The quasineutral limit in the quantum drift-diffusion equations

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1 The quasineutral limit in the quantum drift-diffusion equations Ansgar Jüngel and Ingrid Violet November 8, 006 Abstract The quasineutral limit in the transient quantum drift-diffusion equations in one space dimension is rigorously proved. The model consists of a fourth-order parabolic equation for the electron density, including the quantum Bohm potential, coupled to the Poisson equation for the electrostatic potential. The equations are supplemented with Dirichlet-Neumann boundary conditions. For the proof uniform a priori bounds for the solutions of the semi-discretized equations are derived from so-called entropy functionals. The drift term involving the electrostatic potential is estimated by proving a new bound for the electric energy. Since the electrostatic potential is not an admissible test function, an auxiliary test function has to be carefully constructed. Keywords. Quantum drift-diffusion model, global-in-time existence of weak solutions, entropy estimates, quasi-neutral limit, asymptotic analysis, plasmas, semiconductors. AMS Classification. 35K55, 35B5, 35B45, 35Q40. Introduction In charged particle transport, quasineutrality is a commonly used assumption in order to simplify the model equations. Quasineutrality means that the difference between the concentrations of positive ions and electrons is negligible compared to a reference density. The authors acknowledge partial support from the Project Hyperbolic and Kinetic Equations of the European Union, grant HPRN-CT , and from the DAAD-Procope Program. The first author has been supported by the Deutsche Forschungsgemeinschaft, grants JU359/3 Gerhard-Hess Award and JU359/5 Priority Program Multi-scale Problems. Institut für Mathematik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, Mainz, Germany; juengel@mathematik.uni-mainz.de. Laboratoire de Mathématiques Appliquées, CNRS UMR 660, Université Blaise Pascal Clermont- Ferrand, 6377 Aubière, France; violet@math.univ-bpclermont.fr.

2 Formally, quasineutral models are obtained in the limit as the ratio of the Debye length to a characteristic length tends to zero. Quasineutral models are used, for instance, in semiconductor theory [3] and plasma physics [33]. Recently, quasineutrality has been studied also in quantum models []. An important quantum model are the quantum driftdiffusion equations which are a simple quantum extension of the drift-diffusion model used in both semiconductor and plasma theory see [7] for a derivation and [8, 4] for reviews on macroscopic quantum models. In this paper we analyze rigorously the quasineutral limit in the scaled quantum driftdiffusion equations in one space dimension for the electron density nx, t, the positively charged ion or hole density px, t, and the electrostatic potential V x, t, n t J n,x = 0, J n = ε nlog n xx x + P n n x nv x, p t + J p,x = 0, J p = ε plog p xx x P p p x pv x, λ V xx = n p Cx, x, t Q T = 0, T. 3 Here, J n and J p are the current densities and Cx models fixed charged background ions, usually called the doping concentration. The pressure functions P n and P p are typically of the form P α x = θ α x qα α = n, p for some θ α > 0 and q α. The parameter ε is the scaled Planck constant and λ > 0 is the ratio of the Debye length to the characteristic length e.g., the device diameter. The equations are supplemented with the initial and boundary conditions n = p =, n x = p x = 0, V = V D for x {0, }, t > 0, 4 n, 0 = n I, p, 0 = p I in, 5 where V D x = xu and U R is the applied potential. In the case that the doping vanishes at the boundary, the Dirichlet boundary conditions for n and p express charge neutrality, whereas the Neumann boundary conditions have been employed in numerical simulations of quantum semiconductor devices []. The quantum drift-diffusion model can be derived by the entropy minimization principle from the Wigner-BGK equation in the diffusion limit [7] or from the so-called quantum hydrodynamic equations in the zero-relaxation-time limit [0]. The existence of weak solutions to the stationary equations have been proved in []; the transient equations in one space dimension are analyzed in [3] but only for electrons and isothermal pressure P n n = θ n n. Numerical simulations can be found in [3, 9]. Mathematically, the parabolic equations - are of fourth order. In particular, no maximum principles are available which complicates the analysis [, 3]. In this context, we mention the so-called Derrida-Lebowitz-Speer-Spohn equation [9], obtained from for zero pressure and zero electric field. This equation has recently attracted a lot of attention in the mathematical literature since it possesses several Lyapunov functionals and there are connections to logarithmic Sobolev inequalities see [0] and references therein.

