The Existence theorem of the Stokes-Neumann Problem Nasrin Arab CASA - PowerPoint PPT Presentation

The Existence theorem of the Stokes-Neumann Problem Nasrin Arab CASA Tu / e 28 April 2010 Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 1 / 21

Overview Review 1 Stokes-Neumann problem 2 Solvability of the Stokes equations 3 conformal bijection 4 Main theorem 5 Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 2 / 21

Outline Review 1 Stokes-Neumann problem 2 Solvability of the Stokes equations 3 conformal bijection 4 Main theorem 5 Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 3 / 21

Review Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 4 / 21

The Stokes equation in R 2 Stokes equations � ∆ v − ∇ p = 0 x ∈ G . ∇ · v = 0 Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 5 / 21

The Stokes equation in R 2 Stokes equations � ∆ v − ∇ p = 0 x ∈ G . ∇ · v = 0 we can rewrite it as � ∇ · T = 0 x ∈ G . ∇ · v = 0 Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 5 / 21

The Stokes equation in R 2 Stokes equations � ∆ v − ∇ p = 0 x ∈ G . ∇ · v = 0 we can rewrite it as � ∇ · T = 0 x ∈ G . ∇ · v = 0 where T : = − pI + ∇ v + ( ∇ v ) T Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 5 / 21

General solutions of Stokes equations without regarding boundary conditions Holomorphic representation Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 6 / 21

General solutions of Stokes equations without regarding boundary conditions Holomorphic representation Theorem If p ( x ) , v ( x ) solves the Stokes equation on G , then there exists a pair of analytic functions z �−→ ϕ ( z ) , χ ( z ) on G , such that Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 6 / 21

General solutions of Stokes equations without regarding boundary conditions Holomorphic representation Theorem If p ( x ) , v ( x ) solves the Stokes equation on G , then there exists a pair of analytic functions z �−→ ϕ ( z ) , χ ( z ) on G , such that v 1 + i v 2 = − ϕ + z ϕ ′ + χ ′          − 4 p = T 11 + T 22 = − 8Re ϕ ′ ⇒  p = 2Re ϕ ′         ds ( z ϕ ′ + ϕ + χ ′ )  T · n = 2i d   Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 6 / 21

General solutions of Stokes equations without regarding boundary conditions Holomorphic representation Theorem If p ( x ) , v ( x ) solves the Stokes equation on G , then there exists a pair of analytic functions z �−→ ϕ ( z ) , χ ( z ) on G , such that v 1 + i v 2 = − ϕ + z ϕ ′ + χ ′          − 4 p = T 11 + T 22 = − 8Re ϕ ′ ⇒  p = 2Re ϕ ′         ds ( z ϕ ′ + ϕ + χ ′ )  T · n = 2i d   Vice versa Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 6 / 21

General solutions of Stokes equations without regarding boundary conditions Holomorphic representation Theorem If p ( x ) , v ( x ) solves the Stokes equation on G , then there exists a pair of analytic functions z �−→ ϕ ( z ) , χ ( z ) on G , such that v 1 + i v 2 = − ϕ + z ϕ ′ + χ ′          − 4 p = T 11 + T 22 = − 8Re ϕ ′ ⇒  p = 2Re ϕ ′         ds ( z ϕ ′ + ϕ + χ ′ )  T · n = 2i d   Vice versa The holomorphic representation of a solution by { ϕ, χ } is unique if ϕ (0) = χ (0) = 0 Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 6 / 21

Outline Review 1 Stokes-Neumann problem 2 Solvability of the Stokes equations 3 conformal bijection 4 Main theorem 5 Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 7 / 21

Stokes-Neumann problem Stokes equation with Neumann boundary condition  ∇ · T (x) = 0 , x ∈ G     ∇ · v (x) = 0 , x ∈ G     T (x) · n(x) = f ( x ) , x ∈ ∂ G   Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 8 / 21

Stokes-Neumann problem Stokes equation with Neumann boundary condition  ∇ · T (x) = 0 , x ∈ G     ∇ · v (x) = 0 , x ∈ G     T (x) · n(x) = f ( x ) , x ∈ ∂ G   on the prescribed boundary stress field x �−→ f(x) ∈ R 2 we put condition on f Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 8 / 21

Stokes-Neumann problem Stokes equation with Neumann boundary condition  ∇ · T (x) = 0 , x ∈ G     ∇ · v (x) = 0 , x ∈ G     T (x) · n(x) = f ( x ) , x ∈ ∂ G   on the prescribed boundary stress field x �−→ f(x) ∈ R 2 we put condition on f f ( x ( s )) = d ds { K 1 ( s ) n ( s ) + K 2 ( s ) t ( s ) } Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 8 / 21

