The regularity of the boundary of vortex patches revisited

The regularity of the boundary of vortex patches revisited

Joan Verdera

Abstract

We provide a new argument to show the persistence of boundary smoothness of vortex patches for the vorticity form of Euler’s equation, quite in the spirit of the well-known Bertozzi-Constantin approach. Our argument avoids the use of defining functions and, surprisingly, yields persistence of boundary smoothness of vortex patches for transport equations in the plane given by velocity fields which are convolution of the density with an odd kernel, homogeneous of order −1 and of class C² off the origin. This allows the velocity field to have non-trivial divergence.

AMS 2020 Mathematics Subject Classification: 35Q31,35Q35 (primary);

35Q49, 42B20 (secondary).

Keywords: transport equation, vortex patch, Calder´on-Zygmund singular inte- grals.

1 Introduction

The vorticity form of the Euler equation is

(1)

∂_tω + v · ∇ω = 0,

v(·, t) = ω(·, t) ? ∇^⊥N, ω(z, 0) = ω₀(z),

where ω(z, t) is vorticity at the point z at time t, z = x + iy ∈ C = R², t ∈ R, and N (z) = (1/2π) log |z| is the fundamental solution of the laplacian in the plane. Then

∇^⊥N (z) = (i/2π)z/|z|². It is a deep result of Yudovich [Y] that (1) is well posed in L^∞(R²) ∩ L¹(R²), in the sense that there exists a unique weak solution of (1) for each given initial condition in L^∞(R²) ∩ L¹(R²).

When one takes as initial condition the characteristic function ρ₀ = χ_D0 of a bounded domain D₀, then ω(·, t) = χ_Dt(·), where D_t is a bounded domain. This follows from the

fact that (1) is a transport equation and one refers to such a solution as a vortex patch.

In the eighties the problem of deciding if boundary smoothness persisted for all times was a challenging question. Chemin proved the following [Ch].

Theorem (Chemin, 1993). If D₀ is a bounded simply connected domain with boundary of class C^1+γ, 0 < γ < 1, then the weak solution of (1) with initial condition χ_D0 is of the form ω(z, t) = X_Dt(z) with D_t a simply connected domain of class C^1+γ for all times t ∈ R.

Indeed in [Ch] one proves a more general statement and the proof depends on para- differential calculus. In [BC] a shorter proof, based on classical analysis methods, was devised.

The proof in [BC] takes for granted that a local in time solution exists (see Chapter 8 of [MB]) and then continues by proving an a priori inequality for the relevant quantities giving the smoothness of D_t. These quantities are defined in terms of the defining function for D_t obtained by transporting a given defining function of D₀. The a priori bounds are obtained by resorting to an appropriate commutator, which allows to perform a control in terms of k∇v(·, t)k∞. The second element in the proof, as in [Ch], is a logarithmic estimate for k∇v(·, t)k∞, which follows from a simple property of convolution even homogeneous smooth singular integrals of Calder´on -Zygmund type.

Inspired by conversations with J.Garnett we avoid the use of defining functions and prove directly an a priori inequality for the H¨older seminorm of order γ of the gradient of the local in time solution X(·, t) of the contour dynamics equation (see (16)). This follows by bringing in a commutator, which appears after an application of Whitney’s extension theorem. The proof follows by proving a logarithmic inequality which estimates k∇v(·, t)k∞in terms of k∇X(·, t)kγ,∂D0 and the Lipschitz constant of the inverse of X(·, t).

Our proof is not shorter, nor essentially different from that in [BC] in its basic elements, but it is more direct in that avoids defining functions. It has a surprising consequence: it works not only for the vorticity equation, but for all transport equations of the form

(2)

∂_tω + v · ∇ω = 0,

v(·, t) = ω(·, t) ? k, ω(z, 0) = χ_D0(z),

where k is an odd kernel, homogeneous of degree −1, of class C² off the origin, and D₀ is a simply connected domain with boundary of class C^1+γ. These kernels produce velocity fields v with non-zero divergence and thus the incompressibility assumption in Chemin’s theorem plays no role. For these more general equations a Yudovich theory is not yet available and so uniqueness holds up to now only in the class of patches with smooth boundary.

