The Stokes conjecture for waves with vorticity

Ann. I. H. Poincaré – AN 29 (2012) 861–885

www.elsevier.com/locate/anihpc

The Stokes conjecture for waves with vorticity

Eugen Varvaruca

, Georg S. Weiss

aDepartment of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, UK bDepartment of Mathematics, Heinrich Heine University, 40225 Düsseldorf, Germany

Received 10 October 2011; received in revised form 6 March 2012; accepted 7 May 2012 Available online 23 May 2012

Abstract

We study stagnation points of two-dimensional steady gravity free-surface water waves with vorticity.

We obtain for example that, in the case where the free surface is an injective curve, the asymptotics at any stagnation point is given either by the “Stokes corner flow” where the free surface has acorner of120^◦, or the free surface ends in ahorizontal cusp, or the free surface ishorizontally flatat the stagnation point. The cusp case is a new feature in the case with vorticity, and it is not possible in the absence of vorticity.

In a second main result we exclude horizontally flat singularities in the case that the vorticity is 0 on the free surface. Here the vorticity may have infinitely many sign changes accumulating at the free surface, which makes this case particularly difficult and explains why it has been almost untouched by research so far.

Our results are based on calculations in the original variables and do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity.

©2012 Elsevier Masson SAS. All rights reserved.

1. Introduction

The classical hydrodynamical problem of traveling two-dimensional gravity water waves with vorticity can be described mathematically as a free-boundary problem for a semilinear elliptic equation: given an open connected setΩ in the(x, y)plane and a functionγ of one variable, find a non-negative functionψinΩ such that

ψ= −γ (ψ ) inΩ∩ {ψ >0}, (1.1a)

∇ψ (x, y)²= −y onΩ∩∂{ψ >0}. (1.1b)

The present paper is an investigation by geometric methods of the singularities of the free boundary∂{ψ >0}.

Let us briefly describe, following[4], the connection between problem(1.1)and the nonlinear governing equations of fluid motion. Consider a wave of permanent form moving with constant speed on the free surface of an incompress- ible inviscid fluid, acted on by gravity. With respect to a frame of reference moving with the speed of the wave, the flow is steady and occupies a fixed regionDin the plane. The boundary∂Dof the fluid region contains a part∂_aD

* Corresponding author.

E-mail addresses:e.varvaruca@reading.ac.uk(E. Varvaruca),weiss@math.uni-duesseldorf.de(G.S. Weiss).

0294-1449/$ – see front matter ©2012 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2012.05.001

which is free and in contact with an air region. Under the assumption that the fluid regionDis simply connected, the incompressibility condition shows that the flow can be described by astream functionψ:D→R, so that the relative fluid velocity is(ψ_y,−ψ_x). The Euler equations imply that thevorticityω:= −ψsatisfies

ω_xψ_y=ω_yψ_x inD. (1.2)

It is easy to see that (1.2) is satisfied whenever

ω=γ (ψ ) inD (1.3)

for some (smooth) function γ of variableψ, which will be referred to as avorticity function. (Conversely, under additional assumptions, see[4], (1.2) implies the existence of such a functionγ.) The kinematic boundary condition that the same particles always form the free surface∂aDis equivalent to

ψis locally constant on∂_aD.

Also, in the presence of (1.3), Bernoulli’s Theorem and the fact that on the fluid–air interface∂_aDthe pressure in the fluid equals the constant atmospheric pressure imply that

1

2|∇ψ|²+gyis locally constant on∂_aD,

whereg >0 is the gravitational constant. We therefore obtain, after some normalization, and at least in the case when

∂_aDis connected, that the following equations and boundary conditions are satisfied:

−ψ=γ (ψ ) inD, (1.4a)

ψ=0 on∂_aD, (1.4b)

|∇ψ|²+2gy=0 on∂_aD. (1.4c)

Eqs. (1.4) are usually supplemented by suitable boundary conditions on the rest of the boundary of D, or some conditions on the flow at infinity if the fluid domain is unbounded. Classical types of waves which have received most attention in the literature are periodic and solitary waves of finite depth (in which the fluid domainDhas a fixed flat bottomy= −d, at whichψis constant), and periodic waves of infinite depth (in which the fluid domain extends to y= −∞and the condition limy→−∞∇ψ (x, y)=(0,−c)holds, wherecis the speed of the wave). Conversely, for any vorticity functionγ, any solution of(1.4)gives rise to a traveling free-surface gravity water wave, irrespective of whether D is simply connected or ∂_aD is connected. Problem(1.1)is a local version of problem (1.4), under the additional assumption that ψ >0 in the fluid region, and whereψ has been extended by the value 0 to the air region. In(1.1), the domainΩ is a neighborhood of a point of interest on the fluid–air interface, the fluid regionD is identified with the set {(x, y): ψ (x, y) >0}(in short{ψ >0}) and the fluid–air interface ∂_aD with∂{ψ >0}, while the gravitational constantg has been normalized by scaling. Note that problem(1.1)is also relevant for the description of more general steady flow configurations (for example, the fluid domain could have a non-flat bottom, and there could be some further external forcing acting at the boundary of the fluid region which is not in contact with the air region).