3 The justification of the quasineutral limit in macroscopic models has been first studied in [4] for a nonlinear Poisson equation the ion density being fixed. The limit in the driftdiffusion equations i.e. -3 with ε = 0 has been proved in [3, ] assuming vanishing or at least not sign-changing doping concentrations. Sign-changing doping profiles have been considered in [36]. The quasineutral limit in the steady state Euler-Poisson equations has been investigated in [6, 7, 8, 34], whereas in [6, 4, 5, 35] the time-dependent case has been analyzed. In [3, 7] the limit in the Vlasov-Poisson system has been shown. To our knowledge, no analytical results on the quasineutral limit in fluid-type quantum models are available up to now. In the quasineutral limit λ 0 we obtain formally from 3 n = p and from - n t + ε nlog n xx xx = P nn + P p n xx, x, t > 0, 6 with initial and boundary conditions n =, n x = 0 for x {0, }, n, 0 = n I in, t > 0. 7 In this paper we make the limit rigorous for vanishing doping profile. First we show the existence of weak solutions to -4 for general doping concentrations. In the literature, only results for the unipolar model are available with different boundary conditions [3] or with zero temperature and zero electric field []. Therefore, we include a proof for completeness. Moreover, our proof makes clear which quantities are bounded uniformly in the parameter λ in appropriate norms. More specifically, we show that the entropy n log ndx is nonincreasing with respect to time and that the corresponding entropy production terms provide λ-uniform bounds for log n and log p in L 0, T ; H and for n and p in L 7/ Q T. Also the entropy nlog n dx is nonincreasing in time, providing the uniform bounds n p L Q T cλ, V x L Q T cλ. 8 These estimates are not sufficient to pass to the limit λ 0 in -3. Indeed, the sum of and leads to the drift term in weak formulation Q T n pv x φ x dxdt n p L Q T V x L Q T φ x L Q T c, where φ is some smooth test function and c > 0 a constant independent of λ. Thus, the estimates 8 only show that the above drift term is uniformly bounded; however, we need to prove that it converges to zero as λ 0. The main problem in this limit is that the negative electric field V x is of the order Oλ. Our idea is to derive instead of 8 the estimates n p L Q T cλ, n + pv x L Q T cλ 8/9. 9 3

4 This gives Q T n pv x φ x dxdt n p L Q T n + pv x L Q T φ x L Q T λ /9, and hence, the drift term converges to zero as λ 0. The exponent 8/9 in 9 is connected with the exponents of some Gagliardo-Nirenberg inequalities see Lemma 3. The first bound in 9 is a consequence of the estimate using the entropy n log ndx. The proof of the second bound in 9 is more delicate. It follows from an estimate of the electric energy λ V W xdx if W satisfies the boundary data of V up to first order, i.e. W = V and W x = V x at x {0, }. Since V x 0, t and V x, t are only of the order Oλ, W is of the same order and prevents an appropriate estimate. To solve this problem, we approximate W by a function W δ in such a way that W δ is of the order O + Oδλ in the H norm. Passing to the limit δ 0 then provides the needed estimate in 9. Our main results are the following theorems. Theorem. Let T > 0, U R, C L, and 0 n I, p I L satisfying ni log n I + p I log p I dx + ni log n I + p I log p I dx <. Furthermore, let P n, P p C [0, be nondecreasing and assume that there exist 0 < q < 7/ and C P > 0 such that P α x C P + x q for all x 0, α = n, p. 0 Then there exists a weak solution n, p L 7/ Q T, V L 0, T ; H to -5 such that n, p 0 in Q T, log n, log p L 0, T ; H 0, n t, p t L 0, T ; H 3. The idea of the proof is to use the exponential transformation n = e y and p = e z as in [] since this automatically gives nonnegative particle densities. First we show the existence of weak solutions to a semi-discrete elliptic problem. Appropriate a priori estimates, which are also useful for the quasineutral limit, allow to pass to the limit of vanishing approximation parameter. We stress the fact that, although we employ ideas of [], the existence theorem is needed since first, there is no existence result for the bipolar quantum drift-diffusion model in the literature; and secondly, the approximation argument is needed in the proof of the quasineutral limit due to the lack of regularity of solutions to -5. It is possible to obtain an existence result for more general non-homogeneous boundary data but the proof is very technical; we refer to [8] for a related problem providing the needed mathematical tools. Theorem. Let the assumptions of Theorem hold and let, in addition, Cx 0, q 7/3 and n I = p I in. Let n λ, p λ, V λ be a weak solution in the sense of Theorem 4

