The Viscous Model of Quantum Hydrodynamics in Several Dimensions

Transcription

1 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher The Viscous Model of Quantum Hydrodynamics in Several Dimensions Li Chen Department of Mathematical Sciences, Tsinghua University, Beijing, 00084, P.R. China, Michael Dreher Fachbereich Mathematik und Statistik, P.O.Box D87, Universität Konstanz, Konstanz, Germany, Received Revised We investigate the viscous model of quantum hydrodynamics in one and higher space dimensions. Exploiting the entropy dissipation method, we prove the exponential stability of the thermal equilibrium state in,, and 3 dimensions, provided that the domain is a box. Further, we show the local in time existence of a solution in the one dimensional case; and in the case of higher dimensions under the assumption of periodic boundary conditions. Finally, we discuss the semiclassical limit. Keywords: quantum hydrodynamics; exponential decay; entropy dissipation method; local existence of solutions; semiclassical limit. AMS Mathematics Subject Classification: 35B40, 35Q35, 76Y05. Introduction The flow of charged particles in a semi-conductor can be simulated using different models. Typical examples are the quantum energy transport models QET, the quantum drift diffusion model QDD or the quantum hydrodynamic model QHD. Derivations of the quantum QET and QDD models can be found in Ref. 4. The quantum hydrodynamic models can be derived directly from the Schrödinger Poisson system by WKB wave functions 7 ; or from the collision Wigner equation by the momentum method and closing the system with the quantum thermal equilibrium distribution 5 ; or by the entropy minimization method 9. In this paper, we study the viscous QHD model, a model which is derived from

2 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher Li Chen and Michael Dreher the Wigner equation with the Fokker Planck collision operator: t n div J = ν 0 n, J J t J div n λ V = n Cx, n0, x = n 0 x, T n + n V + ε n J0, x = J 0 x, n n = ν 0 J J τ,. where t, x 0, and is a domain in R d. Our boundary conditions are either ν nt, x = 0, t, x 0,, Jt, x = 0, t, x 0,, ν V t, x = 0, t, x 0,,. where ν denotes the normal derivative, or we assume periodic boundary conditions, i.e., Moreover, we suppose = T d is a d dimensional torus..3 inf x n0 x > 0,.4 n 0 x Cx dx = 0..5 The last condition is necessary; the Poisson equation for V would not be solvable otherwise. The unknown functions in this system are the particle density n = nt, x: [0, R +, the current density J = Jt, x: [0, R d, and the electrostatic potential V = V t, x: [0, R. The given function C = Cx: R is the doping profile of background charges. The scaled physical constants are a viscosity constant ν 0 describing the strength of the collisions, a temperature constant T, the Planck constant ε, the momentum relaxation time τ, and the Debye length λ. All these constants are assumed to be positive. For quantum macroscopic models, some results on local or global existence or long time asymptotics are known. For the QDD model, the existence of weak solutions was shown in Refs., 3, 0; and the semiclassical limit and the long time behaviour were studied in Refs., 3. Concerning the QHD model without viscous terms, the existence of smooth solutions and their long time asymptotics for small initial data were investigated in Refs. 8, 3. It seems that there are less mathematical results for the system.. The authors are only aware of Ref. 6, where the exponential stability of a constant steady state to. in a one-dimensional setting with a certain boundary condition was

3 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher The viscous model of quantum hydrodynamics in several dimensions 3 proved, based on the entropy dissipation method. Most of the difficulties arise from the Bohm potential term Bn = n n, which introduces a third order perturbation to a system which could otherwise be interpreted as a parabolic system coupled to an elliptic equation. In this paper, we follow the approach of Ref. 6 and generalize those results to domains of dimension two and three, see Theorem.. Our key ingredient is an estimate on a certain term containing the Bohm potential, Proposition Appendix A.. Additionally, we are able to prove the local in time existence of sufficiently smooth solutions to.. The proof relies on the observation that the third-order perturbation term has a good sign, which makes standard energy estimates for parabolic systems possible after having introduced a fourth order viscous regularization. Physically spoken, the periodic boundary conditions are of restricted interest; however, they enable us to prove the local in time existence of solutions in a onedimensional setting with boundary conditions. immediately, see Theorem.3. In the course of proving the local existence results of the Theorems. and.3, we will obtain a certain a priori estimate of the solution, from which we then will be investigating the semiclassical limit ε +0, compare Theorem.4.. Main Results Our notations are standard: L p denote the usual Lebesgue spaces, and H k := W k are the L based Sobolev spaces, for k N 0. The brackets, stand for the scalar product in R d, and J J is a d d matrix with entry J k J l at position k, l. We list our results: Theorem. Exponential stability. Let d =,, 3 and = d j= a j, b j be a box. Let the triple n, J, V be a solution to. under the boundary conditions., and suppose that n H 0, t, H L 0, t, H 3, J H 0, t, L L 0, t, H, V H 0, t, H. We assume that C = Cx C 0 > 0 in and inf nt, x > 0. t,x 0,t

