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Bounded vorticity, bounded velocity (Serfati) solutions to the incompressible 2D Euler equations

Helena J. Nussenzveig Lopes IM-Federal University of Rio de Janeiro (UFRJ), BRAZIL HYP 2012, Padova, June 25–29, 2012

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 1 / 26

Collaborators:

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

Collaborators: David Ambrose (Drexel Univ)

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

Collaborators: David Ambrose (Drexel Univ) Jim Kelliher (UC Riverside)

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

Collaborators: David Ambrose (Drexel Univ) Jim Kelliher (UC Riverside) Milton C. Lopes Filho (IM-UFRJ, Rio de Janeiro, Brazil)

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

Introduction

Introduction

Equations for incompressible, non-viscous (ideal) fluid flow in 2D are given by:    ∂tu + (u · ∇)u = −∇p div u = 0.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 3 / 26

Introduction

Introduction

Equations for incompressible, non-viscous (ideal) fluid flow in 2D are given by:    ∂tu + (u · ∇)u = −∇p div u = 0. Above, u = (u1, u2) is a vector field in the plane and p is the scalar pressure. Also, [(u · ∇)u]j =

• i

ui∂xiuj.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 3 / 26

Introduction

Introduction

Equations for incompressible, non-viscous (ideal) fluid flow in 2D are given by:    ∂tu + (u · ∇)u = −∇p div u = 0. Above, u = (u1, u2) is a vector field in the plane and p is the scalar pressure. Also, [(u · ∇)u]j =

• i

ui∂xiuj. These are the Euler equations – they should be supplemented by initial data – u(t = 0) = u0 and – if there is a boundary – by slip boundary conditions → u · ˆ n = 0 on the finite boundary, together with conditions at infinity.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 3 / 26

Introduction

Vorticity is defined by ω ≡ ∂x1u2 − ∂x2u1 ≡ ∇⊥ · u, where ∇⊥ = (−∂x2, ∂x1). It is the only component of curl (u1, u2, 0) ≡ (0, 0, ω).

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 4 / 26

Introduction

Vorticity is defined by ω ≡ ∂x1u2 − ∂x2u1 ≡ ∇⊥ · u, where ∇⊥ = (−∂x2, ∂x1). It is the only component of curl (u1, u2, 0) ≡ (0, 0, ω). Vorticity equation in 2D:

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 4 / 26

Introduction

Vorticity is defined by ω ≡ ∂x1u2 − ∂x2u1 ≡ ∇⊥ · u, where ∇⊥ = (−∂x2, ∂x1). It is the only component of curl (u1, u2, 0) ≡ (0, 0, ω). Vorticity equation in 2D:            ∂tω + u · ∇ω = 0 div u = 0 curl u = ω.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 4 / 26

Introduction

Vorticity is defined by ω ≡ ∂x1u2 − ∂x2u1 ≡ ∇⊥ · u, where ∇⊥ = (−∂x2, ∂x1). It is the only component of curl (u1, u2, 0) ≡ (0, 0, ω). Vorticity equation in 2D:            ∂tω + u · ∇ω = 0 div u = 0 curl u = ω. Above, curl u stands for ∇⊥ · u.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 4 / 26

Introduction

The system    div u = 0 +bdry conditions curl u = ω

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 5 / 26

Introduction

The system    div u = 0 +bdry conditions curl u = ω is elliptic and can, in principle, be explicitly inverted, writing u ≡ K[ω] = ∇⊥(∆)−1ω. Here, K depends on the fluid domain and is given by a Biot-Savart kernel; K[ω] is the Biot-Savart law, integration against K = K(x, y) = ∇⊥

x G(x, y).

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 5 / 26

Introduction

The system    div u = 0 +bdry conditions curl u = ω is elliptic and can, in principle, be explicitly inverted, writing u ≡ K[ω] = ∇⊥(∆)−1ω. Here, K depends on the fluid domain and is given by a Biot-Savart kernel; K[ω] is the Biot-Savart law, integration against K = K(x, y) = ∇⊥

x G(x, y).

If fluid domain is R2 then K[ω] = K ∗ ω with K = K(z) = z⊥ 2π|z|2 .

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 5 / 26

Introduction

The system    div u = 0 +bdry conditions curl u = ω is elliptic and can, in principle, be explicitly inverted, writing u ≡ K[ω] = ∇⊥(∆)−1ω. Here, K depends on the fluid domain and is given by a Biot-Savart kernel; K[ω] is the Biot-Savart law, integration against K = K(x, y) = ∇⊥

x G(x, y).

If fluid domain is R2 then K[ω] = K ∗ ω with K = K(z) = z⊥ 2π|z|2 . Now, K ∈ Lp

loc(R2), 1 ≤ p < 2 and K is q-th power integrable at infinity, with

q > 2. Hence, to calculate K ∗ ω need ω ∈ Lp′ ∩ Lq′ with p′ > 2 and q′ < 2, e.g. ω ∈ L∞ ∩ L1.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 5 / 26

Known well-posedness results

Known well-posedness results

Fluid domains of interest in this talk: R2 and exterior domains.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 6 / 26

Known well-posedness results

Known well-posedness results

Fluid domains of interest in this talk: R2 and exterior domains. If the domain is R2: u0 smooth (Hs(R2), s > 2) then well-posedness (McGrath 1968) → ∃, !, continuous dependence.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 6 / 26

Known well-posedness results

Known well-posedness results

Fluid domains of interest in this talk: R2 and exterior domains. If the domain is R2: u0 smooth (Hs(R2), s > 2) then well-posedness (McGrath 1968) → ∃, !, continuous dependence. ω0 ∈ L∞ ∩ L1 then ∃ and ! (Yudovich 1963, Majda 1986).

