Nonlinear Analysis, Theory, Methods & Applicamns, Vol. 2X. No. I, pp. 87-102, 1997 Copyright SJ 1996 Elsevier Smnce Ltd
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SYMMETRY OF INTERNAL WAVES
LILIANE A. MAIAt Departamento de MatemBtica, Universidade de Brasilia, 70919-900 Brasilia, DF, Brazil
(Received December 1994; received for publication 7 July 1995)
Key words and phrases: Hydrodynamics, symmetry, moving planes, solitary wave solution, maximum principle, Alexandroff method.
A solitary wave is a permanent two-dimensional irrotational flow of a fluid which has a single crest over a horizontal bottom, while the vertical displacement tends to the equilibrium level in both directions at infinity. In proving the existence of a surface solitary wave solution for a fluid of constant density, with speed c near the critical value x@, Friedrichs and Hyers [l] assumed that the wave was symmetric. It was only very recently that Craig and Sternberg  settled this question proving that any supercritical (c > m) surface solitary wave solution is symmetric.
Questions like that of symmetry and monotonicity often arise in studying differential equations, and in particular, problems in hydrodynamics. In their beautiful work Gidas et al.  prove symmetry, and some related properties, of positive solutions of second order elliptic equations, employing various forms of the maximum principle and the Alexandroff method of moving planes. Later, Berestycki and Nirenberg  established monotonicity, symmetry properties and uniqueness for some solutions of semilinear elliptic equations with Dirichlet or Neumann boundary conditions in an infinite cylinder. There they showed, for instance, that a positive wave solution in a smoothly stratified fluid filling an infinite two dimensional strip is symmetric and decreasing to the right hand side of the axis of symmetry. The proof relies strongly on the study of the asymptotic behaviour of the solution near infinity, and that information is obtained with the aid of results on the asymptotic behaviour for linear equations, of Agmon and Nirenberg , and Pazy .
More recently, Li  studied the problem of establishing monotonicity and symmetry of solutions of second order fully nonlinear elliptic equations in unbounded domains, and particularly in infinite cylinders. In order to avoid the analysis of the asymptotic behaviour of the solution, which may become a very difficult task, he introduced a new approach based solely on the maximum principle. This approach enabled him to carry out the moving plane scheme at infinity in a simplified way. He was able to give a different proof of the symmetry of a surface solitary wave as a corollary of his results, avoiding the long and complicated asymptotic analysis found in .
In this paper we will adapt the technique to the problem with internal discontinuities and thus prove symmetry, and monotonicity, of positive solitary wave solutions of the problem of an incompressible, inviscid stratified fluid acted on by gravity, flowing in an infinite
t This work was partially supported by CNPq/Brazil.
88 I.. A.MAIA
two-dimensional strip. Furthermore, assuming that a positive solution u satisfies limxl + .+ u(xr , x2) = 0 and lim SUP~,++~ u(xr , xJ > 0 we prove that this wave solution is monotone increasing in the x, direction in the entire strip. We will assume that the density function pm which describes the vertical density stratification of the fluid at infinity is a positive nonincreasing piecewise C function from bottom to top. The problem is described by a quasilinear partial differential equation with Dirichlet boundary conditions on the top and bottom of the strip, and some internal boundary conditions which originate from the fact that the pressure is continuous across the fluid interfaces, where the jumps in the density occur [cf. (25)-(28) and definition of T]. Our main result in this paper is stated as follows.
THEOREM 1. Let u E Co(F) rl C2(Ti), 1 I i I it + I, be a positive solution of the boundary value problem (25)-(28) for a piecewise continuously differentiable density function pm > 0, with c2 > co. Then either:
(0 limxl + +- ui(xI , x2) = 0 and u is symmetric, that is, there is a vertical axis of symmetry x, = r;l such that for all (x1, x2) E F = Ul+i i=, x1 > q, u(xr , x2) = ~(24 - x1, x2). Further- more, (&4/ax,)(x,, x2) < 0 for x1 > v; or
(ii) lim supXI + +m uI(xr , x2) > 0 and u is monotone increasing in the x1 direction in the entire strip, that is, (&/ax,)(x, , x2) > 0 for all (x1, x2) E UT=+: T.
