Long-time Behaviour of a Model of Rigid Structure Floating in a - PowerPoint PPT Presentation

Long-time Behaviour of a Model of Rigid Structure Floating in a Viscous Fluid G. Vergara-Hermosilla, M. Tucsnak, F. Sueur Universit´ e de Bordeaux CEMRACS19 Geophysical Fluids, Gravity Flows Luminy, August 19, 2019

This work is supported by Marie Sklodowska-Curie Grant agreement number 765579; ConFlex project. Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 2

Outline 1. A Model of rigid structure in a viscous fluid 2. Mittag-Leffler functions 3. Solutions unbounded case 4. Long-time behavior viscous case Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 1

The equilibrium problem Figure 1: Configuration, Image from Tucsnak et al. Notations • µ ≥ 0 is the coefficient of viscosity of the fluid. • E = (0 , a ) ∪ ( b , a + b ). • I = [ a , b ]. Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 2

Equations Return to the Equilibrium Proposition [Tucsnak et al.,2018] If l = a + b , then, the return to the equilibrium problem, the position of the solid is completely determined by the integro-differential equation 1 + ( b − a ) 3 H = − H 0 |I| 2 |E| 2 − ( b − a ) 2  � � ¨ F ∗ ˙ H − µ ( b − a ) ˙ H − ( b − a ) H ,  12 2 ˙ H (0) = H 0 , H (0) = 0 ,  (1) F ( s ) = √ 1 + s µ tanh � � sa and with F such that ˆ √ 1 + s µ . Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 3

Unbounded case Proposition [Tucsnak et al.,2018] If E = ( −∞ , a ) ∪ ( b , ∞ ), then the position of the solid is completely determined by the integro-differential equation � A ¨ H + BF ∗ ˙ H + C ˙ H + DH = 0 , (2) H (0) = ˙ ˙ H (0) = H 0 , H 0 , where A = 1 + ( b − a ) 3 , B = ( b − a ) 2 , C = ( b − a ) µ , D = ( b − a ), and 12 � 1 − e − t � F ( t ) = √ µ µ + D 1 / 2 δ 0 . √ π t 3 2 Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 4

Unbounded case Proposition [Tucsnak et al.,2018] If E = ( −∞ , a ) ∪ ( b , ∞ ), then the position of the solid is completely determined by the integro-differential equation � A ¨ H + BF ∗ ˙ H + C ˙ H + DH = 0 , (2) H (0) = ˙ ˙ H (0) = H 0 , H 0 , where A = 1 + ( b − a ) 3 , B = ( b − a ) 2 , C = ( b − a ) µ , D = ( b − a ), and 12 � 1 − e − t � F ( t ) = √ µ µ + D 1 / 2 δ 0 . √ π t 3 2 Goal: Understand the long-time behavior of solutions H ( t ) for the unbounded case Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 4

Mittag-Leffler functions Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 5

Mittag-Leffler functions Definition The two-parametric Mittag-Leffler function, is the complex-valued function defined by ∞ z k � E α,β ( z ) = with α > 0 , β ∈ C . (3) Γ( α k + β ) k =0 In the case when β = 1 the function is known has the classical Mittag-Leffler function and denoted by E α ( z ). Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 6

Mittag-Leffler functions Definition The two-parametric Mittag-Leffler function, is the complex-valued function defined by ∞ z k � E α,β ( z ) = with α > 0 , β ∈ C . (3) Γ( α k + β ) k =0 In the case when β = 1 the function is known has the classical Mittag-Leffler function and denoted by E α ( z ). Examples • E 1 , 1 ( z ) = e z , • E 2 , 1 ( z ) = cosh( √ z ) , 2 , 1 ( z ) = e z 2 erfc( − z ) . • E 1 Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 6

Mittag-Leffler functions Theorem [Popov-Sedletskii, 2013] For any 0 < α < 1 , β ∈ C , p ∈ N , and z ∈ L 1 /α = { z ∈ C : | arg z | ≤ π α } , the following asymptotics hold p α z (1 − β ) /α e z 1 /α − z − k E α,β ( z ) = 1 � Γ( β − k α ) + R [1] m ( z ; α, β ) , (4) k =1 where the remainder R [1] m admits the estimate  2 Γ( b + 1) e ( 5 π 4 |ℑ β | ) b +2 2  , if b = α ( p + 1) − ℜ β ≥ 0   απ | z | p +1 � �  � R [1] m ( z ; α, β ) � ≤ � � απ ) e ( 5 π 4 |ℑ β | ) 2 (6 +  , if b < 0 .   | z | p +1  (5) The first of estimates in (10) is valid for all z ∈ L 1 /α , and the second under the additional condition | z | ≥ 2. Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 7

