ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION A. Terracina ENTROPY FORMULATION FOR FORWARD-BACKWARD PARABOLIC EQUATION A.Terracina University La Sapienza, Roma, Italy 25/06/2012
ENTROPY FORMULATION FOR Collaboration FORWARD- BACKWARD PARABOLIC EQUATION A. Terracina Corrado Mascia Flavia Smarrazzo Alberto Tesei
ENTROPY FORMULATION FOR Introduction FORWARD- BACKWARD PARABOLIC Forward-backward parabolic equation EQUATION A. Terracina u t = ∆ φ ( u ) (1) where the function φ ∈ Lip loc ( R ) is decreasing in some interval.
ENTROPY FORMULATION FOR Introduction FORWARD- BACKWARD PARABOLIC Forward-backward parabolic equation EQUATION A. Terracina u t = ∆ φ ( u ) (1) where the function φ ∈ Lip loc ( R ) is decreasing in some interval. Example 1 Model of phase separation φ ′ ( u ) > 0 if u ∈ ( −∞ , b ) ∪ ( a , ∞ ) , φ ′ ( u ) < 0 if u ∈ ( b , a ); v B φ ( u ) c b a d u A
Example 2 ENTROPY FORMULATION Model of image processing (1D), Perona–Malik equation FOR FORWARD- u φ ( u ) = 1+ u 2 . BACKWARD PARABOLIC In this case the instability region is unbounded. EQUATION A. Terracina
Example 2 ENTROPY FORMULATION Model of image processing (1D), Perona–Malik equation FOR FORWARD- u φ ( u ) = 1+ u 2 . BACKWARD PARABOLIC In this case the instability region is unbounded. EQUATION Example 3 A. Terracina Model of population dynamic, Padron (Comm. Partial Differential Equations 1998) φ ( u ) = ue − u u ≥ 0.
Example 2 ENTROPY FORMULATION Model of image processing (1D), Perona–Malik equation FOR FORWARD- u φ ( u ) = 1+ u 2 . BACKWARD PARABOLIC In this case the instability region is unbounded. EQUATION Example 3 A. Terracina Model of population dynamic, Padron (Comm. Partial Differential Equations 1998) φ ( u ) = ue − u u ≥ 0. Problems are ill–posed.
Example 2 ENTROPY FORMULATION Model of image processing (1D), Perona–Malik equation FOR FORWARD- u φ ( u ) = 1+ u 2 . BACKWARD PARABOLIC In this case the instability region is unbounded. EQUATION Example 3 A. Terracina Model of population dynamic, Padron (Comm. Partial Differential Equations 1998) φ ( u ) = ue − u u ≥ 0. Problems are ill–posed. Hollig (Trans. Amer. Math. Soc. 83) φ piecewise linear, there is an infinite number of solutions of the Neumann boundary problem. v B ! ! + " c b a d u ! 0 A
ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION A. Terracina IDEA : Introducing a viscous regularization that gives a good formulation in analogy with first order conservation laws
ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION A. Terracina IDEA : Introducing a viscous regularization that gives a good formulation in analogy with first order conservation laws Problem is ill-posed since some relevant physical terms are neglected
ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION A. Terracina Phase transition, Cahn–Hilliard equation u t = ∆( φ ( u ) − δ ∆ u ) .
ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION A. Terracina Phase transition, Cahn–Hilliard equation u t = ∆( φ ( u ) − δ ∆ u ) . An analogous approximation for the Perona–Malik equation (1D)
ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION A. Terracina Phase transition, Cahn–Hilliard equation u t = ∆( φ ( u ) − δ ∆ u ) . An analogous approximation for the Perona–Malik equation (1D) Model of population dynamic, Padron u t = ( φ ( u ) + ǫ u t ) xx .
ENTROPY FORMULATION FOR Phase transition FORWARD- BACKWARD PARABOLIC EQUATION A. Terracina Viscous Cahn–Hilliard Novick Cohen (1988) Gurtin (1996) based on microforce balance u t = ∆( φ ( u ) − δ ∆ u + ǫ u t )
ENTROPY FORMULATION FOR Phase transition FORWARD- BACKWARD PARABOLIC EQUATION A. Terracina Viscous Cahn–Hilliard Novick Cohen (1988) Gurtin (1996) based on microforce balance u t = ∆( φ ( u ) − δ ∆ u + ǫ u t ) In the following δ = 0 , φ is of cubic type.
ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION Novick Cohen-Pego ( Trans. Amer. Math. Soc. 1991) study the A. Terracina viscosity problem u t = ∆ v in Ω × (0 , T ] =: Q T ∂ v ∂ν = 0 ∂ Ω × (0 , T ] (2) in u = u 0 in Ω × { 0 } , where v := φ ( u ) + ǫ u t ( ǫ > 0) , (3) R n is bounded , ∂ Ω regular, T > 0. is the chemical potential , Ω ⊆ I
ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION A. Terracina Partial differential equation in (2) can be rewritten u t = − 1 ǫ ( I − ( I − ǫ ∆) − 1 ) φ ( u ) that corresponds to the Yosida approximation of the operator ∆.
ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION A. Terracina Partial differential equation in (2) can be rewritten u t = − 1 ǫ ( I − ( I − ǫ ∆) − 1 ) φ ( u ) that corresponds to the Yosida approximation of the operator ∆. Moreover v = ( I − ǫ ∆) − 1 φ ( u ) .
ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION A. Terracina Partial differential equation in (2) can be rewritten u t = − 1 ǫ ( I − ( I − ǫ ∆) − 1 ) φ ( u ) that corresponds to the Yosida approximation of the operator ∆. Moreover v = ( I − ǫ ∆) − 1 φ ( u ) . Using the standard theory of ODE in the Banach spaces we have Theorem (Novick Cohen-Pego) Given u 0 ∈ L ∞ (Ω) , ǫ > 0 there exists a unique solution ( u ǫ , v ǫ ) defined in (0 , T ǫ ) , u ǫ ∈ C 1 ([0 , T ǫ ) , L ∞ (Ω)) .
ENTROPY FORMULATION FOR A priori estimates FORWARD- BACKWARD PARABOLIC For every g ∈ C 1 ( R ) such that g ′ ≥ 0 EQUATION A. Terracina � u G ( u ) = g ( φ ( s )) ds + c . 0
ENTROPY FORMULATION FOR A priori estimates FORWARD- BACKWARD PARABOLIC For every g ∈ C 1 ( R ) such that g ′ ≥ 0 EQUATION A. Terracina � u G ( u ) = g ( φ ( s )) ds + c . 0 Then � � − g ′ ( v ǫ ) |∇ v ǫ | 2 + [ G ( u ǫ )] t = div g ( v ǫ ) ∇ v ǫ − 1 � � g ( φ ( u ǫ )) − g ( v ǫ ) ( φ ( u ǫ ) − v ǫ ) . ǫ
ENTROPY FORMULATION FOR A priori estimates FORWARD- BACKWARD PARABOLIC For every g ∈ C 1 ( R ) such that g ′ ≥ 0 EQUATION A. Terracina � u G ( u ) = g ( φ ( s )) ds + c . 0 Then � � − g ′ ( v ǫ ) |∇ v ǫ | 2 + [ G ( u ǫ )] t = div g ( v ǫ ) ∇ v ǫ − 1 � � g ( φ ( u ǫ )) − g ( v ǫ ) ( φ ( u ǫ ) − v ǫ ) . ǫ Integrating in Ω and using boundary condition we have d � G ( u ǫ ( x , t )) dx ≤ 0 dt Ω that gives a priori estimates in L ∞ . Moreover choosing g ( u ) ≡ u we have �� � |∇ v ǫ | 2 + ǫ | ∂ t u ǫ | 2 � dxdt ≤ C 2 . Q T
ENTROPY FORMULATION FOR Entropy formulation FORWARD- BACKWARD PARABOLIC EQUATION In analogy with conservation laws we characterize an entropy solution A. Terracina of problem u t = ∆ φ ( u ) in Ω × (0 , T ] = Q T ∂φ ( u ) = 0 ∂ Ω × (0 , T ] (4) in ∂ν u = u 0 in Ω × { 0 } , as that obtained as limit of the solutions of problem (2) when ǫ → 0 + .
ENTROPY FORMULATION FOR Entropy formulation FORWARD- BACKWARD PARABOLIC EQUATION In analogy with conservation laws we characterize an entropy solution A. Terracina of problem u t = ∆ φ ( u ) in Ω × (0 , T ] = Q T ∂φ ( u ) = 0 ∂ Ω × (0 , T ] (4) in ∂ν u = u 0 in Ω × { 0 } , as that obtained as limit of the solutions of problem (2) when ǫ → 0 + . For every ǫ > 0 and g ∈ C 1 ( R ), g ′ ≥ 0 we have �� � G ( u ǫ ) ψ t − g ( v ǫ ) ∇ v ǫ · ∇ ψ − g ′ ( v ǫ ) |∇ v ǫ | 2 ψ � ≥ 0 (5) Q T for every ψ ∈ C ∞ 0 ( Q T ), ψ ≥ 0.
ENTROPY FORMULATION FOR Entropy formulation FORWARD- BACKWARD PARABOLIC EQUATION In analogy with conservation laws we characterize an entropy solution A. Terracina of problem u t = ∆ φ ( u ) in Ω × (0 , T ] = Q T ∂φ ( u ) = 0 ∂ Ω × (0 , T ] (4) in ∂ν u = u 0 in Ω × { 0 } , as that obtained as limit of the solutions of problem (2) when ǫ → 0 + . For every ǫ > 0 and g ∈ C 1 ( R ), g ′ ≥ 0 we have �� � G ( u ǫ ) ψ t − g ( v ǫ ) ∇ v ǫ · ∇ ψ − g ′ ( v ǫ ) |∇ v ǫ | 2 ψ � ≥ 0 (5) Q T for every ψ ∈ C ∞ 0 ( Q T ), ψ ≥ 0. The idea is to pass in the limit in (5) to characterize an entropy solution of (4).
ENTROPY FORMULATION FOR Plotnikov’s results FORWARD- BACKWARD PARABOLIC Study of the singular limit, Plotnikov (J. Math. Sci. 1993). EQUATION A. Terracina
ENTROPY FORMULATION FOR Plotnikov’s results FORWARD- BACKWARD PARABOLIC Study of the singular limit, Plotnikov (J. Math. Sci. 1993). EQUATION A. Terracina Using previous a priori estimate we deduce that there exist two subsequence { u ǫ n } , { v ǫ n } and a couple ( u , v ) u ∈ L ∞ ( Q T ), v ∈ L ∞ ( Q T ) ∩ L 2 ((0 , T ); H 1 (Ω)) such that for every T > 0: u ǫ n ∗ in L ∞ ( Q T ) , ⇀ u v ǫ n ∗ in L ∞ ( Q T ) , ⇀ v v ǫ n ⇀ v in L 2 ((0 , T ) , H 1 (Ω)) .
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