Multiplication in anisotropic spaces and applications to
quasilinear systems
Jürgen Saal
Operator Semigroups in Analysis: Modern Developments
Bedlewo, 24.-28.4.2017
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
1 / 14
Motivation
Typical nonlinear term in free boundary problems
|∇Γ η|2 · ∂ν u|Γ
Jürgen Saal (HHU Düsseldorf)
?
∈
F := Wp1/2−1/2p (Lp ) ∩ Lp (Wp1−1/p ).
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
2 / 14
Motivation
Typical nonlinear term in free boundary problems
|∇Γ η|2 · ∂ν u|Γ
?
∈
F := Wp1/2−1/2p (Lp ) ∩ Lp (Wp1−1/p ).
Common strategy:
F · F ,→ F
Jürgen Saal (HHU Düsseldorf)
⇒
p > d + 2.
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
2 / 14
Motivation
Typical nonlinear term in free boundary problems
|∇Γ η|2 · ∂ν u|Γ
?
∈
F := Wp1/2−1/2p (Lp ) ∩ Lp (Wp1−1/p ).
Common strategy:
F · F ,→ F
⇒
p > d + 2.
See e.g. papers of: Solonnikov, Denisova, Frolova, Shibata, Prüß, Escher,
Simonett, Wilke, Köhne, Denk, Geißert, Hieber, Sawada, S., ...
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
2 / 14
Motivation
Typical nonlinear term in free boundary problems
|∇Γ η|2 · ∂ν u|Γ
?
∈
F := Wp1/2−1/2p (Lp ) ∩ Lp (Wp1−1/p ).
Common strategy:
F · F ,→ F
⇒
p > d + 2.
See e.g. papers of: Solonnikov, Denisova, Frolova, Shibata, Prüß, Escher,
Simonett, Wilke, Köhne, Denk, Geißert, Hieber, Sawada, S., ...
Note: sufficient was
E · E · F ,→ F
Jürgen Saal (HHU Düsseldorf)
⇒
p>
2(d + 2)
.
5
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
2 / 14
Motivation
Typical nonlinear term in free boundary problems
|∇Γ η|2 · ∂ν u|Γ
?
∈
F := Wp1/2−1/2p (Lp ) ∩ Lp (Wp1−1/p ).
Common strategy:
F · F ,→ F
⇒
p > d + 2.
See e.g. papers of: Solonnikov, Denisova, Frolova, Shibata, Prüß, Escher,
Simonett, Wilke, Köhne, Denk, Geißert, Hieber, Sawada, S., ...
Note: sufficient was
E · E · F ,→ F
⇒
p>
2(d + 2)
.
5
Question: Does there exist a systematic approach to handle nonlinear terms,
even giving optimal range for p ∈ (1, ∞)?
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
2 / 14
Motivation
Typical nonlinear term in free boundary problems
|∇Γ η|2 · ∂ν u|Γ
?
∈
F := Wp1/2−1/2p (Lp ) ∩ Lp (Wp1−1/p ).
Common strategy:
F · F ,→ F
⇒
p > d + 2.
See e.g. papers of: Solonnikov, Denisova, Frolova, Shibata, Prüß, Escher,
Simonett, Wilke, Köhne, Denk, Geißert, Hieber, Sawada, S., ...
Note: sufficient was
E · E · F ,→ F
⇒
p>
2(d + 2)
.
5
Question: Does there exist a systematic approach to handle nonlinear terms,
even giving optimal range for p ∈ (1, ∞)?
Yes, if available: optimal estimates on multiplication and analytic Nemytskij
operators in anisotropic function spaces.
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
2 / 14
Anisotropic function spaces
Let
ν ∈ N, n = (n1 , . . . , nν ) ∈ Nν , Rn = Rn1 × · · · × Rnν
ω = (ω1 , . . . , ων ) ∈ Nν weight vector, ω̇ := lcm { ω1 , . . . , ων }
Anisotropic Sobolev spaces:
(
Wps,ω (Rn ,
X)
:=
kukWps,ω (Rn , X )
:=
n
u ∈ Lp (R , X ) :
X
ν
∂kα u ∈ Lp (Rn , X ), α ∈ Nn0k
)
,
|α| ≤ s/ωk , k = 1, . . . , ν
1/p
X
, s ∈ ω̇ · N0 , 1 ≤ p < ∞.
k∂kα ukpLp (Rn , X )
k=1 |α|≤s/ωk
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
3 / 14
Anisotropic function spaces
Let
ν ∈ N, n = (n1 , . . . , nν ) ∈ Nν , Rn = Rn1 × · · · × Rnν
ω = (ω1 , . . . , ων ) ∈ Nν weight vector, ω̇ := lcm { ω1 , . . . , ων }
