Bounded vorticity, bounded velocity (Serfati) solutions to the - PowerPoint PPT Presentation

Known well-posedness results If ω 0 does not decay at infinity then get existence and uniqueness in periodic setting , i.e., if ω 0 is doubly periodic in the plane. If the domain is an exterior domain: u 0 smooth ( C 1 b ) and the vorticity has some decay ( ω 0 ∈ L 1 , | x | r ω 0 ∈ L 1 , some r > 0) and regularity ( ∇ ω 0 ∈ L ∞ ∩ L 1 ) then well-posedness (Kikuchi 1981). In 1995 Ph. Serfati published announcement in CRAS of ∃ and ! for 2D Euler in R 2 for u 0 ∈ L ∞ such that ω 0 = curl u 0 ∈ L ∞ . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 7 / 26

Known well-posedness results If ω 0 does not decay at infinity then get existence and uniqueness in periodic setting , i.e., if ω 0 is doubly periodic in the plane. If the domain is an exterior domain: u 0 smooth ( C 1 b ) and the vorticity has some decay ( ω 0 ∈ L 1 , | x | r ω 0 ∈ L 1 , some r > 0) and regularity ( ∇ ω 0 ∈ L ∞ ∩ L 1 ) then well-posedness (Kikuchi 1981). In 1995 Ph. Serfati published announcement in CRAS of ∃ and ! for 2D Euler in R 2 for u 0 ∈ L ∞ such that ω 0 = curl u 0 ∈ L ∞ . Proof was terse and incomplete, June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 7 / 26

Known well-posedness results If ω 0 does not decay at infinity then get existence and uniqueness in periodic setting , i.e., if ω 0 is doubly periodic in the plane. If the domain is an exterior domain: u 0 smooth ( C 1 b ) and the vorticity has some decay ( ω 0 ∈ L 1 , | x | r ω 0 ∈ L 1 , some r > 0) and regularity ( ∇ ω 0 ∈ L ∞ ∩ L 1 ) then well-posedness (Kikuchi 1981). In 1995 Ph. Serfati published announcement in CRAS of ∃ and ! for 2D Euler in R 2 for u 0 ∈ L ∞ such that ω 0 = curl u 0 ∈ L ∞ . Proof was terse and incomplete, yet brilliant. June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 7 / 26

Known well-posedness results If ω 0 does not decay at infinity then get existence and uniqueness in periodic setting , i.e., if ω 0 is doubly periodic in the plane. If the domain is an exterior domain: u 0 smooth ( C 1 b ) and the vorticity has some decay ( ω 0 ∈ L 1 , | x | r ω 0 ∈ L 1 , some r > 0) and regularity ( ∇ ω 0 ∈ L ∞ ∩ L 1 ) then well-posedness (Kikuchi 1981). In 1995 Ph. Serfati published announcement in CRAS of ∃ and ! for 2D Euler in R 2 for u 0 ∈ L ∞ such that ω 0 = curl u 0 ∈ L ∞ . Proof was terse and incomplete, yet brilliant. This talk: discussion of Serfati’s work, extension to continuous dependence on initial data and to flow domains exterior to a connected obstacle. June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 7 / 26

Known well-posedness results If ω 0 does not decay at infinity then get existence and uniqueness in periodic setting , i.e., if ω 0 is doubly periodic in the plane. If the domain is an exterior domain: u 0 smooth ( C 1 b ) and the vorticity has some decay ( ω 0 ∈ L 1 , | x | r ω 0 ∈ L 1 , some r > 0) and regularity ( ∇ ω 0 ∈ L ∞ ∩ L 1 ) then well-posedness (Kikuchi 1981). In 1995 Ph. Serfati published announcement in CRAS of ∃ and ! for 2D Euler in R 2 for u 0 ∈ L ∞ such that ω 0 = curl u 0 ∈ L ∞ . Proof was terse and incomplete, yet brilliant. This talk: discussion of Serfati’s work, extension to continuous dependence on initial data and to flow domains exterior to a connected obstacle. This is a report of work in progress. June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 7 / 26

Statement of problem Statement of problem Let D ⊂ R 2 be connected, open, bounded, smooth domain; Ω = R 2 \ D . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 8 / 26

Statement of problem Statement of problem Let D ⊂ R 2 be connected, open, bounded, smooth domain; Ω = R 2 \ D . Let u 0 ∈ L ∞ (Ω; R 2 ) be vector field such that ω 0 = curl u 0 ≡ ∇ ⊥ · u 0 ∈ L ∞ . Suppose also div u 0 = 0 and u 0 · ˆ n = 0 on ∂ D . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 8 / 26

Statement of problem Statement of problem Let D ⊂ R 2 be connected, open, bounded, smooth domain; Ω = R 2 \ D . Let u 0 ∈ L ∞ (Ω; R 2 ) be vector field such that ω 0 = curl u 0 ≡ ∇ ⊥ · u 0 ∈ L ∞ . Suppose also div u 0 = 0 and u 0 · ˆ n = 0 on ∂ D . Problem: existence, uniqueness and continuous dependence of solution to incompressible 2D Euler with initial velocity u 0 . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 8 / 26