Mittag-Leffler functions Theorem [Popov-Sedletskii, 2013] For any 0 < α < 1 , β ∈ C , p ∈ N and z ∈ C \ L 1 /α , z � = 0 , the following asymptotics formulas hold p z − k � Γ( β − k α ) + R [2] E α,β ( z ) = − m ( z ; α, β ) , (6) k =1 where the remainder R [2] m admits the estimate  b +2 2 Γ( b + 1) e ( 3 π 4 |ℑ β | ) 2  , if b ≥ 0   απ | z | p +1 � �  � R [2] m ( z ; α, β ) � ≤ (7) � � απ ) e ( 3 π 4 |ℑ β | ) 2 (6 +   , if b < 0 .   | z | p +1 The first of estimates in (7) is valid for all z ∈ L 1 /α , and the second under the additional condition | z | ≥ 2. Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 8

Laplace transform of Mittag Leffler functions In this work we are interested in the case when α = β > 0 and z = λ t α , with λ ∈ C and t ∈ R . For simplicity we introduce the notation E α ( λ, t ) := E α,α ( λ t α ) . Lemma The Laplace transform of E α ( λ, t ) is given by L [ E α ( λ, t )]( s ) = ( s α − λ ) − 1 , where ℜ s > 0 and | λ s − α | < 1. Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 9

Solutions unbounded case Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 10

Inviscid case If we consider µ = 0 in equation (2), the model is reduced to � A ¨ H + B ˙ H + DH = 0 , H (0) = ˙ ˙ H (0) = H 0 , H 0 . Applying Laplace transform to the equation above, and after simplifications, we obtain As 2 + Bs + D ˆ = H 0 [ As + B ] + A ˙ � � H ( s ) H 0 . Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 11

Solutions of the model, inviscid case Proposition [V-H, Sueur, Tucsnak, 2019] The solution H ( t ) of equation (2) with µ = 0 is given by � �    � B 2 t 4 − A D  B sinh �    B 2  A  t 4 − A D     e − B t  +   H 0 cosh 2 A   � A B 2   2 4 − A D        √ � �  4 A D − B 2 t 2 A sin 2 A  + ˙ √ H 0  . 4 A D − B 2 Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 12

Viscous case Proposition [V-H, Sueur, Tucsnak, 2019] The solution of the equation (2) with µ > 0 is given by 4 � � H ( t ) = e − t � H 0 r i + ˙ H 0 ˙ E 1 2 ( λ i , t ) , r i µ i =1 where each λ i is a root of the polynomial � C − 2 A � λ 2 − B µ λ + A p ( λ ) = A λ 4 + B λ 3 + (8) µ 2 , µ and the parameters r i and ˙ r i depends of the roots. Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 13

Long-time Behavior Viscous Case Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 14

Long-time Behavior Viscous Case In this section we denote by Λ = { λ 1 , λ 2 , λ 3 , λ 4 } the set of roots of the polynomial P ( λ ) in eq. (8). Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 14

Long-time Behavior Viscous Case In this section we denote by Λ = { λ 1 , λ 2 , λ 3 , λ 4 } the set of roots of the polynomial P ( λ ) in eq. (8). Remark Since the coefficients of the polynomial p ( λ ) in eq. (8) admit different signs, we conclude that • Λ ∩ L 1 / 2 � = ∅ , and • Λ ∩ C \ L 1 / 2 � = ∅ . Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 14

Long-time Behavior Viscous Case Proposition [V-H, Sueur, Tucsnak, 2019] Let Θ i = ( H 0 r i + ˙ H 0 ˙ r i ) for i = 1 , 2 , 3 , 4 , and let Λ 1 = L 1 / 2 ∩ Λ. For all p ∈ N , the following asymptotic formula hold: √ t � p � − k 4 � √ √ t ) � Θ i λ i 2 − t te ( λ i Θ i e − t � µ − � � H ( t ) = 2 λ i + R p , µ � 1 2 − k � Γ 2 λ i ∈ Λ 1 i =1 k =1 (9) where the remainder R p admits the estimate p +8 4 Γ(1 + p / 2) µ · 2 1 |R p | ≤ e − t � π |√ t | p +1 · | λ i | p +1 . (10) λ i ∈ Λ Behaviour of a Model of Rigid Structure Floating G. Vergara-Hermosilla, M. Tucsnak, F. Sueur, Universit´ e de Bordeaux 15