Partial regularity results for generalized alpha models of - PowerPoint PPT Presentation

Partial regularity results for generalized alpha models of turbulence Gantumur Tsogtgerel McGill University Joint work with Michael Holst (UCSD) and Evelyn Lunasin (Michigan) SIAM Conference on Analysis of PDE San Diego November 14-17, 2011

Outline Generalized alpha models Basic results Katz-Pavlović result and its extension Further directions Gantumur, Evelyn, Mike Partial regularity for generalized alpha models Nov 14 2 / 8

α models Consider a closed manifold and the Leray projector P on it. Let s − ∆ s = u . NS ∂ t u = ∆ u − P ( u ·∇ u ) ∂ t u = ∆ 2 u − P ( u ·∇ u ) hyperviscous Leray- α ∂ t u = ∆ u − P ( s ·∇ u ) modified Leray- α ∂ t u = ∆ u − P ( u ·∇ s ) Simplified Bardina ∂ t u = ∆ u − P ( s ·∇ s ) NS-Voight ∂ t u = ∆ s − P ( s ·∇ s ) NS- α ∂ t u = ∆ u − P ( s ×∇× u ) NS- ω ∂ t u = ∆ u − P ( u ×∇× s ) ∂ t u = ∆ u − P ( s ·∇ u + u ·∇ s − s ·∇ s −∇ ( ∇ s ·∇ s T )) Clark- α � T � −·∇ � u � � u � � u �� u � ·∇ MHD ∂ t = ∆ + P h h h ·∇ −·∇ h Generalized model: ∂ t u = Au + B ( u , u ) with B ( u , v ) = B 0 ( Mu , Nv ), 〈 B 0 ( u , v ), v 〉 = 0 Gantumur, Evelyn, Mike Partial regularity for generalized alpha models Nov 14 3 / 8

General model ∂ t u = Au + B ( u , u ) with B ( u , v ) = B 0 ( M 1 u , M 2 v ), 〈 B 0 ( u , v ), v 〉 = 0 V a linear space of smooth (tensor) fields, e.g. divergence free fields. A : V → V dissipation operator, e.g. A = ∆ θ B 0 : V × V → V bilinear structure, e.g. B 0 ( u , v ) = u ·∇ v or u ×∇× v M i : V → V smoothing operators, e.g. M i = ( I − ∆ ) − θ i Under certain conditions existence of a global weak solution (e.g. θ + θ 1 > 1 2 ) global regularity (e.g. 4 θ + 4 θ 1 + 2 θ 2 > n + 2 ) inviscid limits, and α → 0 limits finite dimensionality of the flow Partial regularity for 4 θ + 4 θ 1 + 2 θ 2 < n + 2 ? Gantumur, Evelyn, Mike Partial regularity for generalized alpha models Nov 14 4 / 8

Katz-Pavlović idea In R n consider ∂ t u = Au + B ( u , u ) where A = ∆ θ , and B is a bilinear Fourier multiplier. Let P j be Littlewood-Paley projectors, and let j + 2 ˜ � P j ˜ P j = P k , so that P j = P j . k = j − 2 Fix ε > 0 . If λ is a ball of radius 2 ε j 2 − j , let φ λ ∈ C ∞ 0 (2 λ ,[0,1]) satisfy φ λ ≡ 1 on λ , and define P λ = φ λ P j . We have 1 d d t � P λ u � 2 = 〈 P λ Au , P λ u 〉+〈 P λ B ( u , u ), P λ u 〉 2 Gantumur, Evelyn, Mike Partial regularity for generalized alpha models Nov 14 5 / 8

Growth estimates For all large j , and a sufficiently large “neighbourhood” Λ of λ � t � t � P µ u � 2 + O (2 − Nj ) 〈 P µ Au , P µ u 〉 � − 2 2 θ j � � 0 0 µ ∈ Λ µ ∈ Λ Thinking of a Leray- α type model, we have 2 (1 − 2 θ 1 ) k + n 2 ( n 2 j � φ λ P k u � 2 + 2 − 2 θ 1 ) k + j � φ λ P k u � 2 � � � P λ B ( u , u ) � � k ≥ j δ j ≤ k ≤ j + 2 + 2 δ ′ j � � P k u � 2 k ≤ j Corresponding estimates for dyadic models are � P λ B ( u , u ) � � 2 ( n 〈 P λ Au , P λ u 〉 � − 2 2 θ j � P λ u � 2 , 2 + 1 − 2 θ 1 ) j � φ λ P k u � 2 � j − 1 ≤ k ≤ j + 1 The “critical regularity” is P λ u ∼ 2 (2 θ + 2 θ 1 − n 2 − 1) j Gantumur, Evelyn, Mike Partial regularity for generalized alpha models Nov 14 6 / 8

Dyadic model Fix a constant h > 0 , and call λ hopeless if � P λ u � > h 2 (2 θ + 2 θ 1 − n 2 − 1 − ε ) j A point x ∈ R n is hopeless at level j if it is in some hopeless ball of radius 2 ε j 2 − j . Let E be the set of points that are hopeless at infinitely many levels. Then the Hausdorff dimension of E is at most n + 2 + ε − 4 θ − 4 θ 1 . ( n + 2 + ε − 4 θ − 4 θ 1 − 2 θ 2 for the general model) If λ is not hopeless d d t � P λ u � 2 � − 2 2 θ j � P λ u � 2 + 2 (2 θ − ε ) j � φ λ P k u � 2 , � j − 1 ≤ k ≤ j + 1 and one can show that u is regular inside λ (roughly speaking). Gantumur, Evelyn, Mike Partial regularity for generalized alpha models Nov 14 7 / 8

Further directions Ladyzhenskaya’s µ model D = ∇ u +∇ u T , F ( D ) ∼ | D | 2 µ + 1 . ∂ t u = div F ( D ) + B ( u , u ), 2( n + 2) , global regularity for µ ≥ n − 2 n − 2 Weak solution for µ ≥ 4 [Ladyzhenskaya]. For some models in 3D, weak solution for µ ≤ 1 2 , global regularity for µ ≥ 1 10 [Malek et al]. For a dyadic model in 3D, Hausdorff dim of the space singular set is at most 1 − 10 µ 1 − 2 µ [Friedlander-Pavlović]. Space-time singular set For NS, the parabolic Hausdorff dim < 1 [CKN] For µ model, it is ≤ 3 [Seregin] Gantumur, Evelyn, Mike Partial regularity for generalized alpha models Nov 14 8 / 8