Non decaying solutions to the Navier-Stokes equations in the - PowerPoint PPT Presentation

Non decaying solutions to the Navier-Stokes equations in the half-space Yasunori Maekawa (Kyoto University) Hideyuki Miura (Tokyo Institute of Technology) Christophe Prange (CNRS & Universit´ e de Bordeaux) Mathflows, Porquerolles September 5, 2018

Navier-Stokes equations Navier-Stokes equations   ∂ t V + V · ∇ V − ∆ V + ∇ P = 0 , x ∈ Ω , t ∈ (0 , T ) ,  (NS) ∇ · V = 0 , x ∈ Ω , t ∈ [0 , T ) ,   V | t =0 = V 0 , x ∈ Ω . Boundary condition V = 0 on ∂ Ω × (0 , T ) Scaling Ω = R 3 or R 3 + , λ ∈ (0 , ∞ ) , V 0 ,λ ( y ) = λV 0 ( λy ) V λ ( y, s ) = λV ( λy, λ 2 s ) , ∀ y ∈ Ω , s > 0 1 Criticality L 3 ( R 3 ) , ˙ 2 ( R 3 ) scale critical norms H 2 / 27

Navier-Stokes equations Leray-Hopf solutions V is a Leray-Hopf or a finite energy weak solution to (NS) for initial data V 0 ∈ L 2 σ (Ω) if for all T < ∞ , V ∈ L ∞ ((0 , T ); L 2 (Ω)) ∩ L 2 ((0 , T ); H 1 (Ω)) ; V satisfies (NS) in the sense of distributions; V satisfies the global energy inequality for all t ∈ (0 , ∞ ) � � t � � | V ( · , t ) | 2 + 2 |∇ V | 2 ≤ | V 0 | 2 ; Ω 0 Ω Ω we have � V ( · , t ) − V 0 � L 2 (Ω) → 0 , when t → 0 . Regular vs. singular point A point ( x 0 , t 0 ) is a regular point if V is bounded in a parabolic cylinder Q r ( x 0 , t 0 ) for r > 0 , otherwise it is a singular point. 3 / 27

Outline of the talk 1 Motivations 2 Mild solutions and concentration near blow-up 3 Non uniqueness 4 Local energy weak solutions and blow-up of critical norms 4 / 27

Motivation 1: blow-up of critical norms Theorem (Seregin 2012) Let V a Leray-Hopf solution with initial data V 0 ∈ C ∞ c,σ ( R 3 ) . Assume that T > 0 is a blow-up time. Then t → T − . � V ( · , t ) � L 3 ( R 3 ) → ∞ , By contraposition, assume that there exists M ∈ (0 , ∞ ) and t n → T − such that � V ( · , t n ) � L 3 ( R 3 ) ≤ M . Consider � T − t n V n ( y, s ) := λ n V ( λ n y, T + λ 2 n s ) , λ n := . S Then � V n ( · , − S ) � L 3 ≤ M . Remains to see: a priori bounds, convergence to a limit blow-up solution V , V = 0 by backward uniqueness and ε -regularity for smoothness. 5 / 27

Motivation 1: blow-up of critical norms Bounds There exists S ( M ) ∈ (0 , ∞ ) and A ( M ) ∈ (0 , ∞ ) such that � 0 � |∇ V n | 2 dyds ≤ A. x 0 ∈ R 3 � V n ( · , s ) � 2 sup sup L 2 ( B ( x 0 , 1)) + sup x 0 ∈ R 3 s ∈ ( − S, 0) − S B ( x 0 , 1) Convergence V n converges to a V (Blow-up solution) which is a Local Energy Weak Solution (LEWS) to (NS) in R 3 × ( − S, 0) : local energy inequality, weak solution, weak continuity in time, representation formula for the pressure. Strong convergence to initial data: 1 x 0 ∈ R 3 � V ( · , s ) − e ( s + S )∆ V ( · , − S ) � 2 5 . sup L 2 ( B ( x 0 , 1)) ≤ C ( M )( s + S ) At final time V ( · , T ) ∈ L 3 , implies V ( · , 0) = 0 . Liouville Transfer mild decay of V ( · , − S ) ∈ L 3 to V . Implies smoothness of V in R 3 \ B (0 , R ) × ( − S ′ , 0) , S ′ < S and V = 0 by backward uniqueness and unique continuation. 6 / 27