The theory of traveling water waves with vorticity has a long history, whose highlights include the pioneering paper of Gerstner[10], the first rigorous proof of existence of periodic waves of small amplitude by Dubreil-Jacotin [6], and the foundation[4]of Constantin and Strauss, which proved existence of smooth waves of large amplitude for the periodic finite-depth problem. The paper[4]has generated substantial interest and follow-up work on steady water waves with vorticity, see[20]for a survey of recent results.

In this paper we investigate the shape of the free boundary∂{ψ >0}atstagnation points, which are points where the relative fluid velocity(ψ_y,−ψ_x)is the zero vector. The Bernoulli condition (1.1b) shows that such points are on the real axis, while the rest of the free boundary is in the lower half-plane. Stokes[19]conjectured that, in the irrotational caseγ≡0, at any stagnation point the free surface has a (symmetric) corner of 120^◦, and formal asymptotics suggest that the same result might be true also in the general case of waves with vorticityγ ≡0. (See Fig.1.) In the irrotational case, the Stokes conjecture was first proved, under isolatedness, symmetry, and monotonicity assumptions, by Amick, Fraenkel and Toland[3]and Plotnikov[15](see also[21]for a simplification of the proof in[3]), while a geometric proof has recently been given in[23]without any such structural assumptions.

Fig. 1. Stokes corner. Fig. 2. Cusp. Fig. 3. Horizontally flat stagnation point.

In the caseγ ≡0, the only rigorous results available on waves with stagnation points are very recent and require in an essential way symmetry and monotonicity of the free surface: in[22]it was proved that, at stagnation points, a symmetric monotone free boundary has either a corner of 120^◦or a horizontal tangent. Moreover, it was also shown there that, ifγ0 close to the free surface, then the free surface necessarily has a corner of 120^◦. (On the other hand, ifγ (0) <0, there exist very simple examples where the free surface is the real axis, a line of stagnation points.) The existence of waves, with non-zero vorticity, having stagnation points has been obtained in the setting of periodic waves of finite depth over a flat horizontal bottom, in the following cases in the paper[17]submitted simultaneously with the present paper: for any non-positive vorticity functionγ and any period of the wave, and under certain restrictions on the size ofγ and the wave period (roughly speaking, the vorticity has to be sufficiently small and the period sufficiently large) ifγ is positive somewhere. The extreme waves constructed in[17]are obtained as weak limits of large-amplitude smooth waves whose existence was proved by Constantin and Strauss[4], and they are symmetric and monotone. It was shown in[17]that the free surface of any symmetric monotone wave with stagnation points which is a limit of smooth waves cannot have a horizontal tangent at the stagnation points (in particular, the free surface cannot be horizontally flat), irrespective of the vorticity functionγ, and therefore, as a consequence of[22], the free surface of such a wave necessarily has corners of 120^◦at stagnation points.

The present paper is the first study of stagnation points of steady two-dimensional gravity water waves with vor- ticity in the absence of structural assumptions of isolatedness of stagnation points, symmetry and monotonicity of the free boundary, which have been essential assumptions in all previous works. We obtain for example that, in the case when the free surface is an injective curve, the asymptotics at any stagnation point is given either by the “Stokes corner flow” where the free surface has acorner of120^◦, or the free surface ends in ahorizontal cusp(see Fig.2), or the free surface ishorizontally flatat the stagnation point (see Fig.3).

The cusp case is a new feature in the case with vorticity, and it is not possible without the presence of vorticity [23]. It is interesting to point out that Gerstner[10]constructed an explicit example of a steady wave with vorticity whose free surface has avertical cuspat a stagnation point. However, that vertical cusp is due to the fact that in his example the vorticity is infinite at the free surface, while in the present paper we only consider the case of vorticities which are smooth up to the free surface. We conjecture the cusps in our paper — the existence of which is still open

— to be due to the break-down of the Rayleigh–Taylor condition in the presence of vorticity.

The second half of our paper is devoted toexcluding horizontally flat singularitiesin the case that the vorticity is non-negative at the free surface. (Horizontally flat singularities are possible if the vorticity is negative at the free surface.) Of particular difficulty is the case when the vorticity is 0 at the free surface, and may have infinitely many sign changes accumulating there.