For a matter of simplicity of notation and proof, we consider a piecewise density function pa with one jump discontinuity only, and state and prove a simplified version of theorem 1 given by theorem 2. The existence of solitary wave solutions of this problem was proved by Turner  by first showing the existence of periodic internal waves, which are symmetrized, and then taking the limit with periods increasing to m. A particular case is when the density function is a step function, showing that a system of two layers of constant but differing densities will support a solitary wave. The symmetry of the solution of this particular problem of internal solitary waves in an interface between two inmiscible fluids of different constant densities was showed by Craig and Sternberg  also making use of the asymptotic analysis of the solution at infinity in order to start the process of moving planes.
We apply Lis method to our problem in order to avoid the asymptotic analysis at infinity. Li considered positive solutions of the following fully nonlinear elliptic equations
F(x, , Y, u, ui f Uij) = 0 in S, = (-03, +oo) X Q
u(x19Y)9 Ui(XIrY)9 uij + O uniformly in y E Sz as x1 + --co
under the following boundary conditions
Rx,, y, u, u,, Ul, V$) = 0 on as,,
where v is the exterior unit normal to S, at (xl, y), V, is the tangential gradient on aa, and a is a bounded smooth domain in R-. However, we have to adapt Lis results to our needs since we will have internal boundary conditions, which are mixed conditions, different from the top and bottom conditions stated in his theorems. Moreover, for his results he assumes that the boundary conditions are nondegenerate with respect to the variables u and u,, or equivalently, B, and B,,, do not vanish at the same time, and considers, for simplicity, that B, and BUD have the same sign. We show that our internal boundary conditions satisfy slightly different growth properties, but these are sufficient to prove what we need to get started with the method of
Symmetry of internal waves 89
moving planes. Once we get started with that process, the next step is to find the axis of symmetry for the solution. This is possible applying the maximum principle in the various horizontal sectors of the infinite strip, determined by the interfaces of the fluid, together with the Hopf boundary lemma and the Corner-Hopf lemma. The approach in this second step differs from that worked out by Li in the proof of his general theorem. Finally, the symmetry of an internal solitary wave solution in a system of two layers of constant but differing densities follows as a corollary of our more general result.
The outline of this paper is the following: in Section 2 we give a formulation of the problem; we derive the quasilinear elliptic boundary value problem (2)-(5) for a piecewise C density function pm, with only one jump discontinuity for simplicity, and state the main result of the paper in its simplified version in theorem 2. The notation is explained in Section 3. In Section 4, first we prove theorem 2, case (i) of solitary waves, in two steps. Step 1 builds up conditions to get started with the method of moving planes, and in step 2 the process of moving planes is performed up to the point where it is shown that there exists a vertical axis of symmetry for a solution. The remainder of Section 4 is occupied with the proof of case (ii) of theorem 2, the solutions which are monotone in the entire strip, following closely the steps in case (i). Finally, it is shown that the symmetry of a supercritical surface solitary wave solution follows as a corollary of our main result, corollary 5, as well as the symmetry of internal solitary waves in a system of two layers of constant but different densities. The result in its most general form is stated in theorem 1, and proved at the end.