A formal statement of our main result is as follows.

Theorem. If D₀ is a bounded simply connected domain with boundary of class C^1+γ, 0 < γ < 1, then there exists a weak solution of (2) with initial condition χ_D0 of the form ω(z, t) = X_Dt(z) with D_ta simply connected domain of class C^1+γ for all times t ∈ R.

This solution is unique in the class of characteristic functions of domains of class C^1+γ. This has also been proved in any dimension in [CMOV] via the construction of appro- priate defining functions.

2 The a priori estimate

Parametrize the boundary ∂D₀by a mapping h : T → ∂D0, of class C^1+γ(T, R²) satisfying the bilipschitz condition

c⁻¹₀ |u − v| ≤ |h(u) − h(v)| ≤ c0|u − v|, u, v ∈ T

for some positive constant c₀. The mapping h induces a norm in the vector space of C^1+γ functions on the boundary of D₀ with values in the plane defined on X ∈ C^1+γ(∂D₀, R²) by

(3) kXk_1+γ = kXk∞,∂D₀ + k∇Xk_γ,∂D0, where

kXk_∞,∂D0 = sup{|X(α)| : α ∈ ∂D₀} and

k∇Xk_γ,∂D0 = sup{|_du^dX(h(u)) − _du^dX(h(v))|

|u − v|^γ : u, v ∈ T, u 6= v}.

We have used the notation (which is an abuse of language we adopt for the sake of conciseness)

d

duf (u) = d

dθf (h(e^iθ)), u = h(e^iθ),

for each differentiable function f defined on T. The contour dynamics equation is an ordinary differential equation defined in the open set Ω of C^1+γ(∂D₀, R²) consisting of those mappings in C^1+γ(∂D₀, R²) satisfying the bilipschitz condition

(4) 0 < b = b(X) = inf{|X(α) − X(β)|

|α − β| : α, β ∈ ∂D₀, α 6= β}.

For more details on the contour dynamics equation (CDE) in this context the reader may consult the last section.

The CDE has a unique solution X(·, t) ∈ Ω for t ∈ (−T, T ) satisfying the initial condition X(·, 0) = I, where I stands for the identity map on ∂D₀. Then t → X(α, t) is

the trajectory of the particle that at time 0 is at the point α ∈ ∂D₀ and X(∂D₀, t) is a Jordan curve of class C^1+γ which encloses a simply connected domain D_t. The function ω(z, t) = χ_Dt(z) is a weak solution of equation (2).

Thus

d

dtX(α, t) = v(X(α, t), t), X(α, t) = α ∈ R², (5)

is the flow associated with the Lipschitz velocity field v(·, t) = χ_Dt(·, t) ? k and coincides with the solution of the CDE for α ∈ ∂D0. What remains to be proven is that the maximal time of existence of the solution of the CDE is T = ∞ and for that we need an a priori inequality. By (5)

X(h(u), t) = h(u) + Z t

0

v(X(h(u), s), s) ds, u ∈ T.

Hence

(6) d

duX(h(u), t) = dh du(u) +

Z t 0

∇v(X(h(u), s), s) d

duX(h(u), s)

 ds.

Clearly _du^dX(h(u), s) is a tangent vector to ∂D_sat the point X(h(u), s). We would like to show that

∇v(X(h(u), s), s) d

duX(h(u), s)



is a commutator. We need three lemmas. The first works in Rⁿ. The notion of jet is in [S, p. 176], although the term jet is not used there.

Lemma 1. Let Γ be a hypersurface of dimension n − 1 and of class C^1+γ in Rⁿ. Let N (x) = (N₁(x), . . . , N_n(x)) a normal field on Γ of class C^γ(Γ). Then (0, N (x)) is a jet of class C^1+γ(Γ) with constant AkN kγ,Γ, A constant depending only on γ. In other words,

sup

x,y∈Γ, x6=y

|(y − x) · N (x)| ≤ AkN k_γ,Γ|y − x|^1+γ.