Let us briefly state our main result and give a plan of the paper:

Main Result.Letψbe a suitable weak solution of (1.1)(compare to Definition3.2)satisfying ∇ψ (x, y)²Cmax(−y,0) locally inΩ,

let the free boundary∂{ψ >0}be a continuous injective curveσ =(σ₁, σ₂)such thatσ (0)=(x⁰,0), and assume that the vorticity function satisfies either|γ (z)|Cz, orγ (z)0, for allzin a right neighborhood of0.

(i) If the Lebesgue density of the set{ψ >0}at(x⁰,0)is positive, then the free boundary is in a neighborhood of (x⁰,0)the union of twoC¹-graphs of functionsη₁:(x⁰−δ, x⁰] →Randη₂: [x⁰, x⁰+δ)→Rwhich are both continuously differentiable up tox⁰and satisfyη₁(x⁰)=1/√

3andη₂(x⁰)= −1/√ 3.

(ii) Elseσ₁(t) =x⁰in(−t₁, t₁)\ {0},σ₁−x⁰does not change its sign att=0, and

tlim→0

σ₂(t) σ₁(t)−x⁰=0.

If we assume in addition that either{ψ >0}is a subgraph of a function in they-direction or that{ψ >0}is a Lipschitz set, then the set of stagnation points is locally in Ω a finite set, and at each stagnation point (x⁰,0)the statement in(i)holds.

1.1. Plan of the paper

The flow of the paper follows[23]with new aspects and difficulties which we are going to point out:

After gathering some notation in Section2, in Section3we introduce suitable weak solutions and prove a mono- tonicity formula. Consequences of the monotonicity formula (Section4) make a blow-up analysis of singularities possible. The general case (without the injective curve assumption) is stated in Theorem4.5. Different from the zero vorticity case handled in[23], there appears a new case in which the Lebesgue density of the set{ψ >0}is 0. As- suming the free surface to be an injective curve in a neighborhood of the singularity we obtain in Theorem4.6a more precise description: in the new case the free surface formscuspspointing in thex- or−x-direction. As in[23]we are able to show that Stokes corner singularities are isolated points (Section5).

Starting with Section 6, the focus of our analysis is on points at which the set{ψ >0}has full Lebesgue density. In the caseγ (0)=0, an extension of thefrequency formula(Theorem6.7) introduced by the authors in[23]leads here to aBessel differential inequality(see the proof of Theorem6.12) which shows that the right-hand side of the frequency formula is integrable. This part is substantially different from[23]. It is then possible (Sections7–9) to do a blow-up analysis in order to exclude horizontally flat singularities (Theorem10.1). All our results are based on calculations in the original variables.

2. Notation

We denote byχ_Athe characteristic function of a setA. For any real numbera, the notationa⁺stands for max(a,0).

We denote byx·y the Euclidean inner product in Rⁿ×Rⁿ, by|x| the Euclidean norm inRⁿ and byB_r(x⁰):=

{x∈Rⁿ: |x−x⁰|< r}the ball of centerx⁰and radiusr. We will use the notationB_r forB_r(0), and denote byω_nthe n-dimensional volume ofB₁. Also,Lⁿshall denote then-dimensional Lebesgue measure andH^s thes-dimensional Hausdorff measure. Byνwe will always refer to the outer normal on a given surface. We will use functions of bounded variationBV(U ), i.e. functionsf ∈L¹(U )for which the distributional derivative is a vector-valued Radon measure.

Here|∇f|denotes the total variation measure (cf.[12]). Note that for a smooth open setE⊂Rⁿ,|∇χ_E|coincides with the surface measure on∂E.

3. Notion of solution and monotonicity formula

In some sections of the paper we work with an n-dimensional generalization of the problem described in the Introduction. LetΩ be a bounded domain inRⁿwhich has a non-empty intersection with the hyperplane{x_n=0}, in which to consider the combined problem for fluid and air. We study solutionsu, in a sense to be specified, of the problem

u= −f (u) inΩ∩ {u >0},

|∇u|²=x_n onΩ∩∂{u >0}. (3.1)

Note that, compared to the Introduction, we have switched notation fromψ tou and fromγ tof, and we have

“reflected” the problem at the hyperplane{x_n=0}. The nonlinearityf is assumed to be a continuous function with

primitiveF (z)=_z

0f (t ) dt. Since our results are completely local, we do not specify boundary conditions on∂Ω. In view of the second equation in (3.1), it is natural to assume throughout the rest of the paper thatu≡0 inΩ∩ {x_n0}.

We begin by introducing our notion of avariational solutionof problem (3.1).

Definition 3.1(Variational solution).We defineu∈W_loc^1,2(Ω)to be avariational solutionof (3.1) ifu∈C⁰(Ω)∩ C²(Ω∩ {u >0}),u0 inΩ andu≡0 inΩ∩ {x_n0}, and the first variation with respect to domain variations of the functional

J (v):=

Ω

|∇v|²−2F (v)+x_nχ_{v>0} dx

vanishes atv=u, i.e.

