Some explicit solutions of the three-dimensional Euler equations with a free surface

We present a family of radial solutions (given in Eulerian coordinates) to the three-dimensional Euler equations in a fluid domain with a free surface and having finite depth. The solutions that we find exhibit vertical structure and a non-constant vorticity vector. Moreover, the flows described by these solutions display a density that depends on the depth. While the velocity field and the pressure function corresponding to these solutions are given explicitly through (relatively) simple formulas, the free surface defining function is specified (in general) implicitly by a functional equation which is analysed by functional analytic methods. The elaborate nature of the latter functional equation becomes simpler when the density function has a particular form leading to an explicit formula of the free surface. We subject these solutions to a stability analysis by means of a Wentzel–Kramers–Brillouin (WKB) ansatz.


Introduction
The momentum conservation equations together with the equation of mass conservation (proposed by Euler in the middle of the eighteenth century) are widely used today when tackling fluid flow problems. In spite of tremendous progress made over the past two and a half centuries many important questions concerning fluids still remain unanswered. Along the previous lines we note, for instance, the scarcity of explicit solutions. Motivated by this circumstance we set out for a study of the three-dimensional Euler equations with a free surface from the perspective of explicit solutions. While less variations leading to a layering of the flow: fluid layers of different densities organize themselves so that the higher densities are found below lower densities, cf. [13,16,20]. Nevertheless, allowing for variable density significantly complicates the mathematical analysis of an already convoluted problem. Indeed, even in the simpler case of twodimensional gravity water waves (without Coriolis effects) stratified fluids remained to a large extent unapproachable to a thorough mathematical analysis until recently: we refer the reader to [13,16,21,28,[30][31][32]43,44,47,53,55] for a selection of recent advances in the field. On a similar note we would like to remark the very recent results by Escher et al. [23] on stratified water waves exhibiting a very general density distribution allowing for singular gradients.
After a brief presentation of the water wave problem in Sect. 2 we indicate at the beginning of Sect. 3 the ansatz for the velocity field and we subsequently derive the formula for the pressure function satisfying the Euler's equations. Afterwards, from the dynamic condition on the surface we determine an implicit equation for the surface defining function. Functional analytic methods are applied to the latter implicit equation to prove the existence of the surface defining function. Section 4 is concerned with the search for obtaining other (almost) solutions to the water wave problem, the latter task being achieved by means of a perturbation of the basic flow solutions (3.1). Although these perturbed solutions are not explicit they present bounded amplitude. To reach our goal we will avail of the short-wavelength perturbation method for threedimensional flows, devised by Bayly [1], Friedlander and Vishik [24] and Lifschitz and Hameiri [41] and which examines whether the amplitude of the perturbations to the basic flow remains bounded as a function of time. The short-wavelength perturbation method emerged as an essential tool for boosting up the relevance of some recently derived exact solutions in geophysical fluid dynamics [11,[30][31][32][34][35][36]. To be more specific, we start our attempt by searching for perturbations (of the velocity field and of the pressure) in the form of the Wentzel-Kramers-Brillouin (WKB) ansatz, cf. formula (4.1). As it turns out, the components of the amplitude vector of the perturbation to the basic flow satisfy a system of ordinary differential equations. We then prove that the latter system is equivalent to the Hill's equation. A quite involved analysis is then used to show the boundedness of solutions to the Hill's equation.

The three-dimensional Euler equations with free surface boundary conditions
We formulate here the governing equations for free surface water flows. Working in a Cartesian coordinate system of coordinates x, y, z, we assume the water flow to be bounded below by the bed z = −d and above by the free surface z = η(x, y, t), where η is a function that is determined as part of the solution and t denotes the time variable. The guiding principles for water flow propagation refer to mass conservation and Newton's second law of motion. The equation of mass conservation relates the rate of change of density to the field of motion. More precisely, denoting with ρ = ρ(x, y, z, t) the density function the equation of mass conservation has the form where (u, v, w) represents the velocity field. Under the assumption that the water is inviscid, the equations of momentum conservation are where P is the pressure function and g denotes the gravitational constant. While both (2.1), (2.2) are required to hold within the bulk of the fluid, the specification of the water wave problem is completed by the boundary conditions pertaining to the free surface z = η(x, y, t) and to the bed z = −d. These are the kinematic boundary and together with the dynamic boundary condition for some given function (x, y) → p(x, y, t).
In the next section we introduce a family of solutions to the problem (2.2)-(2.5).