Proof. Assume, without loss of generality, that x = 0 and N (0) = (0, . . . , 0, N_n(0)) with N_n(0) > 0. Set δ^−γ = 2^{kN k}_{|N (0)|}^γ,Γ.

If |y| ≥ δ, then

|y · N (0)| ≤ |y||N (0)| = |y|2kN k_γ,Γδ^γ ≤ 2kN k_γ,Γ|y|^1+γ, as required.

If y ∈ B(0, δ) ∩ Γ, then

|N (y) − N (0)| ≤ kN kγ,Γδ^γ = |N (0)|

2 .

Hence the tangent hyperplane to Γ at y forms an angle of less than 30^◦ with the hor- izontal hyperplane. Consequently B(0, δ) ∩ Γ is the graph of a function y_n = ϕ(y⁰), y⁰ = (y₁, . . . , y_n−1), of class C^1+γ, defined on the projection U of B(0, δ) ∩ Γ on {y ∈ Rⁿ: y_n= 0}. Note that U is open in Rⁿ⁻¹ and that, for y ∈ B(0, δ) ∩ Γ, the segment with extremes 0 and y⁰ is contained in U (by the implicit function theorem). It is also clear that |∇ϕ(y)| ≤ 1, y⁰ ∈ U , because of the inclination of tangent hyperplanes with respect to the horizontal hyperplane.

If y ∈ B(0, δ) ∩ Γ we have

|y · N (0)| ≤ |ϕ(y⁰)||N (0)| ≤ sup

|z⁰|≤|y|, z⁰∈U

|∇ϕ(z⁰)|

!

|y⁰||N (0)| ≤ k∇ϕkγ,Γ|y⁰|^1+γ|N (0)|.

Set

M (y⁰) = (−∂₁ϕ(y⁰), . . . , −∂_n−1ϕ(y⁰), 1), y⁰ ∈ U, which is a normal vector to Γ at y = (y⁰, ϕ(y⁰)). Hence

M (y⁰) = N (y)

N_n(y), y ∈ B(0, δ) ∩ Γ, and

k∇ϕk_γ,U = kM k_γ,U =

N (y) N_n(y)

γ,U

. Take y, z ∈ B(0, δ) ∩ Γ. Then

|y − z| = (|y⁰− z⁰|²+ |ϕ(y⁰) − ϕ(z⁰)|²)^1/2 ≤√

2|y⁰ − z⁰|.

One has (7)

N (y)

N_n(y) − N (z) N_n(z)

≤ 1

|N_n(y)||N (y) − N (z)| + |N (z)||N_n(z) − N_n(y)|

|N_n(y)||N_n(z)| . By the definition of δ

|N_n(y)| ≥ |N_n(0)| − |N_n(y) − N_n(0)| ≥ |N (0)| − kN k_γ,Γδ^γ = |N (0)|

2 . Thus the right hand side of (7) is not greater than

2^3+γ/2kN k_γ,Γ

|N (0)||y⁰− z⁰|^γ.

Therefore

|y · N (0)|

|y|^1+γ ≤ k∇ϕk_γ,U|N (0)| ≤ 2^3+γ/2kN k_γ,Γ and the lemma follows with A = 2^3+γ/2.

Lemma 2. Let Γ a Jordan curve of class C^1+γ in the plane and τ (x) = (τ₁(x), τ₂(x)), x ∈ Γ, a tangent vector to Γ at x with τ ∈ C^γ(Γ, R²). Then there exists g(x) = (g₁(x), g₂(x)) ∈ C^γ(R², R²) such that g = τ on Γ, kgk_γ,R² ≤ Ckτ k_γ,Γ for a positive constant C depending only on γ, and div g = 0 in R².

