2.2 Elliptic BVPs of Mixed Order
2.2.2 Ellipticity with Parameter
The solvability and a priori estimate of solutions to scalar, generally nonselfad-joint elliptic boundary value problems under limited smoothness assumptions and under an ellipticity with parameter condition was derived in [7]. The same method was used in [22] and it extended the results of [7] for a scalar case to that for a system involing a discontinuous weight function where the system is elliptic in usual sense(i.e. the highest order of each entry of the matrix operator is a given even integer).
It is pointed out in [22] that the definition of ellipticity with parameter was different from that in [7] since the prolem mentioned in [22] involved a weight function. In this case one allowed the weight function to have discontinuities at certain hypersurfaces lying in Ω, and therefore further transmission boundary conditions had to be supplemented. Very general results for more general elliptic systems of Agmon-Douglis-Nirenberg type were derived in [34] by making minor modifications of the arguments of [22].
We shall use norms depending upon a parameter η ∈ C/{0}. Precisely let s, m be integers satisfying 1≤s ≤m, v ∈Wps(Ω), assume that the boundary
∂Ω of Ω is of class Cm−1,1. Then we define
|||v|||s,p,Ω :=kvkWs
p(Ω)+|η|mskvkLp(Ω). (2.2.2) The vectorsv∈Wps(Ω) have boundary valuesg=v|∂Ωand we denote the space of these boundary values by Ws−
1 p
p (∂Ω) and the infimum is taken over those v ∈ Wps(Ω) for which v|∂Ω =g. Additionally we use norms depending upon a parameterη∈C/{0}:
|||g|||s−1
p,p,∂Ω:=kgk
Ws−
1p p (∂Ω)
+|η|
s−1 p
m kgkLp(∂Ω).
Now let Lbe a closed sector in the complex plane with vertex at the origin, let m, N ∈N, N >1,{sj}N1 ,{mj}N1 ,{rj}
mN 2
1 denote sequences of integers such that sj+mj =m for j= 1,· · · , N and we supposemN is even. Furthermore we assume that 0 = m1 ≤ m2 ≤ · · · ≤ mN and s1 ≥ s2 ≥ · · · ≥ sN. Then we consider the following boundary value problem of ADN type with constant coefficients.
(A(D)−η)u(x) =f(x) in Ω⊂Rn,
Bj(D) =gj(x) on ∂Ω forj = 1,· · · ,mN 2 ,
(2.2.3)
where Ω is a bounded domain in Rn,n ≥2. u(x) = (u1(x),· · ·, uN(x))T and f(x) = (f1(x),· · ·, fN(x))T are N ×1 matrix functions defined in Ω, gj(x) are scalar functions dedined on ∂Ω. The entries Ajk(D) of the N ×N matrix operatorA(D) are linear differential operators of order not exceeding sj+mk and defined to be 0 if sj +mk <0. Bj(D), 1 ≤j ≤ mN
2 , is a 1×N matrix operator whose entries are linear differential operators defined on ∂Ω of order not exceeding rj +mk with rj < m and defined to be 0 if rj +mk < 0. We suppose further thatA(D) has constant coefficients. Then according to [7] and [34] we obtain the following definition and theorem.
Definition 2.2.2. Let L be a closed sector in the complex plane with vertex at the origin, ∂Ω∈C2,1. The boundary problem (2.2.3) will be called elliptic with parameter in L if the following conditions are satisfied.
1. det( ˚A(ξ)−ηI)6= 0 for ξ ∈Rn and η∈ L if |ξ|+|η| 6= 0.
2. Lopatinskii-Shapiro condition holds. More precisely, fixx0 ∈∂Ω, assume that the boundary problem (2.2.3) is rewritten in a local coordinate system associated with x0; i.e., we establish a new coordinate system first by rotation after which the positive xn-axis has the direction of the interior normal to∂Ωatx0and then translation such thatx07→0. LetΦx0 denote the coordinate transformation with Φx0(0) = x0, ξ0 ∈ Rn−1 and η ∈ L, then the boundary problem on the half-line
A(ξ˚ 0, Dn))v(t)−ηv(t) = 0 for t=xn>0, v(t) = 0 at t= 0,
|v(t)| −→0 as t−→ ∞
(2.2.4)
has only the trivial solution for ξ0 ∈Rn−1 and η∈ L if|ξ0|+|η| 6= 0.
Remark 2.2.1. From Proposition 2.2 in [8] it follows that when Condition 1 of Definition 2.2.2 is satisfied, then mN is even. In [34] the matrix valued case with a diagonal discontinuous weight matrix was studied. For this the given region is subdivided into subregions on which the weights are continuous and
the transmission conditions at the boundaries of the subregions are needed. It is redundant in our case since there is no weight function in (2.2.3), i.e. the weight functionswj(x)≡1 for j= 1,· · ·, N.
