A quasineutral type limit for the
Navier Stokes Poisson system with large data
Donatella Donatelli, Pierangelo Marcati
donatell@univaq.it, marcati@univaq.it
Dipartimento di Matematica Pura ed Applicata
Università di L’Aquila
67100 L’Aquila, Italy
Quasineutral limit for Navier Stokes Poisson – p.1/36
Navier Stokes Poisson System in R3
is a simplified model to describe the dynamics of a plasma

λ + div(ρλ uλ ) = 0

∂
ρ

s


λ )γ
∇(ρ
∂s (ρλ uλ )+div(ρλ uλ ⊗ uλ ))+
= µ∆uλ +(µ + ν)∇ div uλ +ρλ ∇V λ

γ


λ2 ∆V λ = ρλ − 1,
x ∈ R3 , s ≥ 0
Quasineutral limit for Navier Stokes Poisson – p.2/36
Navier Stokes Poisson System in R3
is a simplified model to describe the dynamics of a plasma

λ + div(ρλ uλ ) = 0

∂
ρ

s


λ )γ
∇(ρ
∂s (ρλ uλ )+div(ρλ uλ ⊗ uλ ))+
= µ∆uλ +(µ + ν)∇ div uλ +ρλ ∇V λ

γ


λ2 ∆V λ = ρλ − 1,
x ∈ R3 , s ≥ 0
——————————————
ρλ (x, t) is the negative charge density
mλ (x, t) = ρλ (x, t)uλ (x, t) is the current density
uλ (x, t) is the velocity vector density
V λ (x, t) is the electrostatic potential
µ is the shear viscosity and ν is the bulk viscosity
λ is the so called Debye length
Quasineutral limit for Navier Stokes Poisson – p.2/36
Physical background
a charged particle inside a plasma attracts particles with opposite
charge and repels those with the same charge
Quasineutral limit for Navier Stokes Poisson – p.3/36
Physical background
a charged particle inside a plasma attracts particles with opposite
charge and repels those with the same charge
⇓
creation of a net cloud of opposite charge around itself
Quasineutral limit for Navier Stokes Poisson – p.3/36
Physical background
a charged particle inside a plasma attracts particles with opposite
charge and repels those with the same charge
⇓
creation of a net cloud of opposite charge around itself
⇓
the cloud shields the particle’s own charge from external view
Quasineutral limit for Navier Stokes Poisson – p.3/36
Physical background
a charged particle inside a plasma attracts particles with opposite
charge and repels those with the same charge
⇓
creation of a net cloud of opposite charge around itself
Quasineutral limit for Navier Stokes Poisson – p.3/36
Physical background
a charged particle inside a plasma attracts particles with opposite
charge and repels those with the same charge
⇓
creation of a net cloud of opposite charge around itself
the particle’s Coulomb field fall off
as e−r , rather than as 1/r2
Quasineutral limit for Navier Stokes Poisson – p.3/36
Physical background
a charged particle inside a plasma attracts particles with opposite
charge and repels those with the same charge
⇓
creation of a net cloud of opposite charge around itself
the particle’s Coulomb field fall off
as e−r , rather than as 1/r2
infact......let us put a point charge q
Quasineutral limit for Navier Stokes Poisson – p.3/36
V = V (r),
n0 =mean densities of electrons and protons
ne =electron density=n0 eeV /kT
ni =ion density= n0 e−eV /kT
Quasineutral limit for Navier Stokes Poisson – p.4/36
V = V (r), n0 =mean densities of electrons and protons
ne =electron density=n0 eeV /kT ni =ion density= n0 e−eV /kT
n0 e −eV /kT
1
(e
− eeV /kT )
∆V = − (ni − ne ) = −
ǫ0
ǫ0
Quasineutral limit for Navier Stokes Poisson – p.4/36
V = V (r), n0 =mean densities of electrons and protons
ne =electron density=n0 eeV /kT ni =ion density= n0 e−eV /kT
n0 e −eV /kT
1
(e
− eeV /kT )
∆V = − (ni − ne ) = −
ǫ0
ǫ0
potential energy eV ≪ kinetic energy kT
2
n0 e
2n0 e
eV
eV
∆V = −
=
V
1−
−1−
ǫ0
kT
kT
ǫ0 kT
Quasineutral limit for Navier Stokes Poisson – p.4/36
V = V (r), n0 =mean densities of electrons and protons
ne =electron density=n0 eeV /kT ni =ion density= n0 e−eV /kT
n0 e −eV /kT
1
(e
− eeV /kT )
∆V = − (ni − ne ) = −
ǫ0
ǫ0
potential energy eV ≪ kinetic energy kT
2
n0 e
2n0 e
eV
eV
∆V = −
=
V
1−
−1−
ǫ0
kT
kT
ǫ0 kT
λ=
r
ǫ0 kT
= Debye lenght
2
2n0 e
Quasineutral limit for Navier Stokes Poisson – p.4/36
V = V (r), n0 =mean densities of electrons and protons
ne =electron density=n0 eeV /kT ni =ion density= n0 e−eV /kT
n0 e −eV /kT
1
(e
− eeV /kT )
∆V = − (ni − ne ) = −
ǫ0
ǫ0
potential energy eV ≪ kinetic energy kT
2
n0 e
2n0 e
eV
eV
∆V = −
=
V
1−
−1−
ǫ0
kT
kT
ǫ0 kT
λ=
r
ǫ0 kT
= Debye lenght
2
2n0 e


 1 ∂ r2 ∂V = V
r2 ∂r
∂r
λ2
q

V (near r = 0) =
, V (+∞) = 0
4πǫ0 r
qe−r/λ
=⇒ V (r) =
4πǫ0 r
Quasineutral limit for Navier Stokes Poisson – p.4/36
V (r) =
qe−r/λ
4πǫ0 r
λ=
r
ǫ0 kT
2n0 e2
Quasineutral limit for Navier Stokes Poisson – p.5/36
qe−r/λ
r
ǫ0 kT
V (r) =
λ=
4πǫ0 r
2n0 e2
⇒ the plasma beyond a distance λ is essentially shielded from the
effects of the charge
Quasineutral limit for Navier Stokes Poisson – p.5/36
qe−r/λ
r
ǫ0 kT
V (r) =
λ=
4πǫ0 r
2n0 e2
⇒ the plasma beyond a distance λ is essentially shielded from the
effects of the charge
⇒ λ is the “screening” distance or the distance over which the
usual Coulomb field 1/r is killed off exponentially by the
polarization of the plasma
Quasineutral limit for Navier Stokes Poisson – p.5/36
qe−r/λ
r
ǫ0 kT
V (r) =
λ=
4πǫ0 r
2n0 e2
⇒ the plasma beyond a distance λ is essentially shielded from the
effects of the charge
⇒ λ is the “screening” distance or the distance over which the
usual Coulomb field 1/r is killed off exponentially by the
polarization of the plasma
⇒ we don’t expect to find electric fields existing in a plasma over
regions greater in extend than λ
Quasineutral limit for Navier Stokes Poisson – p.5/36
qe−r/λ
r
ǫ0 kT
V (r) =
λ=
4πǫ0 r
2n0 e2
⇒ the plasma beyond a distance λ is essentially shielded from the
effects of the charge
⇒ λ is the “screening” distance or the distance over which the
usual Coulomb field 1/r is killed off exponentially by the
polarization of the plasma
⇒ we don’t expect to find electric fields existing in a plasma over
regions greater in extend than λ
ǫ0 = dielectric constant = 8, 85 · 10−12 C 2 /m2 N
k = Boltzmann constant = 1, 38 · 10−23 N m/K
T = electron temperature = 104 K
n0 = electron density = 1016 m−3
e = electron charge = −1, 6 · 10−19 C
Quasineutral limit for Navier Stokes Poisson – p.5/36
qe−r/λ
r
ǫ0 kT
V (r) =
λ=
4πǫ0 r
2n0 e2
⇒ the plasma beyond a distance λ is essentially shielded from the
effects of the charge
⇒ λ is the “screening” distance or the distance over which the
usual Coulomb field 1/r is killed off exponentially by the
polarization of the plasma
⇒ we don’t expect to find electric fields existing in a plasma over
regions greater in extend than λ
λ ≈ 10−4 m
Quasineutral limit for Navier Stokes Poisson – p.5/36
Navier Stokes Poisson System in R3

λ + div(ρλ uλ ) = 0

∂
ρ

s


λ )γ
∇(ρ
= µ∆uλ +(µ + ν)∇ div uλ +ρλ ∇V λ
∂t (ρλ uλ )+div(ρλ uλ ⊗ uλ ))+

γ


λ2 ∆V λ = ρλ − 1
Quasineutral limit for Navier Stokes Poisson – p.6/36
Navier Stokes Poisson System in R3

λ + div(ρλ uλ ) = 0

∂
ρ

s


λ )γ
∇(ρ
= µ∆uλ +(µ + ν)∇ div uλ +ρλ ∇V λ
∂t (ρλ uλ )+div(ρλ uλ ⊗ uλ ))+

γ


λ2 ∆V λ = ρλ − 1
λ = Debye lenght
↓0
Quasineutral limit for Navier Stokes Poisson – p.6/36
Navier Stokes Poisson System in R3

