GLOBAL EXISTENCE OF SOLUTIONS TO ONE-DIMENSIONAL VISCOUS QUANTUM HYDRODYNAMIC EQUATIONS

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1 GLOBAL EXISENCE OF SOLUIONS O ONE-DIMENSIONAL VISCOUS QUANUM HYDRODYNAMIC EQUAIONS IRENE M. GAMBA, ANSGAR JÜNGEL, AND ALEXIS VASSEUR Abstract. he existence of global-in-time weak solutions to the one-dimensional viscous quantum hydrodynamic equations is proved. he model consists of the conservation laws for the particle density and particle current density, including quantum corrections from the Bohm potential and viscous stabilizations arising from quantum Fokker-Planck interaction terms in the Wigner equation. he model equations are coupled self-consistently to the Poisson equation for the electric potential and are supplemented with periodic boundary and initial conditions. When a diffusion term linearly proportional to the velocity is introduced in the momentum equation, the positivity of the particle density is proved. his term, which introduces a strong regularizing effect, may be viewed as a classical conservative friction term due to particle interactions with the background temperature. Without this regularizing viscous term, only the nonnegativity of the density can be shown. he existence proof relies on the Faedo-Galerkin method together with a priori estimates from the energy functional. 1. Introduction Diffusive corrections in quantum models are of great importance in open quantum systems modeling, for instance, an electron ensemble interacting with a background heat bath. Applications of such systems include quantum semiconductor structures in which non-classical diffusive effects may be relevant in some regimes. Caldeira and Leggett [3] and Diósi [8] have derived closed equations for a dissipative quantum-mechanical system related to quantum Brownian motion. heir approach was later improved by Castella et al. [4] and leads to a Wigner equation with Fokker-Planck-type operator modeling interactions that may take into account basic quantum and classical mechanisms. hus, one may interpret the Wigner-Fokker-Plank equation as a quantum Liouville equation equated to an interaction operator of quantum Fokker-Planck type. Mathematics Subject Classification. 35G5,35Q4,76Y5. Key words and phrases. Viscous quantum hydrodynamics, global existence of solutions, quantum diffusion, semiconductors, Faedo-Galerkin method. I.M. Gamba was partially supported by the NSF grant DMS A. Jüngel acknowledges partial support from the Austrian Science Fund FWF, grant P14 and WK Differential Equations, the German Science Foundation DFG, grant JU 359/7, the Austrian-Croatian Project of the Austrian Exchange Service ÖAD, and the bilateral PROCOPE Project of the German Academic Exchange Service DAAD. his research is part of the ESF program Global and geometrical aspects of nonlinear partial differential equations GLOBAL. A. Vasseur was partially supported by the NSF grant DMS Support from the Institute of Computational Engineering and Sciences and the University of exas Austin is also gratefully acknowledged. 1

2 I. GAMBA, A. JÜNGEL, AND A. VASSEUR Motivated by multi-scale modeling and by the fact that computational approaches for the Wigner-Fokker-Plank equation are expensive, due mainly to its high dimensionality see for example [11] and references therein, associated macroscopic models were derived in an effort to produce asymptotically correct macroscopic reductions. For instance, employing a moment method and a suitable closure condition to the classical Wigner equation in the absence of interactions, quantum hydrodynamic equations are obtained [7, 15]. Another derivation comes from the mixed-state Schrödinger system via the Madelung transform [16]. When particle interactions are taken into account, a non-classical quantum Fokker-Planck interaction operator balances the Wigner equation, with classical and non-classical second-order derivative terms that may be interpreted as viscous terms appearing in the macroscopic model. We refer to [17, 1, 3] for a derivation of a non-classical viscous perturbation due to quantum interactions and to [13, 14] for an analysis of stationary quantum-regularized models with a classical mass-conservative viscous effect. his leads to a broad class of viscous quantum hydrodynamic equations, which are the subject of this paper. he scaled viscous quantum hydrodynamic equations in one space dimension for the particle density ρ, the velocity u, and the electric potential V read as follows: 1 3 ρ t + ρu x = νρ xx, ρu t + ρu + pρ x ρv x δ ρ ρxx ρ x = νρu xx + εu xx ρu τ, λ V xx = ρ Cx, he pressure pρ is assumed to depend on the particle density; typical examples are pρ = p ρ α for some p > and α 1. he function Cx is the doping profile modeling charged background ions in, for instance, semiconductor crystals. he viscosity ν > is related to effects depending also on the scaled Planck constant δ > through the well-known Lindblad condition. his condition guarantees the quantum mechanically correct evolution of the system and the convergence to the classical Fokker-Planck dynamics from stochastic calculus as δ see [3, 8, 7]. he parameter τ > models momentum relaxation time due to classical friction mechanisms, and the parameter ε accounts for classical mass-conservative viscous effects due to classical particle-particle and particle-lattice interactions. Finally, λ > is the scaled Debye length of the device. Equations 1-3 are considered on the one-dimensional torus with size one and are complemented with the initial conditions 4 ρ, = ρ, ρu, = ρ u in. Equation 1 expresses a mass balance law that becomes mass conservative with respect to the effective current density J = ρu νρ x. he second equation is the classical balance equation for the particle current density or momentum ρu including the electric force term ρv x, the relaxation-time term ρu/τ, and the quantum correction with the Bohm potential ρ xx / ρ. he electric potential V is self-consistently given by the Poisson equation 3. In the absence of viscous and quantum effects, i.e. ν = ε = δ =, the above equations represent the hydrodynamic semiconductor equations []. When no viscous effects are present, ν = ε =, we obtain the quantum hydrodynamic equations, studied in, e.g., [13, 14, ]. For more recent papers, we refer to [1, 19, 6, 8, 9].