4 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher 4 Li Chen and Michael Dreher Define an energy as follows: Et = ε n + T n ln n + C 0 + λ C 0 V + J dx. n. Let µ denote the first positive eigenvalue of on with Neumann boundary conditions, and fix { 8T ν0 σ := min ε, µ C 0 µ λ T + C 0, }. τ Then this energy satisfies the inequality t Et σet ε ν 0 c,d n dx + c,d n 4 n 3 dx ν 0 l Jl n dx, n where the numbers c,d and c,d are given in the following table: d 3 c,d 3 c,d Remark.. Physically spoken, the four terms in the energy. are the quantum energy, the thermodynamic entropy, the electric energy and the kinetic energy. We note that the energy of the steady state n, J, V = C 0, 0, 0 is zero, and that E can not become negative. From the above differential inequality of the physical energy we then easily derive decay estimates: Corollary.. Assume that the solution mentioned in Theorem. exists up to

5 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher The viscous model of quantum hydrodynamics in several dimensions 5 t =. Then the function n, J, V decays to the steady state C 0, 0, 0 as follows: ε nt, C0 L e σt E0,. T nt, C 0 L e σt E0,.3 nt, C 0 L p C p,d, ε + e σt E0,.4 T ε ν 0c,d t=0 nt, C0 L Jt, nt, L dt E0,.5 e σt E0,.6 λ V t, L e σt E0,.7 where C p,d, is the norm of the embedding H L p, with p for d =, p < for d =, and p 6 for d = 3. Theorem. Local existence and uniqueness on a torus. Let d be a positive integer and = T d be a torus. Let b be the smallest integer greater than d, and s b be an integer. Suppose n 0 H s+, J 0 H s, C H s and.4,.5. Then the problem. has a solution n, J, V, local in a time interval [0, t ], with the regularity properties n H 0, t, H s L 0, t, H s+, J H 0, t, H s L 0, t, H s+, V H 0, t, H s+.8 L 0, t, H s+4, n, n, J C[0, t ]. The solution is unique and persists as long as nt, and Jt, stay in L and n remains positive. The life span t does not depend on the Sobolev regularity s. Having shown the local existence in the periodic case, we can take advantage from geometric arguments and consider the case R effortlessly: Theorem.3 Local existence and uniqueness in one dimension. Let R be an open and bounded interval. Suppose n 0 H, J 0 H, C L and.4,.5. The initial functions are assumed to satisfy the compatibility conditions ν n 0 x = 0, J 0 x = 0, x.

6 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher 6 Li Chen and Michael Dreher Then the problem. with the boundary conditions. has a local solution n, J, V, with n H 0, t, H L 0, t, H 3, J H 0, t, L L 0, t, H, V H 0, t, H 3 L 0, t, H 5, n, n, J C[0, t ]. This solution is unique and persists as long as nt, and Jt, stay in L and n remains positive. Remark.. The local in time existence in the one-dimensional case can also be proved for solutions of higher regularity provided that the usual compatibility conditions on the initial data are satisfied. Theorem.4 Semiclassical limit. Let be either an open and bounded interval in R or a d-dimensional torus. Suppose that the given data n 0, J 0, C of. satisfy n 0 H b+, J 0 H b, C H b and.4,.5. Let n ε, J ε, V ε, ε > 0, denote the solutions to. with boundary conditions. or.3, existing on a domain [0, t ]. As ε tends to +0, a sub-sequence n ε, J ε, V ε ε then converges to a limit n, J, V, n ε, J ε, V ε n, J, V in C[0, t ], n ε n in L 0, t, H b+, J ε J in L 0, t, H b+, n ε, J ε n, J in L 0, t, H b, which is a solution to the initial value problem t n div J = ν 0 n, J J t J div T n + n V = ν 0 J J n τ, λ V = n Cx, n0, x = n 0 x, J0, x = J 0 x with boundary conditions. or.3, respectively. The regularity of n, J, V is given by n H 0, t, H b L 0, t, H b+, J H 0, t, H b L 0, t, H b+, V H 0, t, H b+ L 0, t, H b+.

7 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher The viscous model of quantum hydrodynamics in several dimensions 7 3. Exponential Stability In this section, we prove Theorem., following an approach of Ref. 6. Rewrite the energy from. as follows: ε Et = n + T n ln n + C 0 + λ C 0 V + J dx n = E + E + E 3 + E 4. We compute the time derivatives. In the sequel, the zeroes denote boundary integrals that vanish due to the boundary conditions: t E = ε ν n t n dσ ε n t n dx = 0 ε Bn t n dx = ε Bn div J + ν 0 n dx. Next, we have t E = T n t ln n dx = T ln n div J + ν 0 n dx C 0 = 0 T n n, J dx T ν n 0 dx n = T n n, J dx 8T ν 0 ε E. Further, we get t E 3 = λ V V t dx = λ V ν V t dσ λ = 0 V n t dx = V div J + ν 0 n dx = 0 + V, J dx + ν 0 V, n dx. Finally, the identity t n J = ν 0 n implies t E 4 = ν 0 + T n V V t dx J, J n J + div n n J J nτ J + T ε n, J V, J divbj B div J n J, J n J n n, J dx n dx + V, J dx ε n J J d σ τ E 4 BJ d σ + ε B div J dx.