A family of solutions exhibiting vertical structure
We first lay out the ansatz giving the explicit formulas of the velocity field and then, availing of the Euler's equations (2.2), we derive the formula for the pressure function. The dynamic surface condition (2.5) will then be used to derive an implicit equation for the surface defining function. The latter equation is analysed by way of the implicit function theorem. More precisely, we will show that there is a unique function describing the shape of the surface, as soon as one applies on the surface a continuous pressure which is a small deviation from the pressure required to maintain a flat surface. In certain scenarios (defined by the choice of the density function) the free surface can also be determined explicitly. To begin with, let us set and w(x, y, z) = 0, (3.1) where f : R → R is such that the functions (x, y) → −y f (x 2 + y 2 ) and (x, y) → x f (x 2 + y 2 ) are differentiable and the density function z → ρ(z) (assumed here to depend only on the depth variable) is positive and differentiable. We would like to note that the velocity field given in (3.1) has a purely radial character with streamlines given by circles x 2 + y 2 = k for some constant k > 0. A computation shows that This means that, in order to find the pressure P we have to solve the system Solving for P in the above system we obtain for some constant c. Clearly, which shows that equation of mass conservation is verified by the velocity field (3.1). We set out to determine the free surface η and to check the boundary conditions (2.3)-(2.5). Using the dynamic surface condition (2.5) the free surface η is determined implicitly by the equation Let us recast the previous equation as the functional equation Note that η 0 ≡ 0 is a solution of (3.6) provided We will resort now to the implicit function theorem to prove the existence of η satisfying (3.6). To this end we denote with C b (R 2 ) the Banach space of continuous and bounded functions defined on R 2 . Note that is a homeomorphism. This implies that for all continuously prescribed surface pressure functions p(x, y), which represent a small deviation from the pressure p 0 required to maintain a flat free surface η 0 ≡ 0, there is a unique continuous η satisfying (3.5). The surface defining function η is in fact differentiable as it can be derived from (3.5) by building the appropriate difference quotients and applying the mean value theorem. We obtain If the given pressure on the free surface is a function of type p(x, y) = G(x 2 + y 2 ) we see from (3.9) that uη x + vη y = 0. The latter implies that the surface kinematic condition (2.3) is satisfied. While, in general, is not possible to establish an explicit formula for the shape of the free surface z = η(x, y), some special choices of density functions will render an easy determination of it. We present below some examples of such density functions, which are decreasing with depth, thus holding physical grounds.
a condition that is easily verified as soon as the imposed pressure at the surface satisfies

Remark 3.2
Taking k = 1 in the formula for ρ from Remark 3.1 we get that η is given as Determining the stability of a flow is of paramount physical importance. Indeed, the stability in the sense defined above is equivalent to finding time-dependent (nearly) solutions to the water wave problem. Here, the term "nearly" refers to the obstruction by which the perturbations of the basic flow (3.1) fail to be exact solutions. More precisely, this obstruction is represented by the quadratic terms (U · ∇)U, often neglectable. However, proving stability is a task that raises serious mathematical challenges. We choose to use the short-wavelength stability method developed by Bayly [1], Friedlander and Vishik [24] and Lifschitz and Hameiri [41]. As it turns out the components of the amplitude vector of the perturbation to the basic flow (u, v, w) from (3.1) satisfy a system of ordinary differential equations. The latter system is proved to be equivalent to the Hill's equation. We then study the boundedness of solutions to the Hill's equation by means of a criterion of Zukovskii, cf. [56].

On the general setting
We consider perturbations U = (U , V , W ) and P of the velocity field (u, v, w) given in (3.1) and of the pressure P given in (3.4), respectively. These perturbations are analyzed along the streamlines of the basic flow (u, v, w). To be more precise, we use for U and P the WKB ansatz A 2 , A 3 ) denotes the amplitude of the perturbation, F is a scalar function (called the phase of the perturbation) and ε > 0 denotes a small parameter. We set as initial condition We say that the basic flow u = (u, v, w) is stable if the amplitude A remains uniformly bounded in time.
From the requirement that the perturbed quantities U + u and P + P satisfy the equations of conservation of mass and of momentum (2.1) and (2.2), respectively, we obtain (by way of neglecting the quadratic term (U · ∇)U) the system Using now the WKB ansatz (4.1) and identifying the coefficients of ε −1 , ε 0 , ε 1 from (4.3) we obtain while the phase F is solution to the equation which, since w = 0, reduces to We proceed to solve (4.7) by the method of characteristics and so obtain the existence of a function (X , and c,c and c 3 are some constants.

