VORTICITY JUMPS IN STEADY WATER WAVES

There exists a large family of water waves with jump discontinuities in the vorticity. These waves travel at a constant speed. They are two-dimensional, periodic, symmetric, and subject to the influence of gravity. Some of them have large amplitudes. Their existence is proven using local and global bifurcation theory, together with elliptic theory of weak solutions with nonlinear boundary conditions.


Introduction.
There is an extensive literature on irrotational waves.For instance, in a key paper [1], J. T. Beale proved the existence of solitary irrotational water waves of small amplitude.Later, in the rotational case, periodic waves of small and large amplitude were proven to exist [3].These periodic waves are symmetric and have vorticity at least of class C 1 .On the other hand, a current can induce sudden changes in vorticity.Wind blowing over a water surface usually induces a thin layer of high vorticity [11].Discontinuities in the vorticity may also occur due to jumps in salinity or temperature.It is only recently that numerical simulations for discontinuous vorticities have been carried out [9,10].They indicate the appearance of flow patterns that are not encountered for flows with continuous vorticity [13].In this paper we survey periodic symmetric two-dimensional waves propagating at constant speed at the surface of a layer of water and announce the existence of flows whose vorticity may be discontinuous.For details, see the paper of A. Constantin and the author [4].
As is well known, the governing equations for steady two-dimensional periodic water waves come in three forms, which we may call the velocity form, the stream function form and the height form (via the partial hodograph transformation).Since we permit our vorticity to be discontinuous, the equations must be expressed in a weak way and the solutions taken in the sense of distributions.The weak height equation, (9) below, turns out to be quite elegant.
Consider a 2D incompressible steady water wave over an impermeable bed {y = −d} and with a free surface {y = η(x)}.For the vorticity ω = v x − u y to be discontinuous, u and v would have to be Lipschitz at best.But because C 0,1 does not enjoy good elliptic estimates, we will work with C α for some 0 < α < 1.The classical Eulerian velocity formulation of a 2D steady water wave is ( Here P atm = atmospheric pressure, g = gravitational acceleration, d = average depth of the water.Say the period = 2π.We can rewrite the Euler equation ( 2) in the weak form The relative stream function ψ is defined by The last equation is the Bernoulli relation.Here and γ is the vorticity function defined on [p 0 , 0] such that ω = γ(ψ).We will assume that u < c throughout the fluid.(6) Because of (6) the partial hodograph (or semi-Lagrangian or Dubreil-Jacotin) transform q = x, p = −ψ, h(q, p) = y + d has nonzero Jacobian and leads to the height formulation Here R is the open rectangle (−π, π) × (p 0 , 0) and we assume the height h is even and of period 2π in the q-variable.The elliptic nondegeneracy condition is equivalent to assumption (6).In turn, (7) has the weak form where Γ(p) = p 0 γ(−s) ds.
Our fundamental assumption will be that Γ ∈ C α , so that the vorticity ω = γ(ψ) might not even be a function!Then we will only have h, ψ, η ∈ C 1+α and u, v ∈ C α .However, the weak forms of the equations do make sense for such functions.In certain places we will need more than this amount of regularity.So for simplicity our main result is stated for γ being a step function.
Theorem 1. Assume that the vorticity function γ is a step function.Let 0 < α < 1. Assume a "local bifurcation" condition (see below).Then there is a connected set C of symmetric periodic solutions u, v ∈ C α , η ∈ C 1+α which contains a laminar flow as well as a sequence for which max u → c.Furthermore, η ∈ C ∞ ; u, v ∈ W 1,r ; P ∈ W 1,r ∩ W 2,r/2 in each period, where r = 2/(1 − α).
In addition, there is a path-connected subset K ⊂ C, with a global continuous parametrization and locally injective C 1 reparametrizations, which contains a laminar flow as well as a sequence for which max u → c.
Several difficulties occur due to the non-regularity of the vorticity, namely: working with non-differentiable functions u, v; extending the classical Schauder estimate to a weak space; proving compactness in the global bifurcation; establishing the nodal properties of weak solutions; applying DeGiorgi-Nash-type estimates with nonlinear boundary conditions; and proving the equivalence of the several formulations for weak solutions.
2. Linear facts.In this short section we use the classical notation u(x) with x ∈ Ω ⊂ R n .When the nonlinear problem ( 9) is linearized around a particular solution, we get a linear elliptic problem in divergence form of the following type: The coefficients are only assumed to be C α .We assume uniform ellipticity on the a ij and uniform obliqueness If the given problem has a solution u ∈ C 1 ( Ω), then this solution has additional regularity, u ∈ C 1+α ( Ω) with All the norms are in Hölder spaces, while | • | 0 denotes the L ∞ norm.The constant C depends only on the bounds of the stated norms of the coefficients and the domain and on the ellipticity and oblique constants.