Statement of problem Statement of problem Let D ⊂ R 2 be connected, open, bounded, smooth domain; Ω = R 2 \ D . Let u 0 ∈ L ∞ (Ω; R 2 ) be vector field such that ω 0 = curl u 0 ≡ ∇ ⊥ · u 0 ∈ L ∞ . Suppose also div u 0 = 0 and u 0 · ˆ n = 0 on ∂ D . Problem: existence, uniqueness and continuous dependence of solution to incompressible 2D Euler with initial velocity u 0 . Fundamental questions: what is a ‘solution’? What happens to the Biot-Savart law? Where do we get ‘uniqueness’? What perturbations are allowed? June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 8 / 26

Related results Related results Taniuchi (2004) gives complete, and very different proof of ∃ , including Serfati flows, in full plane, allowing slightly unbounded initial vorticity (as Yudovich 1995 did in a bounded domain). Uses Littlewood-Paley decomposition and Bony’s paradifferential calculus. Does not generalize to exterior domains. June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 9 / 26

Related results Related results Taniuchi (2004) gives complete, and very different proof of ∃ , including Serfati flows, in full plane, allowing slightly unbounded initial vorticity (as Yudovich 1995 did in a bounded domain). Uses Littlewood-Paley decomposition and Bony’s paradifferential calculus. Does not generalize to exterior domains. Taniuchi, Tashiro, and Yoneda (2010) are concerned with almost periodic flows in the full plane . They prove ∃ and ! assuming u 0 ∈ L ∞ and ω 0 ∈ Y θ ul , for θ = log ( e + q ) ; Y θ ul means “uniformly local" L p -norms grow like θ ( p ) – includes Serfati initial data. Again, proof relies on Littlewood-Paley theory and Bony’s paradifferential calculus; highly non-local proof. June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 9 / 26

Related results Related results Taniuchi (2004) gives complete, and very different proof of ∃ , including Serfati flows, in full plane, allowing slightly unbounded initial vorticity (as Yudovich 1995 did in a bounded domain). Uses Littlewood-Paley decomposition and Bony’s paradifferential calculus. Does not generalize to exterior domains. Taniuchi, Tashiro, and Yoneda (2010) are concerned with almost periodic flows in the full plane . They prove ∃ and ! assuming u 0 ∈ L ∞ and ω 0 ∈ Y θ ul , for θ = log ( e + q ) ; Y θ ul means “uniformly local" L p -norms grow like θ ( p ) – includes Serfati initial data. Again, proof relies on Littlewood-Paley theory and Bony’s paradifferential calculus; highly non-local proof. They also prove continuous dependence, but in B 0 ∞ , 1 . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 9 / 26

Related results Giga, Inui, and Matsui (1999) prove existence and uniqueness of solutions to the Navier-Stokes equations with velocity bounded and uniformly continuous (which includes Serfati initial data). June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 10 / 26

Related results Giga, Inui, and Matsui (1999) prove existence and uniqueness of solutions to the Navier-Stokes equations with velocity bounded and uniformly continuous (which includes Serfati initial data). Cozzi (2009, 2010) proves the vanishing viscosity limit of “viscous Serfati” solutions to inviscid ones in full plane. June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 10 / 26

Related results Giga, Inui, and Matsui (1999) prove existence and uniqueness of solutions to the Navier-Stokes equations with velocity bounded and uniformly continuous (which includes Serfati initial data). Cozzi (2009, 2010) proves the vanishing viscosity limit of “viscous Serfati” solutions to inviscid ones in full plane. Brunelli (2010) → studies full plane flows with velocity growing at infinity. He assumes ω 0 ∈ L ∞ and � 1 | x − y || ω 0 ( y ) | dy < ∞ for some x ∈ R 2 and gets ∃ and ! of ( u , ω, Φ t ) such that | u | grows at most � like | x | at infinity. ( Φ t is the Lagrangian map.) The hypothesis excludes periodic flows. June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 10 / 26

Motivation Motivation Why study vorticity with no decay? June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 11 / 26

Motivation Motivation Why study vorticity with no decay? Main uniqueness result is for vorticity in L ∞ ∩ L 1 (Yudovich 1963), but L 1 hypothesis relevance is to make sense of Biot-Savart law. Uniqueness should be local issue – behavior of vorticity at infinity should not be important. June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 11 / 26

Motivation Motivation Why study vorticity with no decay? Main uniqueness result is for vorticity in L ∞ ∩ L 1 (Yudovich 1963), but L 1 hypothesis relevance is to make sense of Biot-Savart law. Uniqueness should be local issue – behavior of vorticity at infinity should not be important. Assumption of decay of initial vorticity not physically natural – full plane flow is approximate model for flow far from boundaries, where no decay of distant vorticity should be expected. June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 11 / 26

Motivation Motivation Why study vorticity with no decay? Main uniqueness result is for vorticity in L ∞ ∩ L 1 (Yudovich 1963), but L 1 hypothesis relevance is to make sense of Biot-Savart law. Uniqueness should be local issue – behavior of vorticity at infinity should not be important. Assumption of decay of initial vorticity not physically natural – full plane flow is approximate model for flow far from boundaries, where no decay of distant vorticity should be expected. In light of Taniuchi, Tashiro and Yoneda’s work, why revisit Serfati? June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 11 / 26

