = and.7: d [ ] dt ψ t + ψ t ψ tt d ψttd + [w + 3w ψ + φ + 3 ] ψ ψt d

18 A. Jüngel & H.-L. Li 7 = + [ 4 ε ψt + P j g t ψ t d + g t ψ t d w 4 ] ψt d + j ψ t w η t d =I + I 3 + I 4 + I Applying an argument similar to.35, it follows I A + b where we have used ψtd P A j,\e A ψ td + Cδ ψtd, E ψ td ψ td ψ td,.47 based on the facts ψ t, t = ψ t, t =. By.5,., and.47, we have, after integration by parts, I 3 = 4 j 3 ψ t ψ + w ψ tt + ψ t ψ t + w + ψ ψ tt d j 3 ψ t ψt + w + ψψ tt d + Cδ T + δ ψtt + ψt + ψt + ηt d + j w + ψ ψ t ψ tt d + Cδ T + δ + C ep{ c t} ψ tt + ψ t + ψ + ψ td j w 3 ηd + a ψttd + 4 b Elementary computations yield the estimate j ψ t η t d w + ψψ t ψ tt d ψ td. g t, t Cδ T + δ ψ t + ψ tt + ψ t + ψ t + e t,.48 which implies, in view of Cauchy s inequality,. and.47, that I 4 Cδ T + δ ψ t + ψ t + ψ ttd. After a tedious computation, it follows from.6 that g t, t Cδ T + δ ψ t + ψ tt + ψ t + ψ t + η t j + η + w + ψ 3 j 3 η t

19 8 Quantum Euler-Poisson Systems where we have used the equation Cδ T + δ ψ t + ψ + ψ tt + ψ t + ψ t + η t j + η w + ψ w + ψ 3 j 3 ψ tt,.5 4ψ ψ t + w + ψψ tt + w + ψ ψ tt + η t =..5 Using.5,.8,.47,.4, and the fact ψ t, t = ψ t, t =, we can estimate I 5, after integration by parts, as follows: I 5 δ T + δ ψt + ψtt + ψt + ψ + ψ + ψd j + η + w + ψ w + ψ 3 j 3 ψ t ψ tt d [ j + η + w + ψ w + ψ 3 j ] 3 ψ t ψ tt d Cδ T + δ ψ t + ψ tt + ψ t + ψ + ψ + ψ d. Substituting the above estimates for I,..., I 5 into.46, we conclude d [ ] dt ψ t + ψ t ψ tt d + a ψttd + [w + 3w ψ + φ + 3 ] ψ ψt d + A b Cδ T + δ + C ep{ c t} ψtd + P A j,\e A ψtd ψ t + ψ + ψ tt + ψ + ψ t + ψ d η d..5 The net step is to multiply.45 by ψ tt, to integrate the resulting equation over, and to integrate by parts, using ψ tt, t = ψ tt, t =, which yields d [ ψtt + w + 3w ψ + dt φ + 3 ] ψ ψt d + d [ ] dt 4 ε ψt + P j ψt d 3 = w 4 w + ψψt 3 d + ψttd ψ j tt η t d + g t ψ tt d + g t ψ tt d

20 A. Jüngel & H.-L. Li 9 = I 6 + I 7 + I By.5,.5,.,.3,.47, and integration by parts, we have I 6 = 4 j 3 j Cδ T + δ w + ψψ tt ψ tt d w From.48,.7 and.47 it follows j w 3 ψ tt ψt + w + ψψ tt d + ψ t ψ t + w + ψ ψ tt d w j w ψ tt + ψ t + ψ + ψ t + ψ d + C ep{ c t} I 7 Cδ T + δ [ψ t + ψ t + ψ tt]d. Finally, in view of.5,.8,. and integration by parts, it holds I 8 δ T + δ ψt + ψtt + ψt + ψ + ψd j + η w + ψ w + ψ 3 j 3 ψ tt ψ tt d δ T + δ [ψ t + ψ tt + ψ t + ψ + ψ ]d. ψ tt η t d Substituting the estimates for the integrals I 6, I 7 and I 8 into.53 gives d dt + d dt 3 Cδ T + δ [ ψtt + w + 3w ψ + φ + 3 ] ψ ψt d [ 4 ε ψt + P j w + ψψ 3 t d + + C ep{ c t} ψ ttd w 4 ] ψt d ψ t + ψ tt + ψ t + ψ + ψ + ψ d η d, η d..55 Now we add the inequalities.5 and.55, multiplied by + a, to infer d dt [ ] ψ t + ψ t ψ tt + + a ψtt + + a d dt d [w + 3w ψ + φ + 3 ψ ] ψ t d