3 VISCOUS QUANUM HYDRODYNAMIC EQUAIONS 3 he viscous quantum hydrodynamic model for ε = can be derived from the Wigner-Fokker-Planck equation by a moment method [17, 3]. he viscous regularizations arise from the quantum Fokker-Planck interaction operator. More precisely, the part of the scattering operator yielding the non-classical viscous terms is proportional to Q QFP w = νw xx, where wx,k,t is the Wigner function on the position-wave vector space x,k, and ν > depends on the quantum friction. Introducing the moments ρ = wdk R 3 and ρu = wkdk gives R 3 Q QFP wdk = νρ xx, Q QFP wkdk = νρu xx, R 3 R 3 which are the non-classical viscous terms in the quantum fluid system 1-, respectively. In this view, they are not artificial regularizations, but coming from the choice of the quantum interaction operator of Fokker-Plank type in the Wigner equation. he classical diffusive velocity term proportional to ε > can be obtained from a part of the rescaled first moment in the quantum interaction operator Q rqfp w = ε x w, wdk R 3 since Q rqfp wdk = ε w x R 3 R wdk R dk =, 3 3 R3 Q rqfp wk dk = ε wk x R wdk R dk = ε ρu x ρ = εu xx. 3 3 here are only few mathematical results for these viscous quantum hydrodynamic model due to difficulties coming from the third-order derivatives in the quantum correction. he existence of classical solutions to the one-dimensional stationary model with ε = and with physical boundary conditions was shown in [3]. he transient equations are considered in [5, 6, 9], and the local-in-time existence and exponential stability of solutions were proved. Global-in-time solutions in one space dimension are obtained if the initial energy is assumed to be sufficiently small [5]. In [3, 4], numerical solutions of the model and applications to resonant tunneling diodes were presented. We also mention that in the inviscid case ν = ε = there is a recent proof of non-global-in-time existence for a quantum hydrodynamic equation corresponding to a reduced model in the absence of the coupling with the Poisson equation in bounded domains with prescribed data corresponding to high boundary and initial energy [1]. However, no global-in-time existence result without smallness conditions seems to be available up to now for the transient system 1-3. In this paper, we prove such a result, first for the full system with ε > and then, by passing to the limit ε, we obtain a global existence result for the non-classical viscous quantum hydrodynamic model 1-3 with ε =. he main problem of the existence analysis lies in the strongly nonlinear thirdorder differential operator and the dispersive structure of the momentum equation. here are several attempts in the literature to deal with the quantum term. Integrating the stationary momentum equation leads to a second-order differential

4 4 I. GAMBA, A. JÜNGEL, AND A. VASSEUR equation to which maximum principle arguments can be applied [13]. A fourthorder wave equation is obtained after differentiating the equation with respect to the spatial variable. his approach was employed in [5] to prove the existence of global solutions to the quantum hydrodynamic equations with ν =, but only for initial data close to thermal equilibrium. he main idea of [5] was to introduce a bi-laplacian regularization in the viscous model and to employ energy estimates to conclude local existence of solutions. Global existence of solutions to the inviscid model ν = with nonnegative particle density was achieved recently by a wave function polar decomposition method [1]. In this paper, we pursue a different strategy. We employ the Faedo-Galerkin method, introduced by Feireisl in [1] for the analysis of the classical compressible Navier-Stokes equations, applied to 1-3 for ε > with the initial conditions 4. he existence proof relies on the following ideas. First, for given u in a finitedimensional Galerkin space, we solve 1. Since u is given and 1 is parabolic for ε >, a lower bound for the particle density can be concluded from the maximum principle. Classically, this bound depends on the L norm of u x which is prohibitive to set up the fixed-point argument. We prove that the lower bound for ρ depends only on the L norm of u x Lemma 3. In the second step we solve the Poisson equation and then the momentum equation in the Galerkin space, for given ρ, which yields the existence of local-in-time solutions via Banach s fixed-point theorem. A priori bounds and thus global-in-time existence are obtained from the energy inequality defined as follows. Let the enthalpy function h be defined by h y = p y/y for y > and h1 =, and let H be a primitive of h. Furthermore, let the energy, consisting of the internal, kinetic, electric, and quantum energy, be given by 5 Eρ,u = hen we show that 6 Hρ + 1 ρu + λ V x + δ ρ x dx. de dt + ν ρu x + δ ρ xx dx + ε u xdx K, where the constant K > depends only on Cx, ν, and λ. his yields H estimates for ρ and, for fixed ε >, L estimates for u x, needed in the proof of the lower bound for ρ. We consider the one-dimensional equations since we need several times in the proof the embedding H 1 L which is valid in one space dimension only. We comment on the multi-dimensional situation in Remark 6. Our first main result reads as follows. heorem 1. Let >, ε >, and C L. Let the pressure function p C 1 [, be monotone and let the primitive H of the enthalpy satisfy Hy h y 1 for all y and for some h >. Furthermore, let the initial datum ρ,u H 1 L satisfy ρ xdx = Cxdx, ρx η > for x and for some η >, and Eρ,u <. hen there exists a constant η > and