Motivation Motivation Why study vorticity with no decay? Main uniqueness result is for vorticity in L ∞ ∩ L 1 (Yudovich 1963), but L 1 hypothesis relevance is to make sense of Biot-Savart law. Uniqueness should be local issue – behavior of vorticity at infinity should not be important. Assumption of decay of initial vorticity not physically natural – full plane flow is approximate model for flow far from boundaries, where no decay of distant vorticity should be expected. In light of Taniuchi, Tashiro and Yoneda’s work, why revisit Serfati? Local versus non-local; June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 11 / 26

Motivation Motivation Why study vorticity with no decay? Main uniqueness result is for vorticity in L ∞ ∩ L 1 (Yudovich 1963), but L 1 hypothesis relevance is to make sense of Biot-Savart law. Uniqueness should be local issue – behavior of vorticity at infinity should not be important. Assumption of decay of initial vorticity not physically natural – full plane flow is approximate model for flow far from boundaries, where no decay of distant vorticity should be expected. In light of Taniuchi, Tashiro and Yoneda’s work, why revisit Serfati? Local versus non-local; Need new idea to substitute Biot-Savart law; June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 11 / 26

Motivation Motivation Why study vorticity with no decay? Main uniqueness result is for vorticity in L ∞ ∩ L 1 (Yudovich 1963), but L 1 hypothesis relevance is to make sense of Biot-Savart law. Uniqueness should be local issue – behavior of vorticity at infinity should not be important. Assumption of decay of initial vorticity not physically natural – full plane flow is approximate model for flow far from boundaries, where no decay of distant vorticity should be expected. In light of Taniuchi, Tashiro and Yoneda’s work, why revisit Serfati? Local versus non-local; Need new idea to substitute Biot-Savart law; Broader potential applications in Serfati’s key idea (new representation formula). June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 11 / 26

Serfati’s representation formula Serfati’s representation formula Key new idea: consider a smooth cutoff a ε of the origin ( a ε ( z ) = 1 if z ∈ R 2 and | z | small, and vanishes if | z | large) and establish a formula like June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 12 / 26

Serfati’s representation formula Serfati’s representation formula Key new idea: consider a smooth cutoff a ε of the origin ( a ε ( z ) = 1 if z ∈ R 2 and | z | small, and vanishes if | z | large) and establish a formula like � u ( t , x ) = u 0 ( x ) + a ε ( x − y ) K ( x − y )( ω ( t , y ) − ω 0 ( y )) dy + � t � � u ( s , y ) · ∇ y ∇ y [( 1 − a ε ( x − y )) K ( x − y )] ⊥ � − · u ( s , y ) dyds . 0 June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 12 / 26

Serfati’s representation formula Serfati’s representation formula Key new idea: consider a smooth cutoff a ε of the origin ( a ε ( z ) = 1 if z ∈ R 2 and | z | small, and vanishes if | z | large) and establish a formula like � u ( t , x ) = u 0 ( x ) + a ε ( x − y ) K ( x − y )( ω ( t , y ) − ω 0 ( y )) dy + � t � � u ( s , y ) · ∇ y ∇ y [( 1 − a ε ( x − y )) K ( x − y )] ⊥ � − · u ( s , y ) dyds . 0 Why? June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 12 / 26

Serfati’s representation formula Serfati’s representation formula Key new idea: consider a smooth cutoff a ε of the origin ( a ε ( z ) = 1 if z ∈ R 2 and | z | small, and vanishes if | z | large) and establish a formula like � u ( t , x ) = u 0 ( x ) + a ε ( x − y ) K ( x − y )( ω ( t , y ) − ω 0 ( y )) dy + � t � � u ( s , y ) · ∇ y ∇ y [( 1 − a ε ( x − y )) K ( x − y )] ⊥ � − · u ( s , y ) dyds . 0 Why? Because all the terms in this formula converge. June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 12 / 26

Serfati’s representation formula Serfati’s representation formula Key new idea: consider a smooth cutoff a ε of the origin ( a ε ( z ) = 1 if z ∈ R 2 and | z | small, and vanishes if | z | large) and establish a formula like � u ( t , x ) = u 0 ( x ) + a ε ( x − y ) K ( x − y )( ω ( t , y ) − ω 0 ( y )) dy + � t � � u ( s , y ) · ∇ y ∇ y [( 1 − a ε ( x − y )) K ( x − y )] ⊥ � − · u ( s , y ) dyds . 0 Why? Because all the terms in this formula converge. Indeed: a ε K ∈ L 1 and ω ∈ L ∞ so the first integral converges; ∇ y ∇ y [( 1 − a ε ) K ] ∈ L 1 because of the extra decay at infinity coming from taking two derivatives, hence, if u ∈ L ∞ , then the second integral also converges. June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 12 / 26

Serfati’s representation formula Where does this formula come from? June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 13 / 26

Serfati’s representation formula Where does this formula come from? Suppose u smooth, bounded, classical solution of 2D incompressible Euler and ω smooth, bounded, ω 0 compactly supported. Then � u = u ( t , x ) = K ( x − y ) ω ( t , y ) dy . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 13 / 26