Stokes-Neumann problem Stokes equation with Neumann boundary condition  ∇ · T (x) = 0 , x ∈ G     ∇ · v (x) = 0 , x ∈ G     T (x) · n(x) = f ( x ) , x ∈ ∂ G   on the prescribed boundary stress field x �−→ f(x) ∈ R 2 we put condition on f f ( x ( s )) = d ds { K 1 ( s ) n ( s ) + K 2 ( s ) t ( s ) } � K 1 ( s ) ds = 0 ∂ G � { K 1 ( s ) n ( s ) + K 2 ( s ) t ( s ) } ds = 0 ∂ G Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 8 / 21

Stokes-Neumann problem Stokes equation with Neumann boundary condition  ∇ · T (x) = 0 , x ∈ G     ∇ · v (x) = 0 , x ∈ G     T (x) · n(x) = f ( x ) , x ∈ ∂ G   on the prescribed boundary stress field x �−→ f(x) ∈ R 2 we put condition on f f ( x ( s )) = d ds { K 1 ( s ) n ( s ) + K 2 ( s ) t ( s ) } � K 1 ( s ) ds = 0 ∂ G � { K 1 ( s ) n ( s ) + K 2 ( s ) t ( s ) } ds = 0 ∂ G ”What about (non)-uniqueness of the Stokes-Neumann problem” Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 8 / 21

Outline Review 1 Stokes-Neumann problem 2 Solvability of the Stokes equations 3 conformal bijection 4 Main theorem 5 Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 9 / 21

Solvability Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 10 / 21

necessary condition in terms of { ϕ, χ } we want to find analytic ϕ, χ : G −→ C , such that at the boundary ∂ G Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 11 / 21

necessary condition in terms of { ϕ, χ } we want to find analytic ϕ, χ : G −→ C , such that at the boundary ∂ G T · n ( s ) = 2i d ds ( z ( s ) ϕ ′ ( z ( s )) + ϕ ( z ( s )) + χ ′ ( z ( s ))) = − i d dsK ( s )˙ z ( s ) . (1) Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 11 / 21

necessary condition in terms of { ϕ, χ } we want to find analytic ϕ, χ : G −→ C , such that at the boundary ∂ G T · n ( s ) = 2i d ds ( z ( s ) ϕ ′ ( z ( s )) + ϕ ( z ( s )) + χ ′ ( z ( s ))) = − i d dsK ( s )˙ z ( s ) . (1)  z ( s ) ϕ ′ ( z ( s )) + ϕ ( z ( s )) + χ ′ ( z ( s )) = − 1 2 K ( s )˙ z      ϕ (0) = χ (0) = 0   Im ϕ ′ (0) = 0    Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 11 / 21

necessary condition in terms of { ϕ, χ } we want to find analytic ϕ, χ : G −→ C , such that at the boundary ∂ G T · n ( s ) = 2i d ds ( z ( s ) ϕ ′ ( z ( s )) + ϕ ( z ( s )) + χ ′ ( z ( s ))) = − i d dsK ( s )˙ z ( s ) . (1)  z ( s ) ϕ ′ ( z ( s )) + ϕ ( z ( s )) + χ ′ ( z ( s )) = − 1 2 K ( s )˙ z      ϕ (0) = χ (0) = 0   Im ϕ ′ (0) = 0    d z ( s ) = − 1 � � + ϕ ( z ( s ))˙ z ( s ) ϕ ( z ( s )) + χ ( z ( s )) z ( s ) − ϕ ( z ( s ))˙ ¯ 2 K ( s ) ds Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 11 / 21

necessary condition in terms of { ϕ, χ } we want to find analytic ϕ, χ : G −→ C , such that at the boundary ∂ G T · n ( s ) = 2i d ds ( z ( s ) ϕ ′ ( z ( s )) + ϕ ( z ( s )) + χ ′ ( z ( s ))) = − i d dsK ( s )˙ z ( s ) . (1)  z ( s ) ϕ ′ ( z ( s )) + ϕ ( z ( s )) + χ ′ ( z ( s )) = − 1 2 K ( s )˙ z      ϕ (0) = χ (0) = 0   Im ϕ ′ (0) = 0    d z ( s ) = − 1 � � + ϕ ( z ( s ))˙ z ( s ) ϕ ( z ( s )) + χ ( z ( s )) z ( s ) − ϕ ( z ( s ))˙ ¯ 2 K ( s ) ds � K 1 ( s ) ds = 0 ∂ G Nasrin Arab (CASA Tu / e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 11 / 21