Anisotropic Sobolev spaces:
(
Wps,ω (Rn ,
X)
:=
kukWps,ω (Rn , X )
:=
n
u ∈ Lp (R , X ) :
X
ν
∂kα u ∈ Lp (Rn , X ), α ∈ Nn0k
)
,
|α| ≤ s/ωk , k = 1, . . . , ν
1/p
X
, s ∈ ω̇ · N0 , 1 ≤ p < ∞.
k∂kα ukpLp (Rn , X )
k=1 |α|≤s/ωk
Wps,ω (Rn , X ) =
Note:
Tν
k=1
s/ωk
Wp
0
(Rnk , Lp (Rnk , X ))
with nk0 = (n1 , . . . , nk−1 , nk+1 , . . . , nν ) ∈ Nν−1 .
Example:
Here:
2,(2,1)
Wp
(R(1,d) ) = Wp1 (R, Lp (Rd )) ∩ Lp (R, Wp2 (Rd ))
ν = 2, n = (1, d), ω = (2, 1), ω̇ = 2, s = 2.
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
3 / 14
Anisotropic function spaces
Definition (anisotropic function spaces)
Let U ∈ { B, H, W }, ν ∈ N, n = (n1 , . . . , nν ) ∈ Nν , ω = (ω1 , . . . , ων ) ∈ Nν ,
0 ≤ s < ∞, 1 ≤ p < ∞. We set
ν
\
0
Ups,ω (Rn , X ) =
Ups/ωk (Rnk , Lp (Rnk , X ))
k=1
Example:
Here:
1−1/p,(2,1)
Wp
1/2−1/2p
(R(1,d) ) = Wp
1−1/p
(R, Lp (Rd )) ∩ Lp (R, Wp
(Rd ))
ν = 2, n = (1, d), ω = (2, 1), ω̇ = 2, s = 1 − 1/p.
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
4 / 14
Anisotropic function spaces
Definition (anisotropic function spaces)
Let U ∈ { B, H, W }, ν ∈ N, n = (n1 , . . . , nν ) ∈ Nν , ω = (ω1 , . . . , ων ) ∈ Nν ,
0 ≤ s < ∞, 1 ≤ p < ∞. We set
ν
\
0
Ups,ω (Rn , X ) =
Ups/ωk (Rnk , Lp (Rnk , X ))
k=1
Example:
Here:
1−1/p,(2,1)
Wp
1/2−1/2p
(R(1,d) ) = Wp
1−1/p
(R, Lp (Rd )) ∩ Lp (R, Wp
(Rd ))
ν = 2, n = (1, d), ω = (2, 1), ω̇ = 2, s = 1 − 1/p.
the parameters s and p describe smoothness/integrability
ind(Ups,ω (Rn , X )) =
Jürgen Saal (HHU Düsseldorf)
1
ω̇ (s
− ω · n/p) describes the overall regularity
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
4 / 14
Anisotropic function spaces
Definition (anisotropic function spaces)
Let U ∈ { B, H, W }, ν ∈ N, n = (n1 , . . . , nν ) ∈ Nν , ω = (ω1 , . . . , ων ) ∈ Nν ,
0 ≤ s < ∞, 1 ≤ p < ∞. We set
ν
\
0
Ups,ω (Rn , X ) =
Ups/ωk (Rnk , Lp (Rnk , X ))
k=1
Example:
Here:
1−1/p,(2,1)
Wp
1/2−1/2p
(R(1,d) ) = Wp
1−1/p
(R, Lp (Rd )) ∩ Lp (R, Wp
(Rd ))
ν = 2, n = (1, d), ω = (2, 1), ω̇ = 2, s = 1 − 1/p.
the parameters s and p describe smoothness/integrability
ind(Ups,ω (Rn , X )) =
1
ω̇ (s
− ω · n/p) describes the overall regularity
Known results:
Johnson ’95: Mult. in scalar-valued Besov and Triebel-Lizorkin spaces;
Amann ’09: for U ∈ { B, H }: Ups,ω ,→ C0 , if ind(Ups,ω ) > 0;
Li, Sun ’09: embedding thms for anisotropic Sobolev-Lorentz spaces;
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
4 / 14
Multiplication in isotropic function spaces, ω = (1, . . . , 1)
Suppose X1 , . . . , Xm , X HT -Banach spaces with a continuous multiplication
· : X1 × · · · × Xm −→ X
Question: When does there exist a continuous extension
· : Ups11 (Rn , X1 ) × · · · × Upsmm (Rn , Xm ) −→ Ups (Rn , X ) ?
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
(∗)
Bedlewo, 24.-28.4.2017
5 / 14
Multiplication in isotropic function spaces, ω = (1, . . . , 1)
Suppose X1 , . . . , Xm , X HT -Banach spaces with a continuous multiplication
· : X1 × · · · × Xm −→ X
Question: When does there exist a continuous extension
· : Ups11 (Rn , X1 ) × · · · × Upsmm (Rn , Xm ) −→ Ups (Rn , X ) ?
(∗)
Theorem (Amann, 1991)
Let U = W , 0 < s1 , . . . , sm , s < ∞, 1 < p1 , . . . , pm , p < ∞ with
m
1 X 1
s ≤ min { s1 , . . . , sm },
≤
p
pj
j=1
s
ind(Wpjj (Rn ,
and such that for indj :=
Xj )) and ind := ind(Wps (Rn , X )),