0= − d dJ

u

x+φ(x)

=0

=

Ω

|∇u|²−2F (u)

divφ−2∇uDφ∇u+xnχ_{u>0}divφ+χ_{u>0}φn

dx

for anyφ∈C₀¹(Ω;Rⁿ).

Note for future reference that for each open setDΩ there isCD<+∞such thatu+CD is a non-negative Radon measure inD, the support of the singular part of which (with respect to the Lebesgue measure) is contained in the set∂{u >0}: by Sard’s theorem{u=δ} ∩Dis for almost everyδ a smooth surface. It follows that for every non-negativeζ ∈C₀^∞(D)

−

D

∇max(u−δ,0)· ∇ζ−CDζ dx=

D

ζ (χ_{u>δ}u+CD) dx−

D∩∂{u>δ}

ζ∇u·ν dHⁿ⁻¹0, provided that|f (u)|C_DinD. Lettingδ→0 and using thatuis continuous and non-negative inΩ, we obtain

−

D

(∇u· ∇ζ−C_Dζ ) dx0.

Thusu+C_Dis a non-negative distribution inD, and the stated property follows.

Since we want to focus in the present paper on the analysis of stagnation points, we will assume that everything is smooth away fromx_n=0, however this assumption may be weakened considerably by using in{x_n>0}regularity theory for the Bernoulli free boundary problem (see [2] for regularity theory in the case f =0 — which could effortlessly be perturbed to include the case of bounded f — and see [5] for another regularity approach which already includes the perturbation).

Definition 3.2(Weak solution).We defineu∈W_loc^1,2(Ω)to be aweak solutionof (3.1) if the following are satisfied:

uis avariational solutionof (3.1) and the topological free boundary∂{u >0}∩Ω∩{x_n>0}is locally aC^2,α-surface.

Remark 3.3.(i) It follows that in{xn>0}the solution is a classical solution of (3.1).

(ii) For any weak solutionuof (3.1) such that

|∇u|²Cx_n⁺ locally inΩ,

uis a variational solution of (3.1),χ_{u>0}is locally in{x_n>0}a function of bounded variation, and the total variation measure|∇χ_{u>0}|satisfies

r^1/2−n

Br(y)

√x_nd|∇χ_{u>0}|C₀

for allB_r(y)Ωsuch thaty_n=0 (see[23, Lemma 3.4]).

The first tool in our analysis is an extension of the monotonicity formula in[25],[24, Theorem 3.1]to the boundary case. The roots of those monotonicity formulas are harmonic mappings[18,16]and blow-up[14].

Theorem 3.4(Monotonicity formula). Letube a variational solution of (3.1), letx⁰∈Ω such thatx_n⁰=0, and let δ:=dist(x⁰, ∂Ω)/2. Let, for anyr∈(0, δ),

I_x0,u(r)=I (r)=r⁻ⁿ⁻¹

Br(x⁰)

|∇u|²−uf (u)+xnχ_{u>0}

dx, (3.2)

J_x0,u(r)=J (r)=r⁻ⁿ⁻²

∂Br(x⁰)

u²dHⁿ⁻¹, (3.3)

M_x0,u(r)=M(r)=I (r)−3

2J (r) (3.4)

and

K_x0,u(r)=K(r)=r

∂Br(x⁰)

2F (u)−uf (u)

dHⁿ⁻¹+

Br(x⁰)

(n−2)uf (u)−2nF (u)

dx. (3.5)

Then, for a.e.r∈(0, δ), I(r)=r⁻ⁿ⁻²

2r

∂B_r(x⁰)

(∇u·ν)²dHⁿ⁻¹−3

∂B_r(x⁰)

u∇u·ν dHⁿ⁻¹

+r⁻ⁿ⁻²K(r), (3.6)

J(r)=r⁻ⁿ⁻³

2r

∂B_r(x⁰)

u∇u·ν dHⁿ⁻¹−3

∂B_r(x⁰)

u²dHⁿ⁻¹

(3.7)

and

M(r)=2r⁻ⁿ⁻¹

∂Br(x⁰)

∇u·ν−3 2 u r

2

dHⁿ⁻¹+r⁻ⁿ⁻²K(r). (3.8)

Proof. The identity (3.7) can be easily checked directly, being valid for any functionu∈W_loc^1,2(Ω)(not necessarily a variational solution of (3.1)).