5 to -5. Then there exists a subsequence of n λ, p λ, V λ, which is not relabeled, such that, as λ 0, n λ n, p λ n strongly in L 3 Q T, n λ t n t, p λ t n t weakly in L 4/4 0, T ; H 3, log n λ log n, log p λ log n weakly in L 0, T ; H, and the limit function n solves 6-7. Our assumptions avoid boundary and initial layers. We refer to [] for the treatment of boundary layers and to [3] for the analysis of initial layers in the drift-diffusion model cf. [7] and Remark 4. Taking the difference of equations and provides in the limit λ 0 formally an equation for the electrostatic potential, n + pv x x = P n n P p n xx in, V 0, t = 0, V, t = U. However, since V x is of the order Oλ we cannot justify this limit equation rigorously. In the drift-diffusion equations, this is possible under certain assumptions see []. If uniqueness of solutions holds for the problem 6-7, the whole sequence n λ, p λ, V λ converges. However, there is no general uniqueness result for the limit problem. For a uniqueness theorem in the case of vanishing pressure under additional assumptions, we refer to [0]. The proof of Theorems and uses in several places the fact that we consider the one-dimensional equations. An existence proof for the multi-dimensional equations for vanishing pressure and vanishing electric field has been shown in [6] but only using periodic boundary conditions. The treatment of the quantum drift-diffusion model in several dimensions with physically motivated boundary conditions is currently not known. Another interesting limit is the semiclassical limit ε 0. For a result in the stationary equations we refer to []. In [5] the limit has been shown in the transient case with homogeneous Neumann boundary conditions. Our a priori estimates seem to be not sufficient to perform the limit for the boundary conditions 4 see Remark 6. The paper is organized as follows. In section we derive some a priori estimates needed for the existence result and we prove Theorem. The estimates are also useful for the quasineutral limit. Section 3 is devoted to the derivation of additional estimates independent of λ and the proof of Theorem. Finally, for convenience of the reader, we recall the Gagliardo-Nirenberg inequality [9] which are employed several times throughout this paper. Lemma 3. Let m, k N 0 with 0 k m, 0 θ <, and p, q, r, and let R d be a bounded domain with smooth boundary. If both k d p θ m d + θ d q r 5 and p θ q + θ, r