Stability of the flows (3.1) for positive f and f
The purpose of the first part of this section is to show that there are choices for the phase function F for which the right hand side in system (4.4) vanishes. To begin with let us denote now with (x(t), y(t), z(t)) the streamlines of the basic flow (u, v, w) given in (3.1). This means that the equalities hold for all t. Proceeding to solve (4.9) we infer first that there is a constant c such that Consequently, we obtain the existence of constants z 0 , a 1 and a 2 such that the equalities hold for all t. It follows that

F y (t, x(t), y(t), z(t))
= X X(t), Y(t), Z(t) sin(C(t − c)) + Y X(t), Y(t), Z(t) cos(C(t − c)), (4.12) and where (4.14) After picking a certain streamline (x(t), y(t), z(t)) of the basic flow (u, v, w) (by assigning initial values z 0 , a 1 , a 2 for x(t), y(t), z(t) in (4.10)) we choose in the expression of the phase F of the perturbation such that , w(x, y, z) ≡ 0 into the system (4.4) we obtain (taking also into account the choice (4.15)) that the amplitude triple (A 1 , A 2 , A 3 ) satisfies the homogeneous system where More explicitly, denoting A i (t, x(t), y(t), z(t)) =Ã i (t) we obtain from the previous system thatÃ where 2ρ(z 0 ) 3 2 . We will prove in the sequel thatÃ 1 ,Ã 2 are bounded in t. We will be concerned first with the homogeneous system (4.20) which can be equivalently written as (4.22) we see that the previous system can be written as (4.23) EliminatingĀ 2 from above we get thatĀ 1 satisfies the equation (4.25) Upon setting With the substitutionÂ we can convert (4.27) into the Hill's equation Moreover, setting we can convert (4.31) into the Hill's equation Answering the question of boundedness of solutions to the Hill's equation is a result of Zukovskii which we formulate below and refer the reader to [3,56] for details.

Theorem 4.1 [56] Assume that t → Q(t) is a continuous periodic function (of period L). Then all solutions y of the Hill's equation
are bounded if there exists some n ∈ N with the property that the inequality holds for all t.
f (c) be the (principal) period of Q 1 and choose constants a 1 , a 2 in formula (4.10) such that Then the inequality π holds for all t, provided the functions f and f are positive.
Proof Taking into account formula (4.26) we obtain (after tedious calculations and some algebraic manipulations) that (4.36) To check condition (4.34) from Theorem 4.1 we remark that Q 1 is periodic of period π √ ρ(z 0 ) f (c) =: L. Acting on the assumption that f and f are positive functions we clearly have that (4.37) Moreover, we also see that for all t holds , (4.38) where, to obtain the last term in the bracket above, we have used that In the quest to establish the bound on the right hand side of (4.35) we will seek to find a 1 , a 2 (from the formula (4.10) defining the trajectories of the basic flow) such that the inequalities hold for all t. From (4.38) and (4.39) we conclude Inequalities (4.37) and (4.40) can be restated as that is, we have proved (4.35) which guarantees the boundedness of solutions to (4.29).
To ensure that (4.39) holds we impose the conditions Taking into account that we obtain Under the condition we can square in inequality (4.44)and obtain that (4.44) is satisfied precisely when where, we recall, c = a 2 1 + a 2 2 . Obviously, any (a 1 , a 2 ) with (a 2 1 + a 2 2 ) · f (c) f (c) ≤ 1 5 also satisfies the second inequality in the latter system. Analogously, we have Lemma 4.3 Let a 1 , a 2 from formula (4.10) be such that Then the inequality π holds for all t, provided the functions f and f are positive.
Proof A computation similar to the one in Lemma 4.2 shows that (4.47) Undertaking now an analysis similar to the one in Lemma 4.3 by interchanging the role x(t) and y(t) we obtain the claim (4.46).
49) which, after identification of the coefficients of sin f (c)t √ ρ(z 0 ) and cos f (c)t √ ρ(z 0 ) , respectively, is equivalent with the system in the unknowns A, B (4.50) Analogously, we find that the second equation of (4.18) becomes (4.51) Identifying the coefficients of sin f (c)t √ ρ(z 0 ) and cos f (c)t √ ρ(z 0 ) , respectively, we obtain that A and B satisfy the system and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.