Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Some Extensions of the Basic Theory of Evo-Systems. Rainer Picard Department of Mathematics TU Dresden, Germany

Rainer Picard

Evo-Systems

1/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Coupling of Dierent Physical Phenomena. Without coupling, block-diagonal operator matrix:

   ∂t   

V0 . . . . . .

Vn





    +A     

where

  A=  

A0 0

U0 . . . . . .

Un

0

..

···

0 . . .

.

. . .



    =    

···

0 ..



0

.

0

An

f0 . . . . . .

fn

   ,  

     

H = k =0,...,n Hk , since diagonal block entries Ak : D (Ak ) ⊆ Hk → Hk ,k = 0, . . . , n, are skew-self-adjoint. skew-selfadjoint in

L

Rainer Picard

Evo-Systems

2/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Coupling of Dierent Physical Phenomena. The combined material laws now take the simple block diagonal form



  V =  

V0 . . . . . .

Vn





M00 ∂t−1

    =    




..

0

. . . ..

···

M ∂t


−1



0

.

. . . 0

For a proper coupling:

···

0

0

.

0

Mnn ∂t−1




    

U0 . . . . . .

Un

   .  

contains non-zero o-diagonal

block entries



M

∂t−1




  :=   

M0,00 · · · · · · M0,0n . . . . . .

..

. . .

. ..

.

. . .

M0,n0 · · · · · · M0,nn Rainer Picard





     + ∂t−1      Evo-Systems

M1,00 · · · · · · M1,0n . . . . . .

..

. . .

. ..

.

. . .

M1,n0 · · · · · · M1,nn

   .   3/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Coupling of Dierent Physical Phenomena. The combined material laws now take the simple block diagonal form



  V =  

V0 . . . . . .

Vn





M00 ∂t−1

    =    




..

0

. . . ..

···

M ∂t


−1



0

.

. . . 0

For a proper coupling:

···

0

0

.

0

Mnn ∂t−1




    

U0 . . . . . .

Un

   .  

contains non-zero o-diagonal

block entries



M

∂t−1




  :=   

M0,00 · · · · · · M0,0n . . . . . .

..

. . .

. ..

.

. . .

M0,n0 · · · · · · M0,nn Rainer Picard





     + ∂t−1      Evo-Systems

M1,00 · · · · · · M1,0n . . . . . .

..

. . .

. ..

.

. . .

M1,n0 · · · · · · M1,nn

   .   3/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Coupling of Dierent Physical Phenomena. Canonical Form:

Ak =

If



0

Gk

−Gk∗

 ,

0

then, with the unitary permutation matrix

based on

PAP ∗ =

P = (e0 e2 · · · e2n e1 e3 · · · e2n+1 ) , {0, . . . , 2n + 1} → {0, . . . , 2n + 1} k  , k 7→ 1−(−1) (n + 1) + k



0

G

−G ∗ 0

2

we obtain

2

 with

   G =  

G0 0

···

0 ..

0

Rainer Picard

. . .

.

. . .

..

···

0

0

.

0

Gn

   .  

Evo-Systems

4/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Mother and Descendant Mechanism

Initial boundary value problems of classical mathematical physics can be produced from a given mother operator

A

by choosing

suitable projections for constructing descendants.

Rainer Picard

Evo-Systems

5/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Mother and Descendant Mechanism Theorem

Let C : D (C ) ⊆ H0 → H1 be a closed densely dened linear operator, Hk ,k = 0, 1, Hilbert spaces. If Bk : Hk → Xk are continuous linear mappings, Xk Hilbert space,k = 0, 1, such that C ∗ B1∗ densely dened and B0 is a bijection or CB0∗ densely dened and B1 is a bijection. Then



B0 0

0

B1



0

C

−C ∗ 0



B0∗ 0

0

B1∗



is skew-selfadjoint.

Mother anddescendant.

Rainer Picard

Evo-Systems

6/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Mother and Descendant Mechanism Theorem

Let C : D (C ) ⊆ H0 → H1 be a closed densely dened linear operator, Hk ,k = 0, 1, Hilbert spaces. If Bk : Hk → Xk are continuous linear mappings, Xk Hilbert space,k = 0, 1, such that C ∗ B1∗ densely dened and B0 is a bijection or CB0∗ densely dened and B1 is a bijection. Then



B0 0

0

B1



0

C

−C ∗ 0



B0∗ 0

0

B1∗



is skew-selfadjoint.