Serfati’s representation formula Where does this formula come from? Suppose u smooth, bounded, classical solution of 2D incompressible Euler and ω smooth, bounded, ω 0 compactly supported. Then � u = u ( t , x ) = K ( x − y ) ω ( t , y ) dy . Hence, � ∂ t u = ∂ t K ( x − y ) ω ( t , y ) dy � � = ∂ t a ε ( x − y ) K ( x − y ) ω ( t , y ) dy + ( 1 − a ε ( x − y )) K ( x − y ) ∂ t ω dy . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 13 / 26

Serfati’s representation formula Where does this formula come from? Suppose u smooth, bounded, classical solution of 2D incompressible Euler and ω smooth, bounded, ω 0 compactly supported. Then � u = u ( t , x ) = K ( x − y ) ω ( t , y ) dy . Hence, � ∂ t u = ∂ t K ( x − y ) ω ( t , y ) dy � � = ∂ t a ε ( x − y ) K ( x − y ) ω ( t , y ) dy + ( 1 − a ε ( x − y )) K ( x − y ) ∂ t ω dy . Therefore, integrating in time yields � u ( t , x ) = u 0 ( x ) + a ε ( x − y ) K ( x − y )[ ω ( t , y ) − ω 0 ( y )] dy � t � + ( 1 − a ε ( x − y )) K ( x − y ) ∂ t ω dy . 0 June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 13 / 26

Serfati’s representation formula Now, � � ( 1 − a ε ( x − y )) K ( x − y ) ∂ t ω dy = − ( 1 − a ε ( x − y )) K ( x − y ) u · ∇ y ω dy � = − ( 1 − a ε ( x − y )) K ( x − y ) curl y ( u · ∇ y u ) dy . Therefore, June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 14 / 26

Serfati’s representation formula Now, � � ( 1 − a ε ( x − y )) K ( x − y ) ∂ t ω dy = − ( 1 − a ε ( x − y )) K ( x − y ) u · ∇ y ω dy � = − ( 1 − a ε ( x − y )) K ( x − y ) curl y ( u · ∇ y u ) dy . Therefore, integrating by parts carefully, using div u = 0, get � ( 1 − a ε ( x − y )) K ( x − y ) ∂ t ω dy � � u ( s , y ) · ∇ y ∇ y [( 1 − a ε ( x − y )) K ( x − y )] ⊥ � = · u ( s , y ) dy . Finally, substitute to obtain the desired formula – Serfati identity: June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 14 / 26

Serfati’s representation formula Now, � � ( 1 − a ε ( x − y )) K ( x − y ) ∂ t ω dy = − ( 1 − a ε ( x − y )) K ( x − y ) u · ∇ y ω dy � = − ( 1 − a ε ( x − y )) K ( x − y ) curl y ( u · ∇ y u ) dy . Therefore, integrating by parts carefully, using div u = 0, get � ( 1 − a ε ( x − y )) K ( x − y ) ∂ t ω dy � � u ( s , y ) · ∇ y ∇ y [( 1 − a ε ( x − y )) K ( x − y )] ⊥ � = · u ( s , y ) dy . Finally, substitute to obtain the desired formula – Serfati identity: � u ( t , x ) = u 0 ( x ) + a ε ( x − y ) K ( x − y )[ ω ( t , y ) − ω 0 ( y )] dy + � t � � u ( s , y ) · ∇ y ∇ y [( 1 − a ε ( x − y )) K ( x − y )] ⊥ � − · u ( s , y ) dy . 0 June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 14 / 26

Serfati’s strategy for existence Serfati’s strategy for existence Why is this representation formula useful? June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 15 / 26

Serfati’s strategy for existence Serfati’s strategy for existence Why is this representation formula useful? A priori estimates in L ∞ . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 15 / 26

Serfati’s strategy for existence Serfati’s strategy for existence Why is this representation formula useful? A priori estimates in L ∞ . Strategy for existence: start with u 0 ∈ L ∞ such that ω 0 ∈ L ∞ . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 15 / 26

Serfati’s strategy for existence Serfati’s strategy for existence Why is this representation formula useful? A priori estimates in L ∞ . Strategy for existence: start with u 0 ∈ L ∞ such that ω 0 ∈ L ∞ . 1. Construct a sequence { u 0 , N } such that u 0 , N ∈ C ∞ c , u 0 , N = K ∗ ω 0 , N , u 0 , N → u 0 uniformly on compact sets, ω 0 , N → ω 0 in L p on compact sets, for some C > 0, � u 0 , N � L ∞ + � ω 0 , N � L ∞ ≤ C < ∞ for all N . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 15 / 26

Serfati’s strategy for existence Serfati’s strategy for existence Why is this representation formula useful? A priori estimates in L ∞ . Strategy for existence: start with u 0 ∈ L ∞ such that ω 0 ∈ L ∞ . 1. Construct a sequence { u 0 , N } such that u 0 , N ∈ C ∞ c , u 0 , N = K ∗ ω 0 , N , u 0 , N → u 0 uniformly on compact sets, ω 0 , N → ω 0 in L p on compact sets, for some C > 0, � u 0 , N � L ∞ + � ω 0 , N � L ∞ ≤ C < ∞ for all N . 2. Solve 2D Euler with initial velocity u 0 , N ; solution denoted u N . Use Serfati identity to obtain L ∞ estimates for u N , uniform wrt N . Use transport to get uniform L ∞ estimates for ω N = curl u N . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 15 / 26