Proof. The field (−τ₂(x), τ₁(x)) is normal to Γ at each x ∈ Γ. By the previous lemma (0, −τ2(x), τ1(x)) is a C^1+γ jet on Γ with constant Cγkτ kγ,Γ and by Whitney’s extension theorem there is a function ϕ ∈ C^1+γ(R², R) such that

ϕ = 0, ∂₁ϕ = −τ₂, ∂₂ϕ = τ₁, on Γ

and k∇ϕk_γ,R² ≤ C0Cγkτ kγ,Γ, with C0 an absolute constant [S, Theorem 4, p.177]. There- fore

τ (x) = (∂₂ϕ(x), −∂₁ϕ(x)), x ∈ Γ

and g = (∂2ϕ, −∂1ϕ) has no divergence in the plane and kgk_γ,R² ≤ C kτ kγ,Γ, where C = C₀C_γ depends only on γ.

Lemma 3. Let τ (x) = (τ₁(x), τ₂(x)) ∈ C^γ(Γ) be a tangent field on Γ = ∂D_s. Then

(8) ∇v(x, s)(τ (x)), x ∈ Γ,

is the restriction to Γ of the commutator Z

Ds

∇v(x − y, s)(g(x) − g(y)) dy, x ∈ R², where g is the field given by Lemma 2.

Proof. The first component of (8) is

(9) ∂1v1(x, s)τ1(x) + ∂2v1(x, s)τ2(x) = (∂1k1∗ χ)(x)τ1(x) + (∂2k1∗ χ)(x)τ2(x) with χ = χ_Ds. In the sense of distributions

∂₁k₁ = p.v. ∂₁k₁+ c₁δ₀, c₁ = Z

|x|=1

k₁(x)x₁dσ(x)

and

∂₂k₁ = p.v. ∂₂k₁+ c₂δ₀, c₂ = Z

|x|=1

k₁(x)x₂dσ(x).

Let g be the field given by Lemma 2. Since (∂₁k₁ ∗ χ)g₁ = (p.v. ∂₁k₁ ∗ τ )g₁ + c₁χg₁ and ∂₁k₁ ∗ (χg₁) = p.v. ∂₁k₁∗ (χg₁) + c₁χg₁, we get

(∂₁k₁∗ χ)g₁ = (p.v. ∂₁k₁∗ χ)g₁ − p.v. ∂₁k₁∗ (χg₁) + ∂₁k₁∗ (χg₁).

Similarly

(∂₂k₁∗ χ)g₂ = (p.v. ∂₂k₁∗ χ)g₂ − p.v. ∂₂k₁∗ (χg₂) + ∂₂k₁∗ (χg₂).

Note that

∂_ik₁∗ (χg_i) = k₁∗ ∂_i(χg_i) = k₁∗ (g_i∂_iχ + χ∂_ig_i) and so, recalling that g has vanishing divergence,

∂₁k₁∗ (χg₁) + ∂₂k₁∗ (χg₂) = k₁∗ (g · ∇χ + χ div g) = k₁∗ (g · ∇χ) ≡ 0

because g|Γ = τ is tangent to Γ and ∇χ normal to Γ at each point of Γ. Thus if x ∈ Γ =

∂D_s

(∂₁k₁∗ χ)(x)τ₁(x) + (∂₂k₁∗ χ)(x)τ₂(x) = Z

Ds

∂₁k₁(x − y)(g₁(x) − g₁(y) dy +

Z

Ds

∂₂k₁(x − y)(g₂(x) − g₂(y)) dy is a sum of two commutators. One has a similar formula for the second component of

∇v(x, s)(τ (x)) and the lemma follows.

Therefore, by the usual commutator estimate in the H¨older seminorm [BC, Lemma, p.26] or [BGLV, Lemma 3.2, p.3799],

k∇v(x, s)(τ (x))k_γ,Γ≤ Ck∇v(·, s)k_∞kgk_γ,R² ≤ Ck∇v(·, s)k_∞kτ k_γ,Γ. From (6) we see that

d

duX(h(u), t) γ,T

≤

d duh(u)

γ,T

+ C Z t

0

k∇v(·, s)k∞

d

duX(h(u), s) ds γ,T

ds and then, by Gronwall’s lemma, we get the a priori inequality

d

duX(h(u), t) γ,T

≤

d duh(u)

γ,T

exp

 C

Z t 0

k∇v(·, s)k∞ds

 . We also have an a priori inequality for

b(t) ≡ inf

α6=β

X(α, t) − X(β, t)

|α − β|

= 1

sup

α6=β

|α−β|

|X(α,t)−X(β,t)|

= 1

k∇X⁻¹(·, t)k∞

≥ exp



− Z t

0

k∇v(·, s)k_∞ds

 .