λ + div(ρλ uλ ) = 0

∂
ρ

s


λ )γ
∇(ρ
= µ∆uλ +(µ + ν)∇ div uλ +ρλ ∇V λ
∂t (ρλ uλ )+div(ρλ uλ ⊗ uλ ))+

γ


λ2 ∆V λ = ρλ − 1
λ = Debye lenght
↓0
Incompressible Navier Stokes equations
(
∂t u + u · ∇u − ∆u + ∇p = 0
div u = 0
Quasineutral limit for Navier Stokes Poisson – p.6/36
Notations
Nonhomogenous Sobolev Spaces
W
k,p
d
(R ) = (I − ∆)
H k (Rd ) = W k,2 (Rd )
− k2
p
d
L (R )
Homogenous Sobolev Spaces
Ẇ
k,p
d
(R ) = (−∆)
− k2
Lp (Rd )
Ḣ k (Rd ) = W k,2 (Rd )
Quasineutral limit for Navier Stokes Poisson – p.7/36
Notations
Nonhomogenous Sobolev Spaces
W
k,p
d
(R ) = (I − ∆)
− k2
p
Homogenous Sobolev Spaces
d
L (R )
Ẇ
H k (Rd ) = W k,2 (Rd )
k,p
d
(R ) = (−∆)
− k2
Lp (Rd )
Ḣ k (Rd ) = W k,2 (Rd )
Orlicz space
Lp2 (Rd ) = {f ∈ L1loc (Rd ) | |f |χ|f |≤ 1 ∈ L2 (Rd ), |f |χ|f |> 1 ∈ Lp (Rd )}
2
2
Quasineutral limit for Navier Stokes Poisson – p.7/36
Notations
Nonhomogenous Sobolev Spaces
W
k,p
d
(R ) = (I − ∆)
− k2
p
Homogenous Sobolev Spaces
d
L (R )
Ẇ
H k (Rd ) = W k,2 (Rd )
k,p
d
(R ) = (−∆)
− k2
Lp (Rd )
Ḣ k (Rd ) = W k,2 (Rd )
Orlicz space
Lp2 (Rd ) = {f ∈ L1loc (Rd ) | |f |χ|f |≤ 1 ∈ L2 (Rd ), |f |χ|f |> 1 ∈ Lp (Rd )}
2
2
Leray Projectors
Q = ∇∆−1 div
P =I −Q
projection on gradient vector fields
projection on divergence - free vector fields
Quasineutral limit for Navier Stokes Poisson – p.7/36
Existence Incompressible Navier Stokes (Leray ’34)
u satisfies the NS equation in the sense of distribution
Quasineutral limit for Navier Stokes Poisson – p.8/36
Existence Incompressible Navier Stokes (Leray ’34)
u satisfies the NS equation in the sense of distribution
∂ϕ
dxdt
∇u · ∇ϕ − ui uj ∂i ϕj − u ·
∂t
0 Rd
Z T
Z
=
hf, ϕiH −1 ×H01 dxdt +
u0 · ϕdx,
Z TZ
Rd
0
for all ϕ ∈ C0∞ (Rd × [0, T ]), div ϕ = 0 and
div u = 0
in D′ (Rd × [0, T ])
Quasineutral limit for Navier Stokes Poisson – p.8/36
Existence Incompressible Navier Stokes (Leray ’34)
u satisfies the NS equation in the sense of distribution
the following energy inequality hold
Z
Z tZ
1
|u(x, t)|2 dx + µ
|∇u(x, t)|2 dxds
2 Rd
0 Rd
Z
Z t
1
≤
|u0 |2 dx +
hf, uiH −1 ×H01 ds,
for all t ≥ 0.
2 Rd
0
Quasineutral limit for Navier Stokes Poisson – p.8/36
Existence Incompressible Navier Stokes (Leray ’34)
u satisfies the NS equation in the sense of distribution
the following energy inequality hold
Z
Z tZ
1
|u(x, t)|2 dx + µ
|∇u(x, t)|2 dxds
2 Rd
0 Rd
Z
Z t
1
≤
|u0 |2 dx +
hf, uiH −1 ×H01 ds,
for all t ≥ 0.
2 Rd
0
u ∈ L2t Hx1 ∩ Ct (L2w ∩ Lσloc ), 1 ≤ σ < 2,
Ns
q
∂u
2 H −1 + Ls W −1, N s−2 ∩ L Lr
∈
L
t x
t
t x
∂t
Ns
s
2
N
p ∈ Lloc + Lt L s−2 ,
∇p ∈ L2t Hx−1 + Lqt Lrx
Quasineutral limit for Navier Stokes Poisson – p.8/36
Existence for Navier Stokes Poisson

λ + div(ρλ uλ ) = 0

∂
ρ

s


λ )γ
∇(ρ
= µ∆uλ +(µ + ν)∇ div uλ +ρλ ∇V λ
∂t (ρλ uλ )+div(ρλ uλ ⊗ uλ ))+

γ


λ2 ∆V λ = ρλ − 1
ρλ (x, 0) = ρλ0 (x)
ρλ (x, 0)uλ (x, 0) = ρλ0 (x)uλ0 (x)
V λ (x, 0) = V0λ (x)
Quasineutral limit for Navier Stokes Poisson – p.9/36
Existence for Navier Stokes Poisson

λ + div(ρλ uλ ) = 0

∂
ρ

s


λ )γ
∇(ρ
= µ∆uλ +(µ + ν)∇ div uλ +ρλ ∇V λ
∂t (ρλ uλ )+div(ρλ uλ ⊗ uλ ))+

γ


λ2 ∆V λ = ρλ − 1
ρλ (x, 0) = ρλ0 (x)
ρλ (x, 0)uλ (x, 0) = ρλ0 (x)uλ0 (x)
V λ (x, 0) = V0λ (x)
bounded domain Ω, (D. Donatelli, 2003),
Dirichlet boundary condition: uλ = 0 on ∂Ω × (0, T )
Fixed point argument on V λ + compactness arguments
⇓
weak solution
Quasineutral limit for Navier Stokes Poisson – p.9/36
Existence for Navier Stokes Poisson

λ + div(ρλ uλ ) = 0

∂
ρ

s


λ )γ
∇(ρ
= µ∆uλ +(µ + ν)∇ div uλ +ρλ ∇V λ
∂t (ρλ uλ )+div(ρλ uλ ⊗ uλ ))+

γ


λ2 ∆V λ = ρλ − 1
ρλ (x, 0) = ρλ0 (x)
ρλ (x, 0)uλ (x, 0) = ρλ0 (x)uλ0 (x)
V λ (x, 0) = V0λ (x)
whole space Rd , unbounded domains in Rd , (B. Ducomet, E.
Feireisl, H. Petzeltová, and I. Straškraba, 2004)
The proof is based on the analysis of two defect measures
oscp (Ω)[ρn − ρ] = sup(lim sup kTk (ρn ) − Tk (ρ)kLp (Ω) )
n→∞
k≥1
Z
df t[ρn − ρ] =
ρ log ρ(t) − ρ log ρ(t)dx
Ω
where ρn is an approximating sequence of ρλ , Tk (ρ) is a
cut-off function
Quasineutral limit for Navier Stokes Poisson – p.9/36
Existence theorem
Let γ > 3/2, then there exits a global weak solution (ρλ , uλ , V λ ) to
p
γ
λ
∞
2 , uλ ∈ L2 H 1 .
NSP such that ρ − 1 ∈ Lt L2x , ρλ uλ ∈ L∞
L
x
t
t x
The energy inequality holds for almost every t ≥ 0,
Z λ
2
2
1
λ |u |
λ γ λ
ρ
+
(ρ ) + |∇V λ |2 dx
2
γ−1
2
R3
Z tZ +
µ|∇uλ |2 +(ν + µ)| div uλ |2 dxds ≤ C0 .
0
R3
Quasineutral limit for Navier Stokes Poisson – p.10/36
Existence theorem
Let γ > 3/2, then there exits a global weak solution (ρλ , uλ , V λ ) to
p
γ
λ
∞
2 , uλ ∈ L2 H 1 .
NSP such that ρ − 1 ∈ Lt L2x , ρλ uλ ∈ L∞
L
x
t
t x
The energy inequality holds for almost every t ≥ 0,
Z λ
2
2
1
λ |u |
λ γ λ
ρ
+
(ρ ) + |∇V λ |2 dx
2
γ−1
2
R3
Z tZ +
µ|∇uλ |2 +(ν + µ)| div uλ |2 dxds ≤ C0 .
0
R3
The continuity equation is satisfied in the sense of
renormalized solutions, i.e.:
∂t b(ρλ ) + div(b(ρλ )uλ ) + (b′ (ρλ )ρλ − b(ρλ )) div uλ = 0,
for any b ∈ C 1 (R3 ) such that b′ (z) = cons., for any z , z ≥ M .
Quasineutral limit for Navier Stokes Poisson – p.10/36
Existence theorem
Let γ > 3/2, then there exits a global weak solution (ρλ , uλ , V λ ) to
p
γ
λ
∞
2 , uλ ∈ L2 H 1 .
NSP such that ρ − 1 ∈ Lt L2x , ρλ uλ ∈ L∞
L
x
t
t x
The energy inequality holds for almost every t ≥ 0,
Z λ
2
2
1
λ |u |
λ γ λ
ρ
+
(ρ ) + |∇V λ |2 dx
2
γ−1
2
R3
Z tZ +
µ|∇uλ |2 +(ν + µ)| div uλ |2 dxds ≤ C0 .
0
R3
The continuity equation is satisfied in the sense of
renormalized solutions, i.e.:
∂t b(ρλ ) + div(b(ρλ )uλ ) + (b′ (ρλ )ρλ − b(ρλ )) div uλ = 0,
for any b ∈ C 1 (R3 ) such that b′ (z) = cons., for any z , z ≥ M .
The NSP system holds in D′ .
Quasineutral limit for Navier Stokes Poisson – p.10/36

