Scope of the Work













∂tn−divJ =ν04n,

∂_tJ −div

J⊗J n

−T₀∇n+n∇V + ² 2n∇

4√

√ n n

=ν₀4J −J τ, λ²4V =n− C(x), The complete system (1.1.15) is calledthe non-isothermal viscous QHD model.

1.2 Related Analytic Results

Concerning the QHD model without viscous terms J¨ungel and H. Li [49] showed the existence of stationary states for one space dimension. According to this result the exponential decay to a stationary state in one dimensional bounded domains was derived in [50]. H. Li and P.Marcati [62] also investigated the local existence and asymptotic behavior of solutions for several space dimensions.

In case of one dimensional viscous QHD L. Chen and M. Dreher [17] de-rived the local existence and uniqueness with insulating boundary conditions.

A. J¨ungel and J.P. [53] proved the existence of steaty states and obtained the corresponding numerical simulations. Additionally the local existence and ex-ponential decay inR¹ was showed in [32].

For the isothermal viscous QHD in multiple dimensions only a few analytic results are available. In [17] the local existence of solutions in the case of higher dimensions under the assumptions of periodic boundary conditions was proved. Later in [32] the local existence of solutions with more general boundary conditons was obtained.

1.3 Scope of the Work

In this dissertation we analysis the equations of viscous QHD model in multiple dimensions. The work is divided into three major parts.

Part I Local Solutions to the Isothermal Viscous QHD Model

Here we investigate the local existence of solutions to



















∂tn−divJ =ν04n,

∂tJ−div

J⊗J n

−T0∇n+n∇V +² 2n∇

4√

√ n n

=ν04J−J τ, λ²4V =n− C(x), (n, J)(0, x) = (n0, J0)(x),

(1.3.1) subject to the Dirichlet boundary conditions







(n, J, V)(t, x) = (n_Γ, J_Γ, V_Γ) on∂Ω,

n_Γ ∈H^5/2(∂Ω), J_Γ∈H^3/2(∂Ω), V_Γ∈H^3/2(∂Ω),

(1.3.2)

where the following regularity and compatibility conditions











n0 ∈H³(Ω), J0 ∈H²(Ω), (n₀, J₀)|_∂Ω= (n_Γ, J_Γ), (ν04n₀+ divJ0)|_∂Ω= 0.

are also satisfied.

In aspects of mathematics the main difficulties come from the quantum Bohm potential

B(n) := 1 2²4√

√ n n

which brings a third order derivative into the system which otherwise will be considered as a parabolic system coupled to an elliptic equation. Additionally the maximum principle arguments can in general be not applied to third-order equations.

An usual strategy to deal with third-order equations is to introduce a viscous regularization term γ4² (0< γ <1) [17, 32], whereγ4² will vanish asγ →0.

However when one attempts to use this method to extend the results of [17] to a system with more general boundary conditions, one is faced with failure because the estimates of the third-order derivative depends always upon γ, and might converges to ∞ as γ → 0. Instead we will overcome this difficulty by using the a-priori estimates of the BVPs of mixed-order, which was first defined in [5]. Precisely, we check that the corresponding linear matrix operator of (1.3.1) is elliptic with parameter in certain closed sector in the complex plane with vertex at the origin. Then we use the Theorem 2.6 in [34] to abtain the a-priori estimates which provide an estimation of the third order derivative. This will lead to a uniform bounds of the approxiamte solutions.

The last step to prove the local existence and uniqueness is to investigate the converges properties of the approximate solutions in certain Sobolev spaces.

Namely, the sequence of the approximate solutions in a small time interval should be a Cauchy sequence in certain sense. A further analysis will show that the limit is the local-in-time solution we seek.

Part II Global Existence and Asymptotic Behavior on a Torus We use the assumption that the doping profile of background charges is a posi-tive constant (denoted byC₀). The long time behavior for one space dimension and the exponential stability in a muli-dimensional box is obtained in [17], where properties of a box was needed for an estimate of the Bohm potential.

Compared to [17] we consider a more general situation. More precisely, ac-cording to the result of the local existence on a torus in [17] we will extend the local-in-time solution of (1.3.1) on a torus globally in time, and obtain the exponential decay to the steady state (C₀,0,0) when the initial data is close to (C₀,0,0).