21 Quantum Euler-Poisson Systems + + a d [ dt 4 ε ψt a ψ ttd + + A b Cδ T + δ + ep{ c t} ψtd + P w j w 4 ] ψt d [w + 3w ψ + φ + 3 ψ ] ψ t d,\e P A j A ψtd ψ t + ψ tt + ψ + ψ t + ψ + ψ d η d..56 Step 3: combination of the estimates for ψ, ψ t and ψ tt. We combine the estimates.44 and.56 and obtain for some constant β >, using., d dt [ ] ψ + ψt + ψ t ψ + ψ tt + + a ψt + ψtt [w + 3 w ψ + φ + 4 ] ψ ψ d + + a d dt + + a d dt + + a d dt [w + 3w ψ + φ + 3 ] ψ ψt d d [ ] 4 ε ψ + ψt + P j 4 ψ + ψt d + β [ψ + ψt + ψt + ψtt + ψt + ψ]d + β P A j ψ + ψtd,\e C ep{ c t} A provided that δ T + δ is small enough. There eist constants β, β 3 > such that η d,.57 [ β ψ + ψt + ψt + ψtt + ψt + ψ ] d + β P A j,\e A ψ + ψtd [ ] ψ + ψt + ψ t ψ + ψ tt + + a ψt + ψtt d [w + 3 w ψ + φ + 4 ] ψ ψ d + + a + + a [w + 3w ψ + φ + 3 ψ ] ψ t d

22 A. Jüngel & H.-L. Li + + a β 3 β + β 3 β [ 4 ε ψ + ψt + P j [ ψ + ψt + ψt + ψtt + ψt + ψ ] d P A j A ψ + ψtd,,\e Thus, applying Gronwall s inequality to.57, we obtain finally [ ψ + ψt + ψt + ψtt + ψt + ψ ] d +,\E w 4 ] ψ + ψt d P A j A ψ + ψ td C ψ 4 + η 3 ep{ β 3 t},.58 provided that δ T + δ is small enough. The combination of.58,.7,. and.7 gives the assertion.33. Thus, the lemma is proved. We also obtain bounds for higher-order estimates for ψ. Lemma.3 It holds for ψ, η, e XT and < t < T ψt 4 + ψ tt t + ψ ttt t + et 4 +,\E P A j A ψttd C ψ 6 + η 5 ep{ β 4 t},.59 provided that δ T + δ is small enough. Here, C, β 4 > are constants independent of t. Proof: For the proof of the lemma take the time derivative of.45 and estimate similarly as in Lemmas. and.. As the estimates are analogous to those of the proofs of Lemmas.., we omit the details. Proof of Theorem.5. By Theorem., there eists a solution w, j, φ of the IBVP.4.9 for t [, T ]. With the help of Lemmas..3, we infer that the local solution w, j, φ of the IBVP.4.9 satisfies, for t [, T ], w w, j j, φ φ t H 6 H 5 H 4 C ψ 6 + η 5 ep{ Λ t},.6 where C, Λ > are constants independent of t. Choosing the initial data ψ 6 + η 5 so small that C ψ 6 + η 5 < δ T we conclude first, by the Sobolv embedding theorem and.6, that w > in [, ] [, T ], and second, by the usual continuity argument, that w, j, φ eists globally in time and satisfies.9.