Scaling
λ
λ
λ

∂s ρ + div(ρ u ) = 0



λ γ
∇(ρ
)
λ λ
λ λ
λ
∂t (ρ u )+div(ρ u ⊗ u ))+
= µ∆uλ +(µ + ν)∇ div uλ +ρλ ∇V λ

γ


 2
λ ∆V λ = ρλ − 1
Quasineutral limit for Navier Stokes Poisson – p.11/36

Scaling
λ
λ
λ

∂s ρ + div(ρ u ) = 0



λ γ
∇(ρ
)
λ λ
λ λ
λ
∂t (ρ u )+div(ρ u ⊗ u ))+
= µ∆uλ +(µ + ν)∇ div uλ +ρλ ∇V λ

γ


 2
λ ∆V λ = ρλ − 1
t
Time scaling: s =
µ = εµ, ν = εν
ε
t
t
t
1 λ
ε
λ
ε
ε
λ
, u = u x,
, V =V
x,
.
ρ (x, t) = ρ x,
ε
ε
ε
ε
Quasineutral limit for Navier Stokes Poisson – p.11/36

Scaling
λ
λ
λ

∂s ρ + div(ρ u ) = 0



λ γ
∇(ρ
)
λ λ
λ λ
λ
∂t (ρ u )+div(ρ u ⊗ u ))+
= µ∆uλ +(µ + ν)∇ div uλ +ρλ ∇V λ

γ


 2
λ ∆V λ = ρλ − 1
t
Time scaling: s =
µ = εµ, ν = εν
ε
t
t
t
1 λ
ε
λ
ε
ε
λ
, u = u x,
, V =V
x,
.
ρ (x, t) = ρ x,
ε
ε
ε
ε
 ε
ε uε ) = 0
∂
ρ
+
div(ρ

t


ε )γ
ε
∇(ρ
ρ
ε
ε
ε
∂t (ρε uε ) + div(ρε uε ⊗ uε )+
=
µ∆u
+(ν
+
µ)∇
div
u
+
∇V

γε2
ε2

 2
λ ∆V ε = ρε − 1.
Quasineutral limit for Navier Stokes Poisson – p.11/36

Scaling
λ
λ
λ

∂s ρ + div(ρ u ) = 0



λ γ
∇(ρ
)
λ λ
λ λ
λ
∂t (ρ u )+div(ρ u ⊗ u ))+
= µ∆uλ +(µ + ν)∇ div uλ +ρλ ∇V λ

γ


 2
λ ∆V λ = ρλ − 1
t
Time scaling: s =
µ = εµ, ν = εν
ε
t
t
t
1 λ
ε
λ
ε
ε
λ
, u = u x,
, V =V
x,
.
ρ (x, t) = ρ x,
ε
ε
ε
ε
 ε
ε uε ) = 0
∂
ρ
+
div(ρ

t


ε )γ
ε
∇(ρ
ρ
ε
ε
ε
∂t (ρε uε ) + div(ρε uε ⊗ uε )+
=
µ∆u
+(ν
+
µ)∇
div
u
+
∇V

γε2
ε2

 2
λ ∆V ε = ρε − 1.
ε β = λ2 ,
where β > 0
Quasineutral limit for Navier Stokes Poisson – p.11/36
 ε
ε uε ) = 0
∂
ρ
+
div(ρ

t


ε )γ
ε
∇(ρ
ρ
ε
ε
ε
∂t (ρε uε ) + div(ρε uε ⊗ uε )+
=
µ∆u
+(ν
+
µ)∇
div
u
+
∇V

γε2
ε2

 β
ε ∆V ε = ρε − 1.
Renormalized pressure:
ε )γ − 1 − γ(ρε − 1)
(ρ
πε =
ε2 γ(γ − 1)
Quasineutral limit for Navier Stokes Poisson – p.12/36
 ε
ε uε ) = 0
∂
ρ
+
div(ρ

t


ε )γ
ε
∇(ρ
ρ
ε
ε
ε
∂t (ρε uε ) + div(ρε uε ⊗ uε )+
=
µ∆u
+(ν
+
µ)∇
div
u
+
∇V

γε2
ε2

 β
ε ∆V ε = ρε − 1.
Renormalized pressure:
ε )γ − 1 − γ(ρε − 1)
(ρ
πε =
ε2 γ(γ − 1)
Initial conditions:
Z ε
2
|m0 |
ε
β−2 ε 2
π |t=0 +
ε
|V0 | dx ≤ C0 , where
+
ε
2ρ0
R3
ε
p
m
p 0ε ⇀ u0 weakly in L2 (R3 )
ρε uε |t=0 = mε0 mε0 dx ≪ ρε0 dx
ρ0
Quasineutral limit for Navier Stokes Poisson – p.12/36
Main Theorem
Let be (ρε , uε ) a weak solution in R3 , then
(i) ρε −→ 1
strongly in L∞ ([0, T ]; Lk2 (R3 ))
Quasineutral limit for Navier Stokes Poisson – p.13/36
Main Theorem
Let be (ρε , uε ) a weak solution in R3 , then
(i) ρε −→ 1
strongly in L∞ ([0, T ]; Lk2 (R3 ))
(ii) uε ⇀ u weakly in L∞ ([0, T ]; L2 (R3 )) ∩ L2 ([0, T ]; Ḣ 1 (R3 )).
Quasineutral limit for Navier Stokes Poisson – p.13/36
Main Theorem
Let be (ρε , uε ) a weak solution in R3 , then
(i) ρε −→ 1
strongly in L∞ ([0, T ]; Lk2 (R3 ))
(ii) uε ⇀ u weakly in L∞ ([0, T ]; L2 (R3 )) ∩ L2 ([0, T ]; Ḣ 1 (R3 )).
(iii) Quε −→ 0 strongly in L2 ([0, T ]; Lp (R3 )), for any p ∈ [4, 6),
provided β < 1/2.
Quasineutral limit for Navier Stokes Poisson – p.13/36
Main Theorem
Let be (ρε , uε ) a weak solution in R3 , then
(i) ρε −→ 1
strongly in L∞ ([0, T ]; Lk2 (R3 ))
(ii) uε ⇀ u weakly in L∞ ([0, T ]; L2 (R3 )) ∩ L2 ([0, T ]; Ḣ 1 (R3 )).
(iii) Quε −→ 0 strongly in L2 ([0, T ]; Lp (R3 )), for any p ∈ [4, 6),
provided β < 1/2.
(iv) P uε −→ P u = u strongly in L2 ([0, T ]; L2loc (R3 )).
Quasineutral limit for Navier Stokes Poisson – p.13/36
(v) u = P u is a Leray weak solution to the incompressible Navier
Stokes equation
P (∂t u − ∆u + (u · ∇)u) = 0
in D′ ([0, T ] × R3 ),
provided that