{ ind1 , . . . , indm },
if ind1 , . . . , indm ≥ 0

 min m
X
ind ≤
indj ,
otherwise


j=1, indj <0
with strict inequality, if indj = 0 for some j ∈ { 1, . . . , m }. Then (∗) exists.
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
5 / 14
Multiplication in anisotropic function spaces
Suppose X1 , . . . , Xm , X HT -Banach spaces with property α and with a
continuous multiplication
· : X1 × · · · × Xm −→ X
Question: When does there exist a continuous extension
· : Ups11 ,ω (Rn , X1 ) × · · · × Upsmm ,ω (Rn , Xm ) −→ Ups,ω (Rn , X ) ?
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
(∗)
Bedlewo, 24.-28.4.2017
6 / 14
Multiplication in anisotropic function spaces
Suppose X1 , . . . , Xm , X HT -Banach spaces with property α and with a
continuous multiplication
· : X1 × · · · × Xm −→ X
Question: When does there exist a continuous extension
· : Ups11 ,ω (Rn , X1 ) × · · · × Upsmm ,ω (Rn , Xm ) −→ Ups,ω (Rn , X ) ?
(∗)
Theorem (Köhne, S. ’17)
Let U ∈ {B, H}, n, ω ∈ Nν , 0 ≤ s1 , . . . , sm , s < ∞, 1 < p1 , . . . , pm , p < ∞ with
m
1 X 1
s ≤ min { s1 , . . . , sm },
≤
p
pj
j=1
s ,ω
ind(Upjj (Rn ,
and such that for indj :=
Xj )) and ind := ind(Ups,ω (Rn , X )),