8 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher 8 Li Chen and Michael Dreher The two boundary integrals vanish, due to J = 0 on. Concerning the first integral, we can deduce, after partial integration, that n J l J l n J l n dx = Jl n dx. l n l Summing up, we then find t E = ε ν 0 nbn dx 8T ν 0 ε E + ν 0 Jl ν 0 n dx n τ E 4. l V, n dx For the third term, we bring the constance of the doping profile into play: V, n dx = V, n C 0 dx = ν V n C 0 dσ V n C 0 dx = 0 λ n C 0 dx, V, n dx = λ V dx = α 0 λ V dx α 0 n C 0 dx. λ One easily checks that xln x + x, for x > 0, which implies C0 n C 0 n ln n + C 0, C 0 α 0 λ n C 0 dx α 0C 0 λ E, 0 < α 0 <. T If µ denotes the first positive eigenvalue of on with Neumann boundary conditions, then V L µ V L, νv = 0 on. As a consequence, α 0 λ V dx α 0 λ µ V L = α 0µ E 3. Exploiting Proposition Appendix A., we then find t E 8T ν 0 ε E α 0C 0 λ E α 0 µ E 3 T τ E 4 ε ν 0c,d n dx ε ν n 4 0c,d n 3 dx l Jl n dx, n where the numbers c,d and c,d are as in the theorem.

9 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher The viscous model of quantum hydrodynamics in several dimensions 9 We choose α 0 = C 0 µ λ T +C 0, and obtain t E 8T ν 0 µ C 0 ε E µ λ E + E 3 T + C 0 τ E 4 ε ν 0c,d n dx ε ν n 4 0c,d n 3 dx l This completes the proof of Theorem.. Jl n dx. n Proof. [Proof of Corollary.] We directly have Et exp σte0, which gives us. and.7 immediately. Next, it is easy to check that which then yields y y ln y +, 0 < y <, nt, x C 0 nt, x nt, x ln + C 0, C 0 and.3 follows quickly, as well as.4. The remaining estimates are proved similarly. 4. Existence on the Torus The purpose of this section is to prove Theorem.. To this end, we choose numbers γ with 0 < γ <, and consider a family of parabolic initial value problems t n div J = ν 0 n γ n, J J t J div T n + n V + ε n n n = ν 0 J γ J J n τ, λ V = n C γ x, n0, x = n 0 γx, J0, x = Jγx, 0 4. where t, x R. We assume that the functions C γ, n 0 γ, and Jγ 0 belong to C, satisfy the compatibility condition n 0 γ x C γx dx = 0, and converge to C, n 0, J 0 in Sobolev norms as follows, for γ tends to zero: C γ C in H s, n 0 γ n0 in H s+, J 0 γ J 0 in H s. The system 4. is a fourth order nonlinear parabolic system with third order lower terms. It is standard to show that this problem has a unique solution n γ, J γ, V γ C [0, t γ C [0, t γ C [0, t γ,

10 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher 0 Li Chen and Michael Dreher for some t γ > 0. The solution persists as long as n γ stays positive and n γ, J γ, V γ remain bounded. A proof will be sketched in Lemma Appendix B.. Our approach is as follows: shrink the interval [0, t γ to guarantee some boundedness assumptions on n γ, J γ, V γ ; derive uniform in γ a priori estimates of the solutions n γ, J γ, V γ ; show that t γ can not go to zero for γ 0; prove convergence of a sub-sequence of n γ, J γ, V γ γ for γ 0; study the limit of that sub-sequence. Fix a number δ 0 by the conditions 0 < δ 0 < min x n0 x, max x n0 x < δ0. In the following computations, we always assume that δ 0 n γ t, x δ 0. For a multi-index α N d 0, define n γ,α := α x n γ, J γ,α := α x J γ, V γ,α := α x V γ. Then we obtain t J γ,α ν 0 J γ,α + γ J γ,α + τ J γ,α = div x α Jγ J γ + T n γ,α αx n γ V γ ε α x n γ Bn γ. n γ Multiplying this equation with J γ,α, integrating over, performing partial integration, and taking advantage from the periodic boundary conditions, we find t J γ,α L + ν 0 J γ,α L + γ J γ,α L + τ J γ,α L = J γ,α div x α Jγ J γ dx T div J γ,α n γ,α dx n γ J γ,α x α n γ V γ dx ε J γ,α x α n γ Bn γ dx.

11 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher The viscous model of quantum hydrodynamics in several dimensions The integrals on the right-hand side are treated as follows: J γ,α div x α Jγ J γ dx = k J γ,α,l α Jγ,l J γ,k x dx, n γ k,l n γ T div J γ,α n γ,α dx = T t n γ,α ν 0 n γ,α + γ n γ,α nγ,α dx = T t n γ,α L T ν 0 n γ,α L T γ n γ,α L, n γ Bn γ = n γ l n γ n γ l, n γ l ε J γ,α x α n γ Bn γ dx = ε div J γ,α n γ,α dx ε l J γ,α,k α l n γ k n γ x dx 4 4 k,l n γ = ε t n γ,α ν 0 n γ,α + γ n γ,α n γ,α dx ε... dx 4 4 k,l = ε 8 t n γ,α L ε 4 ν 0 n γ,α L ε 4 γ n γ,α L ε... dx. 4 Then it follows that T t n γ,α L + T ν 0 n γ,α L + T γ n γ,α L 4. + ε 8 t n γ,α L + ε 4 ν 0 n γ,α L + ε 4 γ n γ,α L + t J γ,α L + ν 0 J γ,α L + γ J γ,α L + τ J γ,α L = k J γ,α,l x α Jγ,l J γ,k dx J γ,α x α n γ V γ dx k,l n γ ε l J γ,α,k x α l n γ k n γ dx 4 k,l We define an energy: E k t = E 0,...,k := = I,α + I,α + ε 4 I 3,α. α =k n γ T n γ,α L + ε 8 n γ,α L + J γ,α L, k 0, 4.3 k E l. 4.4 l=0 k,l

