Introduction Massive particle Long time behavior Massless particle Long time and small obstacle problem for the 2D NS equations Christophe Lacave IMJ-PRG, University of Paris Diderot (Paris 7), France in collaboration with S. Ervedoza (Toulouse), M. Hillairet (Montpellier) and T. Takahashi (Nancy). Porquerolles, Mathflows 2015, September Long time and small obstacle problem for the 2D NS equations 1 / 32
Introduction Massive particle Long time behavior Massless particle Outline Introduction 1 “Massive” pointwise particle 2 Long time behavior for the unit disk in R 2 3 “Massless” pointwise particle 4 Long time and small obstacle problem for the 2D NS equations 2 / 32
Introduction Massive particle Long time behavior Massless particle Outline Introduction 1 “Massive” pointwise particle 2 Long time behavior for the unit disk in R 2 3 “Massless” pointwise particle 4 Long time and small obstacle problem for the 2D NS equations 3 / 32
Introduction Massive particle Long time behavior Massless particle Navier-Stokes equations Let u ε = ( u ε 1 , u ε 2 ) be the velocity of an incompressible, viscous flow in a domain F ε ( t ) := R 2 \ S ε ( t ) . Let p ε be the pressure. The evolution of such a flow is governed by the Navier-Stokes equations : ∂ t u ε − ν ∆ u ε + u ε · ∇ u ε = −∇ p ε in ( 0 , ∞ ) × F ε ( t ) div ε u = 0 in [ 0 , ∞ ) × F ε ( t ) | x |→∞ | u ε | = 0 lim for t ∈ [ 0 , ∞ ) u ε ( 0 , x ) = u ε in F ε 0 ( x ) 0 We consider the no-slip boundary condition : u ε = 0 on ( 0 , ∞ ) × ∂ S ε 0 if the domain is fixed; Long time and small obstacle problem for the 2D NS equations 4 / 32
Introduction Massive particle Long time behavior Massless particle Navier-Stokes equations Let u ε = ( u ε 1 , u ε 2 ) be the velocity of an incompressible, viscous flow in a domain F ε ( t ) := R 2 \ S ε ( t ) . Let p ε be the pressure. The evolution of such a flow is governed by the Navier-Stokes equations : ∂ t u ε − ν ∆ u ε + u ε · ∇ u ε = −∇ p ε in ( 0 , ∞ ) × F ε ( t ) div ε u = 0 in [ 0 , ∞ ) × F ε ( t ) | x |→∞ | u ε | = 0 lim for t ∈ [ 0 , ∞ ) u ε ( 0 , x ) = u ε in F ε 0 ( x ) 0 We consider the no-slip boundary condition : u ε = 0 on ( 0 , ∞ ) × ∂ S ε 0 if the domain is fixed; Long time and small obstacle problem for the 2D NS equations 4 / 32
Introduction Massive particle Long time behavior Massless particle if the solid moves under the influence of the fluid: u ε ( t , x ) = ( h ε ) ′ ( t ) + ω ε ( t )( x − h ε ( t )) ⊥ for ( t , x ) ∈ [ 0 , ∞ ) × ∂ S ε ( t ) , � m ε ( h ε ) ′′ ( t ) = − Σ ε n ds for t ∈ ( 0 , ∞ ) , ∂ S ε ( t ) � ( x − h ε ( t )) ⊥ · Σ ε n ds J ε ( ω ε ) ′ ( t ) = − for t ∈ ( 0 , ∞ ) . ∂ S ε ( t ) Where Σ ε is the Cauchy stress tensor of the fluid: Σ ε ( t , x ) = − p ε ( t , x ) Id + 2 ν D ( u ε ) , and D ( u ) k , l = 1 � ∂ u k + ∂ u l � 1 ≤ k , l ≤ 2 . 2 ∂ x l ∂ x k The solid is rigid: S ε ( t ) := ( h ε ( t ) − h 0 ) + R ε ( θ ) S ε 0 . Long time and small obstacle problem for the 2D NS equations 5 / 32
Introduction Massive particle Long time behavior Massless particle if the solid moves under the influence of the fluid: u ε ( t , x ) = ( h ε ) ′ ( t ) + ω ε ( t )( x − h ε ( t )) ⊥ for ( t , x ) ∈ [ 0 , ∞ ) × ∂ S ε ( t ) , � m ε ( h ε ) ′′ ( t ) = − Σ ε n ds for t ∈ ( 0 , ∞ ) , ∂ S ε ( t ) � ( x − h ε ( t )) ⊥ · Σ ε n ds J ε ( ω ε ) ′ ( t ) = − for t ∈ ( 0 , ∞ ) . ∂ S ε ( t ) Where Σ ε is the Cauchy stress tensor of the fluid: Σ ε ( t , x ) = − p ε ( t , x ) Id + 2 ν D ( u ε ) , and D ( u ) k , l = 1 � ∂ u k + ∂ u l � 1 ≤ k , l ≤ 2 . 2 ∂ x l ∂ x k The solid is rigid: S ε ( t ) := ( h ε ( t ) − h 0 ) + R ε ( θ ) S ε 0 . Long time and small obstacle problem for the 2D NS equations 5 / 32
Introduction Massive particle Long time behavior Massless particle Well posedness Theorem (Takahashi and Tucsnak, 04) There exists a unique global strong solution ( u ε , p ε , h ε , ω ε ) if u ε 0 ∈ H 1 ( F 0 ) . If u ε 0 belongs to L 2 ( F 0 ) , there exists a unique weak solution. In the case of moving solid, we only consider the case of the disk: hence the regularity of strong solutions is computed through the classical change of unknown: v ε ( t , x ) = u ε ( t , x − h ε ( t )) , p ε ( t , x ) = p ε ( t , x − h ε ( t )) . ˜ which are defined in the fixed domain [ 0 , ∞ ) × ( R 2 \ B ε 0 ) . What is the behavior of the solution when the solids tends to zero : B ε 0 = ε B ( 0 , 1 ) with ε → 0 ? Long time and small obstacle problem for the 2D NS equations 6 / 32