Breafly speaking, we first reformulate the original equations to a fourth-order wave equation. Then we get the a priori estimates. Finally we apply the usual continuity argument to obtain the global existence and large time behavior.

Part III Local Solutions to the Non-isothermal Viscous QHD Model We investigate the local existence and uniqueness of solutions to the non-isothermal viscous QHD model



whereµ is the interaction constant; the (scaled) stress tensor P and the tem-peratureT satisfy

i.e., the (scaled) stress tensor consists of the classical pressure and a quantum

”pressure” term. The energy density is the sum of kinetic energy, thermal energy, and quantum energy.

The initial data and boundary conditions are

(n, J, ne)^T(0) = (n0, J0,(ne)0), (1.3.5) (n, J, ne, V)^T|_∂Ω = (n_Γ, J_Γ,(ne)_Γ, V_Γ)^T (1.3.6) with

x∈Ωinf n0(x)>0, n0 ∈H³(Ω), J0∈H²(Ω), (ne)0 ∈H²(Ω), (1.3.7) n_Γ∈H^5/2(Ω), J_Γ∈H^3/2(Ω),(ne)_Γ ∈H^3/2(Ω), , V_Γ∈H^3/2(Ω), (1.3.8)

(n0, J0,(ne)0)|_∂Ω = (nΓ, JΓ,(ne)Γ), (ν₀4n₀+ divJ₀)|_∂Ω = 0.

)

(1.3.9) By far as we know, it seems that there are hardly any analytic results for this model in multi-dimensions. The system (1.3.3) is more complicated than (1.3.1). The main difficulty is that the evolution equation with respect to ne contains a nonlinear term among which a third-order derivative occurs.

In this part, we study the non-isothermal viscous QHD model (1.3.3) under the assumptions that the given data J0,∇(ne)₀,∇n₀, VΓ are sufficiently small in

(H²(Ω))^d×(L²(Ω))^d×(H²(Ω))^d×H^3/2(Ω))

and C(x) is close to n₀ inL²(Ω) norm. Under these assumptions and with the help of the a-priori estimates fromPart I, we then obtain the uniform bounds of approximate solutions. Finally we analysis the limit to derive a local solution.

Appendices will provide some background material on important calculus, inequalities, functional analysis, Sobolev spaces, etc.

Finally the Bibliography primarily provides a listing of related papers for further informations.

Preliminaries

In this chapter we first present some notations which will be needed in the remainder of the dissertation. Then some basic materials on elliptic boundary value problems of mixed order will be introduced for convenience. Most results are just recalld without proofs, but the relevant references are given in the end.

2.1 Notations and Some Useful Calculus

The symbol Ω will be reserved for a nonempty open set inn-dimensional real Eu-clidean spaceRⁿ. We stipulate that all function vectors are written in columns, for a scalar-valued function the gradient is a row. Given a vector-valued func-tionf:

f : Ω→Rⁿ and matrix-valued functionA

A: Ω→R^n×n

then the following notations are used: 4f =







 4f₁

... 4f_n







 ,

∇f =









∂₁f₁ . . . ∂_nf₁ ... . .. ...

∂1fn . . . ∂nfn









, divA=









div(A₁₁, . . . , A_1n) ...

div(An1, . . . , Ann)







 .

f⊗f is an×nmatrix-valued function with entry f_if_j at position (i, j).

Letf, g be vector-valued functions,φ, ψscalar-valued functions,A a matrix-13

valued function. Then we have the following calculus facts.

(gradf)ij =∂jfi, div(φf) =X

j

∂j(φfj) =X

j

(∂jφ)fj+X

j

φ∂jf =hgradφ, fi+φdivf, (grad(φf))_ij =∂_j(φf_i) = (∂_jφ)f_i+φ∂_jf_i = (f⊗gradφ)_ij +φ(gradf)_ij,

(divA)_j =X

i

∂_iA_ji, (div(f⊗g))_j =X

i

∂_i(f ⊗g)_ji=X

i

∂_i(f_jg_i) =f_jX

i

∂_ig_i+X

i

(∂_if_j)g_i

=fj(divg) +hgradf_j, gi, div(f⊗g) = (divg)f + (gradf)g,

(div(φf⊗g))j =φfj(divg) +φhgradf_j, gi+fjhgradφ, gi, div(φf⊗g) =φ(divg)f+ ((gradφ)g)f +φ(gradf)g (gradhf, gi)_j =∂jhf, gi=h∂_jf, gi+hf, ∂_jgi,

grad(φ(ψ)) =ψ∇φ+φ∇ψ,

4(φ(ψ)) =ψ4φ+φ4ψ+ 2h∇φ,∇ψi.

