Between homogeneous and inhomogeneous Navier-Stokes systems: the - PowerPoint PPT Presentation

Between homogeneous and inhomogeneous Navier-Stokes systems: the issue of stability Liutang Xue joint with Prof. Piotr B. Mucha and Prof. Xiaoxin Zheng Beijing Normal University, China September 3 - Mathflows Confernece 2018 Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 1 / 42

Inhomogeneous Navier-Stokes Equations 3D inhomogeneous Navier-Stokes equations   in R 3 × R + ,  ∂ t ρ + v · ∇ ρ = 0 ,     in R 3 × R + ,  ρ∂ t v + ρ v · ∇ v − ν ∆ v + ∇ p = 0 ,  (INS)  (1)  in R 3 × R + ,  div v = 0 ,      on R 3 . ρ | t = 0 ( x ) = ρ 0 ( x ) , v | t = 0 ( x ) = v 0 ( x ) , describe the motion of incompressible flows of viscous Newtonian fluid with variable density (Lions [Lions96]). x = ( x 1 , x 2 , x 3 ) ∈ R 3 ; ν > 0 the kinematic viscosity (below ν ≡ 1 for brevity); ρ the scalar density field; v = ( v 1 , v 2 , v 3 ) the incompressible velocity vector field. If ρ 0 = const , (INS) system reduces to 3D homogeneous Navier-Stokes equations (Millennium Problem). Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 2 / 42

Research History - I Mathematical properties of (INS) system are almost the same as those of the 3D homogeneous Navier-Stokes equations (see Kazhikhov [Kaz74], Ladyzhenskaya, Solonikov [LS73]). The main difference is found in the issue related to the density. Questions concerned with the low regularity of initial density or the possibility of vacuum states are the subjects of current studies of (INS) system (see e.g. Danchin, Mucha [DanM12,DanM13,DanM17] 1 , Paicu, Zhang et al [HPZ13,PZZ13] 2 , Li [Li17] 3 ). 1 R. Danchin, P . Mucha, [DanM12] Commun. Pure Appl. Math. , 65 (2012), 1458–1480; [DanM13] Arch. Rational Mech. Anal. , 207 (2013), 991–1023; [DanM17] ArXiv:1705.06061 [math.AP]. 2 [HPZ13] Huang, Paicu, Zhang, Arch. Ration. Mech. Anal. , 209 (2), (2013), 631–682; [PZZ13] M. . Zhang, Z. Zhang, Ann. Inst. H. Poincar´ e Anal. Non Lin´ Paicu, P eaire , 32 (2015), no. 4, 813–832. 3 J. Li, J. Differential Equations , 263 (2017), Issue 10, 6512–6536. Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 3 / 42

Stability Between (INS) and (HNS) Our Goal: to find solutions to 3D (INS) system with large velocity field like [ChPZ14] 4 , [PZ15] 5 (where data with slow variable ǫ x 3 ). Main Plan: to study stability issue of 3D (INS) system around the 2D homogeneous Navier-Stokes system with constant density 1. Let us emphasize that such a stability analysis has been well developed for the 3D homogeneous Navier-Stokes equations (see e.g. [CG06] 6 , [ChGP11] 7 , [Muc01] 8 , [KRZ14] 9 ), but has not been pursued for (INS) system. 4 J.-Y. Chemin, M. Paicu, P . Zhang, J. Differential Equations , 256 (2014), no. 1, 223–252. 5 M. Paicu, P . Zhang, Ann. Inst. H. Poincar´ e Anal. Non Lin. , 32 (2015), no. 4, 813–832. 6 J-Y. Chemin, I. Gallagher, Ann. Sci. ´ Ecole Norm. Sup. , 39 (2006), no. 4, 679–698. 7 J.-Y. Chemin, I. Gallagher, M. Paicu, Ann. of Math. , 173 (2011), no. 2, 983–1012. 8 P . B. Mucha, J. Differential Equations , 172 (2001), no. 2, 359–375. 9 I. Kukavica, W. Rusin, M. Ziane, J. Math. Fluid Mech. , 16 (2014), no. 2, 293–305. Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 4 / 42

