On isolated vorticity regions beneath the water surface

We present a class of vorticity functions that will allow for isolated, circular vorticity regions in the background of still water preceding the arrival of a tsunami wave at the shoreline.


Introduction
In two seminal papers [1,2], Constantin and Johnson proposed a model for studying what happens beneath the surface of the ocean before the arrival of a destructive tsunami wave at the shoreline. As opposed to other enterprises, the fluid does not move here irotationally beneath its surface in a global manner, but the water is still with the exception of some isolated, bounded regions where it moves with vorticity. Given the particular character of the vorticity region discussed in [1], the authors envisaged the possibility of more permissive shapes for the boundary of such regions in [2]: the circular vorticity region. The analysis has been put on a firm ground via a dynamical systems approach in the paper [3].
Our intention in this note is to investigate the key features of the technical proof from the latter work and, by relaxing them, to propose a new candidate for the vorticity function in the circular vorticity region. In a loose manner, the new function is a slight deviation from the Constantin-Johnson vorticity.
Local existence I. Assume that the continuous function f from (1) satisfies the following restriction: for any a ≥ 1 there exists η = η(a) ∈ 3, 7 2 such that In particular, for the vorticity function from (3) we have |f (ξ)| ≤ |ξ|+ |ξ| ≤ 1 + η 4 + 1 + η 4 a ≤ ηa whenever a ≥ 1 and η ≥ 28 9 . We claim that the integral equation has a solution in X = C([0, 1], R). The proof of our claim is based on the Schauder fixed point theorem and on the Ascoli-Arzela relative compactness criterion. Notice first that, by means of the L'Hôpital rule, we have lim [ξf (ψ(ξ))] = 0, which means that the function ξ → 1 ξ ξ 0 τ f (ψ(τ )) can be prolongated backwards to zero as a continuous function. Now, consider D the closed ball of radius ηa 4 and center a of the Banach space B = (X, · ∞ ). Obviously, the ball is convex. Let T : D → X be an operator with T (ψ) given by the right-hand member of the equation (7) for any ψ ∈ D. We have the estimates: (i) boundedness of T (D), namely Remark that we have obtained actually that T (D) ⊆ D. (ii) Equicontinuity of T (D), which follows from According to the Ascoli-Arzela criterion, the set T (D) is relatively compact in B, yielding that the operator T transports bounded sets into relatively compact ones. Finally, (iii) the operator T is continuous. Since the function is uniformly continuous, for every ε > 0 there exists δ = δ(ε) > 0 such that, According to the Schauder fixed point theorem, the operator T has a fixed point ψ ∈ D. In particular, This means that ψ ∈ C 1 ([0, 1], R) verifies (7) and, given its smoothness, yields the solution (ψ, ψ ′ ) ∈ C([0, 1], R 2 ) of the system (1) with the starting point (ψ(0), ψ ′ (0)) = (a, 0). In particular, by being an element of D, the function ψ satisfies the inequality Local existence II. We are interested here in the backward existence and uniqueness of the solution to (1). Assume that, in addition to (6), In the particular case of (3), we have Take T ≥ 6 and introduce the system of integral equations and introduce the metric Notice the by-product inequality 11 · ln λ > L 5 4 · 3 ln λ + 1 2 . This is equivalent to 44 ln λ 15 ln λ+2 > L. Since the function x → 44x 15x+2 is increasing in (0, ln 3) and 44 ln 3 15 ln 3+2 > 5 2 > L(a), such a constant λ exists always. Then, M = (D, d) constitutes a complete metric space. Introduce further the operator T : D → X with T (ψ, β) given by the right-hand member of the system (9) for any (ψ, β) ∈ D: T (ψ, β) = (T 1 (ψ, β), T 2 (ψ, β)).
We have the estimates They read as T (D) ⊆ D.
We have and, since  In the particular case when ψ T = a = u 0 = 1 and β T = 0, see e.g., the vorticity function (3), the uniqueness of solution leads to ψ ≡ u 0 in [ √ T 2 − 1, T ]. This remark has an essential consequence: if there exists a non-equilibrium solution (ψ, β) of system (1) that will eventually reach the equilibrium (u 0 , 0) then either the equilibrium is attained in r < 6 "units of time" or it is attained in "infinite time", that is lim r→+∞ (ψ(r), β(r)) = (u 0 , 0). which make sense on a small right-neighborhood of r = 1 via the Peano existence theorem for (1). Since E(r) ≤ E(1) for as long as the solution (ψ, β) exists to the right of 1 (recall that E ′ (r) ≤ 0), the estimate (2) shows that both of the quantities ψ(r), β(r) are bounded on their maximal interval of existence. According to Wintner's non-local existence theorem, this implies the global existence in the future for the solution (ψ, β).
For the sake of contradiction, suppose that lim inf r→+∞ R(r) > 1 + ε. Then, by means of (12), we have (2), observe that the quantity E(r) is bounded from below and monotone non-increasing. This yields lim r→+∞ E(r) ∈ R, and so we have r dr. The convergence of the latter improper integral leads to The function θ being strictly decreasing, we deduce the existence and uniqueness of the increasing, unbounded from above sequences (r + n ) n≥n 0 , (r − n ) n≥n 0 , with r + n > r − n , given by the formulas . Remark also that and, accordingly, r + n−1 < r − n , n ≥ n 0 + 1. We have the estimates Taking into account that we deduce that for any m ≥ n 0 + 1. Notice the by-product inequalities 2η In conclusion, via (15)-(17), we obtain that We have reached a contradiction. Polar coordinates III. Escaping E = 0. The situation depicted in Figure 3 below is inspired by the case of (3). There, see [3], in quadrant I, that is when θ ∈ 0, π 2 +2π·Z, the algebraic curve E(ψ, β) = β 2 − 4 3 |ψ| 3/2 +ψ 2 = 0 is concave, the point (0, 0) is singular and at the "smooth peek" (ψ + , 0), where ψ + = 16 9 , the curvature is 9 4 . The energy of the non-null equilibria is also negative, . A consequence of the concavity reads as follows: in quadrant I, each oblique straight line splits the region E ≤ 0 into two disjoint parts, the "+" and the "−", see Figures 1, 3.