n
o
β = min 1 , −2 + 4
if γ < 2
2 n
γ
o
0 < β < min 1 , κ 3 − 1 , 1 − 2 κ s0 + 7 − 3
if γ ≥ 2
2
q
2
6
3
4
q
where κ > 0, s0 ≥ 3/2 and 4 ≤ q < 6.
Quasineutral limit for Navier Stokes Poisson – p.14/36
D. Donatelli P.Marcati, A quasineutral type limit for the
Navier Stokes Poisson system with large data, Nonlinearity,
21, (2008), 135-148.
Quasineutral limit for Navier Stokes Poisson – p.15/36
D. Donatelli P.Marcati, A quasineutral type limit for the
Navier Stokes Poisson system with large data, Nonlinearity,
21, (2008), 135-148.
References
Quasineutral limit for Euler Poisson system in H s
Cordier and Grenier (’00), Grenier (’96), Cordier, Degond,
Markowich and Schmeiser (’96), Loeper (’05), Peng,
Wang and Yong (’06)
Quasineutral limit for Navier Stokes Poisson – p.15/36
D. Donatelli P.Marcati, A quasineutral type limit for the
Navier Stokes Poisson system with large data, Nonlinearity,
21, (2008), 135-148.
References
Quasineutral limit for Euler Poisson system in H s
Cordier and Grenier (’00), Grenier (’96), Cordier, Degond,
Markowich and Schmeiser (’96), Loeper (’05), Peng,
Wang and Yong (’06)
Quasineutral limit for Navier Stokes Poisson system in H s
Quasineutral limit for Navier Stokes Poisson – p.15/36
D. Donatelli P.Marcati, A quasineutral type limit for the
Navier Stokes Poisson system with large data, Nonlinearity,
21, (2008), 135-148.
References
Quasineutral limit for Euler Poisson system in H s
Cordier and Grenier (’00), Grenier (’96), Cordier, Degond,
Markowich and Schmeiser (’96), Loeper (’05), Peng,
Wang and Yong (’06)
Quasineutral limit for Navier Stokes Poisson system in H s
Wang (’04) and Jiang and Wang (’06)
Quasineutral limit for Navier Stokes Poisson – p.15/36
D. Donatelli P.Marcati, A quasineutral type limit for the
Navier Stokes Poisson system with large data, Nonlinearity,
21, (2008), 135-148.
References
Quasineutral limit for Euler Poisson system in H s
Cordier and Grenier (’00), Grenier (’96), Cordier, Degond,
Markowich and Schmeiser (’96), Loeper (’05), Peng,
Wang and Yong (’06)
Quasineutral limit for Navier Stokes Poisson system in H s
Wang (’04) and Jiang and Wang (’06)
Combined quasineutral and relaxation time limit
Quasineutral limit for Navier Stokes Poisson – p.15/36
D. Donatelli P.Marcati, A quasineutral type limit for the
Navier Stokes Poisson system with large data, Nonlinearity,
21, (2008), 135-148.
References
Quasineutral limit for Euler Poisson system in H s
Cordier and Grenier (’00), Grenier (’96), Cordier, Degond,
Markowich and Schmeiser (’96), Loeper (’05), Peng,
Wang and Yong (’06)
Quasineutral limit for Navier Stokes Poisson system in H s
Wang (’04) and Jiang and Wang (’06)
Combined quasineutral and relaxation time limit
Gasser and Marcati (’01), (’03)
Quasineutral limit for Navier Stokes Poisson – p.15/36
A priori estimate
Z 2
γ
ε
ε
ε
(ρ ) − 1 − γ(ρ − 1) β−2
ε |u |
ε 2
|∇V
| dx
+
+ε
ρ
2
2
ε γ(γ − 1)
R3
Z tZ
ε 2
ε 2
µ|∇u | +(ν + µ)| div u | dxds ≤ C0 .
+
0
R3
Quasineutral limit for Navier Stokes Poisson – p.16/36
A priori estimate
Z 2
γ
ε
ε
ε
(ρ ) − 1 − γ(ρ − 1) β−2
ε |u |
ε 2
|∇V
| dx
+
+ε
ρ
2
2
ε γ(γ − 1)
R3
Z tZ
ε 2
ε 2
µ|∇u | +(ν + µ)| div u | dxds ≤ C0 .
+
0
R3
the convexity of z → z γ − 1 − γ(z − 1)
⇓
ε−1
ρ
k
L
density fluctuation = σ ε =
∈ L∞
2
t
ε
Quasineutral limit for Navier Stokes Poisson – p.16/36
A priori estimate
Z 2
γ
ε
ε
ε
(ρ ) − 1 − γ(ρ − 1) β−2
ε |u |
ε 2
|∇V
| dx
+
+ε
ρ
2
2
ε γ(γ − 1)
R3
Z tZ
ε 2
ε 2
µ|∇u | +(ν + µ)| div u | dxds ≤ C0 .
+
0
R3
the convexity of z → z γ − 1 − γ(z − 1)
⇓
ε−1
ρ
k
L
density fluctuation = σ ε =
∈ L∞
2
t
ε
=⇒
Quasineutral limit for Navier Stokes Poisson – p.16/36
A priori estimate
Z 2
γ
ε
ε
ε
(ρ ) − 1 − γ(ρ − 1) β−2
ε |u |
ε 2
|∇V
| dx
+
+ε
ρ
2
2
ε γ(γ − 1)
R3
Z tZ
ε 2
ε 2
µ|∇u | +(ν + µ)| div u | dxds ≤ C0 .
+
0
R3
the convexity of z → z γ − 1 − γ(z − 1)
⇓
ε−1
ρ
k
L
density fluctuation = σ ε =
∈ L∞
2
t
ε
=⇒
ε
∇u
is bounded in
L2t,x ,
ε
β
−1
2
∇V ε
2
L
is bounded in L∞
x,
t
Quasineutral limit for Navier Stokes Poisson – p.16/36
A priori estimate
Z 2
γ
ε
ε
ε
(ρ ) − 1 − γ(ρ − 1) β−2
ε |u |
ε 2
|∇V
| dx
+
+ε
ρ
2
2
ε γ(γ − 1)
R3
Z tZ
ε 2
ε 2
µ|∇u | +(ν + µ)| div u | dxds ≤ C0 .
+
0
the convexity of z → z γ − 1 − γ(z − 1)
⇓
ε−1
ρ
k
L
density fluctuation = σ ε =
∈ L∞
2
t
ε
=⇒
ε
∇u
uε
R3
is bounded in
L2t,x ,
ε
is bounded in L2t,x ∩ L2t L6x
β
−1
2
∇V ε
2
L
is bounded in L∞
x,
t
σ ε uε
is bounded in L2t Hx−1
Quasineutral limit for Navier Stokes Poisson – p.16/36
Estimate on Quε - Part 1
Since
ρε = εσ ε + 1
we have
Quasineutral limit for Navier Stokes Poisson – p.17/36
Estimate on Quε - Part 1
Since
ρε = εσ ε + 1
we have
Quε = Q(ρε uε ) − εQ(σ ε uε )
Quasineutral limit for Navier Stokes Poisson – p.17/36
Estimate on Quε - Part 1
Since
ρε = εσ ε + 1
we have
Quε = Q(ρε uε ) − εQ(σ ε uε )
| {z }
L2t Hx−1
Quasineutral limit for Navier Stokes Poisson – p.17/36
Estimate on Quε - Part 1
Since
ρε = εσ ε + 1
we have
Quε = Q(ρε uε ) − εQ(σ ε uε )
| {z }
| {z }
→0 ?
but......
L2t Hx−1
Quasineutral limit for Navier Stokes Poisson – p.17/36
Estimate on Quε - Part 1
Since
ρε = εσ ε + 1
we have
Quε = Q(ρε uε ) − εQ(σ ε uε )
| {z }
| {z }
→0 ?
but......
L2t Hx−1
Q(ρε uε ) = ∇∆−1 div(ρε uε )
Quasineutral limit for Navier Stokes Poisson – p.17/36
Estimate on Quε - Part 1
Since
ρε = εσ ε + 1
we have
Quε = Q(ρε uε ) − εQ(σ ε uε )
| {z }
| {z }
→0 ?
but......
Q(ρε uε ) = ∇∆−1 div(ρε uε )
L2t Hx−1
ε∂t σ ε = − div(ρε uε )
Quasineutral limit for Navier Stokes Poisson – p.17/36
Estimate on Quε - Part 1
Since
ρε = εσ ε + 1
we have
Quε = Q(ρε uε ) − εQ(σ ε uε )
| {z }
| {z }
→0 ?
L2t Hx−1
but......
Q(ρε uε ) = ∇∆−1 div(ρε uε )
ε∂t σ ε = − div(ρε uε )
Q(ρε uε ) = ε∇∆−1 ∂t σ ε
Quasineutral limit for Navier Stokes Poisson – p.17/36
Density fluctuation wave equation
1
∂t σ + div(ρε uε ) = 0
ε
1
ε ε
∂t (ρ u ) + ∇σ ε = µ∆uε + (ν + µ)∇ div uε − div(ρε uε ⊗ uε )
ε
ε
1
σ
ε
ε
− (γ − 1)∇π + ∇V + 2 ∇V ε ,
ε
ε
εβ−1 ∆V ε = σ ε .
ε
Quasineutral limit for Navier Stokes Poisson – p.18/36
Density fluctuation wave equation
1
∂t σ + div(ρε uε ) = 0
ε
1
ε ε
∂t (ρ u ) + ∇σ ε = µ∆uε + (ν + µ)∇ div uε − div(ρε uε ⊗ uε )
ε
ε
1
σ
ε
ε
− (γ − 1)∇π + ∇V + 2 ∇V ε ,
ε
ε
εβ−1 ∆V ε = σ ε .
ε
Differentiate in t the “density fluctuation equation”, taking the
divergence of the second equation
ε2 ∂tt σ ε − ∆σ ε = −ε2 div(µ∆uε + (ν + µ)∇ div uε )
+ ε2 div div(ρε uε ⊗ uε ) + ε2 (γ − 1) div ∇π ε
− ε div(σ ε ∇V ε ) − div ∇V ε
Quasineutral limit for Navier Stokes Poisson – p.18/36
changing the time scale: t = ετ
σ̃(x, τ ) = σ ε (x, ετ ),
ρ̃(x, τ ) = ρε (x, ετ )
ũ(x, τ ) = uε (x, ετ ),
Ṽ (x, τ ) = V ε (x, ετ )
Quasineutral limit for Navier Stokes Poisson – p.19/36
changing the time scale: t = ετ
σ̃(x, τ ) = σ ε (x, ετ ),
ρ̃(x, τ ) = ρε (x, ετ )
ũ(x, τ ) = uε (x, ετ ),
Ṽ (x, τ ) = V ε (x, ετ )
∂τ τ σ̃ − ∆σ̃ = −ε2 div(µ∆ũ + (ν + µ)∇ div ũ)
+ ε2 div(div(ρ̃ũ ⊗ ũ) + (γ − 1)∇π̃)
− ε div(σ̃∇Ṽ ) − div ∇Ṽ .
Quasineutral limit for Navier Stokes Poisson – p.19/36
changing the time scale: t = ετ
σ̃(x, τ ) = σ ε (x, ετ ),
ρ̃(x, τ ) = ρε (x, ετ )
ũ(x, τ ) = uε (x, ετ ),
Ṽ (x, τ ) = V ε (x, ετ )
∂τ τ σ̃ − ∆σ̃ = − ε2 div (µ∆ũ + (ν + µ)∇ div ũ)
|
{z
}
L2t Hx−1
+ ε2 div(div (ρ̃ũ ⊗ ũ) + (γ − 1)∇ |{z}
π̃ )
| {z }
∞ 1
1
L∞
t Lx
Lt Lx
-ε div (σ̃∇Ṽ ) − div |{z}
∇Ṽ .
| {z }
∞ 2
1
L∞
t Lx
Lt Lx
Quasineutral limit for Navier Stokes Poisson – p.19/36
Strichartz Estimates
w is a (weak) solution of
2
∂
2w =
− ∆ w = F w(0, ·) = f, ∂t w(0, ·) = g, (x, t) ∈ Rd ×[0, T ]
∂t
Quasineutral limit for Navier Stokes Poisson – p.20/36
Strichartz Estimates
w is a (weak) solution of
2
∂
2w =
− ∆ w = F w(0, ·) = f, ∂t w(0, ·) = g, (x, t) ∈ Rd ×[0, T ]
∂t
sup (kwkḢx1 + kwt kL2x ) ≤ kf kḢx1 + kgkL2x +
t∈[0,T ]
Z
0
T
kF kL2x ds.
Quasineutral limit for Navier Stokes Poisson – p.20/36
Strichartz Estimates
w is a (weak) solution of
2
∂
2w =
− ∆ w = F w(0, ·) = f, ∂t w(0, ·) = g, (x, t) ∈ Rd ×[0, T ]
∂t
sup (kwkḢx1 + kwt kL2x ) ≤ kf kḢx1 + kgkL2x +