{ ind1 , . . . , indm },
if ind1 , . . . , indm ≥ 0

 min m
X
ind ≤
indj ,
otherwise


j=1, indj <0
with strict inequality, if indj = 0 for some j ∈ { 1, . . . , m } and s > 0 and pj = p
for all j for which sj = s, if U = B. Then (∗) exists.
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
6 / 14
Multiplication in anisotropic function spaces
Remark
U = W (= B) is included, if s, s1 , . . . , sm 6∈ N0 ;
Result generalizes to multiplication of the form
· : [U1 ]sp11,ω (Rn , X1 ) × · · · × [Um ]spmm,ω (Rn , Xm ) −→ Ups,ω (Rn , X )
for [U1 ], . . . , [Um ] ∈ {B, H} (under minor constraints on parameters);
s,ω
s,ω
Similar results for more general classes as Bpq
(X ), Fpq
(X ), etc. are in reach.
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
7 / 14
Multiplication in anisotropic function spaces
Remark
U = W (= B) is included, if s, s1 , . . . , sm 6∈ N0 ;
Result generalizes to multiplication of the form
· : [U1 ]sp11,ω (Rn , X1 ) × · · · × [Um ]spmm,ω (Rn , Xm ) −→ Ups,ω (Rn , X )
for [U1 ], . . . , [Um ] ∈ {B, H} (under minor constraints on parameters);
s,ω
s,ω
Similar results for more general classes as Bpq
(X ), Fpq
(X ), etc. are in reach.
Proof makes heavily use of
X -valued multilinear interpolation
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
7 / 14
Multiplication in anisotropic function spaces
Remark
U = W (= B) is included, if s, s1 , . . . , sm 6∈ N0 ;
Result generalizes to multiplication of the form
· : [U1 ]sp11,ω (Rn , X1 ) × · · · × [Um ]spmm,ω (Rn , Xm ) −→ Ups,ω (Rn , X )
for [U1 ], . . . , [Um ] ∈ {B, H} (under minor constraints on parameters);
s,ω
s,ω
Similar results for more general classes as Bpq
(X ), Fpq
(X ), etc. are in reach.
Proof makes heavily use of
X -valued multilinear interpolation
intrinsic norms
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
7 / 14
Multiplication in anisotropic function spaces
Remark
U = W (= B) is included, if s, s1 , . . . , sm 6∈ N0 ;
Result generalizes to multiplication of the form
· : [U1 ]sp11,ω (Rn , X1 ) × · · · × [Um ]spmm,ω (Rn , Xm ) −→ Ups,ω (Rn , X )
for [U1 ], . . . , [Um ] ∈ {B, H} (under minor constraints on parameters);
s,ω
s,ω
Similar results for more general classes as Bpq
(X ), Fpq
(X ), etc. are in reach.
Proof makes heavily use of
X -valued multilinear interpolation
intrinsic norms
operator-valued multiplier results
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
7 / 14
Multiplication in anisotropic function spaces
Remark
U = W (= B) is included, if s, s1 , . . . , sm 6∈ N0 ;
Result generalizes to multiplication of the form
· : [U1 ]sp11,ω (Rn , X1 ) × · · · × [Um ]spmm,ω (Rn , Xm ) −→ Ups,ω (Rn , X )
for [U1 ], . . . , [Um ] ∈ {B, H} (under minor constraints on parameters);
s,ω
s,ω
Similar results for more general classes as Bpq
(X ), Fpq
(X ), etc. are in reach.
Proof makes heavily use of
X -valued multilinear interpolation
intrinsic norms
operator-valued multiplier results
patience and staying power
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
7 / 14
Multiplication in anisotropic function spaces
Remark
U = W (= B) is included, if s, s1 , . . . , sm 6∈ N0 ;