Set

q(t) =

_du^dX(h(u), t) γ,T

b(t)^1+γ , −T < t < T, so that

(10) q(t) ≤ exp

 C

Z t 0

k∇v(·, s)k∞ds



, −T < t < T.

3 The logarithmic inequality

Let K ∈ C¹(R²\{0}) be homogeneous of degree −2 and even. Set T f (x) = p.v.

Z

R²

K(x − y)f (y) dy,

so that T is a convolution, smooth, homogeneous, even Calder´on–Zygmund operator.

Let X ∈ C^1+γ(∂D₀, R²), D₀ a domain of class C^1+γ. Assume that X is bilipschitz onto the image and let b stand for the inverse of Lipschitz norm of the inverse mapping, as in (4). Since X(∂D₀) is a Jordan curve, the simply connected domain D enclosed by the curve satisfies ∂D = X(∂D₀).

Set

T^∗f (x) = sup

ε>0

Z

|y−x|>ε

K(x − y)f (y) dy, x ∈ R². Lemma 4. We have

T^∗(χD)(x) ≤ A



1 + log⁺



|D|^1/2k d

du[X(h(u))]k_γ,T 1 b^1+γ



= A 1 + log⁺ |D|^1/2q(t)

, x ∈ R², where A is a constant depending only on γ and K.

Proof. By an elementary argument [MOV, p.409-410] it is enough to prove the inequality for points x ∈ ∂D. Without loss of generality one can assume that x = 0 = X(0) and that the unitary normal vectors to ∂D₀ and ∂D at 0 are (0, 1). Define δ by

δ^−γ = 1 b^1+γk d

du[X(h(u))]k_γ,T. Then for ε > 0

Z

|y|>ε

K(y)χ_D(y) dy = Z

δ>|y|>ε

K(y)χ_D(y) dy + Z

|D|^1/2>|y|>δ

K(y)χ_D(y) dy

+ Z

|y|>|D|^1/2

K(y)χ_D(y) dy.

(11)

The third term is estimated by Z

|y|>|D|^1/2

1

|y|²χ_D(y) dy ≤ 1 and the second by

2π sup

|y|=1

|K(y)| log |D|^1/2 δ



≤ A log⁺



|D|^1/2k d

du[X(h(u))]k_γ,T 1 b^1+γ

 . It remains to estimate the first term in (11). Set

D_ε,δ = {y ∈ D : ε < |y| < δ}, H⁻= {y = (y₁, y₂) ∈ R² : y₂ < 0} and S_ρ= {y ∈ R², |y| = ρ}, ρ > 0.

Since an even kernel with vanishing integral on the unit circumference has also vanishing integral on half a circumference, we get

Z

Dε,δ

K(y) dy = Z δ

ε

Z

{θ∈S¹:ρθ∈D}

K(θ) dσ(θ) dρ ρ

= Z δ

ε

Z

{θ∈S¹:ρθ∈D}

K(θ) dσ(θ) − Z

S¹∩H⁻

K(θ) dσ(θ) dρ ρ

= Z δ

ε

Z

{θ∈S¹:ρθ∈(D\H⁻)∪(H⁻\D)}

K(θ) dσ(θ) dρ ρ and

     Z

D_ε,δ

K(y) dy     

≤ sup

|θ|=1

|K(θ)|

Z δ 0

σ{θ ∈ S¹ : ρθ ∈ (D\H⁻) ∪ (H⁻\D)}dρ ρ . Since the domains D\H⁻ and H⁻\D are “tangential”

σ{θ ∈ S¹ : ρθ ∈ (D\H⁻) ∪ (H⁻\D)} ≤ A1

ρsup{|X₂(α)| : α ∈ ∂D₀ and |X(α)| = ρ}.