t∈[0,T ]
Z
0
T
kF kL2x ds.
we would like to have
kwkLqt Lrx ≤ kf kḢxγ + kgkḢxγ−1 + kF kLq̃′ Lr̃′
t
x
γ small
Quasineutral limit for Navier Stokes Poisson – p.20/36
Strichartz Estimates
w is a (weak) solution of
2
∂
2w =
− ∆ w = F w(0, ·) = f, ∂t w(0, ·) = g, (x, t) ∈ Rd ×[0, T ]
∂t
sup (kwkḢx1 + kwt kL2x ) ≤ kf kḢx1 + kgkL2x +
t∈[0,T ]
Z
T
kF kL2x ds.
0
we would like to have
kwkLqt Lrx ≤ kf kḢxγ + kgkḢxγ−1 + kF kLq̃′ Lr̃′
t
x
γ small
Strichartz (1977) proved that
kwkL4t,x ≤ kf kḢx1/2 + kgkḢx−1/2 + kF kL4/3
t,x
Quasineutral limit for Navier Stokes Poisson – p.20/36
Strichartz Estimates (Ginibre-Velo (’95), Keel-Tao(’98))
2w(t, x) = F (t, x)
w(0, ·) = f,
∂t w(0, ·) = g,
(x, t) ∈ Rd ×[0, T ]
Quasineutral limit for Navier Stokes Poisson – p.21/36
Strichartz Estimates (Ginibre-Velo (’95), Keel-Tao(’98))
2w(t, x) = F (t, x)
w(0, ·) = f,
∂t w(0, ·) = g,
(x, t) ∈ Rd ×[0, T ]
kwkLqt Lrx + k∂t wkLq Wx−1,r . kf kḢxγ + kgkḢxγ−1 + kF kLq̃′ Lr̃′
t
t
x
Quasineutral limit for Navier Stokes Poisson – p.21/36
Strichartz Estimates (Ginibre-Velo (’95), Keel-Tao(’98))
2w(t, x) = F (t, x)
w(0, ·) = f,
∂t w(0, ·) = g,
(x, t) ∈ Rd ×[0, T ]
kwkLqt Lrx + k∂t wkLq Wx−1,r . kf kḢxγ + kgkḢxγ−1 + kF kLq̃′ Lr̃′
t
(q, r),
t
are
(q̃, r̃)
wave
admissible
x
pairs
if
1/q
1/2
((d−3)/2(d−2),1/2)
q, r, q̃, r̃ ≥ 2
2
1
1
q ≤ (d − 1) 2 − r 1
1
2
≤
(d
−
1)
−
q̃
2
r̃
1
d
d
1
+
=
−
γ
=
q
r
2
q̃ ′ +
1/2
1/r
d
r̃′
−2
Quasineutral limit for Navier Stokes Poisson – p.21/36
1/q
Strichartz d=3
1/2
q, r, q̃, r̃ ≥ 2
1
1
1
≤
−
q
2
r
1
1
1
≤
−
q̃
2
r̃
1
3
3
+
=
q
r
2 −γ =
1/2
1
q̃ ′
+
3
r̃′
−2
1/r
kwkLqt Lrx + k∂t wkLq Wx−1,r . kf kḢxγ + kgkḢxγ−1 + kF kLq̃′ Lr̃′
t
t
x
Quasineutral limit for Navier Stokes Poisson – p.22/36
1/q
Strichartz d=3
1/2
q, r, q̃, r̃ ≥ 2
1
1
1
≤
−
q
2
r
1
1
1
≤
−
q̃
2
r̃
1
3
3
+
=
q
r
2 −γ =
S
1
q̃ ′
+
3
r̃′
−2
1/r
1/2
kwkLqt Lrx + k∂t wkLq Wx−1,r . kf kḢxγ + kgkḢxγ−1 + kF kLq̃′ Lr̃′
t
t
x
Quasineutral limit for Navier Stokes Poisson – p.22/36
1/q
Strichartz d=3
1/2
q, r, q̃, r̃ ≥ 2
1
1
1
≤
−
q
2
r
1
1
1
≤
−
q̃
2
r̃
1
3
3
+
=
q
r
2 −γ =
S
1
q̃ ′
+
3
r̃′
−2
1/r
1/2
kwkLqt Lrx + k∂t wkLq Wx−1,r . kf kḢxγ + kgkḢxγ−1 + kF kLq̃′ Lr̃′ ,
t
t
x
Quasineutral limit for Navier Stokes Poisson – p.22/36
1/q
Strichartz d=3
1/2
q, r, q̃, r̃ ≥ 2
1
1
1
≤
−
q
2
r
1
1
1
≤
−
q̃
2
r̃
1
3
3
+
=
q
r
2 −γ =
S
1
q̃ ′
+
3
r̃′
−2
1/r
1/2
kwkLqt Lrx + k∂t wkLq Wx−1,r . kf kḢxγ + kgkḢxγ−1 + kF kLq̃′ Lr̃′ ,
t
t
x
d = 3 there is a non standard estimate (it follows also by bilinear
estimates, see Strichartz, Sogge, Klainerman and Machedon,
Klainerman and Foschi)
kwkL4t,x + k∂t wkL4 Wx−1,4 .kf kḢx1/2 + kgkḢx−1/2 + kF kL1t L2x .
t
Quasineutral limit for Navier Stokes Poisson – p.22/36
σ̃ =σ̃1 + σ̃2 + σ̃3 + σ̃4 where σ̃1 , σ̃2 , σ̃3 and σ̃4 solve the following
wave equations:
Quasineutral limit for Navier Stokes Poisson – p.23/36
σ̃ =σ̃1 + σ̃2 + σ̃3 + σ̃4 where σ̃1 , σ̃2 , σ̃3 and σ̃4 solve the following
wave equations:
(
(
2σ̃1 = −∆ div ũ = ε2 F1
2σ̃2 = ε2 F2
σ̃1 (x, 0) = σ̃0 ∂τ σ̃1 (x, 0) = ∂τ σ̃0 ,
σ̃2 (x, 0) = ∂τ σ̃2 (x, 0) = 0.
(
(
2σ̃3 = εF3
2σ̃4 = F4
σ̃3 (x, 0) = ∂τ σ̃3 (x, 0) = 0.
σ̃4 (x, 0) = ∂τ σ̃4 (x, 0) = 0.
Quasineutral limit for Navier Stokes Poisson – p.23/36
σ̃ =σ̃1 + σ̃2 + σ̃3 + σ̃4 where σ̃1 , σ̃2 , σ̃3 and σ̃4 solve the following
wave equations:
(
(
2σ̃1 = −∆ div ũ = ε2 F1
2σ̃2 = ε2 F2
σ̃1 (x, 0) = σ̃0 ∂τ σ̃1 (x, 0) = ∂τ σ̃0 ,
σ̃2 (x, 0) = ∂τ σ̃2 (x, 0) = 0.
(
(
2σ̃3 = εF3
2σ̃4 = F4
σ̃3 (x, 0) = ∂τ σ̃3 (x, 0) = 0.
σ̃4 (x, 0) = ∂τ σ̃4 (x, 0) = 0.
we apply
kwkL4τ,x + k∂τ wkL4τ Wx−1,4 . kf kḢx1/2 + kgkḢx−1/2 + kF kL1τ L2x
to w = ∆−1 σ̃1 , F = F1 = div(µ∆ũ + (ν + µ)∇ div ũ) ∈ L2t Hx−2
kσ̃1 kL4τ Wx−2,4 + k∂τ σ̃1 kL4τ Wx−3,4 . kσ̃0 kHx−3/2 + k∂τ σ̃0 kHx−5/2
+ ε3/2 T k div ∆ũ + ∇ div ũkL2τ Hx−2 .
Quasineutral limit for Navier Stokes Poisson – p.23/36
σ̃ =σ̃1 + σ̃2 + σ̃3 + σ̃4 where σ̃1 , σ̃2 , σ̃3 and σ̃4 solve the following
wave equations:
(
(
2σ̃2 = ε2 F2
2σ̃1 = −∆ div ũ = ε2 F1
σ̃1 (x, 0) = σ̃0 ∂τ σ̃1 (x, 0) = ∂τ σ̃0 ,
σ̃2 (x, 0) = ∂τ σ̃2 (x, 0) = 0.
(
(
2σ̃4 = F4
2σ̃3 = εF3
σ̃3 (x, 0) = ∂τ σ̃3 (x, 0) = 0.
σ̃4 (x, 0) = ∂τ σ̃4 (x, 0) = 0.
we apply
kwkL4τ,x + k∂τ wkL4τ Wx−1,4 . kf kḢx1/2 + kgkḢx−1/2 + kF kL1τ L2x to
∆−
s0
−1
2
σ̃2 , s0 ≥ 3/2,
w=
−s0 −2
F = F2 = div(div(ρ̃ũ ⊗ ũ) + ∇π̃) ∈ L∞
H
x
t
kσ̃2 kL4 Wx−s0 −2,4 + k∂τ σ̃2 kL4 Wx−s0 −3,4 .
τ
τ
ε3/2 T k div(div(ρ̃ũ ⊗ ũ) + ∇π̃)kL∞ Hx−s0 −2
τ
Quasineutral limit for Navier Stokes Poisson – p.23/36
σ̃ =σ̃1 + σ̃2 + σ̃3 + σ̃4 where σ̃1 , σ̃2 , σ̃3 and σ̃4 solve the following
wave equations:
(
(
2σ̃1 = −∆ div ũ = ε2 F1
2σ̃2 = ε2 F2
σ̃1 (x, 0) = σ̃0 ∂τ σ̃1 (x, 0) = ∂τ σ̃0 ,
σ̃2 (x, 0) = ∂τ σ̃2 (x, 0) = 0.
(
(
2σ̃3 = εF3
2σ̃4 = F4
σ̃3 (x, 0) = ∂τ σ̃3 (x, 0) = 0.
σ̃4 (x, 0) = ∂τ σ̃4 (x, 0) = 0.
we apply
kwkL4τ,x + k∂τ wkL4 Wx−1,4 . kf kḢx1/2 + kgkḢx−1/2 + kF kL1τ L2x
t
to w = ∆−(s0 +1)/2 σ̃3 , s0 ≥ 3/2,
−s0 −1
F = F3 = εβ−2 div(∇Ṽ ⊗ ∇Ṽ ) + 21 ∇|∇Ṽ |2 ∈ L∞
H
x
t
kσ̃3 kL4 Wx−s0 −1,4 + k∂τ σ̃3 kL4 Wx−s0 −2,4
τ
τ
β−2
. εT kε
1
div(∇Ṽ ⊗ ∇Ṽ ) + ∇|∇Ṽ |2 kL∞ Hx−s0 −1
2 for Navier StokesτPoisson – p.23/36
Quasineutral limit
σ̃ =σ̃1 + σ̃2 + σ̃3 + σ̃4 where σ̃1 , σ̃2 , σ̃3 and σ̃4 solve the following
wave equations:
(
(
2σ̃1 = −∆ div ũ = ε2 F1
2σ̃2 = ε2 F2
σ̃1 (x, 0) = σ̃0 ∂τ σ̃1 (x, 0) = ∂τ σ̃0 ,
σ̃2 (x, 0) = ∂τ σ̃2 (x, 0) = 0.
(
(
2σ̃3 = εF3
2σ̃4 = F4
σ̃3 (x, 0) = ∂τ σ̃3 (x, 0) = 0.
σ̃4 (x, 0) = ∂τ σ̃4 (x, 0) = 0.
we apply
kwkL4τ,x + k∂τ wkL4 Wx−1,4 . kf kḢx1/2 + kgkḢx−1/2 + kF kL1τ L2x
t
to w =
∆−1/2 σ̃4 ,
F = F4 = ε
β
−1
2
−1
H
div(∇V ) ∈ L∞
x
τ
− β2
kσ̃4 kL4τ Wx−1,4 + k∂τ σ̃4 kL4τ Wx−2,4 . ε
T kε
β
−1
2
−1 .
div(∇V )kL∞
τ Hx
Quasineutral limit for Navier Stokes Poisson – p.23/36
Estimate for the density fluctuation
Finally we have the following estimate on σ ε
ε
− 41 + β2
ε
kσ kL4 Wx−s0 −2,4 + ε
β
3
+
4
2
t
t
.ε
β
2
kσ0ε kHx−1
+ε
1+ β2
+ε
1+ β2
k∂t σ ε kL4 Wx−s0 −3,4
β
2
+ ε kmε0 kHx−1
T k div(div(σ ε uε ⊗ uε ) − (γ − 1)∇π ε )kL∞ Hx−s0 −2
t
k div ∆uε + ∇ div uε kL2t Hx−2
ε
1+ β2
−1 + ε
+ T k div ∇V kL∞
t Hx
T kεβ−2 div(σ ε V ε )kL∞ Hx−s0 −1
t
Quasineutral limit for Navier Stokes Poisson – p.24/36
Estimates on Quε - Part 2
Quasineutral limit for Navier Stokes Poisson – p.25/36
Estimates on Quε - Part 2
Q(ρε uε ) = ∇∆−1 div(ρε uε )
ε∂t σ ε = − div(ρε uε )
Q(ρε uε ) = ε∇∆−1 ∂t σ ε
Quasineutral limit for Navier Stokes Poisson – p.25/36
Estimates on Quε - Part 2
Q(ρε uε ) = ∇∆−1 div(ρε uε )
ε ε
Q(ρ u ) = ε
ε
3
+ β2
4
ε∂t σ ε = − div(ρε uε )
1
− β2
4
∇∆
−1
ε
3
+ β2
4
∂t σ ε
k∂t σ ε kL4 Wx−s0 −3,4
t
Quasineutral limit for Navier Stokes Poisson – p.25/36
Estimates on Quε - Part 2
Q(ρε uε ) = ∇∆−1 div(ρε uε )
ε ε
Q(ρ u ) = ε
ε
β
3
+
4
2
ε∂t σ ε = − div(ρε uε )
1
− β2
4
∇∆
−1
ε
3
+ β2
4
∂t σ ε
k∂t σ ε kL4 Wx−s0 −3,4
t
1 β
− >0
4 2
since
1
β≤
2
Quasineutral limit for Navier Stokes Poisson – p.25/36
Young-type estimates
C0∞ (Rd ),
R
α−d ψ
j∈
ψ ≥ 0, Rd jdx = 1, jα (x) =
Then the following Young type inequality hold
kf ∗ jα kLp (Rd ) ≤ Cα
s−d( q1 − p1 )
x
α
.
kf kW −s,q (Rd ) ,
for any p, q ∈ [1, ∞], q ≤ p, s ≥ 0, α ∈ (0, 1).
Quasineutral limit for Navier Stokes Poisson – p.26/36
Young-type estimates
C0∞ (Rd ),
R
α−d ψ
j∈
ψ ≥ 0, Rd jdx = 1, jα (x) =
Then the following Young type inequality hold
kf ∗ jα kLp (Rd ) ≤ Cα
s−d( q1 − p1 )
x
α
.
kf kW −s,q (Rd ) ,
for any p, q ∈ [1, ∞], q ≤ p, s ≥ 0, α ∈ (0, 1).
Moreover for any f ∈ Ḣ 1 (Rd ), one has
kf − f ∗ jα kLp (Rd ) ≤ Cp α1−σ k∇f kL2 (Rd ) ,
where
p ∈ [2, ∞)
if d = 2,
p ∈ [2, 6]
if d = 3 and σ = d
1 1
−
2 p
.
Quasineutral limit for Navier Stokes Poisson – p.26/36
kQuε kL2t Lpx ≤ kQuε − Quε ∗ jα kL2t Lpx + kQuε ∗ jα kL2t Lpx = J1 + J2 ,
Quasineutral limit for Navier Stokes Poisson – p.27/36
kQuε kL2t Lpx ≤ kQuε − Quε ∗ ψα kL2t Lpx + kQuε ∗ ψα kL2t Lpx = J1 + J2 ,
to estimate J1 we use
kf − f ∗ jα kLp (Rd ) ≤ Cp α1−σ k∇f kL2 (Rd ) ,
to get
J1 ≤ α
1−3( 21 − p1 )
k∇uε kL2t L2x .
Quasineutral limit for Navier Stokes Poisson – p.27/36