Result generalizes to multiplication of the form
· : [U1 ]sp11,ω (Rn , X1 ) × · · · × [Um ]spmm,ω (Rn , Xm ) −→ Ups,ω (Rn , X )
for [U1 ], . . . , [Um ] ∈ {B, H} (under minor constraints on parameters);
s,ω
s,ω
Similar results for more general classes as Bpq
(X ), Fpq
(X ), etc. are in reach.
Proof makes heavily use of
X -valued multilinear interpolation
intrinsic norms
operator-valued multiplier results
patience and staying power
But, it was worth it, since
approach only depends on the structure of the nonlinearities
analysis boils down to the computation of the Sobolev indexes
approach yields (often) optimal results (improves p > d + 2 to p > (d + 2)/2)
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
7 / 14
Analytic Nemytskij operators in anisotropic spaces
Theorem (Köhne, S. ’17)
Let n, ω ∈ Nν , U ∈ {B, H}, 0 < s1 , . . . , sm , s < ∞, 1 < p1 , . . . , pm , p < ∞ with
ind(Ups11 ,ω (Rn , X1 )), . . . , ind(Upsmm ,ω (Rn , Xm )) ≥ ind(Ups,ω (Rn , X )) > 0.
Furthermore, for r > 0 let
φ : (−r , r )m −→ R,
X
φ(x) =
aα x α ,
x ∈ Rm , |xj | < r ,
α∈Nm
0
be analytic s.t. φ(0) = 0. Then, there is a ρ > 0 such that the Nemytskij operator
Φ : G −→ Ups,ω (Rn , X ),
Φ(u) = φ ◦ u,
is well-defined and analytic on
o
n
G := (u1 , . . . , um ) ∈ Ups11 ,ω (Rn , X1 ) × . . . × Upsmm ,ω (Rn , Xm ) : max kuj kU sj ,ω ≤ ρ ,
pj
j
and satisfies for some L > 0 the estimate
kΦ(u)kUps,ω ≤ L max kuj kU sj ,ω ,
j
Jürgen Saal (HHU Düsseldorf)
pj
u = (u1 , . . . , um ) ∈ G .
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
8 / 14
A fluid-structure interaction model (FSI)
Model:
ρ(∂t v + v · ∇v )
div v
v
m(∂t , ∂ 0 )
v |t=0
η|t=0
∂t η|t=0
=
=
=
=
=
=
=
−∇q + µ∆v
0
∂tp
ηen
− 1 + |∇0 η|2 enτ T (v , q)ν
v0
η0
η1
t > 0, x ∈ Ω(t)
t > 0, x ∈ Ω(t)
t > 0, x = (x 0 , η(t, x 0 ))
t > 0, x = (x 0 , η(t, x 0 ))
x ∈ Ω(0)
x 0 ∈ Rn−1
x 0 ∈ Rn−1
with (parabolized) structure term
m(∂t , ∂ 0 )η = ∂t2 η + α(∆0 )2 η − β∆0 η − γ∂t ∆0 η.
Literature: Beirão da Veiga, Chambolle, Desjardins, Coutand, Shkoller, Esteban,
Grandmont, Lequeurre, Růžička, Lengeler, Muha, Canić, Hillairet, Takahashi,
Badra, ...
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
9 / 14
Linear theory
Resolvent problem (1D see: Grandmont, Hillairet ’16; Badra, Takahashi ’17):
λu − ∆u + ∇p
div u
u0
λη − u n
p − ∂n u n
where
=
=
=
=
=
0
0
0
0
f + m(λ, ∂ 0 )η
in Rn+ ,
in Rn+ ,
on ∂Rn+ ,
on ∂Rn+ ,
on ∂Rn+ ,
(1)
m(λ, ∂ 0 )η = λ2 η + α(∆0 )2 η − β∆0 η − γλ∆0 η.
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
10 / 14
Linear theory
Resolvent problem (1D see: Grandmont, Hillairet ’16; Badra, Takahashi ’17):
λu − ∆u + ∇p
div u
u0
λη − u n
p − ∂n u n
where
=
=
=
=
=
0
0
0
0
f + m(λ, ∂ 0 )η
in Rn+ ,
in Rn+ ,
on ∂Rn+ ,
on ∂Rn+ ,
on ∂Rn+ ,
(1)
m(λ, ∂ 0 )η = λ2 η + α(∆0 )2 η − β∆0 η − γλ∆0 η.
p
Pr. symbol: P(λ, |ξ 0 |) := λ2 + α|ξ 0 |4 + γλ|ξ 0 |2 + β|ξ 0 |2 |ξ 0 |2 + λ2 λ + |ξ 0 |2 .
m6
Newton polygon:
5/2 •P
PP
•
2
l
N(P)
l
l
l
l
•
Jürgen Saal (HHU Düsseldorf)
2
l•
6
-
Multiplication in anisotropic spaces and applications
j
Bedlewo, 24.-28.4.2017
10 / 14
Maximal regularity
λ ∼ z r , r > 0, with z = |ξ 0 |:

αz 6 ,



2
4
2 2

 (λ + αz + γλz )z ,
λ2 z 2 ,
Pr (λ, z) =

2 2

λ z + λ5/2 ,



λ5/2 ,
r -principle symbols:
Jürgen Saal (HHU Düsseldorf)
0 < r < 2,
r = 2,
2 < r < 4,
r = 4,
r > 4,
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
11 / 14
Maximal regularity
λ ∼ z r , r > 0, with z = |ξ 0 |:

αz 6 ,



2
4
2 2

 (λ + αz + γλz )z ,
λ2 z 2 ,
Pr (λ, z) =

2 2

λ z + λ5/2 ,



λ5/2 ,
r -principle symbols:
0 < r < 2,
r = 2,
2 < r < 4,
r = 4,
r > 4,
Theorem (Denk, S., Seiler ’08 Russian J. Math. Phys.)
Equivalent are:
(i) For every r > 0 we have Pr (λ, z) 6= 0 ((λ, z) ∈ Σθ × Σε ).
√
(ii) ∃λ0 > 0 s.t. P(∂t + λ0 , −∆) : 0 E → 0 F is an isomorphism , where here
0E
= 0 W 3−1/2p
(Lp ) ∩ 0 W 5/2−1/2p
(Hp2 ) ∩ Lp (Wp7−1/p )
p
p
0F
= 0 W 1/2−1/2p
(Lp ) ∩ Lp (Wp1−1/p ).
p
Theorem (Denk, S. ’17)
Linearized FSI model has maximal regularity.
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
11 / 14
Multiplication in anisotropic function spaces
Suppose X1 , . . . , Xm , X HT -Banach spaces with property α and with a
continuous multiplication
· : X1 × · · · × Xm −→ X
Question: When does there exist a continuous extension
· : Ups11 ,ω (Rn , X1 ) × · · · × Upsmm ,ω (Rn , Xm ) −→ Ups,ω (Rn , X ) ?
(∗)
Theorem (Köhne, S. ’17)
Let U ∈ {B, H}, n, ω ∈ Nν , 0 ≤ s1 , . . . , sm , s < ∞, 1 < p1 , . . . , pm , p < ∞ with
m
1 X 1
s ≤ min { s1 , . . . , sm },
≤
p
pj
j=1
s ,ω
ind(Upjj (Rn ,
and such that for indj :=
Xj )) and ind := ind(Ups,ω (Rn , X )),