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 6 / 26

Known well-posedness results

Known well-posedness results

Fluid domains of interest in this talk: R2 and exterior domains. If the domain is R2: u0 smooth (Hs(R2), s > 2) then well-posedness (McGrath 1968) → ∃, !, continuous dependence. ω0 ∈ L∞ ∩ L1 then ∃ and ! (Yudovich 1963, Majda 1986). ω0 ∈ Lp

c, p > 1, u0 ∈ L2 loc then ∃ (DiPerna and Majda 1987).

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 6 / 26

Known well-posedness results

Known well-posedness results

Fluid domains of interest in this talk: R2 and exterior domains. If the domain is R2: u0 smooth (Hs(R2), s > 2) then well-posedness (McGrath 1968) → ∃, !, continuous dependence. ω0 ∈ L∞ ∩ L1 then ∃ and ! (Yudovich 1963, Majda 1986). ω0 ∈ Lp

c, p > 1, u0 ∈ L2 loc then ∃ (DiPerna and Majda 1987).

ω0 ∈ L1

c,

u0 ∈ L2

loc then ∃ (Vecchi and Wu 1993).

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 6 / 26

Known well-posedness results

Known well-posedness results

Fluid domains of interest in this talk: R2 and exterior domains. If the domain is R2: u0 smooth (Hs(R2), s > 2) then well-posedness (McGrath 1968) → ∃, !, continuous dependence. ω0 ∈ L∞ ∩ L1 then ∃ and ! (Yudovich 1963, Majda 1986). ω0 ∈ Lp

c, p > 1, u0 ∈ L2 loc then ∃ (DiPerna and Majda 1987).

ω0 ∈ L1

c,

u0 ∈ L2

loc then ∃ (Vecchi and Wu 1993).

ω0 ∈ BM+,c ∩ H−1 then ∃ (Delort 1991).

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 6 / 26

Known well-posedness results

If ω0 does not decay at infinity then get existence and uniqueness in periodic setting, i.e., if ω0 is doubly periodic in the plane.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 7 / 26

Known well-posedness results

If ω0 does not decay at infinity then get existence and uniqueness in periodic setting, i.e., if ω0 is doubly periodic in the plane. If the domain is an exterior domain: u0 smooth (C1

b) and the vorticity has some decay (ω0 ∈ L1, |x|rω0 ∈ L1,

some r > 0) and regularity (∇ω0 ∈ L∞ ∩ L1) then well-posedness (Kikuchi 1981).

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 7 / 26

Known well-posedness results

If ω0 does not decay at infinity then get existence and uniqueness in periodic setting, i.e., if ω0 is doubly periodic in the plane. If the domain is an exterior domain: u0 smooth (C1

b) and the vorticity has some decay (ω0 ∈ L1, |x|rω0 ∈ L1,

some r > 0) and regularity (∇ω0 ∈ L∞ ∩ L1) then well-posedness (Kikuchi 1981). In 1995 Ph. Serfati published announcement in CRAS of ∃ and ! for 2D Euler in R2 for u0 ∈ L∞ such that ω0 = curl u0 ∈ L∞.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 7 / 26

Known well-posedness results

If ω0 does not decay at infinity then get existence and uniqueness in periodic setting, i.e., if ω0 is doubly periodic in the plane. If the domain is an exterior domain: u0 smooth (C1

b) and the vorticity has some decay (ω0 ∈ L1, |x|rω0 ∈ L1,

some r > 0) and regularity (∇ω0 ∈ L∞ ∩ L1) then well-posedness (Kikuchi 1981). In 1995 Ph. Serfati published announcement in CRAS of ∃ and ! for 2D Euler in R2 for u0 ∈ L∞ such that ω0 = curl u0 ∈ L∞. Proof was terse and incomplete,

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 7 / 26

Known well-posedness results

If ω0 does not decay at infinity then get existence and uniqueness in periodic setting, i.e., if ω0 is doubly periodic in the plane. If the domain is an exterior domain: u0 smooth (C1

b) and the vorticity has some decay (ω0 ∈ L1, |x|rω0 ∈ L1,

some r > 0) and regularity (∇ω0 ∈ L∞ ∩ L1) then well-posedness (Kikuchi 1981). In 1995 Ph. Serfati published announcement in CRAS of ∃ and ! for 2D Euler in R2 for u0 ∈ L∞ such that ω0 = curl u0 ∈ L∞. Proof was terse and incomplete, yet brilliant.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 7 / 26