Additionally let (·,· · ·,·)^T denote the transpose, H₀^k(Ω) the Sobolev space of functions with square integrable weak derivatives of order k, whose trace on the boundary up to the order k−1 are 0, i.e., H₀^k(Ω) consists of all functions from {f ∈L²(Ω)|D^αf ∈L²(Ω) for 0≤ |α| ≤k}with the following property

∂^jf

∂ν^j(x) = 0, x∈∂Ω, j = 0,· · · , k−1,

whereD^αfis the weak (or distributional) partial derivative and letν = (ν₁,· · · , ν_d) denote the unit outward normal vector field on∂Ω, ∂^jf

∂ν^j thej-th (outward) nor-mal derivative of f.

For twon×n matrix-valued functionsA= (aij(x)) andB = (bij(x)) whose entries a_ij(x) and b_ij(x) are square integrable funcions on Ω (a_ij(x), b_ij(x) ∈ L²(Ω), i, j= 1,· · ·, n) let (·,·) denote the function:

L(Cⁿ)×L(Cⁿ)→C, (A, B)7→

n

X

i=1 n

X

j=1

(a_ij, b_ij)_L²_(Ω) (2.1.1) where (·,·)_L2(Ω) is the inner product inL²(Ω).

The norm ofL²(Ω) is denoted byk · k. Letα= (α1,· · ·, αn) be an n-tuple of nonnegative integersα_i, we callαamultiindex and denote byx^α the monomial x^α₁¹· · ·x^α_nⁿ(x∈Rⁿ) with degree|α|=Pn

j=1α_j. Similary, ifD_j =−i ∂

∂_xj, then D:=D1· · ·Dn,

and

D^α =D^α₁¹· · ·D_n^αn,

denotes a differential operator of order|α|. For two multi-indices α and β, we say that β ≤ α provided β_j ≤α_j for 1 ≤ j ≤ n. In this case α−β is also a multi-index, and|α−β|+|β|=|α|. We also denote

α! =α₁!· · ·α_n! and ifβ ≤α,

α β

= α!

β!(α−β)!. We have also the Leibniz formula

D^α(uv)(x) = X

β≤α

α β

D^βu(x)D^α−βv(x)

valid for functionsu andv that are |α|times (weakly) differentiable nearx.

Form∈N0∪ {∞} let

C^m(Ω) ={f : Ω→R|D^αf exists and is continuous for allα ∈Nⁿ₀ such that|α| ≤m}

be the space of all m-times continuously differentiable functions. Let C^m(Ω) denote the topological vector space which consists of all those functions φ ∈ C^m(Ω) for which D^αφ is bounded and uniformly continuous on Ω for all 0≤

|α| ≤m. Notice that D^αφ possesses a unique, bounded, continuous extension to the closure Ω of Ω.

C₀^∞(Ω) ={ϕ∈C^∞(Rⁿ) |suppϕis a compact subset of Ω}.

A sequence {f_j} of functions in C₀^∞(Ω) is said to converge to the function f ∈C₀^∞(Ω) provided

• there existsK bΩ (K ⊂Ω and K is compact) such that the supports of allf_j and f lie in K,

• limj→∞D^αfj(x) =D^αf(x) uniformly for each multi-indexα.

Let X be a Banach space, I an interval in R. Define C(I;X) to be the bounded continuous functions of the form

u:I →X, t7→u(x)∈X

which is equipped with the normkuk_C(I;X)= sup_t∈Iku(t)k_X.The spaceCⁿ(I;X) contains functions whose classical derivatives up to ordernare inC(I;X) where the classical derivative is defined as a limit of difference quotients.

Im Dokument Solutions to the Multi-Dimensional Viscous Quantum Hydrodynamic Equations for Semiconductors (Seite 19-26)