23 Quantum Euler-Poisson Systems 3 Proof of Theorem. The idea of the proof of Theorem. is to linearize the equations.4.6 around the initial state and to construct a sequence of approimate solutions of the linearized problem converging to a solution of the original problem. First we need to study the regularity properties of a certain semilinear fourth-order wave equation. 3. A semilinear fourth-order wave equation Consider the two Hilbert spaces H and L on,, endowed with the scalar products, and, and corresponding norms H = and, respectively. Furthermore, we consider the following initial-value problem on L : u + u + νau + u + Lu = F t, t >, 3. u = u, u = u, 3. where the primes denote derivatives with respect to time, τ, ν > are constants, A = 4 is an operator defined on DA = H H 4 = {u H 4 ; u =, = u =, = }, 3.3 and the operators L and F are given by Lu, v = F t, v = b, tu vd, u, v H, f, tvd, v L, where b, f : [, ] [, T ] R are measurable functions. Related to the operator A, we introduce the coercive, continuous, symmetric bilinear form au, v au, v = ν u v d u, v H. There eist a complete orthonormal family of eigenvectors {r i } i N of L and a family of eigenvalues {µ i } i N such that < µ µ and µ i as i. The family {r i } i N is also orthogonal for au, v on H, i.e. r i, r j = δ ij, ar i, r j = ν Ar i, r j = νδ ij i, j. Using the Faedo-Galerkin method [6, 8], it is possible to prove the eistence of solutions of The result is summarized in the following theorem.

24 A. Jüngel & H.-L. Li 3 Theorem 3. Let T > and assume that F H, T ; L, b C [, T ]; H W,, T ; H. 3.4 Then, if u H 4 H and u H, there eists a solution of satisfying u C[, T ]; H 4 H C [, T ]; H C [, T ]; L. 3.5 Moreover, assume additionally that F H, T ; L C[, T ]; H. Then, if u H 6 H and u H 4 H satisfy νau + Lu F H, it holds u C i [, T ]; H 6 i H C 3 [, T ]; L, i =,,. 3.6 Proof: The eistence of solutions of and the regularity property 3.5 can be shown by applying the Faedo-Galerkin method as in [9]. The regularity property 3.6 follows from 3.5 by considering the problem for the new variable v = u. As the proof is standard, we omit the details. 3. Local eistence In this section we prove Theorem.. For simplicity, we set τ =. We linearize the equations.4.6 around the initial state w, j, φ where φ solves the Poisson equation.6 and.9 with w replaced by w, and prove the local-in-time eistence for the perturbation ψ, η, e = w w, j j, φ φ. For this, we reformulate the original initial-boundary value problem.4.9. It is sufficient to carry out the reformulation for the equations.4,.6 and. because of.5. For given U p = ψ p, η p, e p we obtain the following linearized problems for U p+ = ψ p+, η p+, e p+, p N, writing for the spatial derivative and for the time derivative: { η p+ + η p+ = g 3, U p, 3.7 η p+, =, ψ p+ + ψ p+ + ν ψ 4 p+ + ψ p+ + k, U p ψ p+ = g 4, U p, ψ p+, =, ψ p+, = θ := j, w ψ p+, t = ψ p+, t = ψ p+, t = ψ p+, t =, 3.8 { e p+ = w + ψ p ψ p, e p+, t = e p+, t =, 3.9

25 4 Quantum Euler-Poisson Systems where ν = 4 ε and g 3, U p = 4j [ ] + η p ψ p j + η p w + ψ p w + ψ p P w + ψ p j ] + w + ψ p φ + e p + ε w + ψ p [ w + ψ p w + ψ p k, U p = j + η p w + ψ p, g 4, U p = w + ψ p w + ψ p φ + e p + w + ψ p P w + ψ p 4 ε w + w 4 ε + ψ p ψ p + 3ψ p w + ψ p w + ψ p + j + η p [ ] [ ] w + ψ p w + ψ p 3 w + ψ p j + η p ψ p. 3. We apply an induction argument to prove the eistence of solutions of Lemma 3. Under the assumptions of Theorem., i.e., P C 4,, C H, w, j H 6 H 5 with w > in, and, for some α [ + ε,, with v C [,] < αw 8 w 3. w = min [,] w, 3. there eists a sequence {U i } i= of solutions of in the time interval t [, T ] for some T > which is independent of i, satisfying the regularity properties { ηi C [, T ]; H 3 C [, T ]; H, e i C [, T ]; H 4 H, ψ i C l [, T ]; H 6 l H C 3 [, T ]; L 3.3, l =,,, i N, and the uniform bounds, η it 3 + η i t + e i, e it 4 M, ψ i, ψ i, ψ i, ψ i t H 6 H 4 H L M, η i t 3, ψ i t α w, i, t [, T ], 3.4 where M > is a constant independent of U i i and T. Proof: Step : solution of for p. Obviously, U =,, satisfies Starting with U =,,, we prove the eistence of a solution U =