Known well-posedness results

If ω0 does not decay at infinity then get existence and uniqueness in periodic setting, i.e., if ω0 is doubly periodic in the plane. If the domain is an exterior domain: u0 smooth (C1

b) and the vorticity has some decay (ω0 ∈ L1, |x|rω0 ∈ L1,

some r > 0) and regularity (∇ω0 ∈ L∞ ∩ L1) then well-posedness (Kikuchi 1981). In 1995 Ph. Serfati published announcement in CRAS of ∃ and ! for 2D Euler in R2 for u0 ∈ L∞ such that ω0 = curl u0 ∈ L∞. Proof was terse and incomplete, yet brilliant. This talk: discussion of Serfati’s work, extension to continuous dependence

• n initial data and to flow domains exterior to a connected obstacle.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 7 / 26

Known well-posedness results

If ω0 does not decay at infinity then get existence and uniqueness in periodic setting, i.e., if ω0 is doubly periodic in the plane. If the domain is an exterior domain: u0 smooth (C1

b) and the vorticity has some decay (ω0 ∈ L1, |x|rω0 ∈ L1,

some r > 0) and regularity (∇ω0 ∈ L∞ ∩ L1) then well-posedness (Kikuchi 1981). In 1995 Ph. Serfati published announcement in CRAS of ∃ and ! for 2D Euler in R2 for u0 ∈ L∞ such that ω0 = curl u0 ∈ L∞. Proof was terse and incomplete, yet brilliant. This talk: discussion of Serfati’s work, extension to continuous dependence

• n initial data and to flow domains exterior to a connected obstacle.

This is a report of work in progress.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 7 / 26

Statement of problem

Statement of problem

Let D ⊂ R2 be connected, open, bounded, smooth domain; Ω = R2 \ D.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 8 / 26

Statement of problem

Statement of problem

Let D ⊂ R2 be connected, open, bounded, smooth domain; Ω = R2 \ D. Let u0 ∈ L∞(Ω; R2) be vector field such that ω0 = curl u0 ≡ ∇⊥ · u0 ∈ L∞. Suppose also div u0 = 0 and u0 · ˆ n = 0 on ∂D.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 8 / 26

Statement of problem

Statement of problem

Let D ⊂ R2 be connected, open, bounded, smooth domain; Ω = R2 \ D. Let u0 ∈ L∞(Ω; R2) be vector field such that ω0 = curl u0 ≡ ∇⊥ · u0 ∈ L∞. Suppose also div u0 = 0 and u0 · ˆ n = 0 on ∂D. Problem: existence, uniqueness and continuous dependence of solution to incompressible 2D Euler with initial velocity u0.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 8 / 26

Statement of problem

Statement of problem

Let D ⊂ R2 be connected, open, bounded, smooth domain; Ω = R2 \ D. Let u0 ∈ L∞(Ω; R2) be vector field such that ω0 = curl u0 ≡ ∇⊥ · u0 ∈ L∞. Suppose also div u0 = 0 and u0 · ˆ n = 0 on ∂D. Problem: existence, uniqueness and continuous dependence of solution to incompressible 2D Euler with initial velocity u0. Fundamental questions: what is a ‘solution’? What happens to the Biot-Savart law? Where do we get ‘uniqueness’? What perturbations are allowed?

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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 u0 ∈ L∞ and ω0 ∈ Yθ

ul, for θ = log(e + q); Yθ ul means “uniformly local" Lp-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.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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 u0 ∈ L∞ and ω0 ∈ Yθ

ul, for θ = log(e + q); Yθ ul means “uniformly local" Lp-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 B0

∞,1.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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).

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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 ω0 ∈ L∞ and
• 1

|x − y||ω0(y)| dy < ∞ for some x ∈ R2 and gets ∃ and ! of (u, ω, Φt) such that |u| grows at most like

• |x| at infinity. (Φt is the Lagrangian map.) The hypothesis

excludes periodic flows.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 10 / 26

Motivation

Motivation

Why study vorticity with no decay?

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 11 / 26

Motivation

Motivation

Why study vorticity with no decay? Main uniqueness result is for vorticity in L∞ ∩ L1 (Yudovich 1963), but L1 hypothesis relevance is to make sense of Biot-Savart law. Uniqueness should be local issue – behavior of vorticity at infinity should not be important.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 11 / 26

Motivation

Motivation

Why study vorticity with no decay? Main uniqueness result is for vorticity in L∞ ∩ L1 (Yudovich 1963), but L1 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.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 11 / 26

Motivation

Motivation

Why study vorticity with no decay? Main uniqueness result is for vorticity in L∞ ∩ L1 (Yudovich 1963), but L1 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?

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 11 / 26

Motivation

Motivation

Why study vorticity with no decay? Main uniqueness result is for vorticity in L∞ ∩ L1 (Yudovich 1963), but L1 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;

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 11 / 26

Motivation

Motivation

Why study vorticity with no decay? Main uniqueness result is for vorticity in L∞ ∩ L1 (Yudovich 1963), but L1 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;

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 11 / 26

Motivation

Motivation

Why study vorticity with no decay? Main uniqueness result is for vorticity in L∞ ∩ L1 (Yudovich 1963), but L1 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).