6 then any function f W m,q L r belongs to W k,p, and there exists a constant C > 0 independent of f such that Existence of solutions. A priori estimates f W k,p C f θ W m,q f θ L r. We divide the time interval 0, T ] for some T > 0 in N subintervals t k, t k ] with t k = k, k = 0,..., N, and = T/N is the time step. For given k {,..., N} and y k, z k H 0 we solve the semi-discrete system ey k e y k + ε ey k y k,xx xx = P n x e y k V k,x x ez k e z k + ε ez k z k,xx xx = P p e z k x + e z k V k,x x 3 λ V k,xx = e y k e z k Cx in, 4 for y k, z k H 0, V k V D H, where V D x = xu, x. We introduce the piecewise constant functions y N x, t = y k x, z N x, t = z k x, V N x, t = V k x for x, t t k, t k ], 5 where k =,..., N. First we show that the entropy E k = y k + e z k z k dx is non-increasing. Let y k, z k H 0, V k V D H 0 be a solution to -4. Lemma 4. There exists a constant cλ > 0 which is independent of λ if Cx 0 such that E k + ε k j= yj,xx + zj,xxdx + k λ j= e y j e z j y j z j dx cλe 0. 6 Proof. We employ e y k H 0 as a test function in the weak formulation of to obtain e y k e y k dx + ε yk,xx y k,xy k,xx dx = P ne y k yk,x V k,x y k,x dx. 7 6

7 With the elementary inequality e x + x for x R we can write e y k e y k = e y k e y k + e y k y k y k y k. Since y k,x = 0 on the boundary, the second integral on the left-hand side of 7 becomes ε y k,xx 3 y3k,x x dx = ε y k,xxdx. Thus, it follows from 7, taking into account that P nx 0 by assumption, y k dx + ε y k,xxdx y k dx + V k,x y k,x dx. We obtain a similar equation for z k. Then, adding both inequalities and using the Poisson equation 4, we arrive at E k + ε yk,xx + z k,xxdx E k + V k,x y k,x z k,x dx = E k e z k Cxy λ k z k dx E k e z k y λ k z k dx + λ C L y k + z k dx. Since x e x x for all x R, this yields E k + ε yk,xx + z k,xxdx + e z k y λ k z k dx E k + λ C L E k. Hence, choosing > 0 small enough, we obtain 6. An immediate consequence of the entropy estimate 6 and the Poincaré inequality are the following uniform bounds for the functions y N and z N see 5: y N L 0,T ;L + z N L 0,T ;L cλ, 8 e yn L 0,T ;L + e zn L 0,T ;L cλ, 9 y N L 0,T ;H + z N L 0,T ;H cλ. 0 Again, if Cx 0, the constant cλ does not depend on λ. From these estimates we are able to deduce more uniform bounds. Lemma 5. There exists a constant cλ > 0 which does not depend on λ if Cx 0 such that e yn L 5/ 0,T ;W, + e zn L 5/ 0,T ;W, cλ, where we recall that Q T = 0, T. e yn L 7/ Q T + e zn L 7/ Q T cλ, 7

8 Proof. We employ the Gagliardo-Nirenberg inequality and the estimates 8, 0 to find y N L 5/ 0,T ;W, y x N L 5/ 0,T ;L y N /5 L 0,T ;L yn 4/5 L 0,T ;H cλ. Therefore, with 8 and 9, e yn L 5/ 0,T ;W, c e yn L 5/ 0,T ;L + e yn x L 5/ 0,T ;L c e yn L 5/ 0,T ;L + c e yn L 0,T ;L y x N L 5/ 0,T ;L cλ. This shows. In order to prove we use again the Gagliardo-Nirenberg inequality: T e yn 7/ L 7/ Q T c e yn L e yn 5/ W, dt 0 c e yn L 0,T ;L e yn 5/ cλ. L 5/ 0,T ;W, The bounds for e zn are derived in a similar way. Remark 6. The constants in - depend on ε since the estimates for y N and z N in L 0, T ; H do so. Hence, most of the subsequent bounds also depend on ε. Lemma 7. There exists a constant cλ > 0 depending on λ such that V N L 0,T ;H cλ. 3 Proof. By elliptic estimates, λ V x N L Q T c e yn e zn Cx L 0,T ;H + c e yn e zn Cx L 0,T ;L + cλ, since L injects continuously into H in one space dimension. Finally, we need an estimate for the discrete time derivative. For this, we introduce the shift operator σ N e yn x, t = e y k x, σ N e zn x, t = e z k x for x, t t k, t k ]. 4 Lemma 8. There exists a constant cλ > 0 depending on λ such that for s = min{7/q, 4/} >, e yn σ N e yn L s 0,T ;H 3 + e zn σ N e zn L s 0,T ;H 3 cλ. 5 8