26 A. Jüngel & H.-L. Li 5 ψ, η, e of satisfying The functions g 3, U, g 4, U and k, U only depend on the initial state w, j, φ and satisfy g 3, U =: g 3 H 3, g 4, U =: g 4 H, k, U =: k H 3, t g 3 = t g 4 = t k =, g g 4 + k 3 a I +, 3.5 where a > is some constant and I = w C + w 5 + j The eistence of a solution U = ψ, η, e of the linear system follows from the theory of ordinary differential equations, applied to 3.7, Theorem 3. with f, t = g 4 and b, t = k, applied to 3.8, and elliptic theory, applied to 3.9. The solution U eists on any time interval [, T ], T >, and satisfies 3.3 with T = T and the first two inequalities of 3.4 with i =. We show in the following that U satisfies the last two inequalities of 3.4 for t [, T ], where T > is given by { ln T = min + a I +, να w 4 v C [,] w },. 3.7 a I + a I + We recall that a > is a constant and I and w are given by 3.6 and 3., respectively. It holds να w 4 v C [,] w >, 3.8 since 3. implies 4 v C [,] w < α w 3 From 3.7 we obtain by integrating να w. η t = g 3 and hence, in view of 3.7, t ep{ t s}ds, t [, T ], η t 3 T g 3 3, t [, T ]. 3.9 Multiplying the differential equation in 3.8 by ψ, integrating the resulting equation over,, t for t [, T ] and integrating by parts gives ψ t θ + a T I + e T +a I + ν v C ν [,] w + a T I + α w, 3.

27 6 Quantum Euler-Poisson Systems where we have used θ = w v + w v v C [,] w. 3. This proves the last two bounds in 3.4. Moreover, by the Sobolev embedding theorem, it follows from 3. that ψ t α w, t [, T ]. 3. Now, assume that there eist solutions {U i } p i= p of on the time interval [, T ] where T is given by 3.7, satisfying As above we obtain, for given U p, the eistence of a solution U p+ = ψ p+, η p+, e p+ of in the interval [, T ], satisfying η p+ C [, T ]; H 3 C [, T ]; H, e p+ C [, T ]; H 4 H, ψ p+ C l [, T ]; H 6 l H C 3 [, T ]; L, l =,,. We prove that there eist constants T, T ] and K i > a i =,, 3, 5, 6, 7 independent of {U i } p i=, such that if U p satisfies on [, T ] ψ p t α w, 3.3 ψ p, ψ pt + ψ p t K, 3.4 ψ 3 pt K, ψ 3 p t K, ψ 5 p t K 3, 3.5 η p t 3, η pt K 5, η pt K 6, η pt K 7, 3.6 then U p+ also satisfies on [, T ] ψ p+ t α w, 3.7 ψ p+, ψ p+t + ψ p+t K, ψ p+t K, 3 ψ p+ t K, 5 ψ p+ t K 3, 3.9 η p+ t 3, η p+t K 5, η p+t K 6, η p+t K Notice that it follows from 3.3 and 3.7, employing the boundary conditions in 3.8 and Poincaré s inequality, ψ p t α w, ψ p+ t α w, t [, T ]. 3.3 Step : estimates for g 3, g 4, and k. Let U p satisfy Then a direct computation shows the following estimates for g 3, U p and g 4, U p, for t [, T ], g 3, U p t NI + + K + K 5, 3.3 g 3, U p t NI + + K + K + K + K 3 7, 3.33