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 11 / 26

Serfati’s representation formula

Serfati’s representation formula

Key new idea: consider a smooth cutoff aε of the origin (aε(z) = 1 if z ∈ R2 and |z| small, and vanishes if |z| large) and establish a formula like

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 12 / 26

Serfati’s representation formula

Serfati’s representation formula

Key new idea: consider a smooth cutoff aε of the origin (aε(z) = 1 if z ∈ R2 and |z| small, and vanishes if |z| large) and establish a formula like u(t, x) = u0(x) +

• aε(x − y)K(x − y)(ω(t, y) − ω0(y)) dy+

− t u(s, y) · ∇y∇y[(1 − aε(x − y))K(x − y)]⊥ · u(s, y) dyds.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 12 / 26

Serfati’s representation formula

Serfati’s representation formula

Key new idea: consider a smooth cutoff aε of the origin (aε(z) = 1 if z ∈ R2 and |z| small, and vanishes if |z| large) and establish a formula like u(t, x) = u0(x) +

• aε(x − y)K(x − y)(ω(t, y) − ω0(y)) dy+

− t u(s, y) · ∇y∇y[(1 − aε(x − y))K(x − y)]⊥ · u(s, y) dyds. Why?

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 12 / 26

Serfati’s representation formula

Serfati’s representation formula

Key new idea: consider a smooth cutoff aε of the origin (aε(z) = 1 if z ∈ R2 and |z| small, and vanishes if |z| large) and establish a formula like u(t, x) = u0(x) +

• aε(x − y)K(x − y)(ω(t, y) − ω0(y)) dy+

− t u(s, y) · ∇y∇y[(1 − aε(x − y))K(x − y)]⊥ · u(s, y) dyds. Why? Because all the terms in this formula converge.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 12 / 26

Serfati’s representation formula

Serfati’s representation formula

Key new idea: consider a smooth cutoff aε of the origin (aε(z) = 1 if z ∈ R2 and |z| small, and vanishes if |z| large) and establish a formula like u(t, x) = u0(x) +

• aε(x − y)K(x − y)(ω(t, y) − ω0(y)) dy+

− t u(s, y) · ∇y∇y[(1 − aε(x − y))K(x − y)]⊥ · u(s, y) dyds. Why? Because all the terms in this formula converge. Indeed: aεK ∈ L1 and ω ∈ L∞ so the first integral converges; ∇y∇y[(1 − aε)K] ∈ L1 because of the extra decay at infinity coming from taking two derivatives, hence, if u ∈ L∞, then the second integral also converges.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 12 / 26

Serfati’s representation formula

Where does this formula come from?

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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, ω0 compactly

• supported. Then

u = u(t, x) =

• K(x − y)ω(t, y) dy.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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, ω0 compactly

• supported. Then

u = u(t, x) =

• K(x − y)ω(t, y) dy.

Hence, ∂tu = ∂t

• K(x − y)ω(t, y) dy

= ∂t

• aε(x − y)K(x − y)ω(t, y) dy +
• (1 − aε(x − y))K(x − y)∂tω dy.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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, ω0 compactly

• supported. Then

u = u(t, x) =

• K(x − y)ω(t, y) dy.

Hence, ∂tu = ∂t

• K(x − y)ω(t, y) dy

= ∂t

• aε(x − y)K(x − y)ω(t, y) dy +
• (1 − aε(x − y))K(x − y)∂tω dy.

Therefore, integrating in time yields u(t, x) = u0(x) +

• aε(x − y)K(x − y)[ω(t, y) − ω0(y)] dy

+ t

• (1 − aε(x − y))K(x − y)∂tω dy.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 13 / 26

Serfati’s representation formula

Now,

• (1 − aε(x − y))K(x − y)∂tω dy = −
• (1 − aε(x − y))K(x − y)u · ∇yω dy

= −

• (1 − aε(x − y))K(x − y) curly(u · ∇yu) dy.

Therefore,

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 14 / 26

Serfati’s representation formula

Now,

• (1 − aε(x − y))K(x − y)∂tω dy = −
• (1 − aε(x − y))K(x − y)u · ∇yω dy

= −

• (1 − aε(x − y))K(x − y) curly(u · ∇yu) dy.

Therefore, integrating by parts carefully, using div u = 0, get

• (1 − aε(x − y))K(x − y)∂tω dy

= u(s, y) · ∇y∇y[(1 − aε(x − y))K(x − y)]⊥ · u(s, y) dy. Finally, substitute to obtain the desired formula – Serfati identity:

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 14 / 26

Serfati’s representation formula

Now,

• (1 − aε(x − y))K(x − y)∂tω dy = −
• (1 − aε(x − y))K(x − y)u · ∇yω dy

= −

• (1 − aε(x − y))K(x − y) curly(u · ∇yu) dy.

Therefore, integrating by parts carefully, using div u = 0, get

• (1 − aε(x − y))K(x − y)∂tω dy

= u(s, y) · ∇y∇y[(1 − aε(x − y))K(x − y)]⊥ · u(s, y) dy. Finally, substitute to obtain the desired formula – Serfati identity: u(t, x) = u0(x) +

• aε(x − y)K(x − y)[ω(t, y) − ω0(y)] dy+

− t u(s, y) · ∇y∇y[(1 − aε(x − y))K(x − y)]⊥ · u(s, y) dy.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 14 / 26

Serfati’s strategy for existence

Serfati’s strategy for existence

Why is this representation formula useful?

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 15 / 26

Serfati’s strategy for existence

Serfati’s strategy for existence

Why is this representation formula useful? A priori estimates in L∞.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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 u0 ∈ L∞ such that ω0 ∈ L∞.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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 u0 ∈ L∞ such that ω0 ∈ L∞.