Serfati’s strategy for existence Serfati’s strategy for existence Why is this representation formula useful? A priori estimates in L ∞ . Strategy for existence: start with u 0 ∈ L ∞ such that ω 0 ∈ L ∞ . 1. Construct a sequence { u 0 , N } such that u 0 , N ∈ C ∞ c , u 0 , N = K ∗ ω 0 , N , u 0 , N → u 0 uniformly on compact sets, ω 0 , N → ω 0 in L p on compact sets, for some C > 0, � u 0 , N � L ∞ + � ω 0 , N � L ∞ ≤ C < ∞ for all N . 2. Solve 2D Euler with initial velocity u 0 , N ; solution denoted u N . Use Serfati identity to obtain L ∞ estimates for u N , uniform wrt N . Use transport to get uniform L ∞ estimates for ω N = curl u N . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 15 / 26

Serfati’s strategy for existence 3. Use L ∞ estimate for u N and ω N to prove that u N is uniformly log-Lipschitz. From this get, in standard way, passing to subsequences as needed, uniform convergence (on compacts) of particle trajectories X N = X N ( t , α ) ( dX N / dt = u N ( t , X N ) and X N ( 0 , α ) = α ) to X = X ( t , α ) . Also, easy to show X is measure-preserving. June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 16 / 26

Serfati’s strategy for existence 3. Use L ∞ estimate for u N and ω N to prove that u N is uniformly log-Lipschitz. From this get, in standard way, passing to subsequences as needed, uniform convergence (on compacts) of particle trajectories X N = X N ( t , α ) ( dX N / dt = u N ( t , X N ) and X N ( 0 , α ) = α ) to X = X ( t , α ) . Also, easy to show X is measure-preserving. 4. Define limit vorticity as ω ( t , x ) ≡ ω 0 ( X − 1 ( t , x )) . Show ω N → ω in loc ( dt , L p L ∞ loc ( dx )) , any p < ∞ . Show u N → u uniformly in compacts in space and time. June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 16 / 26

Serfati’s strategy for existence 3. Use L ∞ estimate for u N and ω N to prove that u N is uniformly log-Lipschitz. From this get, in standard way, passing to subsequences as needed, uniform convergence (on compacts) of particle trajectories X N = X N ( t , α ) ( dX N / dt = u N ( t , X N ) and X N ( 0 , α ) = α ) to X = X ( t , α ) . Also, easy to show X is measure-preserving. 4. Define limit vorticity as ω ( t , x ) ≡ ω 0 ( X − 1 ( t , x )) . Show ω N → ω in loc ( dt , L p L ∞ loc ( dx )) , any p < ∞ . Show u N → u uniformly in compacts in space and time. 5. Conclude u , ω satisfy incompressible 2D Euler in distributions, ω constant on particle paths and, also, representation formula remains valid. Also, (limit) u is log-Lipschitz. June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 16 / 26

Serfati’s strategy for existence Illustrate main issues with an outline of proof of Step 2 → uniform L ∞ estimates for u ; work in full plane: June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 17 / 26

Serfati’s strategy for existence Illustrate main issues with an outline of proof of Step 2 → uniform L ∞ estimates for u ; work in full plane: � � u ( t , · ) � L ∞ ≤ � u 0 � L ∞ + sup | a ε ( x − y ) K ( x − y ) || ω ( t , y ) − ω 0 ( y ) | dy x � t � � � � u ( s , y ) · ∇ y ∇ y [( 1 − a ε ( x − y )) K ( x − y )] ⊥ � � � + sup · u ( s , y ) dy � � x � � 0 � t ≤ � u 0 � L ∞ + 2 � ω 0 � L ∞ � a ε K � L 1 + � D 2 � u ( s , · ) � 2 y [( 1 − a ε ) K ] � L 1 L ∞ ds . 0 Now, in the full plane we have June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 17 / 26

Serfati’s strategy for existence Illustrate main issues with an outline of proof of Step 2 → uniform L ∞ estimates for u ; work in full plane: � � u ( t , · ) � L ∞ ≤ � u 0 � L ∞ + sup | a ε ( x − y ) K ( x − y ) || ω ( t , y ) − ω 0 ( y ) | dy x � t � � � � u ( s , y ) · ∇ y ∇ y [( 1 − a ε ( x − y )) K ( x − y )] ⊥ � � � + sup · u ( s , y ) dy � � x � � 0 � t ≤ � u 0 � L ∞ + 2 � ω 0 � L ∞ � a ε K � L 1 + � D 2 � u ( s , · ) � 2 y [( 1 − a ε ) K ] � L 1 L ∞ ds . 0 Now, in the full plane we have � a ε K � L 1 ∼ C ε, June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 17 / 26