5 VISCOUS QUANUM HYDRODYNAMIC EQUAIONS 5 a weak solution ρ, u, V to 1-3 satisfying ρt,x η > for t >, x, V L,;H, ρ t L,;L, ρu t L,;H, ρ L,;H 1 L,;H, u L,;H 1 L,;L, where the lower bound η > depends on ε. he initial conditions 4 are satisfied in the sense of H. he condition Hy h y 1 is satisfied, for instance, if the pressure is given by pρ = p ρ α, where p > and α >. he regularity of ρ and u implies that ρu L,;H 1 and ρu L,;W 1,1. Let ρ ε,u ε be a solution to 1-3 in the sense of the above theorem. In the limit ε we loose the lower bound for ρ ε since it depends on ε. Furthermore, it is not clear how to pass to the limit in ρ ε u ε, since we have only weak convergence of ρ ε u ε in L. Moreover, we loose the control on u ε and obtain results for the current density J = lim ε ρ ε u ε only. In order to overcome these difficulties, we multiply the momentum equation by ρ 3/ ε and pass to the limit ε in the resulting equation. his allows us to control the convective part since ρ 3/ ε ρ ε u ε x = ρε ρ ε u ε x 3 ρ ε x ρ ε u ε. Our second main result is summarized in the following theorem. heorem. Let > and let the assumptions of heorem 1 hold. hen, for ε = there exists a weak solution ρ, J, V to 1-3 with the regularity ρt,x for t >, x, V L,;H, ρ t L,;L, ρ 3/ J t L,;H 1, ρ L,;H 1 L,;H, J L,;H 1, satisfying ρ t + J x = νρ xx and λ V xx = ρ Cx almost everywhere in, and, for all φ L,; H 1, 7 ρ 3/ J t,φ H 1,H 1dt 3 ρρt Jφdxdt J 3 ρ x φ + ρφ x dxdt + pρx ρv x ρ 3/ φdxdt + δ = ν ρ xx 5ρ 3/ ρ x φ + ρ φ x dxdt J x ρ 3 ρ x φ + ρφ x dxdt 1 τ he initial conditions are fulfilled in the following sense: ρ 3/ Jφdxdt. ρ, = ρ in L, ρ 3/ J, = ρ 5/ u in H 1.

6 6 I. GAMBA, A. JÜNGEL, AND A. VASSEUR Equation 7 is the weak formulation of ρ 3/ J t ρ 3/ t J + ρj x 3J ρ x δ ρ ρ xx x + 5δ 8 ρ x ρ xx νρ 3/ J x x + νj x ρ 3/ x + ρ 3/ pρ x ρv x + J =, τ which is obtained from after multiplication of ρ 3/ and setting J = ρu. If the limit density ρ is positive and smooth, we can divide the above equation by ρ 3/ and recover the original formulation. he paper is organized as follows. In the next section, we solve, for ε > and given velocity u, equation 1 for ρ, prove a lower bound for ρ only depending on the L norm of u x, and solve locally in time. In section 3 we show the energy estimates for 5 and infer a global existence result for the nonlinear Faedo-Galerkin problem. heorem 1 is proved in section 4, whereas section 5 is concerned with the proof of heorem. We remark that the a priori estimates derived from the energy functional 5 and its corresponding energy production were already employed in [5, 1, 18].. Linear Faedo-Galerkin approximation In this section, we prove the existence of solutions to the linearized viscous quantum hydrodynamic equations with ε >. Let > and let e n n N be an orthonormal basis of L which is also an orthogonal basis of H 1. For instance, one may take the eigenfunctions of x with eigenvalues µ n >, given by e n x = cosnπx, µ n = 8n π, e n+1 x = sinnπx, µ n+1 = 8n π, n N. Introduce the finite-dimensional space X n = spane,...,e n. We denote by C k, ;Z the space of C k functions on [,] with values in the Banach space Z. Furthermore, let ρ,u C be some initial data satisfying ρ x η > for x and ρ dx = Cxdx. Finally, let v C,;X n be given. We notice that v can be written as n vt,x = λ i te i x, t [,], x, for some λ i t, and we have i=1 v C,;X n = max t [,] i=1 n λ i t. As a consequence, v can be bounded in C,;C k for any k N, and there exists a constant K k > depending on k such that 8 9 v C,;C k K k v C,;L. Now, we define the approximate system. Let ρ be the classical solution to ρ t + ρv x = νρ xx, x, t >, ρ,x = ρ x, x.