• 1. Construct a sequence {u0,N} such that u0,N ∈ C∞

c , u0,N = K ∗ ω0,N,

u0,N → u0 uniformly on compact sets, ω0,N → ω0 in Lp on compact sets, for some C > 0, u0,NL∞ + ω0,NL∞ ≤ C < ∞ for all N.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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 u0 ∈ L∞ such that ω0 ∈ L∞.

• 1. Construct a sequence {u0,N} such that u0,N ∈ C∞

c , u0,N = K ∗ ω0,N,

u0,N → u0 uniformly on compact sets, ω0,N → ω0 in Lp on compact sets, for some C > 0, u0,NL∞ + ω0,NL∞ ≤ C < ∞ for all N.

• 2. Solve 2D Euler with initial velocity u0,N; solution denoted uN. Use

Serfati identity to obtain L∞ estimates for uN, uniform wrt N. Use transport to get uniform L∞ estimates for ωN = curl uN.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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 u0 ∈ L∞ such that ω0 ∈ L∞.

• 1. Construct a sequence {u0,N} such that u0,N ∈ C∞

c , u0,N = K ∗ ω0,N,

u0,N → u0 uniformly on compact sets, ω0,N → ω0 in Lp on compact sets, for some C > 0, u0,NL∞ + ω0,NL∞ ≤ C < ∞ for all N.

• 2. Solve 2D Euler with initial velocity u0,N; solution denoted uN. Use

Serfati identity to obtain L∞ estimates for uN, uniform wrt N. Use transport to get uniform L∞ estimates for ωN = curl uN.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 15 / 26

Serfati’s strategy for existence

• 3. Use L∞ estimate for uN and ωN to prove that uN is uniformly

log-Lipschitz. From this get, in standard way, passing to subsequences as needed, uniform convergence (on compacts) of particle trajectories XN = XN(t, α) (dXN/dt = uN(t, XN) and XN(0, α) = α) to X = X(t, α). Also, easy to show X is measure-preserving.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 16 / 26

Serfati’s strategy for existence

• 3. Use L∞ estimate for uN and ωN to prove that uN is uniformly

log-Lipschitz. From this get, in standard way, passing to subsequences as needed, uniform convergence (on compacts) of particle trajectories XN = XN(t, α) (dXN/dt = uN(t, XN) and XN(0, α) = α) to X = X(t, α). Also, easy to show X is measure-preserving.

• 4. Define limit vorticity as ω(t, x) ≡ ω0(X−1(t, x)). Show ωN → ω in

L∞

loc(dt, Lp loc(dx)), any p < ∞. Show uN → u uniformly in compacts in

space and time.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 16 / 26

Serfati’s strategy for existence

• 3. Use L∞ estimate for uN and ωN to prove that uN is uniformly

log-Lipschitz. From this get, in standard way, passing to subsequences as needed, uniform convergence (on compacts) of particle trajectories XN = XN(t, α) (dXN/dt = uN(t, XN) and XN(0, α) = α) to X = X(t, α). Also, easy to show X is measure-preserving.

• 4. Define limit vorticity as ω(t, x) ≡ ω0(X−1(t, x)). Show ωN → ω in

L∞

loc(dt, Lp loc(dx)), any p < ∞. Show uN → 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.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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:

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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∞ ≤ u0L∞ + sup

x

• |aε(x − y)K(x − y)||ω(t, y) − ω0(y)| dy

+ t sup

x

• u(s, y) · ∇y∇y[(1 − aε(x − y))K(x − y)]⊥

· u(s, y) dy

• ≤ u0L∞ + 2ω0L∞aεKL1 + D2

y[(1 − aε)K]L1

t u(s, ·)2

L∞ ds.

Now, in the full plane we have

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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∞ ≤ u0L∞ + sup

x

• |aε(x − y)K(x − y)||ω(t, y) − ω0(y)| dy

+ t sup

x

• u(s, y) · ∇y∇y[(1 − aε(x − y))K(x − y)]⊥

· u(s, y) dy

• ≤ u0L∞ + 2ω0L∞aεKL1 + D2

y[(1 − aε)K]L1

t u(s, ·)2

L∞ ds.

Now, in the full plane we have aεKL1 ∼ Cε,

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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∞ ≤ u0L∞ + sup

x

• |aε(x − y)K(x − y)||ω(t, y) − ω0(y)| dy

+ t sup

x

• u(s, y) · ∇y∇y[(1 − aε(x − y))K(x − y)]⊥

· u(s, y) dy

• ≤ u0L∞ + 2ω0L∞aεKL1 + D2

y[(1 − aε)K]L1

t u(s, ·)2

L∞ ds.

Now, in the full plane we have aεKL1 ∼ Cε, D2

y[(1 − aε)K]L1 ∼ C

ε .

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 17 / 26

Serfati’s strategy for existence

Hence, the a priori estimate becomes u(t, ·)L∞ ≤ u0L∞ + Cεω0L∞ + C ε t u(s, ·)2

L∞ ds.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 18 / 26

Serfati’s strategy for existence

Hence, the a priori estimate becomes u(t, ·)L∞ ≤ u0L∞ + Cεω0L∞ + C ε t u(s, ·)2

L∞ ds.