kQuε kL2t Lpx ≤ kQuε − Quε ∗ jα kL2t Lpx + kQuε ∗ jα kL2t Lpx
≤α
1−3( 12 − p1 )
k∇uε kL2t L2x + J2 ,
J2 ≤ εkQ(σ ε uε ) ∗ jα kL2t Lpx + kQ(ρε uε ) ∗ jα kL2t Lpx = J2,1 + J2,2 .
Quasineutral limit for Navier Stokes Poisson – p.27/36
kQuε kL2t Lpx ≤ kQuε − Quε ∗ jα kL2t Lpx + kQuε ∗ jα kL2t Lpx
≤α
1−3( 12 − p1 )
k∇uε kL2t L2x + J2 ,
J2 ≤ εkQ(σ ε uε ) ∗ jα kL2t Lpx + kQ(ρε uε ) ∗ jα kL2t Lpx = J2,1 + J2,2 .
to estimate J2,1 we use
kf ∗ jα kLp (Rd ) ≤ Cα
s−d( q1 − p1 )
kf kW −s,q (Rd ) , with s = 1 and q = 2
to get
J2,1 ≤ εα
−1−3( 12 − p1 )
kσ ε uε kL2t Hx−1 .
Quasineutral limit for Navier Stokes Poisson – p.27/36
kQuε kL2t Lpx ≤ kQuε − Quε ∗ jα kL2t Lpx + kQuε ∗ jα kL2t Lpx
≤α
1−3( 12 − p1 )
k∇uε kL2t L2x + J2 ,
J2 ≤ εkQ(σ ε uε ) ∗ jα kL2t Lpx + kQ(ρε uε ) ∗ jα kL2t Lpx
≤ εα
−1−3( 21 − p1 )
kσ ε uε kL2t Hx−1 + J2,2 .
to estimate J2,2 we use
kf ∗jα kLp (Rd ) ≤ Cα
s−d( 1q − p1 )
kf kW −s,q (Rd ) , with s = s0 − 4 and q = 4
to get
J2,2 = ε
≤ε
≤ε
β
1
−
4
2
β
1
−
4
2
β
1
−
4
2
k∇∆
−1
ε
β
3
+
4
2
∂t σ ε ∗ jkL2t Lpx
α
−s0 −4−3( 41 − p1 )
α
−s0 −4−3( 41 − p1 )
kε
T
β
3
+
4
2
1/4
∂t σ ε kL2 Wx−s0 −4,4
t
kε
β
3
+
4
2
∂t σ ε kL4 Wx−s0 −4,4 .
t
Quasineutral limit for Navier Stokes Poisson – p.27/36
kQuε kL2t Lpx ≤ kQuε − Quε ∗ jα kL2t Lpx + kQuε ∗ jα kL2t Lpx
≤α
+ε
1−3( 21 − p1 )
β
1
−
4
2
. Cα
α
ε
k∇u kL2t L2x + εα
−s0 −4−3( 41 − p1 )
1−3( 21 − p1 )
+ CT ε
T
1/4
1
− β2
4
kε
α
−1−3( 12 − p1 )
β
3
+
4
2
kσ ε uε kL2t Hx−1
∂t σ ε kL4 Wx−s0 −4,4
t
−s0 −4−3( 41 − p1 )
Quasineutral limit for Navier Stokes Poisson – p.27/36
kQuε kL2t Lpx ≤ kQuε − Quε ∗ jα kL2t Lpx + kQuε ∗ jα kL2t Lpx
≤α
+ε
1−3( 21 − p1 )
β
1
−
4
2
. Cα
α
ε
k∇u kL2t L2x + εα
−s0 −4−3( 41 − p1 )
1−3( 21 − p1 )
+ CT ε
T
1/4
1
− β2
4
kε
α
−1−3( 12 − p1 )
β
3
+
4
2
kσ ε uε kL2t Hx−1
∂t σ ε kL4 Wx−s0 −4,4
t
−s0 −4−3( 41 − p1 )
choose
α=ε
1−2β
17+s0
Quasineutral limit for Navier Stokes Poisson – p.27/36
kQuε kL2t Lpx ≤ kQuε − Quε ∗ jα kL2t Lpx + kQuε ∗ jα kL2t Lpx
≤α
+ε
1−3( 21 − p1 )
β
1
−
4
2
. Cα
α
ε
k∇u kL2t L2x + εα
−s0 −4−3( 41 − p1 )
1−3( 21 − p1 )
+ CT ε
T
1/4
1
− β2
4
kε
α
−1−3( 12 − p1 )
β
3
+
4
2
kσ ε uε kL2t Hx−1
∂t σ ε kL4 Wx−s0 −4,4
t
−s0 −4−3( 41 − p1 )
choose
α=ε
ε
kQu kL2t Lpx ≤ CT ε
1−2β
17+s0
p−1 1−2β
6p 17+s0
for any p ∈ [4, 6).
Quasineutral limit for Navier Stokes Poisson – p.27/36
kQuε kL2t Lpx ≤ kQuε − Quε ∗ jα kL2t Lpx + kQuε ∗ jα kL2t Lpx
≤α
+ε
1−3( 21 − p1 )
β
1
−
4
2
. Cα
α
ε
k∇u kL2t L2x + εα
−s0 −4−3( 41 − p1 )
1−3( 21 − p1 )
+ CT ε
T
1/4
1
− β2
4
kε
α
−1−3( 12 − p1 )
β
3
+
4
2
kσ ε uε kL2t Hx−1
∂t σ ε kL4 Wx−s0 −4,4
t
−s0 −4−3( 41 − p1 )
choose
α=ε
ε
kQu kL2t Lpx ≤ CT ε
1−2β
17+s0
p−1 1−2β
6p 17+s0
for any p ∈ [4, 6).
⇓
Quε −→ 0
strongly in L2t Lpx , for any p ∈ [4, 6).
Quasineutral limit for Navier Stokes Poisson – p.27/36
Convergence on P uε
convolution techniques
Quasineutral limit for Navier Stokes Poisson – p.28/36
Convergence on P uε
convolution techniques
Lp compactness
kP uε (t + h) − P uε (t)kL2 ([0,T ]×R3 )
Quasineutral limit for Navier Stokes Poisson – p.28/36
Convergence on P uε
convolution techniques
Lp compactness
kP uε (t + h) − P uε (t)kL2 ([0,T ]×R3 )
from now on we set
z ε = uε (t + h) − uε (t) =
Z
t+h
∂s uε (s, x)ds
t
Quasineutral limit for Navier Stokes Poisson – p.28/36
Convergence on P uε
convolution techniques
Lp compactness
kP uε (t + h) − P uε (t)k2L2 =
t,x
=
+
Z TZ
0
dtdx|P z ε |2
R3
Z TZ
dtdx(P z ε ) · (P z ε − P z ε ∗ jα )
0 R3
Z TZ
0
dtdx(P z ε ) · (P z ε ∗ jα )
R3
= I1 + I2
Quasineutral limit for Navier Stokes Poisson – p.28/36
I1 =
Z TZ
dtdx(P z ε ) · (P z ε − P z ε ∗ jα )
0
≤
.
R3
kP z ε kL2t,x kP z ε (t) − (P z ε
αkuε kL2t,x k∇uε kL2t,x .
∗ jα )(t)kL2t,x
Quasineutral limit for Navier Stokes Poisson – p.29/36
Z TZ
I2 =
dtdx(P ρε z ε ) · (P z ε ∗ jα ) + ε
0 R3
= I2,1 + I2,2
Z TZ
dtdx(P σ ε z ε ) · (P z ε ∗ jα )
0 R3
Quasineutral limit for Navier Stokes Poisson – p.30/36
Z TZ
I2 =
dtdx(P ρε z ε ) · (P z ε ∗ jα ) + ε
0 R3
= I2,1 + I2,2
Z TZ
dtdx(P σ ε z ε ) · (P z ε ∗ jα )
0 R3
I2,2 = εkuε kL2 ([0,T ];L4 (R3 )∩L2k/k−1 (R3 )) kσ ε uε kL2 ([0,T ];L4/3 (R3 )+L2k/k+1 ) . ε
Quasineutral limit for Navier Stokes Poisson – p.30/36
Z TZ
I2 =
dtdx(P ρε z ε ) · (P z ε ∗ jα ) + ε
0 R3
= I2,1 + I2,2
Z TZ
dtdx(P σ ε z ε ) · (P z ε ∗ jα )
0 R3
I2,2 = εkuε kL2 ([0,T ];L4 (R3 )∩L2k/k−1 (R3 )) kσ ε uε kL2 ([0,T ];L4/3 (R3 )+L2k/k+1 ) . ε
Z
Z
Z
T
t+h
ε ε
ε
ε
ε
I2,1≤
dt dx
ds(div(ρ u ⊗ u ) + ∆u )(s, x) · (P z ∗ jα )(t, x)
0
R3
t
Z
Z
Z
T
t+h
ε
σ
∇V ε (s, x) · (P z ε ∗ jα )(t, x)
+
dt
dx
dsP
0
ε
R3
t
ε 2 ∞
β−2
1
+kε
|∇V
| kLt L1x
.hk∇uε k2L2 +α−3/2 T 1/2 hk∇uε kL2t,x kρε |uε |2 kL∞
t Lx
t,x
Quasineutral limit for Navier Stokes Poisson – p.30/36
kP uε (t + h) − P uε (t)k2L2 ([0,T ]×R3 ) ≤ C(α + h + hα−3/2 T 1/2 + ε)
Quasineutral limit for Navier Stokes Poisson – p.31/36
kP uε (t + h) − P uε (t)k2L2 ([0,T ]×R3 ) ≤ C(α + h + hα−3/2 T 1/2 + ε)
choose
α = h2/5
Quasineutral limit for Navier Stokes Poisson – p.31/36
kP uε (t + h) − P uε (t)k2L2 ([0,T ]×R3 ) ≤ C(α + h + hα−3/2 T 1/2 + ε)
choose
α = h2/5
kP uε (t + h) − P uε (t)kL2 ([0,T ]×R3 ) ≤ CT (h1/5 + ε1/2 )
Quasineutral limit for Navier Stokes Poisson – p.31/36
kP uε (t + h) − P uε (t)k2L2 ([0,T ]×R3 ) ≤ C(α + h + hα−3/2 T 1/2 + ε)
choose
α = h2/5
kP uε (t + h) − P uε (t)kL2 ([0,T ]×R3 ) ≤ CT (h1/5 + ε1/2 )
⇓
P uε −→ P u,
strongly in L2 (0, T ; L2loc (R3 ))
Quasineutral limit for Navier Stokes Poisson – p.31/36
Passing into the limit
ε
γ
ε
∇(ρ )
ε ε
ε ε
ε
ε
ε ρ
ε
P ∂t (ρ u )+div(ρ u ⊗ u )+
=
µ∆u
+(ν
+
µ)∇
div
u
+
∇V
γε2
ε2
⇓
Quasineutral limit for Navier Stokes Poisson – p.32/36
Passing into the limit
ε
γ
ε
∇(ρ )
ε ε
ε ε
ε
ε
ε ρ
ε
P ∂t (ρ u )+div(ρ u ⊗ u )+
=
µ∆u
+(ν
+
µ)∇
div
u
+
∇V
γε2
ε2
⇓
ε
ρ −1
ε ε
ε ε
ε
ε
ε
∂t P (ρ u ) + P div(ρ u ⊗ u ) = µ∆P u + P
∇V
ε2
as ε → 0..............
Quasineutral limit for Navier Stokes Poisson – p.32/36
Passing into the limit
ε
γ
ε
∇(ρ )
ε ε
ε ε
ε
ε
ε ρ
ε
P ∂t (ρ u )+div(ρ u ⊗ u )+
=
µ∆u
+(ν
+
µ)∇
div
u
+
∇V
γε2
ε2
⇓
ε
ρ −1
ε
ε
ε
ε
ε
ε
ε
∇V
∂t P (ρ u ) + P div(ρ u ⊗ u ) = µ ∆P u + P
ε2
↓
↓
↓
↓
∂t P u
P (u · ∇)u
P ∆u
?
Quasineutral limit for Navier Stokes Poisson – p.32/36
Passing into the limit
ε
γ
ε
∇(ρ )
ε ε
ε ε
ε
ε
ε ρ
ε
P ∂t (ρ u )+div(ρ u ⊗ u )+
=
µ∆u
+(ν
+
µ)∇
div
u
+
∇V
γε2
ε2
⇓
ε
ρ −1
ε
ε
ε
ε
ε
ε
ε
∇V
∂t P (ρ u ) + P div(ρ u ⊗ u ) = µ ∆P u + P
ε2
↓
↓
↓
↓
∂t P u
P (u · ∇)u
P ∆u
?
Case
3
2
≤γ<2
ε
ρ − 1
β
2
1
−1+
+
ε
ε
ε
γ
2
∞ γ k∇V kL∞ L2 . ε
kρ
−
1k
2γ ≤
L
L
x
x
ε2 ∇V ∞ γ+2
t
t
ε2
Lt Lx
Quasineutral limit for Navier Stokes Poisson – p.32/36
Passing into the limit
ε
γ
ε
∇(ρ )
ε ε
ε ε
ε
ε
ε ρ
ε
P ∂t (ρ u )+div(ρ u ⊗ u )+
=
µ∆u
+(ν
+
µ)∇
div
u
+
∇V
γε2
ε2
⇓
ε
ρ −1
ε
ε
ε
ε
ε
ε
ε
∇V
∂t P (ρ u ) + P div(ρ u ⊗ u ) = µ ∆P u + P
ε2
↓
↓
↓
↓
∂t P u
P (u · ∇)u
P ∆u
?
Case
3
2
≤γ<2
ε
ρ − 1
β
2
1
+
−1+
ε
ε
ε
γ
2 −→ 0
∞ γ k∇V kL∞ L2 . ε
kρ
−1k
2γ ≤
L
L
x
x
ε2 ∇V ∞ γ+2
t
t
ε2
Lt Lx
if β < −2 +
4
γ
Quasineutral limit for Navier Stokes Poisson – p.32/36
Energy inequality
We are able to recover the energy inequality if we assume
π ε |t=0 → 0
ε
β
−1
2
∇V0ε → 0
strongly in L∞ ([0, T ]; L1 (R3 )), as ε → 0,
strongly in L∞ ([0, T ]; L2 (R3 )), as ε → 0
Quasineutral limit for Navier Stokes Poisson – p.33/36
Energy inequality
We are able to recover the energy inequality if we assume
π ε |t=0 → 0
ε
Z
β
−1
2
∇V0ε → 0
strongly in L∞ ([0, T ]; L1 (R3 )), as ε → 0,
strongly in L∞ ([0, T ]; L2 (R3 )), as ε → 0
Z TZ
1 2
|u| dx +
µ|∇u|2 dxdt
0 R3
R3 2
Z ε
2
ε |u |
ρ
≤ lim inf
+ π ε + εβ−2 |∇V ε |2 dx
ε→0
2
R3
Z TZ
ε 2
ε 2
+ lim inf
µ|∇u | + (ν + µ)| div u | dxdt
ε→0
0
R3
Z
Z ε
2
|m0 |
1
β−2
ε
ε 2
2
≤ lim inf
ε
|∇V
|
dx
=
+
|u
|
π |t=0 +
0 dx.
0
ε
ε→0
2ρ0
R3 2
R3
Quasineutral limit for Navier Stokes Poisson – p.33/36
Conclusions
Where we can apply the same techniques?
Quasineutral limit for Navier Stokes Poisson – p.34/36
Conclusions
Where we can apply the same techniques?
Artificial compressibility method for the incompressible
Navier Stokes equation, (D.Donatelli, P. Marcati, 2006))

