On smallness condition of initial data for Le JanSznitman cascade of - PowerPoint PPT Presentation

On smallness condition of initial data for Le Jan–Sznitman cascade of the Navier-Stokes equations Tuan Pham Oregon State University October 14, 2019 1/21 Tuan Pham (Oregon State University) October 14, 2019 1 / 21

NSE, mild solutions R d × (0 , ∞ ) ,  ∂ t u − ∆ u + u · ∇ u + ∇ p = 0 in  R d × (0 , ∞ ) , ( NSE ) : div u = 0 in R d . u ( · , 0) = u 0 in  2/21 Tuan Pham (Oregon State University) October 14, 2019 2 / 21

NSE, mild solutions R d × (0 , ∞ ) ,  ∂ t u − ∆ u + u · ∇ u + ∇ p = 0 in  R d × (0 , ∞ ) , ( NSE ) : div u = 0 in R d . u ( · , 0) = u 0 in  Integro-differential equation: � t u ( x , t ) = e ∆ t u 0 − e ∆ s P div [ u ( t − s ) ⊗ u ( t − s )] ds . 0 2/21 Tuan Pham (Oregon State University) October 14, 2019 2 / 21

NSE, mild solutions R d × (0 , ∞ ) ,  ∂ t u − ∆ u + u · ∇ u + ∇ p = 0 in  R d × (0 , ∞ ) , ( NSE ) : div u = 0 in R d . u ( · , 0) = u 0 in  Integro-differential equation: � t u ( x , t ) = e ∆ t u 0 − e ∆ s P div [ u ( t − s ) ⊗ u ( t − s )] ds . 0 Mild solutions – obtained by Picard’s iteration: ≡ v 0 0 v n = U + B ( v n − 1 , v n − 1 ) u = lim v n 2/21 Tuan Pham (Oregon State University) October 14, 2019 2 / 21

NSE, mild solutions R d × (0 , ∞ ) ,  ∂ t u − ∆ u + u · ∇ u + ∇ p = 0 in  R d × (0 , ∞ ) , ( NSE ) : div u = 0 in R d . u ( · , 0) = u 0 in  Integro-differential equation: � t u ( x , t ) = e ∆ t u 0 − e ∆ s P div [ u ( t − s ) ⊗ u ( t − s )] ds . 0 Mild solutions – obtained by Picard’s iteration: ≡ v 0 0 v n = U + B ( v n − 1 , v n − 1 ) u = lim v n 2/21 Tuan Pham (Oregon State University) October 14, 2019 2 / 21

NSE, mild solutions � Global existence and uniqueness in L ∞ t L 2 x for d = 2: Leray (1933). � Local existence and uniqueness in subcritical spaces: Leray (‘34), Kato (‘84),. . . � Global existence in critical spaces for small initial data: Kato (‘84), Koch-Tataru (2001),. . . ? Global existence for arbitrarily large initial data. 3/21 Tuan Pham (Oregon State University) October 14, 2019 3 / 21

NSE, weak solutions Weak formulation = diff. eq. in distribution sense + energy inequality. Energy solutions: Leray ‘34, Hopf ‘51 � t | u ( x , t ) | 2 | u 0 ( x ) | 2 � � � R d |∇ u | 2 dxds ≤ dx + dx 2 2 R d R d 0 4/21 Tuan Pham (Oregon State University) October 14, 2019 4 / 21

NSE, weak solutions Weak formulation = diff. eq. in distribution sense + energy inequality. Energy solutions: Leray ‘34, Hopf ‘51 � t | u ( x , t ) | 2 | u 0 ( x ) | 2 � � � R d |∇ u | 2 dxds ≤ dx + dx 2 2 R d R d 0 Local energy solutions: Scheffer ‘77, CKN ‘82, L-R 2002,. . . ∞ ∞ � | u | 2 � | u | 2 � � � � � � |∇ u | 2 φ dxdt ≤ ( ∂ t φ + ∆ φ ) + + p u ∇ φ dxdt 2 2 0 0 R d R d 4/21 Tuan Pham (Oregon State University) October 14, 2019 4 / 21