Theorem 3. (Hopf-type maximum principle) Let Ω be an open connected set with a C 1+α boundary.Let L be uniformly elliptic of divergence form as in (12) with coefficients in C α such that f ≤ 0. Let u be a solution to Lu = 0 in C 1 ( Ω).Then there is no interior maximum (unless u is a constant), and at any point where the maximum is attained, we have ∂u/∂ν > 0, where ν is the exterior normal.
3. Bifurcation.In this paper we do not discuss local bifurcation but simply state a sufficient condition (see [4]).The condition holds for instance if γ ≤ 0, in which case p 1 = 0. where then there is a local bifurcating curve C loc of solutions of ( 9) - (11).
In order to show that the bifurcation curve can be globally continued, we define the mapping where The choice of the divergence form of Y 1 is crucial in order to be able to use the desired Schauder-type estimate of Theorem 2. We also define Let C δ be the component of S δ that contains the point (Q * , H * ).The following lemmas are required for the global theory.
To prove this lemma, we take a sequence (f n , g n ) converging in Y and let (Q n , h n ) be the corresponding solutions of the PDE with right-hand side (f n , g n ) We must find a subsequence converging (strongly) in R × X.The proof is not straightforward. . .but the key tool is the Schauder estimate.
Indeed, one proves that the linearized problem has a finite-dimensional nullspace and a closed range.But at the bifurcation point, G h (Q * , h * ) has a null space of dimension 1 and an image of codimension 1, so that it has index zero.Hence Then there is a connected set C δ ⊃ C loc in R × X such that one of the following 3 alternatives holds. (i Furthermore, there is a continuous curve C loc ⊂ K δ ⊂ C δ such that each point of K δ has a neighborhood where K δ has an analytic parametrization.Either (i) or (iii) holds for K δ or else K δ is a closed loop.
It is interesting that the two parts of this theorem are proven rather differently.The proof of the first part, involving C δ , is based on the Healey-Simpson degree [7] together with the ideas of Rabinowitz.The second part, involving K δ , follows from the analytic bifurcation theory of Dancer, Buffoni and Toland [2,14,15].
The next theorem states that the water waves are infinitely differentiable along the streamlines.Theorem 6. (Regularity in q) For every solution in C δ and every ≥ 0, we have Proof.Take any solution h along the continuum.Thus G 1 (h) = 0 and G 2 (Q, h) = 0 on T , where h ∈ X, so that h q , h p ∈ C α .Take the difference in q, that is, ξ (q.p) = [h(q + , p) − h(q, p)]/ .
When the equations for h = h(q + , p) and for h = h(q, p) are subtracted, the Γ drops out and we get a uniformly elliptic problem By Theorem 2 we deduce that ξ is bounded in C 1+α .Therefore ξ has a limit in C 1+α , so that h q ∈ C 1+α , ∀ α < α.Now h q ∈ C 1+α ⊂ C 1 satisfies an elliptic PDE with an oblique boundary condition on T , with coefficients in 4. Nodal pattern and bounds.We wish to eliminate the second alternative in the Global Bifurcation Theorem.This is done by tracking the nodes of h q .In order to carry it out, we need some additional regularity in the variable p. Theorem 7. (Additional regularity) Assume that Γ is of class C 1+α for p near 0 and p near p 0 .Then ∂ 2 p h also belongs to C α near both the top T and the bottom B. The same is true for ∂ q ∂ 2 p h for all > 0. The proof is similar to that of Theorem 6 except that we work in thin rectangles near the top and the bottom.Now for h ∈ C δ , we wish to apply the maximum principle to φ ≡ h q .So we differentiate (9) to get the PDE This can be written out more explicitly wherever there are no ∂ 2 p terms.But when we do it, there are the that can only be considered in the weak form.That is, we cannot explicitly take two derivatives with respect to p in the whole domain but only near the top and the bottom.What saves the day is that in the next theorem we require second derivatives with respect to p only near the four corners.
Theorem 8. (Nodal properties) Let R + be the smaller open rectangle from q = 0 to q = π, that is, from crest to trough.Then all solutions (Q, h) ∈ C δ satisfy h q < 0 in R + as well as on its top edge.It also satisfies the strict inequalities: h qq < 0 on the left edge, h qq > 0 on the right edge, h qp < 0 on the bottom edge, as well as strict inequalities at the corners: Then we prove that the inequalities stated in the theorem define a nonempty open set in R × X.Moreover, C δ is a connected set in R × X.Thus if the first part of the theorem were false, there would exist (Q, h) ∈ C δ for which one of the inequalities is an equality although it is a limit of elements for which all the inequalities are strict.The following reasoning concerns this element (Q, h).