28 A. Jüngel & H.-L. Li 7 where g 3, U p t NI + + K + K + K + K 3 + K 5 + K 6 6, 3.34 g 4, U p t NI + + ψ p t 4 + ψ pt 3 + 6ν a ψ p t 4 ψ p t 3.35 NI + + K + K 3, 3.36 g 4, U p t NI + + K + K 5, 3.37 g 4, U p t NI + + K + K + K 5 4, 3.38 g 4, U p t NI + + K + K + K + K 5 + K 7 5, 3.39 a = ma [,] w + ψ p = α w, 3.4 and the estimates for k, U p and e p, t, for t [, T ], k, U p t NI + 3, 3.4 k, U p t NI + + K + K 5 + K 6, 3.4 k, U p t NI + + K + K 5 + K 6 + K 7 3, 3.43 e p 4 + e p, e p NI + + K + K, 3.44 where N > is a constant independent of K i i =,, 3, 5, 6, 7. and Step 3: estimates for η p+. Integration of 3.7 yields η p+, t = t ep{ t s}g 3, U sds, t T T, [, ], 3.45 From we obtain the estimates η p+ C [, T ]; H 3 C [, T ]; H η p+ t T g 3, U p T NI + + K + K 5, η p+ t T g 3, U p T NI + + K + K + K + K 3 7, η p+t T + g 3, U p NT + I + + K + K 5, η p+t T + g 3, U p NT + I + + K + K + K + K 3 7, η p+t 4NT + I + + K + K + K + K 3 7 Thus, η p+ satisfies 3.3 if + NI + + K + K + K + K 3 + K 5 + K 6 6. K 5 =NT + I + + K + K 5, 3.47 K 6 =NT + I + + K + K + K + K 3 7, 3.48

29 8 Quantum Euler-Poisson Systems K 7 =NT + I + + K + K + K + K 3 7, + NI + + K + K + K + K 3 + K 5 + K and if T satisfies T L, 3.5 where L = min { NI + + K + K 5, NI + + K + K + K + K 3 7}. 3.5 ψ p+ Step 4: estimates for ψ p+. We multiply the differential equation in 3.8 by ψ p+, and ψ p+, respectively, integrate the sum of the resulting equations over,, T and integrate by parts. In view of , , , we obtain after tedious computations and where Define ψ p+ t ν v C [,] w + T L 3 e T +NI ψ p+t + ψ t + ν ψ p+t + ν ψ p+t + ψ p+ N I + T L 5 e T L 4, 3.53 L 3 = NI + + K + K 3, 3.54 L 4 = 8 + 6NI + + K + K 5 + K 6 + K 7 3 > + NI +, 3.55 L 5 = NI + + K + K + K + K 5 + K K = NI min {, ν} = NI ma {, ν } Using 3.8 which is a consequence of 3., we see that ψ p+ satisfies ψ p+ t α w, ψ p+, ψ p+t + ψ p+t NI ma {, ν } = K 3.58 if where { T min, L B 4L 3, ln + NI +, ln + NI +, ln, L 4 B = να w 4 v C [,] w. I L 5 }, 3.59 This proves 3.7 and 3.8.

30 A. Jüngel & H.-L. Li 9 To verify 3.9 we employ the differential equation in 3.8 again. We use 3.58, 3.35, 3., and 3.59 to estimate ψ 4 p+ t N ν ψ p+t + ψ p+t + ψ p+ t + k ψ p+t + ν g 4, U p t N ν I K + 64a v C [,] w + T L 3 K. 3.6 Here, we used the fact that 3.5 is also valid for ψ p in [, T ]. From 3. and 3.4 we infer 8a v C [,] w = 8 v C [,] α w w < which implies Thus, choosing K = 8 v C [,] w α w >. 8 α w NI K ν [ α w 8 v C [,] w 3.6 ], where K is defined by 3.57, we obtain from 3.6 and the Sobolev embedding theorem 3 ψ p+ t + 4 ψ p+ t K so long as we choose T = min { L, B L 3, ln + NI +, ln, L 4 I L 5 } We recall that L, L 3, L 4, L 5, and I are given by 3.5, , and 3.6, respectively, N > is a generic constant, and { α w 8 v C B = min [,] w α w ε 6 v C, [,] w } > 64 8 due to 3.. Notice that 3.63 implies 3.5 and Differentiating the differential equation in 3.8 with respect to t, integrating over,, and using 3.58, 3.4 and 3.37, we can estimate 4 ψ p+ as ψ 4 p+t N ν ψ p+t + ψ p+t + ψ p+t + k ψ p+ t Thus, choosing + ν g 4, U p t N ν I + + K + K + K 5 4. K = 4N ν I + + K + K + K 5 4, 3.64