9 Proof. We estimate the semi-discrete equation in the norm of L s 0, T ; H 3. This gives e yn σ N e yn L s 0,T ;H 3 ε e yn y N xx L s 0,T ;H + P n e yn L s 0,T ;H + e yn V x N L s 0,T ;H. The first term on the right-hand side is bounded by Hölder s inequality and 0, : e yn y N xx L s 0,T ;H c e yn y N xx L s 0,T ;L s c e yn L s/ s Q T y N xx L Q T c e yn L 7/ Q T y N xx L Q T cλ, since s/ s 7/ is equivalent to s 4/. For the second term on the above right-hand side we employ the growth condition on the pressure functions and : P n e yn L s 0,T ;H c P n e yn L s 0,T ;L s c + e yn q L 7/ Q T cλ. Finally, the last term on the right-hand side can be estimated by using and 3: e yn V x N L s 0,T ;H c e yn V N x L s 0,T ;L s c e yn L s/ s Q T V x N L Q T cλ. Putting together the three inequalities gives 5. The proof for z N is analogous.. Proof of Theorem First we show that the semi-discrete problem -4 admits a solution. Lemma 9. Under the hypotheses of Theorem there exists a sequence y k, z k, V k H 0 H with V k 0 = 0 and V k = U satisfying -4. Proof. Let y k, z k H 0 be given. Let v, w H and solve first λ V k,xx = e v e w Cx in, V k 0 = 0, V k = U. This problem admits a unique solution V k H. Then we solve in H 0 the linear problems σ ev e y k + ε ev y k,xx xx = σ P n e v x e v V k,x x σ ew e z k + ε ew z k,xx xx = σ P p e w x + e w V k,x x where σ [0, ]. There exists a unique solution y k, z k H 0. This defines the fixed-point operator S : H [0, ] H, v, w, σ y k, z k. Then S is well 9

10 defined and satisfies Sv, w, 0 = 0, 0. Furthermore, it is not difficult to check that S is continuous and, in view of the compact embedding H 0 H, also compact. It remains to show that there is a uniform bound for all fixed points of S,, σ. The estimates of section. establish the case σ =. The estimates for σ < are similar and, in fact, independent of σ. This provides the wanted bound in H and the Leray-Schauder fixed-point theorem can be applied to yield the existence of a solution to -4. Now we are able to prove Theorem. For this, we have to perform the limit 0 in -4. Actually, the uniform bounds and 5 and the compact embedding W, L allow to apply Theorem 5 of [3] Aubin s lemma yielding the existence of a subsequence of e yn and e zn not relabeled such that e yn v, e zn w strongly in L Q T as N or, equivalently, 0. Moreover, again for a subsequence which is not relabeled, y N y, z N z weakly in L 0, T ; H 6 as 0. The bounds 8 and 9 allow to use the same arguments as in the proof of Theorem. in [8] showing that v = e y and w = e z. Since, by, e yn is bounded in L 7/ Q T and e yn e y a.e., the result in [5, Ch..3 and p. 44] yields e yn e y strongly in L Q T. 7 Moreover, the same bound and hypothesis 0 imply that P n e yn is bounded in L s 0, T ; L s for s = 7/q > and hence, by the same argument as before, P n e yn P n e y strongly in L Q T. 8 Finally, the bound 5 gives, up to a subsequence, eyn σ N e yn e y t weakly in L s 0, T ; H 3. 9 The same limits hold for z N. Moreover, by 3, 0 V N V weakly in L 0, T ; H. 30 The limits 6-30 allow to pass to the limit 0 in the weak formulation of, T T σ N e yn φdxdt + ε e yn y xx N φ xx dxdt = eyn T 0 P n e yn φ xx + e yn V N x φ x dxdt for all φ L 0, T ; H 3 H 0. The limit functions satisfy y, z L 0, T ; H 0, V V D L 0, T ; H 0, which shows that the boundary conditions are satisfied. Furthermore, the initial conditions hold in the sense of H 3. This finishes the proof of Theorem. 0 0