12 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher Li Chen and Michael Dreher This energy is related to Sobolev space norms via n γ L T E 0, α =k n γ,α L C ε E k, k, n γ H C k T + ε E 0,...,k. The identity 4. then yields t E k + T ν 0 n γ,α L + ε 4 ν 0 n γ,α L + ν 0 J γ,α L + τ J γ,α L α =k I,α + I,α + ε 4 I 3,α. α =k Next we estimate the integrals I,α, I,α, I 3,α in terms of E 0,...,k. The constants C in the following computations may change from one line to another, and can depend on the order of differentiation k = α, the space dimension d and the lower bound δ 0 of n 0, but are independent of ν 0, γ, ε, τ, and λ. Recall the embedding H b L. We will make free use of the estimates fg H k C f L g H k + f H k f L, k 0, fu H k C u L u H k, k 0, f0 = 0. Then we can conclude that I,α m J γ,α,l L n γ J γ,lj H γ,m α l,m n C J γ,α L γ H α J γ L + n γ L J γ H α J γ L C J γ,α L n γ H α J γ L + J γ H α J γ L C J γ,α L + T E 0,..., α E 0,...,b + E 0,...,b ν 0 4 J γ,α L + C ν 0 E 0,...,b + E 0,...,b + T E 0,..., α. Concerning the second integral, we have I,α J γ,α L n γ V γ H α C J γ,α L nγ L V γ H α + n γ H α V γ L C J γ,α L Vγ H α + + n γ H α V γ H b+ C λ J γ,α L nγ C γ H α + n γ H α n γ C γ H b C λ J γ,α L nγ H α + C γ H α + n γ H α nγ H b + C γ H b

13 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher The viscous model of quantum hydrodynamics in several dimensions 3 C λ J γ,α L + n γ H α + C γ H b + n γ H b + C γ H α C λ E 0,..., α + T E 0,..., α + C γ H b + T E 0,...,b + C γ H α = C λ + T + T C γ H b + T E 0,...,b E 0,..., α + C λ C γ H α. Concerning the last integral, we have I 3,α l J γ,α,k L n γ l n γ k n γ H α k,l k,l l J γ,α,k L l ln n γ k n γ H α C J γ,α L ln nγ H α + n γ L + ln n γ L n γ H α + C J γ,α L n γ H α + n γ L. Fix a number θ d with θ d /, for even d and θ d /3, for odd d. Then we have n γ L C n γ θ d H b+ n γ θ d L, by Lemma Appendix B.3. Together with n γ L δ0, we then get the estimate I 3,α C J γ,α L n γ H α + n γ θ d H b+. Now we distinguish two cases. Case : b + α +. Then we have the interpolation inequalities which imply where we have set ϱ = n γ H b+ C n γ n γ H α + α + b H α + α + b α + b C n γ H α + α + b α + b n γ, H b n γ α + b, H b n γ H α + n γ θ d H C n b+ γ ϱ n H α + γ θ d+ϱ H, b θ d α + b 0,. Altogether, we get I 3,α C J γ,α L n γ ϱ n H α + γ θ d+ϱ H b ϱ C J γ,α L n γ H + n α γ H α If we apply Young s inequality with the exponents, ϱ and ϱ n γ θ d+ϱ H b. to the right-hand

14 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher 4 Li Chen and Michael Dreher side, we deduce that putting k = α ε 4 I ν0 3,α J γ,α L ε ν 0 8Md n γ H k + n γ H k Cν + ϱ 0 ε +ϱ n γ θ d+ϱ H b ν0 ε J ν 0 γ,α L ν 0 8 J γ,α L + ε ν 0 8Md + Cν ϱ 0 ε θ d ϱ 8Md n γ H + n k γ H k Cν + ϱ 0 ε θ θd +ϱ d ε nγ H b n γ H + n k γ H k ε n γ H b θ d +ϱ ϱ Recalling that ϱ = ϱk = θ d k+ b, we see that. ϱ ϱ ε 4 I 3,α ν 0 8 J γ,α L + ε ν 0 8Md n γ H k + Cν 0 + ε T E 0,...,k + Cν ϱ 0 ε k+ b + ε T E 0,...,b + θd ϱ. Case : b + > α +. In this case, we have n γ H α + n γ H b. Applying Young s inequality with the exponents, θ d and θ d to the estimate we get I 3,α C J γ,α L n γ θ d H b+ n γ H b, ε 4 I ν0 θd 3,α J γ,α L ε n γ H Cν b+ 0 ε θ d n γ H b ν 0 8 J γ,α L + C + ε T E 0,...,b + C ν 0 ε θ θ d d ε n γ H b ν 0 8 J γ,α L + C + ε T E 0,...,b θ d + Cν θ d 0 ε + ε T E 0,...,b θ d. Having now the estimates in both cases, we choose the number Md sufficiently large. Then we can conclude that t E k + T ν 0 n γ,α L + ε 8 ν 0 n γ,α L + ν 0 J γ,α L + τ J γ,α L α =k C ν 0 E 0,...,b + E 0,...,b + T E 0,...,k + C λ + T + T C γ H b + T E 0,...,b E 0,...,k + C λ C γ H k + CR,