7 VISCOUS QUANUM HYDRODYNAMIC EQUAIONS 7 he solution satisfies ρ C,;C k for any k N. Furthermore, it holds ρdx = ρ dx = Cxdx. We introduce the operator S : C,;X n C,;C 3 by Sv = ρ. Since v is smooth, the maximum principle shows that ρ = Sv is bounded from above and below, i.e., for v C,;L c, there exist positive constants K c and K 1 c depending on c such that 1 < K c Svt,x K 1 c, t [,], x. Furthermore, since the equation for ρ is linear, there exists K > depending on k and n such that for all v 1, v C,;X n, 11 Sv 1 Sv C,;C k K v 1 v C,;L. We claim that the lower bound for Sv only depends on the L,;L norm of v x, 1 ρ = Sv η = η v x L,;L > in [,]. his result is a consequence of the following lemma whose proof is presented at the end of this section. Lemma 3. Let > and v L,;H 1. Let ρ be the solution to 8-9 with initial datum ρ L satisfying ρ x η > for x. hen there exists a constant η > only depending on ν, ρ, and the L,;L norm of v x such that ρt,x η >, t [,], x. Next, for given ρ = Sv, we wish to solve the following linear problem on X n for u n : ρu n t + ρvu n + pρ x ρv [ρ] x δ ρ ρxx 13 ρ x = νρu n xx + εu n xx ρu n τ, where V [ρ] C,;C is the unique solution to 14 λ V [ρ] xx = ρ Cx in satisfying V dx =. More explicitly, we are looking for a function u n C,; X n verifying, for all test functions φ C 1,;X n with φ, =, ρu n φ t dx + ρvun + pρ φ x dx + ρv [ρ] x φdx δ ρ xx ρφ x dx ρ νρun + εu n φ x xdx 1 ρu n φdx = ρ u φ, dx. τ For given ρ N η = {ρ L 1 : inf x ρ η > }, we introduce the following family of operators, following [1]: M[ρ] : X n Xn, M[ρ]u,w = ρuwdx, u,w X n. hese operators are symmetric and positive definite with the smallest eigenvalue inf M[ρ]w,w = inf ρw dx inf ρx > η, w L =1 w L =1 x

8 8 I. GAMBA, A. JÜNGEL, AND A. VASSEUR employing the bound 1. Hence, as we are working in finite dimensions, the operators are invertible with M 1 [ρ] LX n,x n η 1, where LX n,x n is the set of bounded linear mappings from X n to X n. Moreover, similar as in [1], it holds: 15 M 1 [ρ 1 ] M 1 [ρ ] LX n,x n Kn,η ρ 1 ρ L 1 for all ρ 1, ρ N η. With these notations, we can rephrase problem 13 as an ordinary differential equation on the finite-dimensional space X n : d 16 M[ρt]un t = N[v,u n t], t >, M[ρ ]u n = M[ρ ]u, dt where N[v,u n ],φ = ρvu n + pρ + δ ρxx x + ρv [ρ] x ρ x + νρu n xx + εu n xx 1 τ ρu n φdx, φ X n. Recall that ρ = Sv C,;C 3 is bounded from below, so the above integral is well defined. he operator N[v, ], defined for every t [,] as an operator from X n to X n, is continuous in time. hen, standard theory of finitedimensional systems of differential equations provides the existence of a unique C 1 solution of 16. In other words, there exists a unique solution u n C 1,;X n to 13. Proof of Lemma 3. We introduce the function Lt,x = ln 1 t t,x + ρ which is a solution to where Since we obtain vs, ydyds, L t νl xx = v x ṽl x νl x, ṽ = v vdx. 1 ṽl x = ν ṽ νl x ṽ 4ν + νl x, L t νl xx v x + ṽ 4ν. he idea is to show an upper bound for L which only depend on η, ν, and the L -norm of v x and from which the lower bound for ρ follows. his is achieved by estimating the solution ψ to a certain parabolic problem and using the comparison principle to obtain L ψ. We introduce the functions ψ 1, which is a solution to ψ 1 t νψ 1 xx = v x, x, t >, ψ 1,x =, x,

9 VISCOUS QUANUM HYDRODYNAMIC EQUAIONS 9 and ψ, which solves ψ t νψ xx = ṽ, x, t >, 4ν 1 ψ 1,x = L,x = ln ρ x, x. First, notice that ṽ x = v x and ṽ dx =. Hence, by the Poincaré inequality in one space dimension, ṽ L1,;L = his shows that ṽ L dt ṽ x L dt = v x L,;L. 17 ψ t,x 1 4ν v x L L + ln 1 η, t >,x. Multiplying the equation for ψ 1 by ψ 1 xx and integrating over, we find that 1 d ψ 1 dt xdx + ν ψ 1 xxdx = v x ψ 1 xx dx 1 v 4ν xdx + ν ψ 1 xxdx, from which we conclude that ψ 1 x L,;L 1 ν v x L,;L. Finally, for any t >, the integral of ψ 1 t, over vanishes, and an application of the Poincaré inequality then gives 18 ψ 1 L,;L 1 ν v x L,;L. Consider now the sum ψ = ψ 1 + ψ which is a solution to ψ t νψ xx = v x + ṽ, x, t >, 4ν 1 ψ,x = L,x = ln ρ x, x. hen, by the comparison principle, L ψ in [,], and, together with 18 and 17, we obtain for any t > and x : Lt,x 1 ν v x L,;L + ln 1 η. his leads to ρt,x η exp 1 ν v x L,;L, for every x, t >.