For small positiveκandη_κ(t):=max(0,min(1,^r−_κ^t)), we take after approximationφ_κ(x):=η_κ(|x−x⁰|)(x−x⁰) as a test function in the definition of a variational solution. We obtain

0=

Ω

|∇u|²−2F (u)+x_nχ_{u>0}

nη_κx−x⁰+η_κx−x⁰x−x⁰dx

−2

Ω

|∇u|²η_κx−x⁰+ ∇u· x−x⁰

|x−x⁰|∇u· x−x⁰

|x−x⁰|ηx−x⁰x−x⁰ dx

+

Ω

η_κx−x⁰x_nχ_{u>0}dx.

Passing to the limit asκ→0, we obtain, for a.e.r∈(0, δ), 0=n

Br(x⁰)

|∇u|²−2F (u)+x_nχ_{u>0} dx−r

∂Br(x⁰)

|∇u|²−2F (u)+x_nχ_{u>0} dHⁿ⁻¹ +2r

∂Br(x⁰)

(∇u·ν)²dHⁿ⁻¹−2

Br(x⁰)

|∇u|²dx+

Br(x⁰)

x_nχ_{u>0}dx. (3.9)

Also observe that letting→0 in

B_r(x⁰)

∇u· ∇max(u−,0)¹⁺dx=

B_r(x⁰)

f (u)max(u−,0)¹⁺dx+

∂B_r(x⁰)

max(u−,0)¹⁺∇u·ν dHⁿ⁻¹

for a.e.r∈(0, δ), we obtain the integration by parts formula

Br(x⁰)

|∇u|²−uf (u) dx=

∂Br(x⁰)

u∇u·ν dHⁿ⁻¹ (3.10)

for a.e.r∈(0, δ).

Note also that

I(r)= −(n+1)r⁻ⁿ⁻²

Br(x⁰)

|∇u|²−uf (u)+xnχ_{u>0} dx

+r⁻ⁿ⁻¹

∂Br(x⁰)

|∇u|²−uf (u)+x_nχ_{u>0}

dHⁿ⁻¹. (3.11)

Using (3.9) and (3.10) in (3.11), we obtain (3.6). Finally, (3.8) follows immediately by combining (3.6) and (3.7). 2 4. Densities

From now on we assume Assumption 4.1.Letusatisfy

|∇u|²Cx_n⁺ locally inΩ.

Remark 4.2.Note that Assumption4.1implies that u(x)C₁

x_n⁺3/2

and that in the casex_n⁰=0, r⁻ⁿ⁻²K(r)C2

√1 r,

whereC2depends onx⁰but is locally uniformly bounded.

Remark 4.3.Unfortunately the combination of vorticity and gravity makes it hard to obtain the estimate

|∇u|²+2F (u)−x_n⁺0 (4.1)

related to the Rayleigh–Taylor condition in the time-dependent problem, but the weaker estimate Assumption4.1has been verified under certain assumptions in[22].

We first show that the functionM_x0,uhas a right limitM_x0,u(0+), of which we derive structural properties.

Lemma 4.4.Letube a variational solution of (3.1)satisfying Assumption4.1. Then:

(i) Letx⁰∈Ω be such thatx_n⁰=0. Then the limitM_x0,u(0+)exists and is finite.(Note thatu=0 in{x_n=0}by assumption.)

(ii) Letx⁰∈Ωbe such thatx_n⁰=0, and let0< r_m→0+asm→ ∞be a sequence such that the blow-up sequence u_m(x):=u(x⁰+r_mx)

r_m^3/2

(4.2) converges weakly inW_loc^1,2(Rⁿ)to a blow-up limitu₀. Thenu₀is a homogeneous function of degree3/2, i.e.

u₀(λx)=λ^3/2u₀(x) for anyx∈Rⁿandλ >0.

(iii) Letu_mbe a converging sequence of (ii). Thenu_mconverges strongly inW_loc^1,2(Rⁿ).

(iv) Letx⁰∈Ωbe such thatx⁰_n=0. Then M_x0,u(0+)= lim

r→0+r⁻ⁿ⁻¹

Br(x⁰)

x_n⁺χ_{u>0}dx,

and in particularM_x0,u(0+)∈ [0,+∞). Moreover,M_x0,u(0+)=0implies thatu0=0inRⁿfor each blow-up limitu0of (ii).

(v) The functionx→M_x,u(0+)is upper semicontinuous in{x_n=0}.

(vi) Letu_mbe a sequence of variational solutions of (3.1)with nonlinearityf_min a domainΩ_m, where Ω₁⊂Ω₂⊂ · · · ⊂Ω_m⊂Ω_m+1⊂ · · · and

∞ m=1

Ω_m=Rⁿ,

such thatu_m converges strongly tou₀inW_loc^1,2(Rⁿ),χ_{um>0} converges weakly inL²_loc(Rⁿ)toχ₀, andf_m(u_m) converges to0locally uniformly inRⁿ. Thenu₀is a variational solution of (3.1)with nonlinearityf =0inRⁿ and satisfies the monotonicity formula(withf =0), but withχ_{u0>0}replaced byχ₀. Moreover, for eachx⁰∈Rⁿ such thatx_n⁰=0, and all instances ofχ_{u0>0}replaced byχ₀,

M_x0,u0(0+)lim sup

m→∞ M_x0,um(0+).