Mother anddescendant.

Rainer Picard

Evo-Systems

6/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Mother and Descendant Mechanism Theorem

Let C : D (C ) ⊆ H0 → H1 be a closed densely dened linear operator, Hk ,k = 0, 1, Hilbert spaces. If Bk : Hk → Xk are continuous linear mappings, Xk Hilbert space,k = 0, 1, such that C ∗ B1∗ densely dened and B0 is a bijection or CB0∗ densely dened and B1 is a bijection. Then



B0 0

0

B1



0

C

−C ∗ 0



B0∗ 0

0

B1∗



is skew-selfadjoint.

Mother anddescendant.

Rainer Picard

Evo-Systems

6/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Mother and Descendant Mechanism Theorem

Let C : D (C ) ⊆ H0 → H1 be a closed densely dened linear operator, Hk ,k = 0, 1, Hilbert spaces. If Bk : Hk → Xk are continuous linear mappings, Xk Hilbert space,k = 0, 1, such that C ∗ B1∗ densely dened and B0 is a bijection or CB0∗ densely dened and B1 is a bijection. Then



B0 0

0

B1



0

C

−C ∗ 0



B0∗ 0

0

B1∗



is skew-selfadjoint.

Mother anddescendant.

Rainer Picard

Evo-Systems

6/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Mother and Descendant Mechanism Theorem

Let C : D (C ) ⊆ H0 → H1 be a closed densely dened linear operator, Hk ,k = 0, 1, Hilbert spaces. If Bk : Hk → Xk are continuous linear mappings, Xk Hilbert space,k = 0, 1, such that C ∗ B1∗ densely dened and B0 is a bijection or CB0∗ densely dened and B1 is a bijection. Then



B0 0

0

B1



0

C

−C ∗ 0



B0∗ 0

0

B1∗



is skew-selfadjoint.

Mother anddescendant.

Rainer Picard

Evo-Systems

6/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Mother and Descendant Mechanism Examples: Dirichlet boundary condition

˚: A := G =∇



0

G

−G ∗



0

tensor order (or degree;Stufe) symmetric/alternating 3-dimensional order 0, 1



acoustics

order 1, 2

symmetric

elastics

order 1, 2

alternating

electrodynamics

descend in space dimension vanishing trace condition (divergence-free; incompressible Stokes equation)

Rainer Picard

Evo-Systems

7/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Mother and Descendant Mechanism Examples: Dirichlet boundary condition

˚: A := G =∇



0

G

−G ∗



0

tensor order (or degree;Stufe) symmetric/alternating 3-dimensional order 0, 1



acoustics

order 1, 2

symmetric

elastics

order 1, 2

alternating

electrodynamics

descend in space dimension vanishing trace condition (divergence-free; incompressible Stokes equation)

Rainer Picard

Evo-Systems

7/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Mother and Descendant Mechanism Examples: Dirichlet boundary condition

˚: A := G =∇



0

G

−G ∗



0

tensor order (or degree;Stufe) symmetric/alternating 3-dimensional order 0, 1



acoustics

order 1, 2

symmetric

elastics

order 1, 2

alternating

electrodynamics

descend in space dimension vanishing trace condition (divergence-free; incompressible Stokes equation)

Rainer Picard

Evo-Systems

7/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Mother and Descendant Mechanism Examples: Dirichlet boundary condition

˚: A := G =∇



0

G

−G ∗



0

tensor order (or degree;Stufe) symmetric/alternating 3-dimensional order 0, 1



acoustics

order 1, 2

symmetric

elastics

order 1, 2

alternating

electrodynamics

descend in space dimension vanishing trace condition (divergence-free; incompressible Stokes equation)

Rainer Picard

Evo-Systems

7/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Mother and Descendant Mechanism Examples: Dirichlet boundary condition

˚: A := G =∇



0

G

−G ∗



0

tensor order (or degree;Stufe) symmetric/alternating 3-dimensional order 0, 1



acoustics

order 1, 2

symmetric

elastics

order 1, 2

alternating

electrodynamics

descend in space dimension vanishing trace condition (divergence-free; incompressible Stokes equation)