10 1 I. GAMBA, A. JÜNGEL, AND A. VASSEUR 3. Solution of the nonlinear approximate problem In this section, we show that there exists a solution to the system 8-9 and 13 on. More precisely, we prove the following result. Proposition 4. Let the assumptions of heorem 1 hold and let the initial data be smooth with positive particle density. hen there exists a solution ρ,u n C,;C 3 C 1,;X n to 8-9 and 13, with v = u n and ρ = ρ n = Su n, satisfying the following estimates: ρ n t,x ηε >, t [,], x, ρ n L,;H 1 + ρ n L,;H K, ρ n u n L,;L + ρ n u n x L,;L K, ε u n x L,;L K, V [ρ n ] L,;H 1 K, where ηε > depends on ε, the initial data and the L norm of Cx, and K > only depends on ν, λ, the initial data, and Cx. he potential V [ρ n ] is defined by 14 with ρ = ρ n. Proof. Integrating 16 over,t, we can write the problem as the following nonlinear equation: t u n t = M 1 [Su n t] M[ρ ]u + N[u n,u n s]ds in X n. aking into account 11 and 15, this equation can be solved with the fixed-point theorem of Banach, at least on a short time interval [, ], where, in the space C, ;X n. In fact, we obtain even u n C 1, ;X n. We have to show that we can choose =. It is sufficient to prove that u n is bounded in X n on the whole interval [, ]. his is achieved by employing the energy estimate. We multiply 8 by φ = hρ n V [ρ n ] u n/ δ / ρ n xx / ρ n, use the test function u n in 13, with v = u n and ρ = ρ n, and add both equations. his leads to u n = ρ n t hρ n ρ n t + ρ nu n t u n dx + ρn t V [ρ n ] ρ n u n x V [ρ n ] + νρ n xx V [ρ n ] ρ n V [ρ n ] x u n dx + ρn u n x hρ n + pρ n x u n dx δ ρ n xx ρn xx ρ n xx ρ n u n x + ρ n u n + ρ n t dx ρn ρn x ρn + 1 ρ nu n x u n + ρ n u n x u n dx + ν ρ n xx hρ n + 1 ρ n xx u n + δ ρ ρn xx n xx ρ n u n xx u n dx ρn ε u n u n xx dx + 1 ρ n u τ ndx = I I 8.

11 VISCOUS QUANUM HYDRODYNAMIC EQUAIONS 11 Notice that at this point, we need a pointwise solution to 8 such that this equation can be multiplied by φ. If 8 was solved in a Galerkin space only, we could not use φ as a test function since it is not admissible. On the other hand, u n is an admissible test function for the Galerkin equation 13. We estimate the above expression integral by integral. he first integral can be reformulated as I 1 = t Hρ + 1 ρ nu n dx, where we recall that H is a primitive of h. he second integral becomes, after integrating by parts and employing the Poisson equation, I = λ V [ρ n ] xxt V [ρ n ] + νρ n V [ρ n ] xx dx λ = tv [ρ n ] x + νλ ρ n ρ n Cx dx. Integrating by parts in the first member of the third integral, we see that I 3 = ρn u n h ρ n ρ n x + p ρ n ρ n x u n dx =, since, by definition, p ρ n = ρ n h ρ n. Again by integrating by parts, the fourth integral simplifies to I 4 = δ ρ n t ρ n xx dx = δ t ρ n xdx. he fifth integral vanishes since, in view of the periodic boundary conditions, I 5 = 1 ρ n u 3 n x dx =. Integrating by parts in the sixth integral gives I 6 = ν h ρ n ρ n x ρ n x u n u n x + δ ρ n xx + δ 3 + ρ n u n x u n x dx = ν Gρ n x + ρ n u n x + δ ρ n xx δ 4 ρ n 4 x where G y = h y, y. Summarizing, we obtain t Hρ n + 1 ρ nu n + λ V [ρ n] x + δ ρ n x 4 + ν + ε dx ρ n 3 x x ρn dx, Gρ n x + ρ n u n x + δ ρ n xx δ 4 ρ n 4 x dx n u xdx + 1 ρ n u τ ndx Cx dx. = νλ ρ n ρ n Cx ν 4λ From this estimate and the assumption Hy h y 1, the uniform bounds -3 follow. Using 1 and, we infer the lower bound 19. hen, the estimate 1 shows that u n is bounded in L,;L with a bound which