Proof. (i) By Remark4.2, u(x)C₁

x_n⁺3/2

locally inΩ

and r⁻ⁿ⁻²K(r)C₂r^−1/2 for eachx⁰∈Ωsatisfyingx_n⁰=0. (4.3) Thusr→r⁻ⁿ⁻²K(r)is integrable at such pointsx⁰, and from Theorem3.4we infer that the functionM_x0,u has a finite right limitM_x0,u(0+).

(ii) For each 0< σ <∞the sequenceu_mis by assumption bounded inC^0,1(B_σ). For any 0< < σ <∞, we write the identity (3.8) in integral form as

2 σ

r⁻ⁿ⁻¹

∂Br(x⁰)

∇u·ν−3 2 u r

2

dHⁿ⁻¹dr=M(σ )−M()− σ

r⁻ⁿ⁻²K(r) dr. (4.4)

It follows by rescaling in (4.4) that 2

B_σ(0)\B(0)

|x|⁻ⁿ⁻³

∇um(x)·x−3 2um(x)

2

dx

M(r_mσ )−M(r_m)+

r_mσ

r_m

r⁻ⁿ⁻²K(r)dr→0 asm→ ∞, which yields the desired homogeneity ofu₀.

(iii) In order to show strong convergence ofu_minW_loc^1,2(Rⁿ), it is sufficient, in view of the weakL²-convergence of∇u_m, to show that

lim sup

m→∞

Rⁿ

|∇um|²η dx

Rⁿ

|∇u0|²η dx

for eachη∈C₀¹(Rⁿ). Letδ:=dist(x⁰, ∂Ω)/2. Then, for eachm,umis a variational solution of um= −rm^1/2f

rm^3/2um

inBδ/r_m∩ {um>0},

|∇um|²=xn onBδ/r_m∩∂{um>0}. (4.5)

Sinceu_mconverges tou₀locally uniformly, it follows from (4.5) thatu₀ is harmonic in{u₀>0}. Also, using the uniform convergence, the continuity ofu₀and its harmonicity in{u₀>0}we obtain as in the proof of (3.10) that

Rⁿ

|∇u_m|²η dx= −

Rⁿ

u_m

∇u_m· ∇η−r_m^1/2f r_m^3/2u_m

η dx

→ −

Rⁿ

u₀∇u₀· ∇η dx=

Rⁿ

|∇u₀|²η dx

asm→ ∞. It therefore follows thatu_mconverges tou0strongly inW_loc^1,2(Rⁿ)asm→ ∞.

(iv) Let us take a sequencer_m→0+such thatu_mdefined in (4.2) converges weakly inW_loc^1,2(Rⁿ)to a functionu₀. Note that by the definition of a variational solution,u_m andu₀ are identically zero inx_n0. Using (iii) and the homogeneity ofu₀, we obtain that

mlim→∞M_x0,u(r_m)=

B1

|∇u₀|²dx−3 2

∂B1

u²₀dHⁿ⁻¹+ lim

r→0+r⁻ⁿ⁻¹

Br(x⁰)

x_n⁺χ_{u>0}dx

= lim

r→0+r⁻ⁿ⁻¹

Br(x⁰)

x_n⁺χ_{u>0}dx.

ThusM_x0,u(0+)0, and equality implies that for eachτ >0,u_mconverges to 0 in measure in the set{x_n> τ}as m→ ∞, and consequentlyu0=0 inRⁿ.

(v) For eachδ >0 we obtain from the monotonicity formula (Theorem3.4), Remark4.2as well as the fact that lim_x→x⁰Mx,u(r)=M_x0,u(r)forr >0, that

M_x,u(0+)M_x,u(r)+C₂√

rM_x0,u(r)+δ

2M_x0,u(0+)+δ, if we choose for fixedx⁰firstr >0 and then|x−x⁰|small enough.

(vi) The fact thatu0is a variational solution of (3.1) and satisfies the monotonicity formula in the sense indicated follows directly from the convergence assumption. The proof for the rest of the claim follows by the same argument as in (v). 2

In the two-dimensional case, we identify the possible values ofM_x0,u(0+), and classify the blow-up limits atx⁰in terms of the value ofM_x0,u(0+), which leads to the proof of asymptotic homogeneity of the solution.