Motivation 2: large forward self-similar solutions 1 t V ( x Forward self-similar V ( x, t ) = t , 1) . √ √ Initial data − 1 homogeneous: for example V 0 ( x ) = ( − x 2 | x | 2 , x 1 | x | 2 , 0) belongs to � L 2 uloc ( R 3 ) := v ∈ L 2 loc : x 0 ∈ R 3 � v � L 2 ( B ( x 0 , 1)) < ∞ sup � x 0 →∞ and � v � L 2 ( B ( x 0 , 1)) − → 0 . Jia, Sverak 2014 For scale-invariant divergence-free V 0 ∈ C ∞ ( R 3 \ { 0 } ) and any scale invariant LEWS V , we have V ( · , 1) ∈ C ∞ and C ( α, V 0 ) | ∂ α ( V ( x, 1) − V 0 ( x )) | ≤ ∀| α | ≥ 0 . (1 + | x | ) 3+ | α | , A priori estimate coming from local in space near initial time regularity result for the LEWS V . 7 / 27

Motivation 3: singular Burgers vortex Moffatt (2000) found blow-up solutions to (NS) which have a similar structure to the regular Burgers vortex:   � T − t U G �� − x 1 T − t x ′ � µ − 1 µ − 1 µ , x ′ = ( x 1 , x 2 ) ⊤ ,   + α V sB ( t, x ) = − x 2 T − t 2 x 3 �� T − t x ′ � Ω sB ( t, x ) = ∇ × V SB ( t, x ) = α µ − 1 µ − 1 T − t G where     0 U G ( X ′ ) = 1 − e − | X ′| 2 − X 2   4   , 0 G ( X ′ ) = X 1   2 π | X ′ | 2 4 π e − | X ′| 2 0 1 4 α ∈ R : circulation at infinity. and µ > 1 : magnitude of the strain Blow-up in backward self-similar form, but out of Cafarelli-Kohn-Nirenberg (1982) or Necas-Ruzicka-Sverak (1997), Tsai (1998). Stability of blow-up solution: Maekawa, Miura, P. 2018. 8 / 27

Linear theory: resolvent estimates Resolvent problem  x ∈ R 3 λU − ∆ U + ∇ P = f , + ,   x ∈ R 3 (R) ∇ · U = 0 , + ,   U | x 3 =0 = 0 . Theorem (Maekawa, Miura, P. 2017) For all ε > 0 , λ ∈ S π − ε , q ∈ (1 , ∞ ) , there exists C ( ε, q ) ∈ (0 , ∞ ) , for all f ∈ L q uloc,σ ( R 3 + ) , there is a unique solution to the Stokes resolvent R →∞ problem with �∇ ′ P � L 1 ( | x ′ | < 1 ,R<x d <R +1) − → 0 and 1 2 �∇ U � L q | λ |� U � L q uloc + | λ | uloc ≤ C � f � L q uloc , � � 1 2 log | λ | �∇ 2 U � L q 1 + e − c | λ | uloc + �∇ P � L q uloc ≤ C � f � L q q � = ∞ . uloc , Desch, Hieber, Pr¨ uss 2001; Abe, Giga, Hieber 2015 9 / 27

Linear theory: proof of the resolvent estimates Main source of inspiration: Desch, Hieber, Pr¨ uss 2001. Decompose U = U D.L. + U nonloc : for all ξ ∈ R 2 , x 3 > 0 , � ∞ 1 ( e − ω λ ( ξ ) | x 3 − z 3 | − e − ω λ ( ξ )( x 3 + z 3 ) ) � � U D.L. ( ξ, x 3 ) = f ( ξ, z 3 ) dz 3 2 ω λ ( ξ ) 0 � ∞ nonloc ( ξ, x 3 ) = − iξ ( e − ω λ ( ξ ) | x 3 − z 3 | − e − ω λ ( ξ )( x 3 + z 3 ) ) e −| ξ | z 3 � � U ′ P 0 ( ξ ) dz 3 2 ω λ ( ξ ) 0 � ∞ | ξ | ( e − ω λ ( ξ ) | x 3 − z 3 | − e − ω λ ( ξ )( x 3 + z 3 ) ) e −| ξ | z 3 � � U nonloc, 3 ( ξ, x 3 ) = P 0 ( ξ ) dz 3 , 2 ω λ ( ξ ) 0 � λ + | ξ | 2 and for ξ � = 0 where ω λ ( ξ ) := � ∞ P 0 ( ξ ) = − ω λ ( ξ ) + | ξ | e − ω λ ( ξ ) z 3 � � f 3 ( ξ, z 3 ) dz 3 . | ξ | 0 Integration by parts: � ∞ U nonloc ( ξ, x 3 ) ≃ 1 λ ( e −| ξ | x 3 − e − ω λ ( ξ ) x 3 ) ξ ⊗ ξ e − ω λ ( ξ ) y 3 � � f ′ ( ξ, y 3 ) dy 3 | ξ | 0 10 / 27