12 1 I. GAMBA, A. JÜNGEL, AND A. VASSEUR depends on ε. ogether with the Lipschitz estimates 11 and 15, this allows us to apply the fixed-point theorem recursively until =. We end this section by proving some estimates uniform in n and ε. Lemma 5. he following estimates holds: t ρ n L,;L + ρ n L 6,;W 1,6 K, ρ n L,;H 1 + ρ n L,;H K, t ρ n u n L,;H K, ρ α n t ρ n u n L,;H 1 K, for all α 1/, where K > is independent of n and ε. Proof. By the Galiardo-Nirenberg inequality with θ = 1/3, we have 9 ρ n x 6 L 6,;L 6 K ρ n x 6θ H 1 ρ n x 61 θ K ρ n 4 L,;H 1 L dt ρ n H dt K, taking into account the bound. his shows that ρ n is bounded in L 6,; W 1,6. he function ρ n solves 8-9, with v = u n, written as t ρ n = ρ n ρn u n x ρ n u n ρ n x + ν ρ n ρ n xx + ν ρ n x. In view of, 1, and 9, we infer that t ρ n L,;L. Furthermore, by 9, ρ n xx = ρ n ρ n xx + ρ n x is bounded in L,;L. We claim that t ρ n u n is bounded in L,;H. We have to verify that all terms in 13, with ρ = ρ n and v = u n, except t ρ n u n lie in this space. his is clear for the terms pρ n x, ρ n V [ρ n ] x, εu n xx, and ρ n u n /τ. Furthermore, ρ n u n = ρ n u n is bounded in L,;L 1, by 1, such that ρ n u n x is bounded in L,;W 1,1 L,;H ; νρ n u n xx = ν ρ n u n ρ n x + ρ n u n x x is bounded in L,;H 1 ; and ρn xx ρ n ρn x = ρn ρ n xx x ρ n x ρ n xx is bounded in L,;H 1. his shows the claim. Next, let α 1/. We want to show that ρ α n t ρ n u n is bounded in L,; H 1. he term ρ α nρ n u n x = ρ α 1/ n ρ n x ρ n u n + ρ α n ρn u n ρn u n x is bounded in L,;L 1 and hence also in L,;H 1. Notice that we have used here that α 1/. If α only, the bound depends on ε through the lower bound of ρ n. Furthermore, ρ α npρ n x, ρ α+1 V [ρ n ] x, and ρ α+1 u n /τ are bounded in L,;L. he first term of ρ α+1 ρn xx n = ρ α+1/ n ρ n xx ρn x x α + 1ρα n ρ n x ρ n xx

13 VISCOUS QUANUM HYDRODYNAMIC EQUAIONS 13 is bounded in L,;H 1, the second term in L,;L 1, so the sum is bounded in L,;H 1. Similarly, the sequences ερ α nu n xx = ρ α nεu n x n ρ n x εu n x, ρ α nρ n u n xx = ρ α+1/ n x αρα 1/ ρn u n x + ρ α 1/ n ρ n xx ρn u n x α 1ρα n ρ n x ρn u n x are bounded in L,;H Proof of heorem 1 In this section, we perform the limit n, for fixed ε >, in the system 8-9, 13, and 14, with ρ = ρ n and v = u n. In view of 5, 6, and the compactness of the embeddings H 1 L and H H 1, the Aubin lemma provides the existence of a subsequence of ρ n not relabeled such that, as n, ρ n ρ ρ n ρ t ρ n t ρ strongly in L,;H 1 and L,;L, weakly in L,;H, weakly in L,;L. Since ρ n is bounded from below, ρ n x converges weakly up to a subsequence to ρ x in L,;L. Moreover, since ε > is fixed, by 1 and, u n converges weakly to a function u in L,;H 1. hese results show that t ρ n + ρ n u n x νρ n xx t ρ + ρu x νρ xx weakly in L 1,;L and that pρ n x δ ρ ρn xx n νρ n u n xx εu n xx + 1 ρn x τ ρ nu n converges weakly in L 1,;H 1 to pρ x δ ρ ρxx νρu xx εu xx + 1 ρ x τ ρu. We also have V [ρ n ] V [ρ] in L,;H. In order to pass to the limit in the convection term, we observe first that ρ n u n ρu weakly* in L,;L since ρ n converges strongly in L,;L and u n converges weakly* in L,;L. On the other hand, taking into account 7 and the bound for ρ n u n in L,;H 1, Aubin s lemma implies that ρ n u n ρu strongly in L,;L. hus, ρ n u n ρu weakly* in L 1,;L. hus, passing to the limit n in 13, with ρ = ρ n and v = u n, shows that ρ,u,v [ρ] is a solution to 1-3 for ε >. his finishes the proof of heorem 1.