Theorem 4.5(Two-dimensional case). Letn=2, letube a variational solution of (3.1)satisfying Assumption4.1, letx⁰∈Ω be such thatx₂⁰=0, and suppose that

r^−3/2

Br(x⁰)

√x2d|∇χ_{u>0}|C0

for allr >0such thatB_r(x⁰)Ω. Then the following hold:

(i) M(0+)∈

0,

B1

x₂⁺χ_{x:π/6<θ <5π/6}dx,

B1

x₂⁺dx

.

(ii) IfM(0+)=

B1x₂⁺χ_{x:π/6<θ <5π/6}dx, then u(x⁰+rx)

r^3/2 →

√2

3 ρ^3/2cos 3

2

min

max

θ,π 6

,5π

6

−π 2

asr→0+, strongly inW_loc^1,2(R²)and locally uniformly onR², wherex=(ρcosθ, ρsinθ ).

(iii) IfM(0+)∈ {0,

B₁x₂⁺dx}, then u(x⁰+rx)

r^3/2 →0 asr→0+,

strongly inW_loc^1,2(R²)and locally uniformly onR².

Proof. Consider a blow-up sequenceu_mas in Lemma4.4(ii), wherer_m→0+, with blow-up limitu₀. Because of the strong convergence ofu_mtou0inW_loc^1,2(R²)and the compact embedding fromBVintoL¹,u0is a homogeneous solution of

0=

R²

|∇u₀|²divφ−2∇u₀Dφ∇u₀ dx+

R²

(x₂χ₀divφ+χ₀φ₂) dx (4.6)

for anyφ∈C₀¹(R²;R²), whereχ₀is the strongL¹_loc-limit ofχ_{um>0}along a subsequence. The values of the function χ₀are almost everywhere in{0,1}, and the locally uniform convergence ofu_mtou₀implies thatχ₀=1 in{u₀>0}.

The homogeneity ofu₀and its harmonicity in{u₀>0}show that each connected component of{u₀>0}is a cone with vertex at the origin and of opening angle 120^◦. Sinceu=0 in{x₂0}, this shows that{u₀>0}has at most one connected component. Note also that (4.6) implies thatχ₀is constant in each open connected setG⊂ {u₀=0}^◦that does not intersect{x₂=0}.

Consider first the case when{u₀>0}is non-empty, and is therefore a cone as described above. Letzbe an arbitrary point in∂{u₀>0} \ {0}. Note that the normal to∂{u₀>0}has the constant valueν(z)inB_δ(z)∩∂{u₀>0}for some δ >0. Plugging inφ(x):=η(x)ν(z)into (4.6), whereη∈C₀¹(B_δ(z))is arbitrary, and integrating by parts, it follows that

0=

∂{u0>0}

−|∇u₀|²+x₂(1− ¯χ₀)

η dH¹. (4.7)

Hereχ¯0denotes the constant value ofχ0in the respective connected component of{u0=0}^◦∩ {x2 =0}. Note that by Hopf’s principle,∇u0·ν =0 onBδ(z)∩∂{u0>0}. It follows therefore thatχ¯0 =1, and hence necessarilyχ¯0=0.

We deduce from (4.7) that |∇u0|²=x2 on ∂{u0>0}. Computing the solutionu0 of the corresponding ordinary differential equation on∂B1yields that

u0(x)=

√2

3 ρ^3/2cos 3

2

min

max

θ,π 6

,5π

6

−π 2

, wherex=(ρcosθ, ρsinθ ), and thatM(0+)=

B1x₂⁺χ_{x:π/6<θ <5π/6}dxin the case under consideration.

Consider now the caseu₀=0. It follows from (4.6) thatχ₀is constant in{x₂>0}. Its value may be either 0 in which caseM(0+)=0, or 1 in which caseM(0+)=

B1x₂⁺dx.

Since the limit M(0+) exists, the above proof shows that it can only take one of the three distinct values {0,

B1x⁺₂χ_{x:π/6<θ <5π/6}dx,

B1x₂⁺dx}. The above proof also yields, for each possible value ofM(0+), the ex- istence of auniqueblow-up limit, as claimed in the statement of the theorem. 2

Under the assumption that the free boundary is locally an injective curve, we now derive its asymptotic behavior as it approaches a stagnation point.

Fig. 4. Stokes corner. Fig. 5. Full density singularity.

Fig. 6. Left cusp. Fig. 7. Right cusp.