15 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher The viscous model of quantum hydrodynamics in several dimensions 5 where the remainder term R equals ν 0 + ε T E 0,...,k + ν ϱk 0 ε k+ b + ε T E 0,...,b + θd ϱk in case of k b ; and for k < b we have R = + ε T E 0,...,b + ν θ d 0 ε + ε T E 0,...,b θ d. For simplicity, we only discuss the case s = b. The other cases s > b run similarly. Summing up for k = 0,..., b, we find the energy estimate t E 0,...,b + T ν 0 n γ H + ε b 8 ν 0 n γ H + ν 0 b J γ H + b τ J γ H b C + E 0,...,b + E 0,...,b + + ν0 + ε T ν 0 ν 0 T + λ + λ T + λ T C γ H b + λ T E 0,...,b E 0,...,b + C λ C γ H b + Cν θ d 0 ε + ε T E 0,...,b θ d + Cν +θ d θ d 0 ε + ε T E 0,...,b θ d + Cν 3+θ d θ d 0 ε 4 + ε T +θd θ E 0,...,b d. 4.5 Observe that the right-hand side does not depend on γ. It exists a number t > 0 such that E 0,...,b C [0, t ] and E 0,...,b t E 0,...,b 0 for 0 t t. On the left-hand side, we have a term J γ H b. Shrinking the interval [0, t ] if necessary, we can arrange that t and div J γ L 0,t,H b. By Lemma Appendix B., we can show that n γ t, n γ t, L C, α, ν 0 t t α C t t α, 0 t, t t γ, div J γ L [0,t γ],h b + n 0 H γ b+ where 0 < α <. Note that the right-hand side does not depend on γ, 0 < γ <. From this we can find a lower bound of the time the solution n γ needs to touch the boundary of the interval [δ 0, δ0 ]. We list the results obtained so far: We have determined a number t > 0 with the property that, for each γ with 0 < γ <, we have a solution n γ, J γ, V γ C [0, t ] C [0, t ] C [0, t ], satisfying δ 0 n γ t, x δ 0 for all t, x [0, t ].

16 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher 6 Li Chen and Michael Dreher These functions satisfy the a priori estimates n γ L 0,t,H b+ C, J γ L 0,t,H b C, n γ L 0,t,H b C, J γ L 0,t,H b C. These constants C may depend on T, ν 0, ε, λ, and δ 0, but not on γ. Because the function C γ belongs to H b, we have V γ L 0,t,H b C. From the differential equations, we then obtain the uniform in γ estimates on the time derivatives: t n γ L 0,t,H b 4 + tj γ L 0,t,H b 4 C. The embedding H b+ H b is compact. Therefore, the Aubin Lemma implies that a sub-sequence which we will not relabel of n γ γ converges in the space C[0, t ], H b to a limit function n. The sequence n γ γ is bounded in L 0, t, H b+. By interpolation, we get the strong convergences And we have the weak convergences n γ n in C[0, t ], H b+ δ, δ > 0, n γ n in L 0, t, H b+ δ, δ > 0. n γ n in L 0, t, H b+, By a similar reasoning, we can show n γ n in L 0, t, H b+. J γ J in C[0, t ], H b δ, δ > 0, J γ J in L 0, t, H b+ δ, δ > 0, J γ J J γ J in L 0, t, H b+, in L 0, t, H b. Especially, we have the uniform convergences n γ, n γ, J γ n, n, J in CQ, where we have put Q := 0, t. In particular, n and J satisfy the initial conditions n0, x = n 0 x and J0, x = J 0 x. The convergence of n γ γ yields the convergence of V γ γ, too: V γ V in C[0, t ], H b+ δ, δ > 0. Finally, we show that n, J, V is a solution to.. By the above reasoning, the identity λ V t, x = nt, x Cx, t, x Q,

17 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher The viscous model of quantum hydrodynamics in several dimensions 7 is obvious. Take a function ϕ C0 Q. By the usual arguments, we find ϕt n γ + ν 0 n γ ϕ + γn γ ϕ dx dt = J γ ϕ dx dt. Q Q We send γ to zero and find ϕ t n + ν 0 n ϕ dx dt = J ϕ dx dt, Q Q ϕ t n + ϕ ν 0 n div J dx dt = 0. Q We conclude that the function n has distributional time derivative t n = ν 0 n + div J. We study the terms of the J equation: Jγ J γ J J div div n n γ T n γ T n in CQ, n γ V γ n V in CQ, n γ Bn γ n Bn J γ J in L 0, t, L, in L 0, t, L, in L 0, t, L. Similarly as for n, we can compute the distributional time derivative of J, and we will see that n, J, V solve.. To complete the proof of Theorem., we have to check the uniqueness of the solution: let n, J, V and n, J, V be two solutions with regularity as in.8. Put Then we obtain the system n = n n, J = J J, V = V V. t n ν 0 n = div J, t J ν 0 J + τ J T n + ε n = R R, J j J j R j = div n j n j V j + l n j n j l n j, j =,, λ V = n, with vanishing initial values for n and J. Multiplying the second equation with l

