The heat equation in a non-cylindrical domain governed by a - PowerPoint PPT Presentation

The heat equation in a non-cylindrical domain governed by a subdifferential inclusion Jos´ e Alberto Murillo Hern´ andez Universidad Polit´ ecnica de Cartagena Spain J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 1 / 20

Outline 1 The heat equation in time-varying domains. 2 Morphological shape equations: a way to describe evolving sets. 3 The heat equation in a time-varying domain described by a subdifferential inclusion. 4 Conclusions. J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 1 / 20

Outline 1 The heat equation in time-varying domains. 2 Morphological shape equations: a way to describe evolving sets. 3 The heat equation in a time-varying domain described by a subdifferential inclusion. 4 Conclusions. J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 1 / 20

Outline 1 The heat equation in time-varying domains. 2 Morphological shape equations: a way to describe evolving sets. 3 The heat equation in a time-varying domain described by a subdifferential inclusion. 4 Conclusions. J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 1 / 20

Outline 1 The heat equation in time-varying domains. 2 Morphological shape equations: a way to describe evolving sets. 3 The heat equation in a time-varying domain described by a subdifferential inclusion. 4 Conclusions. J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 1 / 20

Outline 1 The heat equation in time-varying domains. 2 Morphological shape equations: a way to describe evolving sets. 3 The heat equation in a time-varying domain described by a subdifferential inclusion. 4 Conclusions. J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 1 / 20

Cylindrical problems  u t ( x , t ) − div ( a ( x ) ∇ u ( x , t )) = f ( x , t ) , in Q    u = 0 , on Σ (CP)   u ( x , 0) = u 0 ( x ) , x ∈ Ω  where J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 2 / 20

Cylindrical problems  u t ( x , t ) − div ( a ( x ) ∇ u ( x , t )) = f ( x , t ) , in Q    u = 0 , on Σ (CP)   u ( x , 0) = u 0 ( x ) , x ∈ Ω  where R N +1 Q = Ω × ]0 , T [ ⊂ I J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 2 / 20

Cylindrical problems  u t ( x , t ) − div ( a ( x ) ∇ u ( x , t )) = f ( x , t ) , in Q    u = 0 , on Σ (CP)   u ( x , 0) = u 0 ( x ) , x ∈ Ω  where R N +1 Q = Ω × ]0 , T [ ⊂ I R N spatial domain Ω ⊂ I J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 2 / 20

Cylindrical problems  u t ( x , t ) − div ( a ( x ) ∇ u ( x , t )) = f ( x , t ) , in Q    u = 0 , on Σ (CP)   u ( x , 0) = u 0 ( x ) , x ∈ Ω  where R N +1 Q = Ω × ]0 , T [ ⊂ I R N spatial domain Ω ⊂ I Σ = { ( x , t ) : 0 ≤ t < T , x ∈ Γ } J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 2 / 20

Non-cylindrical problems  u t ( x , t ) − div ( a ( x ) ∇ u ( x , t )) = f ( x , t ) , in Q    u = 0 , on Σ (NCP)   u ( x , 0) = u 0 ( x ) , x ∈ Ω 0  where J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 2 / 20

Non-cylindrical problems  u t ( x , t ) − div ( a ( x ) ∇ u ( x , t )) = f ( x , t ) , in Q    u = 0 , on Σ (NCP)   u ( x , 0) = u 0 ( x ) , x ∈ Ω 0  where R N +1 Q = { ( x , t ) : 0 < t < T , x ∈ Ω t } ⊂ I J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 2 / 20

Non-cylindrical problems  u t ( x , t ) − div ( a ( x ) ∇ u ( x , t )) = f ( x , t ) , in Q    u = 0 , on Σ (NCP)   u ( x , 0) = u 0 ( x ) , x ∈ Ω 0  where R N +1 Q = { ( x , t ) : 0 < t < T , x ∈ Ω t } ⊂ I R N , spatial domain changing along time Ω t ⊂ I J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 2 / 20