We have plenty of regularity in q but not so much in p.Let φ = ∂ q h.Then φ ∈ C 1+α and it satisfies an explicit equation of the form (with b j (q, p)φ j + c(q, p)φ = 0 on T.
We have h q = φ ≤ 0 in R+ .By Theorem 3, φ < 0 in R + and we have the usual Hopf inequalities on ∂R + .It remains to prove that φ < 0 on top, as well as inequalities at the four corners: φ q > 0 at the top right corner, φ q < 0 at the top left corner, φ qp > 0 at the bottom right corner, and φ qp < 0 at the bottom left corner.
If φ = 0 at some point on the top {0 < q < π, p = 0}, then at that point we also have by simple calculus that φ q = 0.By the differentiated boundary condition we have φ p = 0, which contradicts the Hopf condition φ p > 0. Therefore φ < 0 on the top.
At the top right corner (π, 0) we will use Serrin's edge point theorem [12].It is valid here because it is a local theorem in a neighborhood of the corner point, and in such a neighborhood we have sufficient regularity.By oddness we have h q = h qqq = h qp = h qpp = 0 at (π, 0).Here we have used the extra local regularity of the solution near the corner.Since h qq > 0 on the right side, we have by continuity that h qq ≥ 0 at the corner.So suppose h qq = 0 at the corner and we seek a contradiction.Take the boundary condition on the top and differentiate twice with respect to q.The result is many terms, but only one term that does not have a factor of h q or h qq or h qp so we get h p h qqp = 0, whence h qqp = 0. Hence all the first and second derivatives of φ = h q vanish at (π, 0).This contradicts Serrin's theorem.Therefore h qq (π, 0) > 0. The other three corners are treated similarly.Theorem 8 implies that the second alternative of Theorem 5 cannot occur, nor can K δ be a closed loop.Now we wish to simplify the first alternative of Theorem 5.One of the possibilities is that along the continuum C δ , the solutions h become unbounded in C 1+α ( R).
Proof.On the contrary, suppose that |h p | 0 + Q is bounded along the continuum C δ .We have to prove a C 1+α bound on h, uniformly along C δ .
by definition of p 0 .Thus h is also bounded along C δ .Next we consider h q = φ, which satisfies the elliptic divergence-form equation with coefficients in C α .By Theorem 3 there is no interior maximum.But on the bottom B, φ = 0, while on the top T we have Next we claim that h ∈ W 2,r and thus also in C 1+α .(The norms are taken over R).Indeed, since we already know that h ∈ C 1+α and also that h qq , h qp ∈ C α ⊂ L r , it suffices to prove that h pp ∈ L r .We already know that h pp ∈ C α near the top and bottom boundaries of R. Formally, we just use the partial differential equation written in non-divergence form to express h pp in terms of γ and h qq , h qp , h p , h q , h, all of which are in L r or better.We can make this argument rigorous by using difference quotients in the variable p.
It remains to prove that sup C δ |∇h| α < ∞.The solutions are sufficiently regular but we must show that the estimate is uniform over the continuum.The estimate is local.In the interior of R (away from the top and the bottom), we use the DeGiorgi-Nash estimate applied to both h q and h p ; see Theorem 8.24 in [6].In a thin strip near the bottom boundary B where h = 0, we know that u is of class C 2 and we use the global Ladyzhenskaya-Uralsteva estimate applied to h; see Theorem 12.7 in [6].In a thin strip near the top boundary T with its nonlinear boundary condition, we know by Theorem 7 that u is of class C 2 and we use the generalized global Ladyzhenskaya-Uralsteva estimate applied to h; see Lemma 2.4 in [8].The conclusion of these three estimates is that there exists µ > 0 such that sup In order to improve µ to the given α, we first consider h q , to which we apply Theorem 2, thus obtaining the estimate sup C δ |h q | 1+µ < ∞.We then employ the PDE (7) where each of the last three terms are bounded in L r along the continuum, or better.Therefore we obtain the estimate sup where ∇ 2 denotes all the second derivatives.

5.