18 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher 8 Li Chen and Michael Dreher J, integrating over, and performing partial integration gives t J L + ν 0 J L + τ J L + T n div J dx ε n div J dx = J R R dx, t J L + ν 0 J L + τ J L + T t n L + T ν 0 n L = + ε t n L + ε 4 ν 0 n L J R R dx. Now it is standard to estimate J R R dx C J j L, n j L, n j L, V j L J L J L + J L n L + V L + J L n H. We apply Young s inequality and find T t n L + ε t n L + t J L C J L + n H. An application of Gronwall s lemma then yields n 0, J 0, which concludes the proof of Theorem.. 5. Existence in One Dimension In this section, we prove Theorem.3. We consider. and its viscous regularization 4.. Put = 0, L. Our goal is to follow the proof of Theorem. with s = b =. Therefore, we choose the approximations C γ, n 0 γ and J γ 0 from the proof of Theorem. in such a way that subject to the boundary conditions C γ C in L, n 0 γ n 0 in H, J 0 γ J 0 in H, j xc γ x = 0, x, j, j xn 0 γx = 0, x, j, x j J γ 0 x = 0, x, j = 0, j. Next, we extend these functions to the interval L, L via C γ x := C γ x, n 0 γ x := n 0 γx, J 0 γ x := J 0 γx,

19 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher The viscous model of quantum hydrodynamics in several dimensions 9 for x 0, L. Then we observe that C γ, n 0 γ, and Jγ 0 satisfy periodic boundary conditions on the interval := L, L. We construe as a torus, and have C γ, n 0 γ, Jγ 0 C. We obtain a fourth order nonlinear parabolic system with third order lower terms. It is standard to show that this problem has a unique and smooth local in time solution n γ, J γ, V γ. The function ñ γ t, x, J γ t, x, Ṽγt, x := n γ t, x, J γ t, x, V γ t, x, defined for t, x [0, t γ ], is a solution, too. This is the step where we use d =. Then the uniqueness of the solution implies n γ t, x = n γ t, x, J γ t, x = J γ t, x, V γ t, x = V γ t, x, for t, x [0, t γ ]. Following the proof of Theorem., we send γ to zero, and have the convergence of a sub-sequence of n γ, J γ, V γ γ to a solution n, J, V of the system. on [0, t ]. Clearly, the functions n and V must be even, and J must be odd, which guarantees the boundary conditions.. The uniqueness can be shown in the same way as for Theorem.. Now the proof of Theorem.3 is complete. 6. Semiclassical Limit Finally, we show Theorem.4. We go back to the proof of Theorem.. Integrating 4.5 over [0, t ] and choosing t small enough, we can arrange that the solutions n ε,γ, J ε,γ, V ε,γ to 4. fulfil the a priori estimate sup E 0,...,b t + t [0,t ] E 0,...,b 0. t t=0 T ν 0 n ε,γ Hb + ε 8 ν 0 n ε,γ H + ν 0 b J ε,γ H dt b The number t only depends on the initial energy E 0,...,b 0, a bound of C γ H b, and the constants ν 0, λ, T, ε 0, where ε 0 is an upper bound of ε, 0 < ε ε 0. The constant t is independent of γ and ε itself. This gives us the possibility to first send γ to zero, and then ε. We start with the uniform in γ and ε estimates n ε,γ C[0,t ],H b + ε n ε,γ L 0,t,H b+ + J ε,γ L 0,t,H b + n ε,γ L 0,t,H b+ + ε n ε,γ L 0,t,H b+ + J ε,γ L 0,t,H b+ C. We know that the limit n ε, J ε, V ε solves. and satisfies the corresponding inequality n ε C[0,t ],H b + ε n ε L 0,t,H b+ + J ε L 0,t,H b + n ε L 0,t,H b+ + ε n ε L 0,t,H b+ + J ε L 0,t,H b+ C.

20 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher 0 Li Chen and Michael Dreher Making use of n 0 H b+ and the maximal regularity property of the parabolic operator t ν 0, we even get n ε L 0,t,H b+ C. Moreover, the functions n ε are bounded from below, n ε t, x δ 0 > 0 for t, x [0, t ]. A careful analysis of the differential equations for n ε and J ε reveals the uniform in ε bounds t n ε L 0,t,H b + tj ε L 0,t,H b C. We can apply Aubin s Lemma Corollary 4 in Ref., and find a converging subsequence n ε, J ε n, J in C[0, t ] and in C[0, t ], H b δ, δ > 0. A direct consequence then is V ε V in C[0, t ], H b. Additionally, we have the weak convergences n ε n in L 0, t, H b+, n ε, J ε n, J in L 0, t, H b. J ε J in L 0, t, H b+, Now fix Q := 0, t and choose a test function ϕ C0 Q. Then we have ϕ t n ε ν 0 ϕ n ε ϕ div J ε dx dt = 0. Q Sending ε to +0 we get n t ν 0 n = div J with distributional derivatives. This equation then gives us t n L 0, t, H b. Next, after choosing a R d -valued test function ϕ C0 Q, we can write ϕ t J ε + J ε J ε ϕ + div ϕt n ε + ϕn ε V ε dx dt Q n ε = ε ϕ n ε + ε l ϕ ln ε n ε + ν 0 ϕ J ε n ε τ ϕj ε dx dt. Observing l l ϕ ln ε n ε n ε l dx dt C ϕ δ L Q n ε L Q, 0 we can send ε to +0, and it follows that J J ϕ t J + ϕ div T n + n V ν 0 J + τ n J dx dt = 0. Q