NSE, weak solutions Weak formulation = diff. eq. in distribution sense + energy inequality. Energy solutions: Leray ‘34, Hopf ‘51 � t | u ( x , t ) | 2 | u 0 ( x ) | 2 � � � R d |∇ u | 2 dxds ≤ dx + dx 2 2 R d R d 0 Local energy solutions: Scheffer ‘77, CKN ‘82, L-R 2002,. . . ∞ ∞ � | u | 2 � | u | 2 � � � � � � |∇ u | 2 φ dxdt ≤ ( ∂ t φ + ∆ φ ) + + p u ∇ φ dxdt 2 2 0 0 R d R d � Global existence ? Uniqueness, smoothness 4/21 Tuan Pham (Oregon State University) October 14, 2019 4 / 21

NSE, weak solutions Partial regularity: Let u 0 ∈ L 2 . How big is the set of singular points S ⊂ R d × (0 , ∞ )? 5/21 Tuan Pham (Oregon State University) October 14, 2019 5 / 21

NSE, weak solutions Partial regularity: Let u 0 ∈ L 2 . How big is the set of singular points S ⊂ R d × (0 , ∞ )? 2 d H 1 ( R d ) ֒ d − 2 ( R d ) → L d = 2: S = ∅ (Leray ‘33). d = 3: H 1 par ( S ) = 0 (CKN ‘82). d = 4: H 2 par ( S ) = 0 (Dong-Gu 2014, Wang-Wu ‘14). d = 5 (stationary): S = ∅ (Struwe 1995). d = 6 (stationary): H 2 ( S ) = 0 (Dong-Strain 2012). 5/21 Tuan Pham (Oregon State University) October 14, 2019 5 / 21

Fourier transformed Navier-Stokes (FNS) � t � u ( ξ, t ) = e −| ξ | 2 t ˆ e −| ξ | 2 s | ξ | ˆ u 0 ( ξ )+ c 0 R d ˆ u ( η, t − s ) ⊙ ξ ˆ u ( ξ − η, t − s ) d η ds 0 where a ⊙ ξ b = − i ( e ξ · b )( π ξ ⊥ a ). 6/21 Tuan Pham (Oregon State University) October 14, 2019 6 / 21

Fourier transformed Navier-Stokes (FNS) � t � u ( ξ, t ) = e −| ξ | 2 t ˆ e −| ξ | 2 s | ξ | ˆ u 0 ( ξ )+ c 0 R d ˆ u ( η, t − s ) ⊙ ξ ˆ u ( ξ − η, t − s ) d η ds 0 where a ⊙ ξ b = − i ( e ξ · b )( π ξ ⊥ a ). Normalization to (FNS): LJS ‘97, Bhattacharya et al (2003) e − t | ξ | 2 χ 0 ( ξ ) χ ( ξ, t ) = � t � e − s | ξ | 2 | ξ | 2 + R d χ ( η, t − s ) ⊙ ξ χ ( ξ − η, t − s ) H ( η | ξ ) d η ds 0 u / h and H ( η | ξ ) = h ( η ) h ( ξ − η ) where χ = c 0 ˆ . | ξ | h ( ξ ) h : majorizing kernel , i.e. h ∗ h = | ξ | h . 6/21 Tuan Pham (Oregon State University) October 14, 2019 6 / 21

Cascade structure of FNS 7/21 Tuan Pham (Oregon State University) October 14, 2019 7 / 21

Cascade structure of FNS Define a stochastic multiplicative functional recursively as � χ 0 ( ξ ) if T 0 > t , X FNS ( ξ, t ) = X (1) FNS ( W 1 , t − T 0 ) ⊙ ξ X (2) FNS ( ξ − W 1 , t − T 0 ) T 0 ≤ t . if 7/21 Tuan Pham (Oregon State University) October 14, 2019 7 / 21

Closed form of X FNS Consider the following event: On this event, X FNS ( ξ, t ) = ( χ 0 ( W 11 ) ⊙ W 1 χ 0 ( W 12 )) ⊙ ξ χ 0 ( W 2 ) . 8/21 Tuan Pham (Oregon State University) October 14, 2019 8 / 21

Closed form of X FNS Consider the following event: On this event, X FNS ( ξ, t ) = ( χ 0 ( W 11 ) ⊙ W 1 χ 0 ( W 12 )) ⊙ ξ χ 0 ( W 2 ) . Three ingredients: clocks, branching process, product. Cascade structure = clocks + branching process. 8/21 Tuan Pham (Oregon State University) October 14, 2019 8 / 21