Theorem 4.6(Curve case). Letn=2, letube a weak solution of (3.1)satisfying Assumption4.1, and letx⁰∈Ω be such that x₂⁰=0. Suppose in addition that ∂{u >0} is in a neighborhood of x⁰ a continuous injective curve σ:(−t₀, t₀)→R²such thatσ=(σ₁, σ₂)andσ (0)=x⁰. Then the following hold:

(i) If M(0+)=

B1x₂⁺χ_{x:π/6<θ <5π/6}dx, then (cf. Fig.4) σ₁(t) =x₁⁰ in(−t₁, t₁)\ {0} and, depending on the parametrization, either

t→lim0+

σ2(t)

σ₁(t)−x₁⁰= 1

√3 and lim

t→0−

σ2(t)

σ₁(t)−x₁⁰= − 1

√3, or

t→lim0+

σ2(t)

σ₁(t)−x₁⁰= − 1

√3 and lim

t→0−

σ2(t)

σ₁(t)−x₁⁰= 1

√3. (ii) IfM(0+)=

B₁x₂⁺dx, then(cf. Fig.5)σ1(t) =x₁⁰in(−t1, t1)\ {0},σ1−x₁⁰changes sign att=0and

tlim→0

σ₂(t) σ₁(t)−x₁⁰=0.

(iii) IfM(0+)=0, then(cf. Fig.6and Fig.7)σ1(t) =x₁⁰in(−t1, t1)\ {0},σ1−x₁⁰does not change its sign att=0, and

tlim→0

σ₂(t) σ₁(t)−x₁⁰=0.

Proof. We may assume thatx₁⁰=0. Moreover, for eachy∈R²we define argy as the complex argument ofy, and we define the sets

L±:=

θ₀∈ [0, π]:there ist_m→0±such that argσ (t_m)→θ₀asm→ ∞ . Step 1:BothL₊andL₋are subsets of{0, π/6,5π/6, π}.

Indeed, suppose towards a contradiction that a sequence 0 =tm→0, m→ ∞exists such that argσ (tm)→θ0∈ (L+∪L−)\ {0, π/6,5π/6, π}, letrm:= |σ (tm)|and let

u_m(x):=u(r_mx) rm^3/2

.

For eachρ >0 such thatB˜:=B_ρ(cosθ₀,sinθ₀)satisfies

∅ = ˜B∩

(x,0):x∈R

∪

(x,|x|/√

3):x∈R ,

we infer from the formula for the unique blow-up limitu₀(see Theorem4.5) that the signed measure u_m(B)˜ →u₀(B)˜ =0 asm→ ∞.

On the other hand, u_m= −r_m^1/2f

r_m^3/2u_m +√

x₂H¹∂_{um>0},

which implies, sinceB˜∩∂{u_m>0}contains a curve of length at least 2ρ−o(1), that 0←um(B)˜ c(θ0, ρ)−C1r_m^1/2 asm→ ∞,

wherec(θ₀, ρ) >0, a contradiction. Thus the property claimed in Step 1 holds.

Step 2: It follows thatσ₁(t) =0 for all sufficiently smallt =0. Now a continuity argument yields that bothL₊ andL₋are connected sets. Consequently

₊:= lim

t→0+argσ (t )

exists and is contained in the set{0, π/6,5π/6, π}, and ₋:= lim

t→0−argσ (t )

exists and is contained in the set{0, π/6,5π/6, π}.

Step 3: In the case M(0+)=

B1x₂⁺χ_{x:π/6<θ <5π/6}dx, we know now from the formula for u₀ that u0(B1/10(√

3/2,1/2)) >0 and thatu0(B1/10(−√

3/2,1/2)) >0. It follows that the set {₊, ₋}contains both π/6 and 5π/6. But then the sets{₊, ₋}and{π/6,5π/6}must be equal, and the fact thatu=0 onx2=0 implies case (i) of the theorem.

Step 4: In the case M(0+)∈ {0,

B1x₂⁺dx}, we have that u₀(B_1/10(±√

3/2,1/2))=0, which implies that ₊, ₋∈ {/ π/6,5π/6}. Thus₊, ₋∈ {0, π}. Using the fact that u=0 onx₂=0, we obtain in the case₊ =₋ that M(0+)=

B1x₂⁺dx and in the case₊=₋thatM(0+)=0. Together, the last two properties prove case (ii) and case (iii) of the theorem. 2

Remark 4.7. In[23]we used a strong version of the Rayleigh–Taylor condition (which is always valid in the case of zero vorticity) in order to prove that the cusps of case (iii) are not possible. Unfortunately we do not have the Rayleigh–Taylor condition (4.1) in the case with non-zero vorticity, and the method of[23]breaks down here. Still weconjecturethat the cusps in case (iii) arenotpossible when assuming the Rayleigh–Taylor condition.

5. Partial regularity at non-degenerate points

Definition 5.1(Stagnation points).Letu be a variational solution of (3.1). We callS^u:= {x∈Ω:x_n=0 andx∈

∂{u >0}}the set ofstagnation points.

Throughout the rest of this section we assume thatn=2.