21 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher The viscous model of quantum hydrodynamics in several dimensions We then deduce that t ν 0 J = R with some R L 0, t, H b L 0, t, H b. By a maximal regularity argument, we then have t J L 0, t, H b. Appendix A. An Estimate of the Bohm Potential The main result of this section is the following estimate. Proposition Appendix A.. Let d =, or 3. Assume that the domain R d is a bounded box, = d j= a j, b j. We assume that a function n H satisfies the following conditions: inf nx > 0, A. x ν nx = 0, x. A. Then we have the estimates from above Bn n dx 5 n dx + 8 n 4 n 3 in all dimensions, and the estimates from below n dx + 7 n n 3 dx : d = 3, Bn n dx 3 n dx + n 4 4 n 3 dx : d =. In case of d =, we have Bn n dx = n dx + n 4 4 n 3 dx. dx A.3 Proof. The estimate from above is a direct consequence of Young s inequality and Bn n = n + n n n. The statement in case of d = follows by partial integration. We start the proof of A.3 with some observations: Due to the embedding H C for d 3, the condition A. is meaningful. Every function n from H satisfying A. and A. can be approximated by a sequence n γ γ of functions n γ H 3 that satisfy ν n γ x = 0 on and inf x n γ x inf x nx. The both sides of A.3 are continuous mappings from the positive cone of H into R. Therefore, we can additionally assume that n H 3. Moreover, we will need the following fact: if p H 3 with ν p = 0 on, then also ν p = 0 on. This is the place where we need the assumption that is a box.

22 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher Li Chen and Michael Dreher Put px := n /4 x. The assumptions A. and A. guarantee p H 3 and ν p = 0 on. Then we have Bn n = n/ n = 8 p p + 3 p 4 + 4p p p. n / The last term is the delicate one, since its sign is unknown. To obtain the estimates from below, we perform partial integration repeatedly: p p p dx = p 4 dx + p p dx = p 4 dx + p p dx + ν p p p ν p dσ = p 4 dx + p k k j p dx + 0 j,k = p 4 dx + p k j p dx + p 3, p dx j,k 3 = p 4 dx + p k j p dx j,k + ν p 3 3 p dσ p 3 p dx 3 = p 4 dx + p k j p dx j,k + 0 p p dx p p p dx. The last integral is the same as on the left-hand side. Plugging the resulting expression into the integral Bn n dx, we then find Bn n dx = 8 4 p j k p p p + 5 p 4 dx. 3 j,k Using Young s inequality ab a + b, we get the estimate 4 j,k j k p p 3 p in case of d = 3, and, consequently, Bn n dx 8 p p + 5 p 4 dx. 9 Now we have which gives p = 4p p + 4 p 4 + 8p p p 8p p + 8 p 4, Bn n dx p dx = n dx p 4 dx n 4 n 3 dx.

23 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher The viscous model of quantum hydrodynamics in several dimensions 3 This proof works also for d =, and the constants can be improved a bit. Appendix B. Some Technicalities Let = T d be a torus, and consider the problem { t u ν 0 u + γ u = f, t, x 0, T, where ν 0 > 0 and γ 0. u0, x = u 0 x, x, Lemma Appendix B.. Any smooth solution to this initial value problem satisfies the estimate ut ut L C, α, ν 0, T t t α f L 0,T,H b + u 0 Hb + γ u0 H b+ uniformly in γ, where 0 t < t T, 0 < α <, and the constant C remains bounded for T 0. Proof. Apply to this equation, multiply with u, and integrate over : t u L + ν 0 l u dx + γ u L = f u dx l d f L ν + ν 0 l u dx. 0 l If denotes the matrix of all second derivatives, then we obtain the estimates u L 0,T,L u 0 L + d f L ν 0,T,L, 0 u u L 0,T,L 0 L ν + d f L 0,T,L. 0 The key information here is that the constants do neither depend on T nor on γ. Apply to the equation, multiply with u, integrate over, perform partial integration, apply Young s inequality to the right hand side, integrate over the time interval: u L 0,T,L + ν 0 u L 0,T,L + γ u L 0,T,L u 0 L + γ f L 0,T,L. Finally, using the equation we get t u L 0,T,L 3ν0 u L 0,T,L + 3 γ u L 0,T,L + 3 f L 0,T,L Cd ν 0 u 0 L + γ u 0 L + f L 0,T,L. ν 0

24 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher 4 Li Chen and Michael Dreher The differential equation and the boundary condition do not change after differentiating the equation with respect to x. Therefore, we are allowed to replace L everywhere by H m in the above four estimates. Assume 0 t < t T. Then we have ut ut L C ut ut H b α C u α L 0,T,H b ut ut α H b C u α L 0,T,H b u α t t=t t u H b dt C t t α L 0,T,H b tu α L 0,T,H b C, α, ν 0, T t t α f L 0,T,H b + u 0 H b + γ u0 H b+. Here, we have made use of the standard estimate u L 0,T,L u 0 L + T f L 0,T,L. Next, we consider fourth-order parabolic systems with nonlocal nonlinear lower order terms { t Y t, x + γ Y t, x = F {Y }, t, x 0, T, α Y 0, x = Y 0 x, x, B. where is a smooth d-dimensional manifold without boundary and F comprises local or nonlocal nonlinear terms of at most third order. Lemma Appendix B.. Assume that Y 0 C, and that F is defined for functions Y C taking values in a tubular neighbourhood of the graph of Y 0, and satisfies estimates F {Y } H k C F,k Y L Y H k+3 + Y L, k 0, F {Y } F {Z} L C F,0 Y L + Y H 3 + Y L + Z H 3 Y Z H 3, with some continuous and increasing functions C F,k. Then there is a constant T > 0 such that B. has a unique solution Y C [0, T ]. Proof. Consider first the linear case t Y t, x + γ Y t, x = Gt, x, Y 0, x = Y 0 x. B. Multiplying B. with Y and integrating the resulting equation over [0, T ] we quickly get Y L 0,T,L Y 0 L + T G L 0,T,L, B.3 Y L 0,T,L T Y 0 L + T G L 0,T,L. B.4