Introduction Massive particle Long time behavior Massless particle Well posedness Theorem (Takahashi and Tucsnak, 04) There exists a unique global strong solution ( u ε , p ε , h ε , ω ε ) if u ε 0 ∈ H 1 ( F 0 ) . If u ε 0 belongs to L 2 ( F 0 ) , there exists a unique weak solution. In the case of moving solid, we only consider the case of the disk: hence the regularity of strong solutions is computed through the classical change of unknown: v ε ( t , x ) = u ε ( t , x − h ε ( t )) , p ε ( t , x ) = p ε ( t , x − h ε ( t )) . ˜ which are defined in the fixed domain [ 0 , ∞ ) × ( R 2 \ B ε 0 ) . What is the behavior of the solution when the solids tends to zero : B ε 0 = ε B ( 0 , 1 ) with ε → 0 ? Long time and small obstacle problem for the 2D NS equations 6 / 32
Introduction Massive particle Long time behavior Massless particle Large literature on such problems For Euler: Iftimie, Lopes Filho, Nussenzveig Lopes (03); Lopes Filho (07); C.L. (09,12); Glass, C.L., Sueur (14,15?); Glass, Munnier, Sueur (15?). For Navier-Stokes: Iftimie, Lopes Filho, Nussenzveig Lopes (06,09); Iftimie, Kelliher (09); C.L. (09, 15). In porous medium: Cioranescu, Murat (79); Sanchez-Palencia (80); Tartar (80); Conca (87); Allaire (90); Mikelic, Paoli (99); Lions, Masmoudi (05), Bonnaillie-Noel, C.L., Masmoudi (15); C.L., Masmoudi (15?). Long time and small obstacle problem for the 2D NS equations 7 / 32
Introduction Massive particle Long time behavior Massless particle Outline Introduction 1 “Massive” pointwise particle 2 Long time behavior for the unit disk in R 2 3 “Massless” pointwise particle 4 Long time and small obstacle problem for the 2D NS equations 8 / 32
Introduction Massive particle Long time behavior Massless particle Assumptions: m ε i = m 1 J ε i = ε 2 J 1 i > 0 i > 0; and u ε L 2 ( R 2 ) ; 0 ⇀ u 0 in | ℓ ε ε | r ε 0 | ≤ C , 0 | ≤ C . Theorem For any T > 0 we have ∗ u ε L ∞ ( 0 , T ; L 2 ( R 2 )) ∩ L 2 ( 0 , T ; H 1 0 ( R 2 )) ⇀ u in where u is the weak solution of the Navier-Stokes equations associated to u 0 . Long time and small obstacle problem for the 2D NS equations 9 / 32
Introduction Massive particle Long time behavior Massless particle Scheme of the proof 1) energy estimate: � t � � � R 2 ρ ε | u ε | 2 + 4 ν R 2 | D ( u ε ) | 2 ≤ 0 ( x ) | 2 . R 2 ρ ε ( x ) | u ε 0 Then up to a subsequence, ∗ u ε L ∞ ( 0 , T ; L 2 ( R 2 )) ∩ L 2 ( 0 , T ; H 1 ( R 2 )) ⇀ u in h ε → h uniformly in [ 0 , T ] , with h ∈ W 1 , ∞ ( 0 , T ) . We need a strong compactness on the support of the test function. Problem of compatibility of test function Long time and small obstacle problem for the 2D NS equations 10 / 32
Introduction Massive particle Long time behavior Massless particle Scheme of the proof 1) energy estimate: � t � � � R 2 ρ ε | u ε | 2 + 4 ν R 2 | D ( u ε ) | 2 ≤ 0 ( x ) | 2 . R 2 ρ ε ( x ) | u ε 0 Then up to a subsequence, ∗ u ε L ∞ ( 0 , T ; L 2 ( R 2 )) ∩ L 2 ( 0 , T ; H 1 ( R 2 )) ⇀ u in h ε → h uniformly in [ 0 , T ] , with h ∈ W 1 , ∞ ( 0 , T ) . We need a strong compactness on the support of the test function. Problem of compatibility of test function Long time and small obstacle problem for the 2D NS equations 10 / 32
Introduction Massive particle Long time behavior Massless particle Scheme of the proof 2) modified test function: Proposition Let T > 0 and ϕ ∈ C ∞ c ([ 0 , T ) × R 2 ) with div ϕ = 0 . For any η > 0 , there exists ϕ η ∈ W 1 , ∞ ([ 0 , T ); H 1 0 ( R 2 )) satisfying c div ϕ η = 0 in [ 0 , T ) × Ω , h ( t ) , η ϕ η ≡ 0 � � t ∈ ( 0 , T ) , x ∈ B , 2 ∗ ϕ η L ∞ ( 0 , T ; H 1 (Ω)) , ⇀ ϕ ∗ ∂ t ϕ η L ∞ ( 0 , T ; L 2 (Ω)) . ⇀ ∂ t ϕ Long time and small obstacle problem for the 2D NS equations 11 / 32
Recommend
More recommend