{ ind1 , . . . , indm },
if ind1 , . . . , indm ≥ 0

 min m
X
ind ≤
indj ,
otherwise


j=1, indj <0
with strict inequality, if indj = 0 for some j ∈ { 1, . . . , m } and s > 0 and pj = p
for all j for which sj = s, if U = B. Then (∗) exists.
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
12 / 14
Estimating the nonlinear terms of FSI
Set J = (0, T ), Q := J × Rn−1 .
(∂t η − ∆0 η)
·
∂n u
Wp2−1/p,(2,1) (Q) · Hp1,(2,1) (Q, Lp (R+ ))
|
{z
}
{z
}
|
ind1 =1− n+2
2p
ind2 = 12 − n+1
2p
0,(2,1)
∈ Hp
(Q, Lp (R+ )) = Lp (J × Rn+ )
,→ Hp0,(2,1) (Q, Lp (R+ ))
{z
}
|
ind=− n+1
2p
is fullfilled, if max{ind1 , ind2 } ≥ 0, 0 6= ind ≤ ind1 + ind2
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
⇔
p ≥ (n + 2)/3.
Bedlewo, 24.-28.4.2017
13 / 14
Estimating the nonlinear terms of FSI
Set J = (0, T ), Q := J × Rn−1 .
(∂t η − ∆0 η)
·
0,(2,1)
∈ Hp
∂n u
Wp2−1/p,(2,1) (Q) · Hp1,(2,1) (Q, Lp (R+ ))
|
{z
}
{z
}
|
ind1 =1− n+2
2p
ind2 = 12 − n+1
2p
(Q, Lp (R+ )) = Lp (J × Rn+ )
,→ Hp0,(2,1) (Q, Lp (R+ ))
{z
}
|
ind=− n+1
2p
is fullfilled, if max{ind1 , ind2 } ≥ 0, 0 6= ind ≤ ind1 + ind2
∇0 η
Wp3−1/p,(2,1) (Q)
|
{z
ind1 = 32 − n+2
2p
}
∂n u|∂Rn+
·
Wp1−1/p,(2,1) (Q)
{z
ind2 = 21 − n+2
2p
is fullfilled, if 0 6= ind ≤ ind1 , ind1 > 0
Jürgen Saal (HHU Düsseldorf)
1−1/p,(2,1)
·
|
⇔
⇔
∈ Wp
}
,→
p ≥ (n + 2)/3.
(Q)
Wp1−1/p,(2,1) (Q)
|
{z
ind= 21 − n+2
2p
}
p > (n + 2)/3.
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
13 / 14
Estimating the nonlinear terms of FSI
Set J = (0, T ), Q := J × Rn−1 .
(∂t η − ∆0 η)
·
0,(2,1)
∈ Hp
∂n u
Wp2−1/p,(2,1) (Q) · Hp1,(2,1) (Q, Lp (R+ ))
|
{z
}
{z
}
|
ind1 =1− n+2
2p
ind2 = 12 − n+1
2p
(Q, Lp (R+ )) = Lp (J × Rn+ )
,→ Hp0,(2,1) (Q, Lp (R+ ))
{z
}
|
ind=− n+1
2p
is fullfilled, if max{ind1 , ind2 } ≥ 0, 0 6= ind ≤ ind1 + ind2
∇0 η
Wp3−1/p,(2,1) (Q)
|
{z
ind1 = 32 − n+2
2p
}
1−1/p,(2,1)
·
∂n u|∂Rn+
·
Wp1−1/p,(2,1) (Q)
|
{z
ind2 = 21 − n+2
2p
is fullfilled, if 0 6= ind ≤ ind1 , ind1 > 0
⇔
⇔
∈ Wp
}
,→
p ≥ (n + 2)/3.
(Q)
Wp1−1/p,(2,1) (Q)
|
{z
ind= 21 − n+2
2p
}
p > (n + 2)/3.
Theorem (Denk, S. ’17)
Let p > (n + 2)/3. Then FSI model has a unique maximal solution.
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
13 / 14
Thank you !
Jürgen Saal (HHU Düsseldorf)
Multiplication in anisotropic spaces and applications
Bedlewo, 24.-28.4.2017
14 / 14