14 14 I. GAMBA, A. JÜNGEL, AND A. VASSEUR Remark 6. he construction of approximate solutions of section can be generalized to the multi-dimensional quantum hydrodynamic equations 3 ρ t + divρu = ν ρ, 31 ρu t + divρu u + pρ ρ V δ n ρ n 3 = ν ρu + ε u ρu τ, λ V = ρ Cx, x d. Indeed, let X n be a finite-dimensional space defined, for instance, by the span of the first n+1 eigenfunctions of on L d. Further, let ρ n be the classical solution to 3 with u replaced by some given function v C,;X n, and let V [ρ n ] be the unique solution to 3. By the maximum principle, ρ n is strictly positive with a bound depending on the L norm of divv. Finally, we can define u n to be the solution to 31, projected on X n, in the sense of section. he nonlinear finitedimensional problem then is solved by employing Banach s fixed-point theorem, giving a local-in-time solution u n C, ;X n on the time interval [, ]. he energy estimate 4 can also be generalized to the multi-dimensional problem see, e.g., [5]: t Hρ n + 1 ρ n u n + λ V [ρ n] + δ ρ n dx d + ν Gρn + ρ n u n + δ ρ n log ρ n dx d + ε u n dx + 1 ρ n u n dx τ d d ν 4λ d Cx dx, where denotes the l norm of a matrix and log ρ n the Hessian of log ρ n. By the estimate 1.3 of [] also see Proposition A.1 in [5], ρ n log ρ n dx c ρ n dx, d d for some constant c >, which provides a uniform L, ;H d estimate for ρn. Moreover, ρ n is uniformly bounded in L, ;H 1 d. Provided that the lower bound for ρ n is independent of n and only depends on the L, ;L d norm of u n, it is possible to perform the limit n. he most difficult parts are the limits in the third-order expression and the convective term. Similarly as in the one-dimensional case, we can prove uniform bounds for ρ n and ρ n u n which enable us to apply Aubin s lemma, thus providing the strong convergence of these sequences and allowing us to pass to the limit n. he problem, however, is to prove the lower bound for ρ n only depending on the L, ;L d norm of u n. Indeed, the proof of Lemma 3 relies on certain L estimates for ψ i and Sobolev embeddings which are only valid in one space dimension, and we are not able to extend them to the multi-dimensional situation. 5. Proof of heorem Let ρ ε,u ε,v [ρ ε ] be the solution to 1-3, for ε >, constructed in the previous section. In this section, we will perform the limit ε.

15 VISCOUS QUANUM HYDRODYNAMIC EQUAIONS 15 By the ε-independent estimates 5 and 6, the Aubin lemma gives the existence of a subsequence again not relabeled such that ρ ε ρ ρ ε ρ t ρ ε t ρ strongly in L,;H 1 and in L,;L, weakly in L,;H, weakly in L,;L. Furthermore, by, up to a subsequence, ρε ρ weakly* in L,;H 1 and weakly in L,;H. By 1, ρ ε u ε x = ρ ε x ρε u ε + ρ ε ρε u ε x is bounded in L,;L, and hence, ρ ε u ε is bounded in L,;H 1. herefore, ρ ε u ε J weakly in L,;H 1. hus, by 5 and 6, letting ε in the mass conservation equation 8 with ρ = ρ ε and v = u ε yields ρ t + J x = νρ xx in L,;L. In order to let ε in the momentum equation, we need to multiply 13 by ρ 3/ ε. he reason is that we cannot control u ε but only ρ ε u ε which makes it difficult to pass to the limit in ρ ε u ε x. he L,;H 1 bound for ρ ε u ε together with 7 implies that, by Aubin s lemma, ρ ε u ε J strongly in L,;L. hus, for any test function φ L,;H 1, as ε, ρ 3/ ε ρ ε u ε x φdx = 3 ρε x φ + ρ ε φ x ρε u ε dx 3 ρx φ + ρφ x J dx. By 8, ρ 3/ ε ρ ε u ε t is bounded in L,;H 1. Hence, also ρ 5/ ε u ε t = ρ 3/ ε ρ ε u ε t + 3 ρ ερ ε t ρ ε u ε is bounded in this space and we infer that 33 ρ 5/ ε u ε t ρ 3/ J t weakly in L,;H 1.

16 16 I. GAMBA, A. JÜNGEL, AND A. VASSEUR Using ρ 3/ ε φ with φ L,;H 1 as a test function in the weak formulation of, it holds = + ν ρ 3/ ε ρ ε u ε t φdxdt + 1 τ = K K 7. Employing 33, we have K 1 = ρ 5/ ε V [ρ ε ] x φdxdt + δ ρ 5/ ρ ε u ε x ρ 3/ φ x + ε ε ρ 5/ ε u ε φdxdt ρε u ε + pρ ε ρ 3/ ε φ x dxdt ε u ε t,φ H 1,H 1dt 3 ρ 3/ J t,φ H 1,H 1dt 3 ρ ε xx ρ 5/ φ x dxdt ρε u ε x φ x dxdt ρ 3/ ε ρ ε t u ε φdxdt ε ρρt Jφdxdt. For the second integral, we obtain K = ρε u ε + ρ ε pρ ε 3 ρ ε x φ + ρ ε φ x dxdt J + ρpρ 3 ρ x φ + ρφ x dxdt, since ρ ε u ε converges strongly in L 1,;L and ρ ε x converges weakly* in L,;L. Furthermore, V [ρ ε ] x converges weakly* in L,; H 1 to V [ρ]: K 3 ρ 5/ V [ρ] x φdxdt. he fourth integral can be written as K 4 = δ 5 ρ ε xx ρ ερ ε x φ + ρ εφ x dxdt. Since ρ ε converges strongly in L,;L and in L,;H 1, it follows that K 4 δ ρ xx 5ρ 3/ ρ x φ + ρ φ x dxdt. he weak convergence of ρ ε u ε in L,;H 1 and the strong convergences of ρε in L,;L and of ρ ε x in L,;L imply that K 5 = ν ν 3 ε u ε x ρ J x 3 ρε ρ ε x φ + ρ 3/ ε φ x dxdt ρρx φ + ρ 3/ φ x dxdt = ν J x ρ 3/ φ x dxdt.