Perturbed System Let v 2d = ( v 2d 1 , v 2d 2 , v 2d 3 ) be a three component 2D vector which solves  ∂ t v 2d + v 2d h · ∇ h v 2d − ∆ h v 2d + ∇ p 2d = 0 ,      ∇ h · v 2d (HNS)  h = 0 , (2)     v 2d | t = 0 ( x h ) = v 2d 0 ( x h ) , where x h = ( x 1 , x 2 ) ∈ R 2 , ∇ h := ( ∂ x 1 , ∂ x 2 ) , ∇ := ( ∂ x 1 , ∂ x 2 , ∂ x 3 ) , ∆ h := ∂ 2 x 1 + ∂ 2 x 2 . For solution ( v , ρ, p ) of (INS) system and solution ( v 2d , p 2d ) of 2D (HNS), set w ( t , x ) := v ( t , x ) − v 2d ( t , x h ) , q ( t , x ) := p ( t , x ) − p 2d ( t , x h ) , h ( t , x ) := ρ ( t , x ) − 1 , we obtain the following perturbed system   h t + v · ∇ h = 0 ,      w t + v · ∇ w − ∆ w + ∇ q = F ,   (3)   div w = 0 ,     w | t = 0 = w 0 , h | t = 0 = h 0 , F := − h ( v 2d ) t − h w t − h ( v · ∇ w ) − ρ ( w h · ∇ h v 2d ) − h ( v 2d h · ∇ h v 2d ) . (4) Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 5 / 42

Main Result - I Our first result concerns the flow in the whole space with regular initial density. Theorem 1 (Mucha, Xue, Zheng, 2018) 0 ∈ L 2 ∩ ˙ B 3 − 2 / p Let p > 3 , v 2d ( R 2 ) , ρ 0 − 1 ∈ L 2 ∩ L ∞ ( R 3 ) , ∇ ρ 0 ∈ L 3 ( R 3 ) and p , p ( R 3 ) . There exist c 0 , C ′ > 0 such that if ( ρ 0 , v 0 ) satisfies 0 ∈ L 2 ∩ ˙ B 2 − 2 / p v 0 − v 2d p , p � � � � e C ′ ( 1 + � v 2d 0 � 4 L 2 ) − C ′ 0 � 4 p � ρ 0 − 1 � L 2 ∩ L ∞ + � v 0 − v 2d � v 2d 0 � L 2 ∩ ˙ ≤ c 0 exp + 1 , B 2 − 2 / p B 2 − 2 / p L 2 ∩ ˙ p , p p , p (5) then we have a unique global-in-time solution ( ρ, v ) to the system (INS) which satisfies � ρ − 1 � L ∞ ( 0 , ∞ ; L 2 ∩ L ∞ ( R 3 )) ≤ � ρ 0 − 1 � L 2 ∩ L ∞ ( R 3 ) , � v − v 2d � L 2 ∩ ˙ + � ( v − v 2d ) t , ∇ 2 ( v − v 2d ) , ∇ ( p − p 2d ) � L p ( R 3 × ( 0 , ∞ )) ≤ C c 0 , sup B 2 − 2 / p p , p t < ∞ �∇ ρ ( t ) � L 3 ( R 3 ) ≤ �∇ ρ 0 � L 3 ( R 3 ) e ( 1 + T ) C ( h 0 , w 0 , v 2d 0 ) , sup for any T > 0 . t ∈ [ 0 , T ] where v 2d = ( v 2d 1 , v 2d 2 , v 2d 3 ) is the unique solution to 2D (HNS) system. Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 6 / 42