11 3 The quasi-neutral limit 3. A priori estimates For the quasi-neutral limit λ 0 we need additional estimates. We recall that the condition Cx 0 implies that the uniform bounds 6- are independent of λ. Lemma 0. There exists a constant c > 0 independent of λ such that e yn / e zn / L Q T cλ. 3 Proof. The entropy estimate 6 gives e yn e zn y N z N dxdt cλ. Q T Then the assertion follows if we can show that x y x ylog x log y for all x, y 0. 3 This inequality can be seen as follows. It is sufficient to consider x y > 0. Then 3 is equivalent to x y x + y log x y and Thus we only need to prove that x/y x y = log x/y + x + y z z + log z for all z. x y. But this is a consequence of log z z z /z + for z, thus proving the lemma. The following estimates are derived from the boundedness of the entropy E k = y k + e z k z k + dx > 0. Lemma. The following estimate holds: E k + ε k j= e y j yj,xx + e z j zj,xxdx + k λ j= e y j e z j dx E 0. 33

12 Proof. We employ the test function y k H0 in the weak formulation of to obtain e y k y k dx + ε e y k y k,xxdx = P ne y k e y k yk,x V k,x e y k y k,x dx. 34 The convexity of x e x implies that e x e y e y x y 0 and hence, e y k y k e y k y k + e y k y k y k e y k + e y k Thus it follows e y k y k dx + ε = e y k y k e y k y k. e y k y k,xxdx e y k y k dx + V k,x x dx. A similar inequality holds for z k. Adding both inequalities and then employing the Poisson equation 4 gives E k + ε yk,xx + e z k z k,xxdx E k e z k dx. λ This gives the assertion. From Lemma immediately follows that and therefore, for sufficiently small λ > 0, e yn e zn L Q T cλ V N x L Q T c + λ e yn e zn L Q T cλ. In the L 3 norm the exponent in λ is smaller as shown in the following lemma. Lemma. There exists a constant c > 0 independent of λ such that Proof. By Hölder s inequality, e yn e zn L 3 Q T cλ /9, 35 V N L 3 0,T ;W,3 cλ 6/9. 36 e yn e zn L 3 Q T e yn e zn /9 L Q T eyn e zn 7/9 L 7/ Q T cλ/9, employing and 33, which shows 35. The estimate 36 is a consequence from 35: This finishes the proof. V N xx L 3 Q T = λ e yn e zn L 3 Q T cλ 6/9.

13 The following lemma is our key result. Lemma 3. There exists a constant cε > 0 independent of λ such that, for sufficiently small λ > 0, e yn / + e zn / V N x L Q T cλ 8/9. 37 Proof. The key idea is to define a special extension W k x of the boundary data such that W k V k H0 becomes an admissible test function in the weak formulation of -3. The problem is that V k,x 0 and V k,x are unbounded as λ 0. Therefore, we need to take special care in the definition of W k. We define δv k,x 0 U x 3 δ + δu Vk,x 0 x δ + δvk,x 0 x : x [0, δ] δ xu : x [δ, δ] W k x = δu V k,x x δ 3 + δv k,x U x δ δvk,x x δ + U : x [ δ, ]. This function is continuously differentiable, is an element of H and satisfies W k 0 = 0, W k = U, W k,x 0 = V k,x 0, W k,x = V k,x. Let W N, t = W k if t t k, t k ]. We claim that, for sufficiently small λ > 0, Indeed, by elliptic estimates and 35, we have W N W x N L 3 Q T cδλ 6/9, 38 W xx N L 3 Q T cλ 6/9. 39 x L 3 Q T c + δ V x N 0, L 3 0,T + δ V x N, L 3 0,T c + δ V N L 3 0,T ;W, c + δ V N L 3 0,T ;W, c + δλ e yn e zn L 3 0,T ;L cδλ 6/9. This shows 38. In order to prove 39, we use 36: W N xx L 3 Q T c + V x N 0, L 3 0,T + V x N, L 3 0,T c + V N L 3 0,T ;W, cλ 6/9. Now we employ W k V k H0 as a test function in -3 and take the difference of the resulting equations to obtain e z k e z k W k V k dx + ε y k,xx e z k z k,xx W k V k xx dx 40 = P n P p e z k W k V k xx dx + + e z k V k,x W k V k x dx. 3