1
 ε
ε
ε
ε
ε
∂t u + ∇p = µ∆u − (u · ∇) u − (div uε )uε
2
ε∂ pε + div uε = 0
t
Quasineutral limit for Navier Stokes Poisson – p.34/36
Conclusions
Where we can apply the same techniques?
Artificial compressibility method for the incompressible
Navier Stokes equation, (D.Donatelli, P. Marcati, 2006))

1
 ε
ε
ε
ε
ε
∂t u + ∇p = µ∆u − (u · ∇) u − (div uε )uε
2
ε∂ pε + div uε = 0
t
The hyperbolicity of the approximation provides dispersive
estimates
The convergence in the whole space will be obtained via
dispersion and not via compactness (Temam)
Quasineutral limit for Navier Stokes Poisson – p.34/36
Conclusions
Where we can apply the same techniques?
Artificial compressibility method for the incompressible
Navier Stokes equation, (D.Donatelli, P. Marcati, 2006))

1
 ε
ε
ε
ε
ε
∂t u + ∇p = µ∆u − (u · ∇) u − (div uε )uε
2
ε∂ pε + div uε = 0
t
The hyperbolicity of the approximation provides dispersive
estimates
The convergence in the whole space will be obtained via
dispersion and not via compactness (Temam)
ε∂tt
pε − ∆pε
=
−∆ div uε + div
(uε
·
∇) uε
+
1
ε )uε
(div
u
2
Quasineutral limit for Navier Stokes Poisson – p.34/36
Where we can apply the same techniques?
Quasineutral limit for Navier Stokes Poisson – p.35/36
Where we can apply the same techniques?
Artificial compressibility method for the Navier - Stokes Fourier System (D.Donatelli 2007)