Choose ε = t

0 u(s, ·)2 L∞ ds

1/2. Then get, after use of Young’s inequality,

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 18 / 26

Serfati’s strategy for existence

Hence, the a priori estimate becomes u(t, ·)L∞ ≤ u0L∞ + Cεω0L∞ + C ε t u(s, ·)2

L∞ ds.

Choose ε = t

0 u(s, ·)2 L∞ ds

1/2. Then get, after use of Young’s inequality, u(t, ·)2

L∞ ≤ 2u02 L∞ + Cω0L∞

t u(s, ·)2

L∞ ds.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 18 / 26

Serfati’s strategy for existence

Hence, the a priori estimate becomes u(t, ·)L∞ ≤ u0L∞ + Cεω0L∞ + C ε t u(s, ·)2

L∞ ds.

Choose ε = t

0 u(s, ·)2 L∞ ds

1/2. Then get, after use of Young’s inequality, u(t, ·)2

L∞ ≤ 2u02 L∞ + Cω0L∞

t u(s, ·)2

L∞ ds.

The desired L∞-estimate for u follows, hence, by Gronwall’s lemma.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 18 / 26

Exterior domain

Exterior domain

What about exterior domains? Assume domain exterior to unit disk.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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(x − y) ← → KΩ(x, y) = K(x − y) − K(x − y∗), where y∗ = y/|y|2.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 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(x − y) ← → KΩ(x, y) = K(x − y) − K(x − y∗), where y∗ = y/|y|2. Second, (hindsight from previous exterior domain work) know KΩ(x, y) + K(x) ∼ K(x − y) (K(x) = x⊥/(2π|x|2)), not KΩ ∼ K(x − y).

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 19 / 26

Exterior domain

Denote J = J(x, y) = KΩ(x, y) + K(x).

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 20 / 26

Exterior domain

Denote J = J(x, y) = KΩ(x, y) + K(x). Downside: K(x, ·) vanishes at boundary, but J(x, ·) does not.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 20 / 26

Exterior domain

Denote J = J(x, y) = KΩ(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:

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 20 / 26

Exterior domain

Denote J = J(x, y) = KΩ(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) = u0(x) +

• Ω

aε(x − y)J(x, y)(ω(t, y) − ω0(y)) dy − t

• Ω

(u(s, y) · ∇y) ∇⊥

y [(1 − aε(x − y))J(x, y)] · u(s, y) dy ds

+ K(x) 2 t

• Γ

|u(s, y)|2 ∇aε(x − y) · dσ(y) ds.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 20 / 26

Uniqueness

Uniqueness

How about uniqueness?

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

Uniqueness

Uniqueness

How about uniqueness? Serfati’s strategy: assume two solutions u1, u2, same initial data. Let X1 and X2 be respective flow maps. Show X1 = X2. This implies u1 = u2.

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

Uniqueness

Uniqueness

How about uniqueness? Serfati’s strategy: assume two solutions u1, u2, same initial data. Let X1 and X2 be respective flow maps. Show X1 = X2. This implies u1 = u2. Start by estimating, using the Serfati identity,

• dX1

dt − dX2 dt

• = |u1(t, X1) − u2(t, X2)|.

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

Uniqueness

Uniqueness

How about uniqueness? Serfati’s strategy: assume two solutions u1, u2, same initial data. Let X1 and X2 be respective flow maps. Show X1 = X2. This implies u1 = u2. Start by estimating, using the Serfati identity,

• dX1

dt − dX2 dt

• = |u1(t, X1) − u2(t, X2)|.

Use modulus of continuity (log-Lipschitz) of u. Denote µ = µ(h) ∼ |h|| log h|.

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

Uniqueness

Uniqueness

How about uniqueness? Serfati’s strategy: assume two solutions u1, u2, same initial data. Let X1 and X2 be respective flow maps. Show X1 = X2. This implies u1 = u2. Start by estimating, using the Serfati identity,

• dX1

dt − dX2 dt

• = |u1(t, X1) − u2(t, X2)|.

Use modulus of continuity (log-Lipschitz) of u. Denote µ = µ(h) ∼ |h|| log h|. Write h = h(t) = X1(t, ·) − X2(t, ·)L∞.

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

Uniqueness

Uniqueness

How about uniqueness? Serfati’s strategy: assume two solutions u1, u2, same initial data. Let X1 and X2 be respective flow maps. Show X1 = X2. This implies u1 = u2. Start by estimating, using the Serfati identity,

• dX1

dt − dX2 dt

• = |u1(t, X1) − u2(t, X2)|.

Use modulus of continuity (log-Lipschitz) of u. Denote µ = µ(h) ∼ |h|| log h|. Write h = h(t) = X1(t, ·) − X2(t, ·)L∞. Write J(t) = u1(t, X1(t, ·)) − u2(t, X2(t, ·))L∞, and M(t) = t

0 J(s) ds. After many

estimates arrive at

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

Uniqueness

Uniqueness

How about uniqueness? Serfati’s strategy: assume two solutions u1, u2, same initial data. Let X1 and X2 be respective flow maps. Show X1 = X2. This implies u1 = u2. Start by estimating, using the Serfati identity,

• dX1

dt − dX2 dt

• = |u1(t, X1) − u2(t, X2)|.