14 The first integral on the left-hand side can be estimated by means of the Poisson equation 4 and Young s inequality: = λ e z k e z k W k V k dx = λ V k V k x V k W k x dx V k W k xdx λ V k W k x V k W k x dx + λ W k W k x V k W k x dx λ V k W k xdx λ V k W k xdx + λ W k W k x V k W k x dx. Applying Young s inequality to the last integral in 40 gives + e z k V k,x W k V k x dx + e z k V k,xdx + Thus, summation over k in 40 yields λ N W N V xdx + N k= N V 0 W 0 xdx λ λ k= + e z k V k,xdx W k W k x V k W k x dx + e z k W k,xdx. + N N + e z k W k,xdx + ε y k,xx + e z k z k,xx V k W k xx dx k= k= N P n P p e z k V k W k xx dx k= = I + + I 5. In the following, we write the integrals I,..., I 5 in terms of y N, z N, V N, and W N. For the first integral I we notice that our assumption on the initial data gives λ V 0,xx = e y 0 e z 0 = 0 in which, together with the boundary conditions V 0 0 = 0, V 0 = U, shows that V 0 is a linear function and in particular independent of λ. Thus, also W 0,x does not depend on λ and For I we use 38: I λ N W x L Q T I λ V x N, 0 L + λ W x N, 0 L c. V N x L Q T + W x N L Q T 4 cδλ /9 λ + δλ 6/9 cδ λ 7/9,

15 choosing δ λ 7/9. Taking into account and 38 gives I 3 e yn L 3 Q T + e zn L 3 Q T W x N L 3 Q T cδ λ 3/9, and an application of Hölder s inequality and, 33, 36, and 39 yield I 4 ε e yn / y xx N L Q T e yn / L 6 Q T + e zn / z xx N L Q T e yn / L 6 Q T V xx N L 3 Q T + W xx N L 3 Q T cλ 6/9. We proceed with the integral I 5 which we estimate using the growth condition on P n and P p and 36, 39: V I 5 P n e yn L 3/ Q T + P n e yn N L 3/ Q T xx L 3 Q T + W xx N L 3 Q T c + e yn q + L 3q/ Q T ezn q L 3q/ Q T λ 6/9 cλ 6/9, since 3q/ 7/ is equivalent to our assumption q 7/3. The above estimates yield, for sufficiently small λ > 0, e yn + e zn V N xdxdt c + λ 6/9 + δ λ 3/9 + δ λ 7/9 Q T Letting δ 0 then gives the assertion. Remark 4. In order to avoid an initial time layer we have assumed that n I = p I. The above proof shows that it is enough to require that n I p I H is of the order Oλ /9. Indeed, the estimate shows that I cλ 6/9 holds. λ V 0,x L c + λ e y 0 e z 0 H cλ 6/9 Remark 5. The assumption q 7/3 can be improved to q < 5/ by more technical effort. Indeed, this condition is only needed in the computation of the integral I 5. In order to show how I 5 can be estimated assuming only q < 5/, we proceed as follows. By the same arguments as in the proof of Lemma, we can derive e yn e zn L r Q T cλ θ, V N L r 0,T ;W,r cλ θ for < r < 7/ and θ = 7 r/3r 0,. Then I 5 c + e yn q + L qr/r Q T ezn q L qr/r Q T λ θ cλ θ, since qr/r 7/ is equivalent to q < 5/. This yields e yn + e zn V N x L Q T cλ θ/, which is sufficient for the proof of Theorem. However, the proof of Lemma 6 below becomes more involved. Therefore, and since the improvement is only marginal, we have assumed the stronger condition q 7/3. 5