Serfati’s strategy for existence Illustrate main issues with an outline of proof of Step 2 → uniform L ∞ estimates for u ; work in full plane: � � u ( t , · ) � L ∞ ≤ � u 0 � L ∞ + sup | a ε ( x − y ) K ( x − y ) || ω ( t , y ) − ω 0 ( y ) | dy x � t � � � � u ( s , y ) · ∇ y ∇ y [( 1 − a ε ( x − y )) K ( x − y )] ⊥ � � � + sup · u ( s , y ) dy � � x � � 0 � t ≤ � u 0 � L ∞ + 2 � ω 0 � L ∞ � a ε K � L 1 + � D 2 � u ( s , · ) � 2 y [( 1 − a ε ) K ] � L 1 L ∞ ds . 0 Now, in the full plane we have � a ε K � L 1 ∼ C ε, y [( 1 − a ε ) K ] � L 1 ∼ C � D 2 ε . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 17 / 26

Serfati’s strategy for existence Hence, the a priori estimate becomes � t � u ( t , · ) � L ∞ ≤ � u 0 � L ∞ + C ε � ω 0 � L ∞ + C � u ( s , · ) � 2 L ∞ ds . ε 0 June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 18 / 26

Serfati’s strategy for existence Hence, the a priori estimate becomes � t � u ( t , · ) � L ∞ ≤ � u 0 � L ∞ + C ε � ω 0 � L ∞ + C � u ( s , · ) � 2 L ∞ ds . ε 0 �� t � 1 / 2 . Then get, after use of Young’s inequality, 0 � u ( s , · ) � 2 Choose ε = L ∞ ds June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 18 / 26

Serfati’s strategy for existence Hence, the a priori estimate becomes � t � u ( t , · ) � L ∞ ≤ � u 0 � L ∞ + C ε � ω 0 � L ∞ + C � u ( s , · ) � 2 L ∞ ds . ε 0 �� t � 1 / 2 . Then get, after use of Young’s inequality, 0 � u ( s , · ) � 2 Choose ε = L ∞ ds � t � u ( t , · ) � 2 L ∞ ≤ 2 � u 0 � 2 � u ( s , · ) � 2 L ∞ + C � ω 0 � L ∞ L ∞ ds . 0 June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 18 / 26

Serfati’s strategy for existence Hence, the a priori estimate becomes � t � u ( t , · ) � L ∞ ≤ � u 0 � L ∞ + C ε � ω 0 � L ∞ + C � u ( s , · ) � 2 L ∞ ds . ε 0 �� t � 1 / 2 . Then get, after use of Young’s inequality, 0 � u ( s , · ) � 2 Choose ε = L ∞ ds � t � u ( t , · ) � 2 L ∞ ≤ 2 � u 0 � 2 � u ( s , · ) � 2 L ∞ + C � ω 0 � L ∞ L ∞ ds . 0 The desired L ∞ -estimate for u follows, hence, by Gronwall’s lemma. June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 18 / 26

Exterior domain Exterior domain What about exterior domains? Assume domain exterior to unit disk. June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 19 / 26

Exterior domain Exterior domain What about exterior domains? Assume domain exterior to unit disk. First, no longer have convolution in Biot-Savart law: → K Ω ( x , y ) = K ( x − y ) − K ( x − y ∗ ) , K ( x − y ) ← where y ∗ = y / | y | 2 . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 19 / 26

Exterior domain Exterior domain What about exterior domains? Assume domain exterior to unit disk. First, no longer have convolution in Biot-Savart law: → K Ω ( x , y ) = K ( x − y ) − K ( x − y ∗ ) , K ( x − y ) ← where y ∗ = y / | y | 2 . Second, (hindsight from previous exterior domain work) know K Ω ( x , y ) + K ( x ) ∼ K ( x − y ) ( K ( x ) = x ⊥ / ( 2 π | x | 2 ) ), not K Ω ∼ K ( x − y ) . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 19 / 26

Exterior domain Denote J = J ( x , y ) = K Ω ( x , y ) + K ( x ) . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 20 / 26

Exterior domain Denote J = J ( x , y ) = K Ω ( x , y ) + K ( x ) . Downside: K ( x , · ) vanishes at boundary, but J ( x , · ) does not. June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 20 / 26

Exterior domain Denote J = J ( x , y ) = K Ω ( x , y ) + K ( x ) . Downside: K ( x , · ) vanishes at boundary, but J ( x , · ) does not. Still, arrive at new – and just as useful – Serfati identity; note boundary integral: June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 20 / 26

Exterior domain Denote J = J ( x , y ) = K Ω ( x , y ) + K ( x ) . Downside: K ( x , · ) vanishes at boundary, but J ( x , · ) does not. Still, arrive at new – and just as useful – Serfati identity; note boundary integral: u ( t , x ) � = u 0 ( x ) + a ε ( x − y ) J ( x , y )( ω ( t , y ) − ω 0 ( y )) dy Ω � t � ( u ( s , y ) · ∇ y ) ∇ ⊥ − y [( 1 − a ε ( x − y )) J ( x , y )] · u ( s , y ) dy ds 0 Ω � t + K ( x ) � | u ( s , y ) | 2 ∇ a ε ( x − y ) · d σ ( y ) ds . 2 0 Γ June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 20 / 26