Use modulus of continuity (log-Lipschitz) of u. Denote µ = µ(h) ∼ |h|| log h|. Write h = h(t) = X1(t, ·) − X2(t, ·)L∞. Write J(t) = u1(t, X1(t, ·)) − u2(t, X2(t, ·))L∞, and M(t) = t

0 J(s) ds. After many

estimates arrive at M′(t) ≤ C(1 + t + M(t))µ(M(t)) + CM(t).

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

Uniqueness

Uniqueness

How about uniqueness? Serfati’s strategy: assume two solutions u1, u2, same initial data. Let X1 and X2 be respective flow maps. Show X1 = X2. This implies u1 = u2. Start by estimating, using the Serfati identity,

• dX1

dt − dX2 dt

• = |u1(t, X1) − u2(t, X2)|.

Use modulus of continuity (log-Lipschitz) of u. Denote µ = µ(h) ∼ |h|| log h|. Write h = h(t) = X1(t, ·) − X2(t, ·)L∞. Write J(t) = u1(t, X1(t, ·)) − u2(t, X2(t, ·))L∞, and M(t) = 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

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

Uniqueness

Uniqueness

How about uniqueness? Serfati’s strategy: assume two solutions u1, u2, same initial data. Let X1 and X2 be respective flow maps. Show X1 = X2. This implies u1 = u2. Start by estimating, using the Serfati identity,

• dX1

dt − dX2 dt

• = |u1(t, X1) − u2(t, X2)|.

Use modulus of continuity (log-Lipschitz) of u. Denote µ = µ(h) ∼ |h|| log h|. Write h = h(t) = X1(t, ·) − X2(t, ·)L∞. Write J(t) = u1(t, X1(t, ·)) − u2(t, X2(t, ·))L∞, and M(t) = 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

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

Uniqueness

Uniqueness

How about uniqueness? Serfati’s strategy: assume two solutions u1, u2, same initial data. Let X1 and X2 be respective flow maps. Show X1 = X2. This implies u1 = u2. Start by estimating, using the Serfati identity,

• dX1

dt − dX2 dt

• = |u1(t, X1) − u2(t, X2)|.

Use modulus of continuity (log-Lipschitz) of u. Denote µ = µ(h) ∼ |h|| log h|. Write h = h(t) = X1(t, ·) − X2(t, ·)L∞. Write J(t) = u1(t, X1(t, ·)) − u2(t, X2(t, ·))L∞, and M(t) = 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 = ⇒ X1 ≡ X2.

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

Weak solution

Weak solution

Introduce the Serfati space S, of divergence-free vector fields tangent to the boundary with the norm uS = uL∞ + curl uL∞.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 22 / 26

Weak solution

Weak solution

Introduce the Serfati space S, of divergence-free vector fields tangent to the boundary with the norm uS = uL∞ + curl uL∞. Definition We say u ∈ L∞

loc(R+; S), is a weak solution if

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 22 / 26

Weak solution

Weak solution

Introduce the Serfati space S, of divergence-free vector fields tangent to the boundary with the norm uS = uL∞ + curl uL∞. Definition We say u ∈ L∞

loc(R+; S), is a weak solution if

1

the incompressible 2D Euler equations hold, in distributions, against div-free test functions;

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 22 / 26

Weak solution

Weak solution

Introduce the Serfati space S, of divergence-free vector fields tangent to the boundary with the norm uS = uL∞ + curl uL∞. Definition We say u ∈ L∞

loc(R+; S), is a weak solution if

1

the incompressible 2D Euler equations hold, in distributions, against div-free test functions;

2

the Serfati identity holds for at least one cutoff function, a;

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 22 / 26

Weak solution

Weak solution

Introduce the Serfati space S, of divergence-free vector fields tangent to the boundary with the norm uS = uL∞ + curl uL∞. Definition We say u ∈ L∞

loc(R+; S), is a weak solution if

1

the incompressible 2D Euler equations hold, in distributions, against div-free test functions;

2

the Serfati identity holds for at least one cutoff function, a;

3

vorticity is transported by the flow;

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 22 / 26

Weak solution

Weak solution

Introduce the Serfati space S, of divergence-free vector fields tangent to the boundary with the norm uS = uL∞ + curl uL∞. Definition We say u ∈ L∞

loc(R+; S), is a weak solution if

1

the incompressible 2D Euler equations hold, in distributions, against div-free test functions;

2

the Serfati identity holds for at least one cutoff function, a;

3

vorticity is transported by the flow;

4

Velocity has a spatial log-Lipschitz MOC uniformly over finite time;

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 22 / 26

Weak solution

Weak solution

Introduce the Serfati space S, of divergence-free vector fields tangent to the boundary with the norm uS = uL∞ + curl uL∞. Definition We say u ∈ L∞

loc(R+; S), is a weak solution if

1

the incompressible 2D Euler equations hold, in distributions, against div-free test functions;

2

the Serfati identity holds for at least one cutoff function, a;

3

vorticity is transported by the flow;

4

Velocity has a spatial log-Lipschitz MOC uniformly over finite time; Theorem Let Ω be a smooth domain exterior to a connected and bounded set. Let u0 ∈ S. Then there exists one and at most one weak solution of Euler in Ω with initial velocity u0.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 22 / 26

Continuous dependence

Continuous dependence

What about “Continuous dependence on initial data”?