16 Lemma 6. There exists a constant c > 0 independent of λ such that for s = 4/4, e yn + e zn σ N e yn + e zn L s 0,T ;H 3 c. Recall that σ N is the shift operator defined in 4. Proof. We estimate the sum of equations and 3: eyn + e zn σ N e yn + e zn L s 0,T ;H 3 ε eyn y xx N + e zn z N xx L s Q T + P n e yn + P p e zn L s Q T + e yn e z V N x L s Q T. The first term on the right-hand side is bounded by 0 and : e yn y N xx L s Q T e yn L s/ s Q T y N xx L Q T e yn L /0 Q T y N xx L Q T c, and similarly for the expression for z N. Taking into account the growth assumption on P n and we find P n e yn L s Q T c + e yn q L q Q T c, and analogously for z N. For the drift term we need Lemma 0 and : e yn / e zn / L /0 Q T e yn / e zn / 8/9 L Q T eyn / e zn / /9 L 7/ Q T cλ8/9. This yields, together with Lemma 3, e yn e zn V N x L s Q T e yn / e zn / L /0 Q T e yn / + e zn / V N x L Q T cλ 8/9 λ 8/9 = c. Putting together the above bounds gives the assertion. 3. Proof of Theorem The results of section. allow to pass to the limit 0 in the uniform estimates of the previous section. This yields weak solutions y λ, z λ, and V λ satisfying the equations e yλ t + ε eyλ y λ xx xx = P n e yλ x e yλ V λ x, 4 e zλ t + ε ezλ z λ xx xx = P p e zλ x + e zλ V λ x, 4 6

17 the boundary and initial conditions 4-5 and the following uniform bounds: e yλ L 5/ 0,T ;W, + e zλ L 5/ 0,T ;W, c, e yλ + e zλ t L 4/4 0,T ;H 3 c, y λ L 0,T ;H + z λ L 0,T ;H c, e yλ L 7/ Q T + e zλ L 7/ Q T c as well as, by Lemmas 0 and 3, e yλ / e zλ / L Q T cλ, e yλ / + e zλ / V λ x L Q T cλ 8/9. 43 Thus, Aubin s lemma and the arguments of section. show the existence of a subsequence not relabeled such that, as λ 0, e yλ e y, e zλ e y strongly in L 3 Q T and weakly in L 7/ Q T, y λ y, z λ y weakly in L 0, T ; H, e yλ + e zλ t e y t weakly in L 4/4 0, T ; H 3. These convergence results imply for all sufficiently smooth φ, as λ 0, T T e yλ + e zλ t, φ H 3,H 3dt e y t, φ H 3,H 3dt, 0 0 e yλ y xx λ + e zλ z xx λ φ xx dxdt e y y xx φ xx dxdt, Q T Q T P n e yλ + P p e zλ φ xx dxdt P n e y + P p e y φ xx dxdt. Q T Q T The delicate integral is the expression containg the drift term. Here we need 43: e yλ e zλ V x λ φ x dxdt Q T e yλ / e zλ / L Q T e yλ / + e zλ / V x λ L Q T φ x L Q T cλ λ 8/9 cλ /9 0. These results allow to pass to the limit in the sum of the equations 4 and 4, e yλ + e zλ ε t, φ H 3,H3dt + e yλ Q T = P n e yλ + P p e zλ φ xx dxdt + e yλ Q T Q T T 0 which proves Theorem. 7 y λ xx + e zλ z λ xx φ xx dxdt e zλ V x λ φ x dxdt,

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