Linear theory: proof of the resolvent estimates Estimates singularity at 0 and decay for kernel � R d − 1 e ix ′ · ξ � e −| ξ | x 3 − e − ω λ ( ξ ) x 3 � s λ ( x ′ , x 3 , z 3 ) := 1 e − ω λ ( ξ ) z 3 ξ ⊗ ξ | ξ | dξ. λ Pointwise estimate There exist c ( ε ) , C ( ε ) ∈ (0 , ∞ ) such that for all λ ∈ S π − ε , x ′ ∈ R 2 , z 3 , x 3 > 0 , 1 e − c | λ | 2 z 3 Cx 3 | s λ | ≤ � �� � ( x 3 + z 3 + | x ′ | ) 2 1 1 2 ( x 3 + z 3 + | x ′ | ) 2 ( x 3 + z 3 ) 1 + | λ | 1 + | λ | 11 / 27

Linear theory: proof of the resolvent estimates � � ∞ s λ ( x ′ − z ′ , x 3 , z 3 ) f ′ ( z ′ , z 3 ) dz 3 dz ′ I ( f ′ )( x ′ , x 3 ) := R 2 0 Convolution estimates in horizontal direction: � � � � � I [ f ′ ]( · , y 3 ) � L p ((0 , 1) 2 ) ≤ + i |≤ 2 , max | α ′ i + β ′ i |≥ 3 , max | α ′ i + β ′ max | α ′ max | α ′ i |≤ 2 i |≤ 2 � � 1 � n +1 � ∞ � � s ′ λ ( · , x 3 , z 3 ) � L s ( α ′ +(0 , 1) 2 ) � f ′ ( · , z 3 ) � L q ( β ′ +(0 , 1) 2 ) dz 3 + 0 n n =1 where 1 2 z 3 Ce − c | λ | � s λ ( · , x 3 , z 3 ) � L s ( R 2 ) ≤ p ) . 2 ( x 3 + z 3 ))( x 3 + z 3 ) 2( 1 q − 1 1 1 2 (1 + | λ | | λ | q = p = 1 excluded, also noticed in Desch, Hieber, Pr¨ uss 2001: � I ( f ′ ) � L 1 , ∞ x 3 ((0 , 1); L 1 ((0 , 1) 2 ) ≤ | λ | − 1 � f ′ � L 1 uloc ( R 2 )) . x 3 ((0 , 1); L 1 12 / 27

Linear theory: semigroup estimates Theorem (Maekawa, Miura, P. 2017) For q ∈ (1 , ∞ ) , let A the Stokes operator in L q uloc,σ ( R 3 + ) . Then − A generates a bounded analytic semigroup in L q uloc,σ ( R 3 + ) . Moreover, for 1 ≤ q < p ≤ ∞ or 1 < q = p ≤ ∞ , there is C ( d, p, q ) ∈ (0 , ∞ ) , � � p ) + 1 uloc ≤ Ct − | α | t − 3 2 ( 1 q − 1 �∇ α e − t A f � L p � f � L q uloc , t ∈ (0 , ∞ ) , | α | ≤ 1 , 2 Abe, Giga 2013, 2014 (compactness method) Solonnikov 2003, Maremonti, Starita 2003 (Green tensor) 13 / 27

Mild solutions: bilinear estimates For θ ∈ (0 , 2) , 2 − θ P ∇· ( U ⊗ V ) = ∂ α ( U β V γ )+( − ∆ ′ ) 2 G θ, ≥| λ | 2 ( U ⊗ V ) + G ≤| λ | 2 ( U ⊗ V ) 1 1 For 1 ≤ q < p ≤ ∞ or 1 < q = p ≤ ∞ and 0 ≤ 1 q − 1 p < 1 3 � � � p ) � � � 2 ( 1 3 q − 1 ≤ C | λ | − 1 − θ � G θ, ≥| λ | 2 ( U ⊗ V ) � 1 + | λ | � U ⊗ V � L q uloc , 2 1 L p � � uloc � � 1 2 � U ⊗ V � L q � G ≤| λ | 2 ( U ⊗ V ) � ≤ C | λ | uloc . 1 L q uloc Estimates for Oseen’s kernel Let 1 < q ≤ p ≤ ∞ or 1 ≤ q < p ≤ ∞ . Then for | α | ≤ 1 and for all t ∈ (0 , ∞ ) , 2 � � t − 3 2 ( 1 q − 1 p ) + 1 uloc ≤ Ct − 1+ α �∇ α e − t A P ∇ · ( U ⊗ V ) � L p � U ⊗ V � L q uloc , 2 � � uloc ≤ Ct − 1 �∇ e − t A P ∇ · ( U ⊗ V ) � L q � U · ∇ V � L q uloc + � V · ∇ U � L q . uloc 14 / 27