Rainer Picard

Evo-Systems

7/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Mother and Descendant Mechanism Examples: Dirichlet boundary condition

˚: A := G =∇



0

G

−G ∗



0

tensor order (or degree;Stufe) symmetric/alternating 3-dimensional order 0, 1



acoustics

order 1, 2

symmetric

elastics

order 1, 2

alternating

electrodynamics

descend in space dimension vanishing trace condition (divergence-free; incompressible Stokes equation)

Rainer Picard

Evo-Systems

7/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Mother and Descendant Mechanism Examples: Dirichlet boundary condition

˚: A := G =∇



0

G

−G ∗



0

tensor order (or degree;Stufe) symmetric/alternating 3-dimensional order 0, 1



acoustics

order 1, 2

symmetric

elastics

order 1, 2

alternating

electrodynamics

descend in space dimension vanishing trace condition (divergence-free; incompressible Stokes equation)

Rainer Picard

Evo-Systems

7/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Mother and Descendant Mechanism Examples: Dirichlet boundary condition

˚: A := G =∇



0

G

−G ∗



0

tensor order (or degree;Stufe) symmetric/alternating 3-dimensional order 0, 1



acoustics

order 1, 2

symmetric

elastics

order 1, 2

alternating

electrodynamics

descend in space dimension vanishing trace condition (divergence-free; incompressible Stokes equation)

Rainer Picard

Evo-Systems

7/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Abstract grad-div Systems. For abstract grad − div systems the spatial operator form

A=



0

G

−G ∗ 0

A

is still of the

 ,

but here



G = 

(

Gk not

systems

G1 . . .

Gn

   : D ( G ) ⊆ H0 → H1 ⊕ · · · ⊕ Hn

closable, in the standard case of grad − div or Gk = ∂k ), i.e. we have that the range space is a

necessarily

Gk = ˚ ∂k

direct sum of Hilbert spaces.

Rainer Picard

Evo-Systems

8/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Abstract grad-div Systems.



G = 

With

G

also

Gk

G1



. . .

 ,

Gn

can be considered as a continuous mapping from

D (G ) (as a Hilbert space with respect to the graph inner product; D (G ) ⊆ H0 ⊆ D (G )0 Gelfand triple ) to Hk ,k = 1, . . . , n. 0 0 Identifying Hk with Hk we have as continuous duals Gk : Hk → D (G ) ,

k=

1

, . . . , n.

G ∗ ⊆ G1 · · · Gn

It is

    x1 .  ∗ D (G ) =   ..  ∈ H1 ⊕ · · · ⊕ Hn |  and

xn




  x1
 ..    G1 · · · Gn  .  =

Rainer Picard

xn

Evo-Systems

  

∑ Gk xk ∈ H0  .

k =1,...,n



9/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Abstract grad-div Systems.



G = 

With

G

also

Gk

G1



. . .

 ,

Gn

can be considered as a continuous mapping from

D (G ) (as a Hilbert space with respect to the graph inner product; D (G ) ⊆ H0 ⊆ D (G )0 Gelfand triple ) to Hk ,k = 1, . . . , n. 0 0 Identifying Hk with Hk we have as continuous duals Gk : Hk → D (G ) ,

k=

1

, . . . , n.

G ∗ ⊆ G1 · · · Gn

It is

    x1 .  ∗ D (G ) =   ..  ∈ H1 ⊕ · · · ⊕ Hn |  and

xn




  x1
 ..    G1 · · · Gn  .  =

Rainer Picard

xn

Evo-Systems

  

∑ Gk xk ∈ H0  .

k =1,...,n



9/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Abstract grad-div Systems.



G = 

With

G

also

Gk

G1



. . .

 ,

Gn

can be considered as a continuous mapping from

D (G ) (as a Hilbert space with respect to the graph inner product; D (G ) ⊆ H0 ⊆ D (G )0 Gelfand triple ) to Hk ,k = 1, . . . , n. 0 0 Identifying Hk with Hk we have as continuous duals Gk : Hk → D (G ) ,

k=

1

, . . . , n.