Figure 3
A solution (ψ, β) of (1) starting from (a, 0) for some very great a > 1 will reach the ring 1 + ε < R < 1 + δ in finite time r. Since ψ + > 1 ≥ u 0 , the positive quantities ε, δ can be taken small enough, meaning such that ψ + > 1 + δ > 1 + ε > (1 + c) 1/ν ≥ 1, for the ring to intersect the region E < 0 of the phase plane as presented in Figure 3. The decay of E(r), detailed previously, has the following consequences: if the point (ψ(T ), β(T )), for some great T > 1, is on the algebraic curve E = 0 then the trajectory will cross (transversally) from outside to inside the algebraic curve; if the trajectory will intersect the algebraic curve in its (smooth) peek (ψ + , 0) then, again, the trajectory will enter the right "lobe" of region E < 0.
Suppose, for the sake of contradiction, that the solution (ψ, β) remains for ever outside the region E ≤ 0, that is lim inf r→+∞ E(r) > 0. As in Figure 3, this means its trajectory will intersect the horizontal axis outside the interval [ψ − , ψ + ]. The estimates (13), (14), namely show that the trajectory will rotate around the origin O infinitely many times and it will also intersect the ring infinitely many times. Taking into account the possible positions of the points from the trajectory that lie inside the ring, we have four (in symmetrical positions) "siblings" of the situation depicted in quadrant I. There exist, accordingly, two increasing, unbounded from above sequences (r + n ) n≥n 0 , (r − n ) n≥n 0 , with r + n > r − n , given by the formulas for some great integer n 0 . As before, since 2π − (θ 0 − θ 1 ) > 0, we have r + n−1 < r − n when n ≥ n 0 + 1. Notice also that R(r) ≥ 1 + ε > 1 whenever r ∈ [r − n , r + n ], see [3,Eq. (3.18)]. When the solution escapes from quadrant I while remaining outside the region E ≤ 0, its possible trajectories in quadrant IV read as in Figure 4. Repeating the argumentation from Polar coordinates II, we reach a contradiction. So, lim r→+∞ E(r) ≤ 0, which means that the solution (ψ, β) will enter the region E ≤ 0 in "finite time".

Figure 4
Let us return to the estimates (13), (14). Introduce ζ ∈ (0, 1 − λ) such that r ++ > , r ++ . The "+/−" splitting of the region E ≤ 0 displayed in Figure 3 implies that, if the solution (ψ, β) will not leave quadrant I ever again, then its trajectory will cross from outside to inside the algebraic curve E = 0 but it will not encounter the origin O as the angle θ is decreasing with non-null "angular velocity" θ ′ ! In conclusion, if the solution (ψ, β) has not reached the origin O in a short period of the time r then the only possible "entry windows" are quadrants II and IV. See Figure 4.
Transversality of flow on the vertical axis. Assume again that the energy of the non-null equilibria is negative, whenever ψ(r) = 0. Take r ⋆ > 1 such a zero of the function ψ. Applying the local inversion theorem, we deduce the existence of a C 1 -function r = r(ψ) with r : (−ε ⋆ , ε ⋆ ) → (r ⋆ −δ ⋆ , r ⋆ +δ ⋆ ) ⊂ (1, +∞) for some small ε ⋆ , δ ⋆ > 0. We have now a new ODE for characterizing the other function β = β(r(ψ)) = β(ψ), namely The function β → 1 β being locally Lipschitzian in R − {0}, the solution of this new ODE exhibits continuous dependence of the data. The same issue has been dealt with in [3, Eq. (3.12)] by means of polar coordinates. The topological arguments from [3, Section 3] can now be applied verbatim to establish that there exists some a > 1 great enough in order to have a solution (ψ, β) that will reach the null equilibrium O in finite time r.