In (2.2.4) just the pullback of the original operator with respect to the new coordinate system (after rotation and translation) needs to be considered. Thus according to theorem 2.6 in [34] the definition above gives just the sufficient conditions of the following result.
Theorem 2.2.1. [34] Assume the boundary problem (2.2.3) is elliptic with parameter in the sector L. Then there exists a η0 = η0(p) > 0 such that for η∈ L with|η| ≥η0, the boundary problem (2.2.3) admits a unique solution u= (u1,· · ·, uN)∈QN
j=1Wpmj+m(Ω)for anyf = (f1,· · ·, fN)∈QN
j=1Wpmj(Ω) andg= (g1,· · · , gmN
2 )∈Q
mN 2
j=1Wm−rj−
1 p
p (∂Ω), and the a priori estimate
N
X
j=1
|||uj|||mj+m,p,Ω ≤c
N
X
j=1
|||fj|||mj,p,Ω+
mN 2
X
j=1
|||gj|||m−r
j−1
p,p,∂Ω
(2.2.5) holds, where the constantc does not depend upon f, g, η and we use the norms depending on a parameter η ∈ C/{0}, namely let v ∈ Wps(Ω), g ∈ Ws−
1
p p(∂Ω) where sis an integer satisfying 1≤s≤m,
|||v|||s,p,Ω:=kvkWs
p(Ω)+|η|mskvkLp(Ω), (2.2.6)
|||g|||s−1
p,p,∂Ω:=kgk
Ws−
1p p (∂Ω)
+|η|
s−1 p
m kgkLp(∂Ω). (2.2.7)
Local Solutions to the
Isothermal Viscous QHD
Concerning the inviscid model (the viscosity constant is zero) of quantum hy-drodynamics several results are derived. For instance in [62] the asymptotic behavior for a constant profile of background charges in a torus1is derived. For viscous model only a few results are available. The local existence and unique-ness of solutions for one-dimensional interval with insulating boundary condi-tions and for higher dimensions on a torus were derived in [17]. Afterwards M.
Dreher proved the local existence and uniqueness of solutions of viscous model with a general boundary conditions in [32] where a method of introducing a vis-cous regularization term was used. In this chapter we study the local existence of solutions to viscous quantum hydrodynamics
∂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),
(3.0.1) where the initial data (n0, J0) satisfies
n0 ∈H3(Ω), J0 ∈(H2(Ω))d, (3.0.2) and the boundary conditions read
(n, J, V)(t, x) = (nΓ, JΓ, VΓ) on ∂Ω,
nΓ∈H5/2(∂Ω), JΓ∈H3/2(∂Ω), VΓ∈H3/2(∂Ω),
(3.0.3)
1In geometry, a torus (pl. tori) is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle. PDEs on a torus satisfy the periodic boundary conditions.
21
i.e. the Dirichlet boundary conditions (3.0.3) are given. Furthermore the fol-lowing compatibility conditions are also required:
( (n0, J0)|∂Ω= (nΓ, JΓ), (ν04n0+ divJ0)|∂Ω= 0.
(3.0.4) Here (t, x)∈(0,∞)×Ω denotes the temporal and spatial variables, Ω⊂Rd, d= 1,2,3 with boundary∂Ω. The unknown functions aren,J,V which model the particle density, the momentum and the electrostatic potential respectively.
The given function C(x) models the given profile of background charges. The scaled constants are the temperature T0, the Planck constant , the Debye lengthλ, and a viscosity constantν0, the momentum relaxation timeτ. J⊗J denotes the tensor product with components JiJk and the i-th component of the convective term div
J ⊗J n
equals
d
X
k=1
∂
∂xk JiJk
n
.
The idea to derive the local existence result is first to linearize the system (3.0.1) and then to study the corresponding linear system. Next construct approximate solutions (nk, Jk, Vk) from a fixed-point procedure, which are ex-pected to converge to a solution of the original problem (n, J, V). Namely we shall derive a contraction mapping and then use Banach fixed point theorem.
For this a uniform estimate of the approximate solutions (nk, Jk, Vk) which is the key task is indispensable.
3.1 Linearization of the Original Problem
A reformulation of the Bohm potential term is derived in [32] as follows 2
2n∇
4√
√ n n
= 2
4∇4n−2div (∇√
n)⊗(∇√ n)
. (3.1.1)
Putting U := (n, J)T,J := (J1, . . . , Jd)T we can reformulate the equations for nand J from (3.0.1) as
∂tU+A(∂x)U + 0
G
= 0, (3.1.2)
where
A(∂x) :=
−ν04 −div
−T0∇+2
4∇4 −ν04+τ−1
, (3.1.3)
G:=−div
J ⊗J n
+n∇V −2div (∇√
n)⊗(∇√ n)
. (3.1.4)
By this reformulation we have successfully linearized (3.0.1), i.e., separated the linear term A(∂x) and the nonlinear term G where G has lower order than A(∂x).