Uniqueness Uniqueness How about uniqueness? June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 21 / 26

Uniqueness Uniqueness How about uniqueness? Serfati’s strategy: assume two solutions u 1 , u 2 , same initial data. Let X 1 and X 2 be respective flow maps. Show X 1 = X 2 . This implies u 1 = u 2 . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 21 / 26

Uniqueness Uniqueness How about uniqueness? Serfati’s strategy: assume two solutions u 1 , u 2 , same initial data. Let X 1 and X 2 be respective flow maps. Show X 1 = X 2 . This implies u 1 = u 2 . Start by estimating, using the Serfati identity, � � dX 1 dt − dX 2 � � � = | u 1 ( t , X 1 ) − u 2 ( t , X 2 ) | . � � dt � June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 21 / 26

Uniqueness Uniqueness How about uniqueness? Serfati’s strategy: assume two solutions u 1 , u 2 , same initial data. Let X 1 and X 2 be respective flow maps. Show X 1 = X 2 . This implies u 1 = u 2 . Start by estimating, using the Serfati identity, � � dX 1 dt − dX 2 � � � = | u 1 ( t , X 1 ) − u 2 ( t , X 2 ) | . � � dt � Use modulus of continuity (log-Lipschitz) of u . Denote µ = µ ( h ) ∼ | h || log h | . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 21 / 26

Uniqueness Uniqueness How about uniqueness? Serfati’s strategy: assume two solutions u 1 , u 2 , same initial data. Let X 1 and X 2 be respective flow maps. Show X 1 = X 2 . This implies u 1 = u 2 . Start by estimating, using the Serfati identity, � � dX 1 dt − dX 2 � � � = | u 1 ( t , X 1 ) − u 2 ( t , X 2 ) | . � � dt � Use modulus of continuity (log-Lipschitz) of u . Denote µ = µ ( h ) ∼ | h || log h | . Write h = h ( t ) = � X 1 ( t , · ) − X 2 ( t , · ) � L ∞ . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 21 / 26

Uniqueness Uniqueness How about uniqueness? Serfati’s strategy: assume two solutions u 1 , u 2 , same initial data. Let X 1 and X 2 be respective flow maps. Show X 1 = X 2 . This implies u 1 = u 2 . Start by estimating, using the Serfati identity, � � dX 1 dt − dX 2 � � � = | u 1 ( t , X 1 ) − u 2 ( t , X 2 ) | . � � dt � Use modulus of continuity (log-Lipschitz) of u . Denote µ = µ ( h ) ∼ | h || log h | . Write h = h ( t ) = � X 1 ( t , · ) − X 2 ( t , · ) � L ∞ . Write � t J ( t ) = � u 1 ( t , X 1 ( t , · )) − u 2 ( t , X 2 ( t , · )) � L ∞ , and M ( t ) = 0 J ( s ) ds . After many estimates arrive at June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 21 / 26

Uniqueness Uniqueness How about uniqueness? Serfati’s strategy: assume two solutions u 1 , u 2 , same initial data. Let X 1 and X 2 be respective flow maps. Show X 1 = X 2 . This implies u 1 = u 2 . Start by estimating, using the Serfati identity, � � dX 1 dt − dX 2 � � � = | u 1 ( t , X 1 ) − u 2 ( t , X 2 ) | . � � dt � Use modulus of continuity (log-Lipschitz) of u . Denote µ = µ ( h ) ∼ | h || log h | . Write h = h ( t ) = � X 1 ( t , · ) − X 2 ( t , · ) � L ∞ . Write � t J ( t ) = � u 1 ( t , X 1 ( t , · )) − u 2 ( t , X 2 ( t , · )) � L ∞ , and M ( t ) = 0 J ( s ) ds . After many estimates arrive at M ′ ( t ) ≤ C ( 1 + t + M ( t )) µ ( M ( t )) + CM ( t ) . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 21 / 26

Uniqueness Uniqueness How about uniqueness? Serfati’s strategy: assume two solutions u 1 , u 2 , same initial data. Let X 1 and X 2 be respective flow maps. Show X 1 = X 2 . This implies u 1 = u 2 . Start by estimating, using the Serfati identity, � � dX 1 dt − dX 2 � � � = | u 1 ( t , X 1 ) − u 2 ( t , X 2 ) | . � � dt � Use modulus of continuity (log-Lipschitz) of u . Denote µ = µ ( h ) ∼ | h || log h | . Write h = h ( t ) = � X 1 ( t , · ) − X 2 ( t , · ) � L ∞ . Write � t J ( t ) = � u 1 ( t , X 1 ( t , · )) − u 2 ( t , X 2 ( t , · )) � L ∞ , and M ( t ) = 0 J ( s ) ds . After many estimates arrive at M ′ ( t ) ≤ C ( 1 + t + M ( t )) µ ( M ( t )) + CM ( t ) . = ⇒ M ( t ) ≡ 0 by Osgood’s lemma June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 21 / 26