Take |X(α)| = ρ < δ. If α = h(u) and 0 = h(u₀), then

|X₂(α)| = |X₂(h(u)) − X₂(h(u₀))| ≤ k d

du[X₂(h(u))]k_γ,T|u − u₀|^1+γ

≤ k d

du[X(h(u))]k_γ,T 1

b^1+γ |X(α)|^1+γ

= k d

du[X(h(u))]k_γ,T 1

b^1+γ ρ^1+γ.

Thus

     Z

Dε,δ

K(y) dy     

≤ Ak d

du[X(h(u))]k_γ,T 1 b^1+γ

Z δ 0

ρ^γ−1dρ

= A k d

du[X(h(u))]k_γ,T 1 b^1+γ

δ^γ γ = A.

Having at our disposition the a priori estimate of section 2 and the logarithmic estimate it is a standard matter to complete the proof. We remark that ∇v(·, t) is a 2 × 2 matrix with entries which are either constant multiples of χ_Dt or even homogeneous smooth Calder´on -Zygmund operators applied to χDt. Inserting the a priori estimate (10) into the logarithmic inequality we obtain

k∇v(·, t)k∞ ≤ C + C Z t

0

k∇v(·, s)k∞ds, t ∈ (−T, T ).

The factor |D|^1/2 in the inequality of Lemma 4 causes no trouble because of the standard estimate

k∇X(·, t)k_∞≤ exp Z t

0

k∇v(·, s)k_∞ds, t ∈ (−T, T ).

Gronwall’s lemma then yields

k∇v(·, t)k∞≤ C exp(Ct), t ∈ (−T, T ).

Hence, for t ∈ (−T, T ), d

du[X(h(u), t)]k_γ,T ≤ C exp(C exp(Ct)), and

b(t) ≥ C⁻¹exp(−C exp(Ct)).

Consequently T = ∞.

4 Appendix: local in time existence for the CDE

Our first remark is that if k is an odd kernel, homogeneous of degree −1, differentiable off the origin, then

(12) k = ∂₁(x₁k) + ∂₂(x₂k), x = (x₁, x₂) ∈ R²\ {0}.

This follows straightforwardly from Euler’s theorem on homogeneous functions.

Assume that ω(z, t) = χ_Dt(z) is a weak solution of the general equation (2). The velocity field is

v(·, t) = χ_Dt ? k = χ_Dt ? ( ∂

∂x(xk) + ∂

∂y(yk))

= ∂

∂xχDt ? (xk) + ∂

∂yχDt ? (yk).

Since ∇χ_Dt = −~ndσ = idz, where dσ is the arc-length measure on the curve ∂D_t and dz = dz_∂Dt, we have

∂

∂xχ_Dt = −n₁dσ = −dy and ∂

∂yχ_Dt = −n₂dσ = dx.

Setting z = x + iy and w = u + iv we get v(z, t) = −

Z

∂Dt

(x − u)k(z − w) dv + Z

∂Dt

(y − v)k(z − w) du

= Z

∂Dt

k(z − w)h−i(z − w) · dwi, (13)

where h·, ·i denotes scalar product in the plane.

The flow mapping is the solution of the ODE d

dtX(α, t) = v(X(α, t), t), X(α, t) = α ∈ R². (14)

Substituting the expression (13) for the velocity field in (14) and making the change of variables w = X(β, t) we get

d

dtX(α, t) = v(X(α, t), t) = Z

∂Dt

k(X(α, t) − w)h−i(X(α, t) − w), dwi,

= Z

∂D0

k(X(α, t) − X(β, t))h−i(X(α, t) − X(β, t)), ∇X(β, t)(dβ)i.

In the last line β is a generic point of the curve ∂D₀ and dβ the standard differential form associated with the curve. Then

∇X(β, t)(dβ) = ∂X₁

∂β₁(β, t) dβ₁+ ∂X₁

∂β₂(β, t) dβ₂, ∂X₂

∂β₁(β, t) dβ₁+ ∂X₂

∂β₂(β, t) dβ₂

 .