FNS: mild solutions, cascade solutions e − t | ξ | 2 χ 0 ( ξ ) χ ( ξ, t ) = � t � e − s | ξ | 2 | ξ | 2 + R d χ ( η, t − s ) ⊙ ξ χ ( ξ − η, t − s ) H ( η | ξ ) d η ds 0 Mild solution: γ 0 ≡ 0 e − t | ξ | 2 χ 0 + ¯ γ n = B ( γ n − 1 , γ n − 1 ) χ = lim γ n 9/21 Tuan Pham (Oregon State University) October 14, 2019 9 / 21

FNS: mild solutions, cascade solutions e − t | ξ | 2 χ 0 ( ξ ) χ ( ξ, t ) = � t � e − s | ξ | 2 | ξ | 2 + R d χ ( η, t − s ) ⊙ ξ χ ( ξ − η, t − s ) H ( η | ξ ) d η ds 0 Mild solution: γ 0 ≡ 0 e − t | ξ | 2 χ 0 + ¯ γ n = B ( γ n − 1 , γ n − 1 ) χ = lim γ n Cascade solution ( ∼ LJS 1997): χ ( ξ, t ) = E ξ, t X FNS 9/21 Tuan Pham (Oregon State University) October 14, 2019 9 / 21

FNS: mild solutions, cascade solutions e − t | ξ | 2 χ 0 ( ξ ) χ ( ξ, t ) = � t � e − s | ξ | 2 | ξ | 2 + R d χ ( η, t − s ) ⊙ ξ χ ( ξ − η, t − s ) H ( η | ξ ) d η ds 0 Mild solution: γ 0 ≡ 0 e − t | ξ | 2 χ 0 + ¯ γ n = B ( γ n − 1 , γ n − 1 ) χ = lim γ n Cascade solution ( ∼ LJS 1997): χ ( ξ, t ) = E ξ, t X FNS Two issues: (1) stochastic explosion and (2) existence of expectation . 9/21 Tuan Pham (Oregon State University) October 14, 2019 9 / 21

✶ Explosion Branching process may never stop, potentially making X FNS not well-defined. Property of cascade structure, not of product. Depending only on the majorizing kernel h and the clocks. 10/21 Tuan Pham (Oregon State University) October 14, 2019 10 / 21

✶ Explosion Branching process may never stop, potentially making X FNS not well-defined. Property of cascade structure, not of product. Depending only on the majorizing kernel h and the clocks. 3D self-similar cascade h dilog ( ξ ) = C | ξ | − 2 : stochastic explosion a.s. (Dascaliuc, Pham, Thomann, Waymire 2019) 3D Bessel cascade h b ( ξ ) = C | ξ | − 1 e −| ξ | : no-explosion a.s. (Orum, Pham 2019) 10/21 Tuan Pham (Oregon State University) October 14, 2019 10 / 21

Explosion Branching process may never stop, potentially making X FNS not well-defined. Property of cascade structure, not of product. Depending only on the majorizing kernel h and the clocks. 3D self-similar cascade h dilog ( ξ ) = C | ξ | − 2 : stochastic explosion a.s. (Dascaliuc, Pham, Thomann, Waymire 2019) 3D Bessel cascade h b ( ξ ) = C | ξ | − 1 e −| ξ | : no-explosion a.s. (Orum, Pham 2019) We bypass the explosion problem by defining instead χ ( ξ, t ) = E ξ, t [ X FNS ✶ S > t ] , where S is the shortest path. 10/21 Tuan Pham (Oregon State University) October 14, 2019 10 / 21

Existence of expectation It may happen that E ξ, t [ | X FNS | ✶ S > t ] = ∞ . � X FNS ( ξ, t ) ✶ S > t = χ 0 ( W s ) ( finite product ) s ∈V 0 ( ξ, t ) 11/21 Tuan Pham (Oregon State University) October 14, 2019 11 / 21

Existence of expectation It may happen that E ξ, t [ | X FNS | ✶ S > t ] = ∞ . � X FNS ( ξ, t ) ✶ S > t = χ 0 ( W s ) ( finite product ) s ∈V 0 ( ξ, t ) This issue depends on both cascade structure and the product. 11/21 Tuan Pham (Oregon State University) October 14, 2019 11 / 21