31 3 Quantum Euler-Poisson Systems and using Sobolev embedding theorem, we have 3 ψ p+t + 4 ψ p+t K 3.65 Differentiating the differential equation in 3.8 once and twice with respect to, integrating over,, and employing 3.58, 3.4, and 3.37, we can estimate 5 ψ p+ and 6 ψ p+ as ψ 5 p+ t + ψ 6 p+ t N ν ψ p+t + ψ p+t + ψ p+ t + N ν k ψ p+t + ν g 4, U pt K 3, where we choose K 3 = 6N ν I + + K + K + K Now we choose the constants K i as follows. Let K be given by 3.57, K by 3.6, K 5 by 3.47, K by 3.64, K 3 by 3.66, K 6 by 3.48, and K 7 by 3.49 with ν = ε /4. The constant T is determined by This shows that ψ p+, η p+ satisfies for t [, T ]. Step 5: end of the proof. The uniform bounds for e p+ C [, T ]; H 4 of 3.9 follow from similar computations as those needed to derive 3.44, where the inde p is replaced by p +. By induction, we conclude that {U i } p= eists uniformly in [, T ] with T given by 3.63 and satisfies uniformly for The proof of Lemma 3. is complete. M = ma {K, K, K, K 3, K 5, K 6, K 7 }. Remark 3.3 It follows from the last two inequalities of 3.4 that ψ i, t + w αw >, i For the proof of Theorem., we observe that after a tedious computation similarly as in the proof of Lemma 3., we can obtain the the following estimates η p+ η p C,T ;H + ψ p+ ψ p C i,t ;H 4 i + e p+ e p C,T ;H T αn, M η p η p C,T ;H + ψ p ψ p C i,t ;H 4 i, i =,,, 3.68

32 A. Jüngel & H.-L. Li 3 for any T T. Here αn, M is a function of N, M. Taking T such that { } b T < min αn, M, T, b,, 3.69 then it follows from 3.68: p= + η p+ η p C,T ;H + e p+ e p C,T ;H ψ p+ ψ p C i,t ;H 4 i C, i =,,, 3.7 p= with C > a constant. Proof of Theorem.. By Lemma 3. and 3.7, the sequence {U p } p= satisfies , 3.67, and 3.7 uniformly in [, T ] with T T. Applying the Ascoli-Arzela theorem and the Aubin-Lions lemma to {U p } p=, it follows that there eists U = ψ, η, e satisfying η C [, T ]; H 3, e C [, T ]; H 4, ψ C i [, T ]; H 6 i H C 3 [, T ]; L, i =,,, and there is a subsequence {U p j, U p j+ } j= with p j + p j+ such that ψ pj +, ψ pj η pj +, η pj e pj +, e pj j ψ strongly in C i [, T ]; H 6 i σ, i =,,, j η strongly in C [, T ]; H 3 σ, j e strongly in C [, T ]; H 4 σ, for any σ >. It is not difficult to verify that U is a solution of and satisfies 3.67 where U p is replaced by U. Setting w = w + ψ >, j = j + η, φ = φ + e, we see that j C[, T ]; H 5 and w, j, φ is a local-in-time solution of the IBVP.4.9. The uniqueness can be proven similarly as the estimates 3.7. The proof of Theorem. is complete. Acknowledgments: A.J. acknowledges partial support from the TMR project Asymptotic Methods in Kinetic Theory, grant ERB-FMBX-CT97-57, from the Gerhard-Hess Award of the Deutsche Forschungsgemeinschaft DFG, grant JU359/3-, from the Priority Program Multi-scale Problems of the DFG, grant JU359/5-, and from the AFF Project of the University of Konstanz, grant 4/. H.L. acknowledges partial support from the Austrian Chinese Scientific Technical Collaboration Agreement, from CTS of Taiwan, and from the Wittgenstein Award of P. A. Markowich, funded by the Austrian FWF.

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