In fact ∇X(β, t) can be understood more intrinsecally as the differential DX(β, t) of the mapping X(·, t) as a differentiable mapping from the differentiable curve ∂D₀ onto the differentiable curve ∂D_t. This mapping takes a tangent vector to ∂D₀ at the point β into a tangent vector to ∂D_t at the point X(β, t). The point is that this operation involves only first derivatives of X(·, t) on ∂D₀.

Consider the Banach space C^1+γ(∂D₀, R²) endowed with the norm (3) and the open set Ω consisting of those mappings in C^1+γ(∂D₀, R²) satisfying the bilipschitz condition (4). For each X ∈ Ω define

(15) F (X)(α) = Z

∂D0

k(X(α) − X(β))h−i(X(α) − X(β)), ∇X(β)(dβ)i, α ∈ ∂D₀.

The CDE is the autonomous ODE in Ω

(16) dX

dt = F (X), X(·, 0) = I, where I is the identity mapping on ∂D0.

Of course one has to check that F (X) belongs to C^1+γ(∂D₀, R²) for each X ∈ Ω. After that one wants to apply the existence and uniqueness theorem of Picard, and for that one needs to check that F is locally Lipschitz in Ω.

All these facts are verified routinely and depend essentially on the fact that the kernel appearing in (15) is of class C²(R²\ {0}) and homogeneous of degree 0. The reader may consult [MB, Chapter 8] for the case of the vorticity equation. For instance, when you take a derivative in α in (15) to prove that F (X) is in C^1+γ(∂D₀, R²), then you get an expression of the form

(17)

Z

∂D0

H(X(α) − X(β))L(∇X(β), dβ),

where H(·) is a kernel of homogeneity −1 of class C¹(R²\ {0}) and L(∇X(β), dβ) a linear expression in ∂X_j/∂β_k and dβ_j. Hence (17) may be understood as a (non-convolution) singular integral on the curve ∂D₀ applied to a linear combination of the ∂X_j/∂β_k, which are functions satisfying a H¨older condition of order γ. Then the result is a function of α in the same space.

An analogous situation appears in checking that F is locally Lipschitz in Ω.

The reader may also consult [BGLV], where full details are provided in a similar context.

Acknowledgements. The author acknowledges support by 2017-SGR-395 (Generalitat de Catalunya), MTM2016-75390 (Mineco) and PID2020-112881GB-100.

References

[BC] A. L. Bertozzi and P. Constantin, Global regularity for vortex patches, Comm.

Math. Phys., 152(1):19–28, 1993.

[BGLV] A.L. Bertozzi, J.B. Garnett, T. Laurent and J. Verdera, The Regularity of the Boundary of a Multidimensional Aggregation Patch, SIAM J. Math. Anal., 48(6), 3789–3819, 2016.

[CMOV] J.C. Cantero, J. Mateu, J. Orobitg and J. Verdera, The regularity of the bound- ary of vortex patches for some non-linear transport equations, arXiv 2103.05356.

[Ch] J. Y. Chemin, Persistance de structures g´eom´etriques dans les fluides incompress- ibles bidimensionnels, Ann. Sci. ´Ecole Norm. Sup. (4), 26(4):517–542, 1993.

[MB] A. J. Majda and A. L. Bertozzi. Vorticity and incompressible flow, volume 27 of Cambridge Texts in Applied Mathematics. Cambridge University Press, Cam- bridge, 2002.

[MOV] J. Mateu, J. Orobitg and J. Verdera, Extra cancellation of even Calder´on- Zygmund operators and quasiconformal mappings, Journal de Math´ematiques Pures et Appliqu´es, 91(4):402–431, 2009.

[R] R. Radu, Existence and regularity for vortex patch solutions of the 2D incom- pressible Euler equations, to appear in Indiana Math. Journal.

[S] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, 1970.

[Y] V. I. Yudovich, Non-stationary flow of an ideal incompressible liquid, USSR Computational Mathematics and Mathematical Physics, 3(6):1407–1456, 1963.

J. Verdera,

Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, Centre de Recerca Matem`atica, Barcelona, Catalonia.

E-mail: [email protected]