25 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher The viscous model of quantum hydrodynamics in several dimensions 5 Applying to B., multiplying with Y and integrating over, we see t Y L + γ Y = G Y dx L γ Y + L γ G L, Y L 0,T,L + γ Y L 0,T,L Y 0 L + γ G L 0,T,L. B.5 We interpolate B.5 with B.3 and with B.4, and conclude that Y L 0,T,H + Y L 0,T,H 3 C 0 Y 0 H + T / G L 0,T,L, with C 0 = C 0 γ,, T 0, 0 < T T 0, which can be lifted to Y L 0,T,H k+ + Y L 0,T,H k+3 C k Y 0 H k+ + T / G L 0,T,H k. For b being the smallest integer greater than d/, we choose k = maxb, and find an estimate for Y in the spaces L 0, T and L 0, T, H 3. Now let us be given the composition operator F. For a moment we assume F to be defined everywhere. Now it is standard to construct an iteration scheme of Picard-Lindelöf type, to exploit the above estimates of solutions to B., and to show that this iteration scheme converges in L 0, T, H 3 to a solution Y provided that T is chosen small enough. The maximal regularity of the system B. as expressed in B.5 then shows Y C [0, T ]. We can also obtain an estimate of t Y in L 0, T. Next, let us be given the composition operator F, defined in a tubular neighbourhood of the graph of Y 0. We can extend F outside this neighbourhood and then follow the above proof. The estimate on t Y guarantees that the found solution indeed solves the problem B. provided that the time interval is short. For completeness, we give a proof of an interpolation estimate exploited in the proof of Theorem.. Lemma Appendix B.3. Let d N + and b be the smallest integer greater than d, and be a bounded domain in R d with smooth boundary or a bounded d-dimensional manifold. Put { θ d = b + d : d even, + κ = κ + 3 : d odd with κ > 0 and θ d <. Then the following interpolation inequality holds: u L C u θ d H b+ u θ d L.

26 January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher 6 Li Chen and Michael Dreher Proof. Let r be huge. We have H b+ = F, b+ and Lr = Fr, 0 as well as the complex interpolation [ F s p, F ] s, p = F s, θ p, s, = θs + θs, p = θ p + θ p provided that < p, p < and 0 θ. Details on the Triebel-Lizorkin spaces Fp,q s can be found in Ref.. In our case, s = θ d s + θ d s = b + θ d and /p = θ d / + θ d /r. It is easy to check that b + θ d > dθ d /, hence s > d/p for large r, which gives us the embedding F s p, C. Since is bounded, we also have L r L, which completes the proof. Acknowledgements Both authors have been supported by the Deutsche Forschungsgemeinschaft DFG, grant number 446 CHV 3/70, and the National Natural Science Foundation of China NSFC, grant number We express our gratitude to Tsinghua University of Beijing and Konstanz University for hospitality; and thank Robert Denk for helpful discussions. References. J. Carrillo, A. Jüngel, P. Markowich, G. Toscani, and A. Unterreiter. Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatsh. Math., 33: 8, 00.. L. Chen, X. Chen, and H. Jian. Isentropic quantum drift diffusion model. preprint, L. Chen and Q. Ju. Existence of weak solution and semiclassical limit for quantum drift diffusion model. to appear in ZAMP, P. Degond, F. Méhats, and C. Ringhofer. Quantum energy transport and drift diffusion models. J. Stat. Phys., 8:65 667, C. L. Gardner. The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math., 54:409 47, M. Gualdani, A. Jüngel, and G. Toscani. Exponential decay in time of solutions of the viscous quantum hydrodynamic equations. Appl. Math. Lett., 68:73 78, A. Jüngel. Quasi-hydrodynamic semiconductor equations. Progress in Nonlinear Differential Equations and their Applications. 4. Basel: Birkhäuser., A. Jüngel and H.-L. Li. Quantum Euler Poisson systems: global existence and exponential decay. Quart. Appl. Math., 6: , A. Jüngel and D. Matthes. A derivation of the isothermal quantum hydrodynamic equations using entropy minimization. Z. Angew. Math. Mech., 85:806 84, A. Jüngel and R. Pinnau. A positivity-preserving numerical scheme for a nonlinear fourth order parabolic system. SIAM J. Num. Anal., 39: , 00.. J. Simon. Compact sets in the space L p 0, T ; B. Ann. Mat. Pura Appl., IV. Ser., 46:65 96, H. Triebel. Theory of function spaces II. Monographs in Mathematics. 84. Basel etc.: Birkhäuser Verlag., B. Zhang and J. Jerome. On a steady-state quantum hydrodynamic model for semiconductors. Nonlin. Anal., 6: , 996.