∂t u + u · ∇u + ∇p = µ∆u
∂t θ + u · ∇θ = κ∆θ

div u = 0 u(x, 0) = u , θ(x, 0) = θ .
0
0
Quasineutral limit for Navier Stokes Poisson – p.35/36
Where we can apply the same techniques?
Artificial compressibility method for the Navier - Stokes Fourier System (D.Donatelli 2007)


∂t u + u · ∇u + ∇p = µ∆u
∂t θ + u · ∇θ = κ∆θ

div u = 0 u(x, 0) = u , θ(x, 0) = θ .
0
0

1
ε
ε
ε
ε
ε
ε ε


∂
u
+
∇p
=
µ∆u
−
(u
·
∇)
u
−
)u
(div
u

 t
2
1
ε
ε
ε
ε ε
∂
θ
+
u
·
∇θ
=
κ∆θ
−
)θ
(div
u
t


2

ε∂ pε + div uε = 0.
t
Quasineutral limit for Navier Stokes Poisson – p.35/36
Extensions
Quasineutral limit for Navier Stokes Poisson – p.36/36
Extensions
Model for plasma physics that takes into account the
temperature effects and balance equation
Quasineutral limit for Navier Stokes Poisson – p.36/36
Extensions
Model for plasma physics that takes into account the
temperature effects and balance equation
Bipolar models for semiconductors
Quasineutral limit for Navier Stokes Poisson – p.36/36
Extensions
Model for plasma physics that takes into account the
temperature effects and balance equation
Bipolar models for semiconductors
Navier Stokes flow outside a convex body K
Ω = R3 −K
we need to use Strichartz estimate on exterior domain
(Smith and Sogge (’99), Burq (’02), Metcalf (’04))
Quasineutral limit for Navier Stokes Poisson – p.36/36