G ∗ ⊆ G1 · · · Gn

It is

    x1 .  ∗ D (G ) =   ..  ∈ H1 ⊕ · · · ⊕ Hn |  and

xn




  x1
 ..    G1 · · · Gn  .  =

Rainer Picard

xn

Evo-Systems

  

∑ Gk xk ∈ H0  .

k =1,...,n



9/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Non-Autonomous Evo-Systems

(∂0 M0 (m0 ) + M1 (m0 ) + A)U = F , where

M0 , M1 ∈ L∞s (R; L(H )),

the space of strongly measurable

bounded functions with values in

L(H ).

M0 :

Properties for (a) (b)

M0 (t ) M0 (t )

(t ∈ R), non-negative (t ∈ R),

is selfadjoint is

(c) the mapping

M0

(d) there exists a set

x ∈H

is Lipschitz-continuous,

N ⊆R

of measure zero such that for each

the function

R \ N 3 t 7→ M0 (t )x is dierentiable.

Rainer Picard

Evo-Systems

10/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Non-Autonomous Evo-Systems

(∂0 M0 (m0 ) + M1 (m0 ) + A)U = F , where

M0 , M1 ∈ L∞s (R; L(H )),

the space of strongly measurable

bounded functions with values in

L(H ).

M0 :

Properties for (a) (b)

M0 (t ) M0 (t )

(t ∈ R), non-negative (t ∈ R),

is selfadjoint is

(c) the mapping

M0

(d) there exists a set

x ∈H

is Lipschitz-continuous,

N ⊆R

of measure zero such that for each

the function

R \ N 3 t 7→ M0 (t )x is dierentiable.

Rainer Picard

Evo-Systems

10/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Non-Autonomous Evo-Systems

(∂0 M0 (m0 ) + M1 (m0 ) + A)U = F , where

M0 , M1 ∈ L∞s (R; L(H )),

the space of strongly measurable

bounded functions with values in

L(H ).

M0 :

Properties for (a) (b)

M0 (t ) M0 (t )

(t ∈ R), non-negative (t ∈ R),

is selfadjoint is

(c) the mapping

M0

(d) there exists a set

x ∈H

is Lipschitz-continuous,

N ⊆R

of measure zero such that for each

the function

R \ N 3 t 7→ M0 (t )x is dierentiable.

Rainer Picard

Evo-Systems

10/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Non-Autonomous Evo-Systems

(∂0 M0 (m0 ) + M1 (m0 ) + A)U = F , where

M0 , M1 ∈ L∞s (R; L(H )),

the space of strongly measurable

bounded functions with values in

L(H ).

M0 :

Properties for (a) (b)

M0 (t ) M0 (t )

(t ∈ R), non-negative (t ∈ R),

is selfadjoint is

(c) the mapping

M0

(d) there exists a set

x ∈H

is Lipschitz-continuous,

N ⊆R

of measure zero such that for each

the function

R \ N 3 t 7→ M0 (t )x is dierentiable.

Rainer Picard

Evo-Systems

10/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Non-Autonomous Evo-Systems We require:

_

c0 >

^

,ρ0 >0 t ∈R\N ,ρ ≥ρ0

1 ˙ 0 (t ) + symM1 (t ) ≥ c0 . ρ M0 (t ) + M

0

2

Theorem

Let A : D (A) ⊆ H → H be skew-selfadjoint and M0 , M1 ∈ L∞ (R; L(H )). Under these assumptions the operator ∂0 M0 (m0 ) + M1 (m0 ) + A is continuously invertible in Hρ ,0 (R; H ) for each ρ ≥ ρ0 . A norm bound for the inverse is1/c0 . Moreover, we get that (∂0 M0 (m0 ) + M1 (m0 ) + A)∗ = M0 (m0 ) ∂0∗ + M1 (m0 )∗ − A.

The solution operator (∂0 M0 (m0 ) + M1 (m0 ) + A)−1 is causal in Hρ ,0 (R, H ) for each ρ ≥ ρ0 . Rainer Picard

Evo-Systems

11/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Further Extensions

ResearchGate & arXiv

Rainer Picard

Evo-Systems

12/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Thanks!

Thank you very much for showing your interest and for attending this course!

Rainer Picard

Evo-Systems

13/14

Coupling of Dierent Physical Phenomena Mother and Descendant Mechanism Abstract grad-div Systems Non-Autonomous Evo-Systems Further Extensions

Thanks!

A cordial word of thanks to the organizers of this event from Khovd and Ulaanbataar for a tremendous job well done.

Rainer Picard

Evo-Systems

14/14