17 VISCOUS QUANUM HYDRODYNAMIC EQUAIONS 17 Finally, the estimate shows that K 6, and K 7 1 τ ρ 3/ Jφdxdt. his proves that ρ,j,v solves the system 1-3 for ε = and for smooth initial data. A standard approximation procedure gives the result for initial data ρ,u H 1 L with positive particle density and finite energy. heorem is now proven. References [1] P. Antonelli and P. Marcati. On the finite energy weak solutions to a system in quantum fluid dynamics. o appear in Commun. Math. Phys., 9. [] G. Baccarani and M. Wordeman. An investigation of steady state velocity overshoot effects in Si and GaAs devices. Solid State Electr , [3] A. Caldeira and A. Leggett. Path integral approach to quantum Brownian motion. Physica A , [4] F. Castella, L. Erdös, F. Frommlet, and P. Markowich. Fokker-Planck equations as scaling limits of reversible quantum systems. J. Stat. Phys. 1, [5] L. Chen and M. Dreher. he viscous model of quantum hydrodynamics in several dimensions. Math. Models Meth. Appl. Sci. 17 7, [6] L. Chen and M. Dreher. Viscous quantum hydrodynamics and parameter-elliptic systems. Preprint, University of Konstanz, Germany, 8. [7] P. Degond and C. Ringhofer. Quantum moment hydrodynamics and the entropy principle. J. Stat. Phys. 11 3, [8] L. Diósi. On high-temperature Markovian equation for quantum Brownian motion. Europhys. Letters 1993, 1-3. [9] M. Dreher. he transient equations of viscous quantum hydrodynamics. Math. Meth. Appl. Sci. 31 8, [1] E. Feireisl. Dynamics of Viscous Compressible Fluids. Oxford University Press, 4. [11] I. M. Gamba, M. Gualdani, and R. Sharp. Adaptive discontinuous Galerkin scheme for the Wigner-Fokker-Planck Equation. Submitted for publication, 8. [1] I. M. Gamba, M. Gualdani, and P. Zhang. On the blowing up of solutions to the quantum hydrodynamic equations in a bounded domain. o appear in Monatsh. Math., 9. [13] I. M. Gamba and A. Jüngel. Positive solutions to singular and third order differential equations for quantum fluids. Arch. Rat. Mech. Anal , [14] I. M. Gamba and A. Jüngel. Asymptotic limits in quantum trajectory models. Commun. Part. Diff. Eqs. 7, [15] C. Gardner. he quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math , [16] I. Gasser, P. Markowich, D. Schmidt, and A. Unterreiter. Macroscopic theory of charged quantum fluids. In: Mathematical Problems in Semiconductor Physics, Pitman Res. Notes Math. Ser. 34, Longman, Harlow, [17] M. Gualdani and A. Jüngel. Analysis of the viscous quantum hydrodynamic equations for semiconductors. Europ. J. Appl. Math. 15 4, [18] M. Gualdani, A. Jüngel, and G. oscani. Exponential decay in time of solutions of the viscous quantum hydrodynamic equations. Appl. Math. Lett. 16 3, [19] F. Huang, H.-L. Li, and A. Matsumura. Existence and stability of steady-state of onedimensional quantum hydrodynamic system for semiconductors. J. Diff. Eqs. 5 6, 1-5. [] A. Jüngel. A steady-state quantum Euler-Poisson system for semiconductors. Commun. Math. Phys , [1] A. Jüngel. ransport Equations for Semiconductors. Lecture Notes in Physics 773. Springer, Berlin, 9. [] A. Jüngel and D. Matthes. he Derrida-Lebowitz-Speer-Spohn equation: existence, nonuniqueness, and decay rates of the solutions. SIAM J. Math. Anal. 39 8,

18 18 I. GAMBA, A. JÜNGEL, AND A. VASSEUR [3] A. Jüngel and J.-P. Milišić. Physical and numerical viscosity for quantum hydrodynamics. Commun. Math. Sci. 5 7, [4] A. Jüngel and S. ang. Numerical approximation of the viscous quantum hydrodynamic model for semiconductors. Appl. Numer. Math. 56 6, [5] H.-L. Li and P. Marcati. Existence and asymptotic behavior of multi-dimensional quantum hydrodynamic model for semiconductors. Commun. Math. Phys. 45 4, [6] H.-L. Li, G. Zhang, and K. Zhang. Algebraic time decay for the bipolar quantum hydrodynamic model. Math. Models Meth. Appl. Sci. 18 8, [7] G. Lindblad. On the generators of quantum mechanical semigroups. Commun. Math. Phys , [8] S. Nishibata and M. Suzuki. Initial boundary value problems for a quantum hydrodynamic model of semiconductors: asymptotic behaviors and classical limits. J. Diff. Eqs. 44 8, [9] G. Zhang, H.-L. Li, and K. Zhang. Semiclassical and relaxation limits of bipolar quantum hydrodynamic model for semiconductors. J. Diff. Eqs. 45 8, Department of Mathematics, University of exas at Austin, X 7871, USA address: gamba@math.utexas.edu Institute for Analysis and Scientific Computing, Vienna University of echnology, Wiedner Hauptstr. 8-1, 14 Wien, Austria address: juengel@anum.tuwien.ac.at Department of Mathematics, University of exas at Austin, X 7871, USA address: vasseur@math.utexas.edu