Non-cylindrical problems  u t ( x , t ) − div ( a ( x ) ∇ u ( x , t )) = f ( x , t ) , in Q    u = 0 , on Σ (NCP)   u ( x , 0) = u 0 ( x ) , x ∈ Ω 0  where R N +1 Q = { ( x , t ) : 0 < t < T , x ∈ Ω t } ⊂ I R N , spatial domain changing along time Ω t ⊂ I Σ = { ( x , t ) : 0 ≤ t < T , x ∈ Γ t } J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 2 / 20

Cylindrical and non-cylindrical domains R N +1 spatio-temporal domain Q ⊂ I J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 3 / 20

Cylindrical and non-cylindrical domains R N +1 spatio-temporal domain Q ⊂ I Cylindrical case J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 3 / 20

Cylindrical and non-cylindrical domains R N +1 spatio-temporal domain Q ⊂ I Cylindrical case Non-cylindrical case J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 3 / 20

Describing the evolution of Ω t : Velocity method Usually Ω t is assumed to be generated by the flow of a nonautonomous vector field R N − R N V : [0 , T ] × I → I J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 4 / 20

Describing the evolution of Ω t : Velocity method Usually Ω t is assumed to be generated by the flow of a nonautonomous vector field R N − R N V : [0 , T ] × I → I That is Ω t = T t (Ω 0 ), where  ∂ T t ( x ) = V ( t , T t ( x ))   ∂ t  T 0 ( x ) = x  J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 4 / 20

Describing the evolution of Ω t : Velocity method Usually Ω t is assumed to be generated by the flow of a nonautonomous vector field R N − R N V : [0 , T ] × I → I That is Ω t = T t (Ω 0 ), where  ∂ T t ( x ) = V ( t , T t ( x ))   ∂ t  T 0 ( x ) = x  Cannarsa, Da Prato & Zol´ esio (1989, 1990) Zol´ esio (2004) Burdzy, Chen & Sylvester (2004) etc... J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 4 / 20

Describing the evolution of Ω t : Velocity method J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 5 / 20

Solving (NCP) The classical procedure to solve (NCP) is: � Show that T t is a diffeomorphism for any t , Lipschitz w.r. to t . � Use T t to transform the heat equation into a parabolic one with variable coefficients defined in the reference cylinder Ω 0 × ]0 , T [. � Establish the existence of solution ω ( x , t ) of the parabolic problem. � The map u ( x , t ) = ω ( T − 1 ( x ) , t ) provides the solution of the original t non-cylindrical problem. J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 6 / 20

Solving (NCP) The classical procedure to solve (NCP) is: � Show that T t is a diffeomorphism for any t , Lipschitz w.r. to t . � Use T t to transform the heat equation into a parabolic one with variable coefficients defined in the reference cylinder Ω 0 × ]0 , T [. � Establish the existence of solution ω ( x , t ) of the parabolic problem. � The map u ( x , t ) = ω ( T − 1 ( x ) , t ) provides the solution of the original t non-cylindrical problem. J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 6 / 20

Solving (NCP) The classical procedure to solve (NCP) is: � Show that T t is a diffeomorphism for any t , Lipschitz w.r. to t . � Use T t to transform the heat equation into a parabolic one with variable coefficients defined in the reference cylinder Ω 0 × ]0 , T [. � Establish the existence of solution ω ( x , t ) of the parabolic problem. � The map u ( x , t ) = ω ( T − 1 ( x ) , t ) provides the solution of the original t non-cylindrical problem. J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 6 / 20

Solving (NCP) The classical procedure to solve (NCP) is: � Show that T t is a diffeomorphism for any t , Lipschitz w.r. to t . � Use T t to transform the heat equation into a parabolic one with variable coefficients defined in the reference cylinder Ω 0 × ]0 , T [. � Establish the existence of solution ω ( x , t ) of the parabolic problem. � The map u ( x , t ) = ω ( T − 1 ( x ) , t ) provides the solution of the original t non-cylindrical problem. J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 6 / 20

Solving (NCP) The classical procedure to solve (NCP) is: � Show that T t is a diffeomorphism for any t , Lipschitz w.r. to t . � Use T t to transform the heat equation into a parabolic one with variable coefficients defined in the reference cylinder Ω 0 × ]0 , T [. � Establish the existence of solution ω ( x , t ) of the parabolic problem. � The map u ( x , t ) = ω ( T − 1 ( x ) , t ) provides the solution of the original t non-cylindrical problem. J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 6 / 20