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 23 / 26

Continuous dependence

Continuous dependence

What about “Continuous dependence on initial data”? Theorem Suppose initial velocities, u0

1 and u0 2, with vorticities, ω0 1 and ω0 2, such that

u0

1 − u0 2 lies in

Sp := {u ∈ (L∞(Ω))2 : div u = 0, u · n = 0, ω ∈ Lp(Ω)} for some p in (2, ∞], with · Sp = · L∞ + ω(·)Lp. Then, for all sufficiently small s0 = u0

1 − u0 2Sp,

u1(t) − u2(t)L∞ ≤ s0eCt + Ct(s0t)e−Ct(2+t)[log Ct + s0te−Ct(2+t)] [C(2 + t)eCt + 1], where C and Ct depend on the initial data and on p, with Ct a continuous function of time.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 23 / 26

Concluding remarks

Concluding remarks

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 24 / 26

Concluding remarks

Concluding remarks

• 1. To extend results to more general exterior domains (exterior of bounded,

smooth, connected domain in the plane) bring in conformal maps. Works beautifully.

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 24 / 26

Concluding remarks

Concluding remarks

• 1. To extend results to more general exterior domains (exterior of bounded,

smooth, connected domain in the plane) bring in conformal maps. Works beautifully.

• 2. Should try to improve continuous dependence result so as to have same

norm comparison, not velocity stable in L∞ in terms of initial perturbation in S. Recall Taniuchi et alli, continuous dependence in B0

∞,1. But only for full plane and only by Littlewood-Paley...

H.J. Nussenzveig Lopes (IM-UFRJ) Serfati solutions to incompressible 2D Euler June 25th, 2012 24 / 26

Concluding remarks

• 3. One natural question:

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

Concluding remarks

• 3. One natural question: can we characterize those vorticities which “are

Serfati"? I.e., bounded vorticities which are the curl of a bounded velocity? Note that, if ω0 ≡ 1 then u0 is not bounded, hence it is not Serfati.

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

Concluding remarks

• 3. One natural question: can we characterize those vorticities which “are

Serfati"? I.e., bounded vorticities which are the curl of a bounded velocity? Note that, if ω0 ≡ 1 then u0 is not bounded, hence it is not Serfati.

• 4. Some examples of Serfati flows:

(i) All smooth (doubly) periodic flows are Serfati.

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

Concluding remarks

• 3. One natural question: can we characterize those vorticities which “are

Serfati"? I.e., bounded vorticities which are the curl of a bounded velocity? Note that, if ω0 ≡ 1 then u0 is not bounded, hence it is not Serfati.

• 4. Some examples of Serfati flows:

(i) All smooth (doubly) periodic flows are Serfati. Adding a compactly supported function to the vorticity does not change this.

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

Concluding remarks

• 3. One natural question: can we characterize those vorticities which “are

Serfati"? I.e., bounded vorticities which are the curl of a bounded velocity? Note that, if ω0 ≡ 1 then u0 is not bounded, hence it is not Serfati.

• 4. Some examples of Serfati flows:

(i) All smooth (doubly) periodic flows are Serfati. Adding a compactly supported function to the vorticity does not change this. (ii) u(x) = x⊥/|x| has vorticity 1/|x|, hence Serfati. Note that Biot-Savart law cannot be used.

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

Concluding remarks

• 3. One natural question: can we characterize those vorticities which “are

Serfati"? I.e., bounded vorticities which are the curl of a bounded velocity? Note that, if ω0 ≡ 1 then u0 is not bounded, hence it is not Serfati.

• 4. Some examples of Serfati flows:

(i) All smooth (doubly) periodic flows are Serfati. Adding a compactly supported function to the vorticity does not change this. (ii) u(x) = x⊥/|x| has vorticity 1/|x|, hence Serfati. Note that Biot-Savart law cannot be used. (iii) Serfati vorticity with no decay: vorticity is characteristic function of a strip lying outside/”above" obstacle, velocity interpolates linearly from 0 ”below" strip to constant parallel to and ”above" strip. For example, if the strip is {2 < x2 < 3} then

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

Concluding remarks

• 3. One natural question: can we characterize those vorticities which “are

Serfati"? I.e., bounded vorticities which are the curl of a bounded velocity? Note that, if ω0 ≡ 1 then u0 is not bounded, hence it is not Serfati.

• 4. Some examples of Serfati flows:

(i) All smooth (doubly) periodic flows are Serfati. Adding a compactly supported function to the vorticity does not change this. (ii) u(x) = x⊥/|x| has vorticity 1/|x|, hence Serfati. Note that Biot-Savart law cannot be used. (iii) Serfati vorticity with no decay: vorticity is characteristic function of a strip lying outside/”above" obstacle, velocity interpolates linearly from 0 ”below" strip to constant parallel to and ”above" strip. For example, if the strip is {2 < x2 < 3} then u = u(x) =    (1, 0) if x2 > 3, (x2 − 2, 0) if 2 < x2 < 3, (0, 0) if x2 < 2, is Serfati in the exterior of the unit disk.

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

Concluding remarks

Thank you!

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