Main Result - II Our second global result removes this extra regularity condition of density. Theorem 2 (Mucha, Xue, Zheng, 2018) B 4 − 2 / p 0 ∈ L 2 ∩ ˙ Let p > 3 , v 2d ( R 2 ) , ρ 0 − 1 ∈ L 2 ∩ L ∞ ( R 3 ) and p , p ( R 3 ) . There exist two generic constants c 0 , C ′ > 0 such 0 ∈ L 2 ∩ ˙ B 2 − 2 / p v 0 − v 2d p , p that if ( ρ 0 , v 0 ) satisfies � � � � e C ′ ( 1 + � v 2d 0 � 4 L 2 ) − C ′ 0 � 4 p � ρ 0 − 1 � L 2 ∩ L ∞ + � v 0 − v 2d � v 2d 0 � L 2 ∩ ˙ ≤ c 0 exp + 1 , B 2 − 2 / p L 2 ∩ ˙ B 2 − 2 / p p , p p , p (6) then we have a unique global-in-time solution ( ρ, v ) to the system (INS) which obeys the uniform estimates � ρ − 1 � L ∞ ( 0 , ∞ ; L 2 ∩ L ∞ ( R 3 )) ≤ � ρ 0 − 1 � L 2 ∩ L ∞ ( R 3 ) , (7) � v − v 2d � L 2 ∩ ˙ + � ( v − v 2d ) t , ∇ 2 ( v − v 2d ) , ∇ ( p − p 2d ) � L p ( R 3 × ( 0 , ∞ )) ≤ C c 0 . (8) sup B 2 − 2 / p p , p t < ∞ Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 7 / 42

Remark 1 Theorems 1 and 2 say that if the initial perturbation is small (5) , then the solutions exist globally in time and they are close to the ones of 2D (HNS) system. Hence the physical interpretation is the following: all solutions to 2D (HNS) are stable globally in time in the inhomogeneous Navier-Stokes system regime, provided that perturbation of density is close to a constant in the L 2 ∩ L ∞ -norm. It means the higher norms of the density have no influence of our issue of stability. Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 8 / 42

Density Patch Problem Remark 2 Let ρ 0 = 1 D c + ( 1 − η ) 1 D = 1 − η 1 D with η a small constant, and D ⊂ R 3 a bounded, simple, connected set. According to Theorem 2, if ( ρ 0 , v 0 ) satisfies � � � � e C ′ ( 1 + � v 2d 0 � 4 0 � 4 p L 2 ) | η | ( 1 + |D| 1 / 2 ) + � v 0 − v 2d − C ′ � v 2d 0 � L 2 ∩ ˙ ≤ c 0 exp + 1 , B 2 − 2 / p L 2 ∩ ˙ B 2 − 2 / p p , p p , p then the (INS) system generates a unique global solution ( ρ, v ) which satisfies (7) - (8) . In particular, we have for γ ∈ ( 0 , 1 − 2 / p ] , p > 3 , � v � L p ( 0 , ∞ ; C 1 ,γ ( R 3 )) ≤ � v 2d � L p ( 0 , ∞ ; C 1 ,γ ( R 2 )) + C � v − v 2d � L p ( 0 , ∞ ; W 2 p ( R 3 )) < ∞ . It guarantees that if initial boundary ∂ D is C 1 ,γ -regular, then its evolution ∂ D ( t ) = X v ( t , ∂ D ) remains C 1 ,γ -regular, with X v ( t , · ) the flow of v. For more results on density patch problem of (INS) system, e.g. considering ρ 0 = ρ 1 1 D + ρ 2 1 D c , with ρ 1 , ρ 2 > 0 constants, one can see Danchin, Zhang [DanZ17], Gancedo, Garcia-Juarez [GanGJ18], Liao, Zhang [LZ16] etc. Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 9 / 42

Unsolved Problem Remark 3 It is not clear that 2D (INS) system is globally stable in the 3D (INS) regime, if perturbation of initial density is small in the norm of L 2 ∩ L ∞ ( R 3 ) . For example, for the data ρ 0 = ρ 1 1 D + ρ 2 1 D c + small perturbation in L 2 ∩ L ∞ ( R 3 ) , v 0 = v 2d 0 + small w 0 , with ρ 1 , ρ 2 > 0 constants, it is not clear that the 3D (INS) system will has a unique global solution. Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 10 / 42