Uniqueness Uniqueness How about uniqueness? Serfati’s strategy: assume two solutions u 1 , u 2 , same initial data. Let X 1 and X 2 be respective flow maps. Show X 1 = X 2 . This implies u 1 = u 2 . Start by estimating, using the Serfati identity, � � dX 1 dt − dX 2 � � � = | u 1 ( t , X 1 ) − u 2 ( t , X 2 ) | . � � dt � Use modulus of continuity (log-Lipschitz) of u . Denote µ = µ ( h ) ∼ | h || log h | . Write h = h ( t ) = � X 1 ( t , · ) − X 2 ( t , · ) � L ∞ . Write � t J ( t ) = � u 1 ( t , X 1 ( t , · )) − u 2 ( t , X 2 ( t , · )) � L ∞ , and M ( t ) = 0 J ( s ) ds . After many estimates arrive at M ′ ( t ) ≤ C ( 1 + t + M ( t )) µ ( M ( t )) + CM ( t ) . = ⇒ M ( t ) ≡ 0 by Osgood’s lemma = ⇒ J ≡ 0 June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 21 / 26

Uniqueness Uniqueness How about uniqueness? Serfati’s strategy: assume two solutions u 1 , u 2 , same initial data. Let X 1 and X 2 be respective flow maps. Show X 1 = X 2 . This implies u 1 = u 2 . Start by estimating, using the Serfati identity, � � dX 1 dt − dX 2 � � � = | u 1 ( t , X 1 ) − u 2 ( t , X 2 ) | . � � dt � Use modulus of continuity (log-Lipschitz) of u . Denote µ = µ ( h ) ∼ | h || log h | . Write h = h ( t ) = � X 1 ( t , · ) − X 2 ( t , · ) � L ∞ . Write � t J ( t ) = � u 1 ( t , X 1 ( t , · )) − u 2 ( t , X 2 ( t , · )) � L ∞ , and M ( t ) = 0 J ( s ) ds . After many estimates arrive at M ′ ( t ) ≤ C ( 1 + t + M ( t )) µ ( M ( t )) + CM ( t ) . = ⇒ M ( t ) ≡ 0 by Osgood’s lemma = ⇒ J ≡ 0 = ⇒ X 1 ≡ X 2 . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 21 / 26

Weak solution Weak solution Introduce the Serfati space S , of divergence-free vector fields tangent to the boundary with the norm � u � S = � u � L ∞ + � curl u � L ∞ . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 22 / 26

Weak solution Weak solution Introduce the Serfati space S , of divergence-free vector fields tangent to the boundary with the norm � u � S = � u � L ∞ + � curl u � L ∞ . Definition We say u ∈ L ∞ loc ( R + ; S ) , is a weak solution if June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 22 / 26

Weak solution Weak solution Introduce the Serfati space S , of divergence-free vector fields tangent to the boundary with the norm � u � S = � u � L ∞ + � curl u � L ∞ . Definition We say u ∈ L ∞ loc ( R + ; S ) , is a weak solution if the incompressible 2D Euler equations hold, in distributions, against 1 div-free test functions; June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 22 / 26

Weak solution Weak solution Introduce the Serfati space S , of divergence-free vector fields tangent to the boundary with the norm � u � S = � u � L ∞ + � curl u � L ∞ . Definition We say u ∈ L ∞ loc ( R + ; S ) , is a weak solution if the incompressible 2D Euler equations hold, in distributions, against 1 div-free test functions; the Serfati identity holds for at least one cutoff function, a ; 2 June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 22 / 26

Weak solution Weak solution Introduce the Serfati space S , of divergence-free vector fields tangent to the boundary with the norm � u � S = � u � L ∞ + � curl u � L ∞ . Definition We say u ∈ L ∞ loc ( R + ; S ) , is a weak solution if the incompressible 2D Euler equations hold, in distributions, against 1 div-free test functions; the Serfati identity holds for at least one cutoff function, a ; 2 vorticity is transported by the flow; 3 June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 22 / 26

Weak solution Weak solution Introduce the Serfati space S , of divergence-free vector fields tangent to the boundary with the norm � u � S = � u � L ∞ + � curl u � L ∞ . Definition We say u ∈ L ∞ loc ( R + ; S ) , is a weak solution if the incompressible 2D Euler equations hold, in distributions, against 1 div-free test functions; the Serfati identity holds for at least one cutoff function, a ; 2 vorticity is transported by the flow; 3 Velocity has a spatial log-Lipschitz MOC uniformly over finite time; 4 June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 22 / 26

Weak solution Weak solution Introduce the Serfati space S , of divergence-free vector fields tangent to the boundary with the norm � u � S = � u � L ∞ + � curl u � L ∞ . Definition We say u ∈ L ∞ loc ( R + ; S ) , is a weak solution if the incompressible 2D Euler equations hold, in distributions, against 1 div-free test functions; the Serfati identity holds for at least one cutoff function, a ; 2 vorticity is transported by the flow; 3 Velocity has a spatial log-Lipschitz MOC uniformly over finite time; 4 Theorem Let Ω be a smooth domain exterior to a connected and bounded set. Let u 0 ∈ S. Then there exists one and at most one weak solution of Euler in Ω with initial velocity u 0 . June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 22 / 26

Continuous dependence Continuous dependence What about “Continuous dependence on initial data”? June 25 th , 2012 H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler 23 / 26