Return to the physical formulation.We will now return to the stream function formulation and then to the velocity formulation (1)-(3).We have proven that our solutions h belong to W 2,r where r = 2/(1 − α).We need to recover first ψ and η and then u, v and P .The free surface is easily identified as By Theorem 6, η ∈ C ∞ .Let F = 1/h p .For a fixed x ∈ R we solve the ordinary differential equation ψ y (x, y) = −1/h p (x, −ψ(x, y)) (16) with initial data ψ y (x, η(x)) = 0. Since 1/h p ∈ C α (R), Peano's theorem yields the existence of a local solution.Since h p ≥ δ > 0 throughout R for some δ > 0, any solution y → ψ(x, y) is strictly increasing as y decreases and can be continued until it reaches the value −p 0 at some y(x) < η(x).Differentiating the expression with respect to y ∈ [y(x), η(x)] and using (16), we see that the expression y By the implicit function theorem, h p > 0 and (17) ensure the uniqueness of the solution to (16).Thus for any x ∈ R we can define ψ(x, y) on the whole interval [−d, η(x)], and To show that the constructed stream function ψ satisfies (5), observe that we already know that ψ = −p 0 on y = −d, and ψ = 0 on y = η(x).Differentiating ψ, we obtain By inspection we see that ψ ∈ W 2,r (D η ).The Bernoulli condition is immediate from (10).To obtain the PDE for ψ, notice that and It follows immediately that To return to the velocity formulation, we define u = c + ψ y and v = −ψ x , which belong to W 1,r , and It is obvious that u x + v y = 0. Furthermore, using the chain rule (see below) on Γ(−ψ), These are the Euler equations (2).Moreover, ∆P = −2u y v x − v 2 y − u 2 x ∈ L r/2 so that P ∈ W 2,r/2 .
We now justify the chain rule ∇Γ(−ψ) = −Γ (−ψ)∇ψ, which was used above.(General composition results in Sobolev spaces require a Lipschitz condition Γ ∈ W 1,∞ to ensure the validity of the chain rule.Our assumption Γ ∈ W 1,r is less restrictive because we can take advantage of the condition that ψ y < 0.) Performing the change of variables q = x, p = −ψ(x, y), we have and In particular, for ω ∈ L r we have We have γ ∈ L r so that Γ ∈ W 1,r ⊂ C α .Since ψ ∈ W 2,r ⊂ C 1+α , we have Γ(−ψ) ∈ C α .For any test function φ, we have In fact, we can prove an equivalence of the three formulations, always assuming u < c or ψ y < 0 or h p > 0. (iii) the height formulation for h ∈ W 2,r per (R × (p 0 , 0)) and Γ ∈ W 1,r [p 0 , 0].We now continue the proof of Theorem 1.We define C and K to be the unions of C δ and K δ as δ → 0. We conclude from the previous considerations that either Q → ∞, or max R {h p } → ∞, or the equation loses ellipticity somewhere.Then we deduce (just as in the smooth case analyzed in [3]) that either sup R u n → c or inf R u n → −∞.
Next we claim that there is a lower bound for u at the crest: To prove this claim, we use (9) on the crest line q = 0 to get The right side is positive in view of the nodal pattern established in Theorem 8 and since h q = 0 at q = 0. We integrate along the crest line from p to 0 to get Solving this inequality for h p , we obtain Similarly, integrating along the trough line q = π we obtain On the other hand, since v = 0 at the crest and at the trough, from (5) we evaluate Q at both points to obtain where the function is strictly convex on (−2Γ min , ∞) with its minimum at λ = λ 0 .. Now ( 22) implies (21).As already pointed out, P ∈ W ψ).We now make the additional assumption that γ is bounded by a constant M .Thus ∆(P − M 2 (y + d) 2 /2) ≤ 0. By the weak maximum principle for weak solutions [6], the minimum of P − M 2 (y + d) 2 /2 is attained on the flat bed or on the free surface.Because the flux p 0 is fixed, there must be points y n for which u n (0, y n ) → c.This means that in any case there are waves in C arbitrarily close to critical (u close to c).Similar reasoning applies to the curve K. 6. Final remarks.
6.1.Numerical simulation of waves of large amplitude.Due to the lack of explicit exact solutions for gravity waves of large amplitude in finite depth, we resort to numerics (see [9,10]).We illustrate the results for period T=2π, gravity g = 9.8 and flux p 0 = −2.We depict two outcomes for a surface layer of vorticity strength γ = 10 above an irrotational layer.In the top figure a surface layer of "thickness" 25 % (where 25 % means that in the (q, p) coordinates the layer goes a quarter of the way down from the top of the rectangle R).In the bottom figure, the surface layer has "thickness" 35 %.On the right are the computed bifurcation diagrams (maximum amplitude versus Q).On the left are the pictures of almost stagnant waves in the moving plane; that is, they are waves that are very near the end of the bifurcation curve K.It is interesting that in the top figure the stagnation occurs at the crest, while in the bottom figure it occurs internally where the streamlines are most separated.The vertical scale refers to the elevation above the flat bed.

Lemma 3 .
Let γ ∈ L r with r = 2 1−α and let γ ∈ C α for p near 0 and p near p 0

Figure 1 .
Figure 1.Numerical simulations for constant positive vorticity adjacent to the free surface: almost stagnant waves on the left and the bifurcation diagrams on the right for two case studies.