2. FORMULATION OF THE PROBLEM
We consider an incompressible, inviscid fluid acted on by gravity and restrict attention to a two dimensional flow confined to and filling a region S = ((x, u) 1 --co < x < 00, -h < y < 1 - h]. The acceleration of gravity has magnitude g in the negative y direction. We further assume that the fluid is heterogeneous and nondiffusive for simplicity of the problem. We are interested in waves of permanent form which are moving from right to left with speed c > 0. The flow is steady when viewed in a reference frame which moves with the wave. Further details about the physical problem may be found in [lo, 111. We further assume that in the original spatial coordinates the fluid is at rest at x = --00 and connects to a parallel flow at +oo. Moreover, the density function p, which describes the stratification of the fluid, approaches a prescribed density pm as x -+ --03 which is nonincreasing in y, and satisfies p-(-h) > ~~(1 - h) > 0. For simplicity we shall study a piecewise continuous density
such that p:(O) > p;(O). Under the assumptions for the fluid velocity q = (q,, q2) that div q = 0 (volume incompressibility), and q * Vp = 0 (density does not diffuse), it follows that p is constant along streamlines, and a pseudo-stream-function I,Y is defined by the formula
P12q = (WY> --WA (1)
the subscripts denoting partial derivatives, thus p is a function of v, p(v). We shall consider only flows for which no reversal occurs, that is q, > 0 (l/, > 0). Closely following the description in [lo], it is possible to obtain from the Euler equations of the problem a semilinear elliptic equation for y (cf. [lo, (2.23)]), and an elliptic boundary value problem. An additional
90 L. A. MAIA
requirement is that on the dividing streamline (w = 0), where the jump in the density occurs, the pressure be continuous. Thus, an interior free boundary arises (cf. [l 11). The assumption that w, > 0 means that for such functions one can solve for y as a function of the spatial coordinate x and the material coordinate y. We obtain an alternative formulation of the problem, the so-called semi-Lagrangian description. Thus, the semilinear equation is replaced by a singular quasilinear equation for y(x, w). Since y(x, ~(x, y)) = y one has the relations
Yx + Y$Yx = 0 Y$Yy = 1.
We define the pseudo-stream function at -co by
Y(y) = c --> -\ip,o ds I, 0
and denote by its inverse function Y(Y) so that Y(Y(y)) = y for -h < y < 1 - h. Moreover, the streamline coordinate ly can be normalized to coincide with the height function as the horizontal coordinate x approaches -co. One final change of variables is performed letting x1 = x and x2 = Y(w), and
a,, $1 = Y(Xl 3 Wx,)) - x2
so that u represents the deviation at a horizontal position x, of the streamline which has height x, at --oo. Note that the assumption of no reversability implies au/ax, + 1 > 0 (cf. [lo, (3. lo)]). In describing the problem in the new coordinates we set
fi = af/api, i = 1,2; J. = g/c2; T = R x (-h, 1 - h); and use the summation convention. A calculation produces an eigenvalue problem for the pair (A, u)
PzxX2)fiWf) = -w(x2)u+ in T+ - in T-
u-h,, -h) = u+(xl, 1 - h) = 0 xl E R (3)
lim u+(xI ,x2) = 0. (4) XI--m
Here T- = R x (-A, 0), T+ = R x (0, 1 - h) and f are used to denote values or limits taken within T'. The internal boundary condition, which states that the pressure is continuous across the fluid interface, is translated in these new coordinates into
pJO)(f-(VU-) - Au-) - p;(o)(f,(vu+) - Au) = 0 on x2 = 0. (5)
Note that on the dividing streamline u(xr , 0) = y(x, , 0). Under this formulation, a solution of the problem is a pair (A, u) satisfying (2)-(4) and the internal boundary condition (5), where u is continuous on r and C2 in T.
Symmetry of internal waves 91
Our goal is to show that a positive flow u associated to the boundary value problem (2)-(5) and, in addition satisfying lim,, _ +co zP(xl, x2) = 0, also called wave of elevation, is symmetric with respect to some vertical axis x1 = fl whenever A is in some appropriate range (i.e. A < A,,, where A, is a positive real number yet to be specified). We shall also prove that u is a monotonic function of the x1 variable, decreasing for xi > 8. Otherwise, if lim SUP,~++~ u&(x1, x,) > 0, we show that u is monotone increasing in the x1 direction in the entire strip.
We will see [cf. (S)-(lo)] that the linearization of (2) at u = 0
- ( $ #&,(x2) & 24 + $ &x2) $ u
> = -Ap~(x,)u
1 1 2 2
u = 0 at x2 = 1 - h, -h, and u-(x,, 0) = u+(xr , 0) in xi E R, together with the linearized internal boundary condition
Pzw2 - Prncw;~ = am) - P,(O))U
on x, = 0, has a lowest positive eigenvalue & corresponding to a velocity c0 = (g/&,)12, a so-called critical velocity. With this notation and in this semi-Lagrangian formulation of the problem we are now ready to state our main result.
THEOREM 2. Let u E Co(T) II C(T) be a positive solution of the boundary value problem (2)-(5) with c2 > cz, then either:
6) lim,, - +m u(xi ,x2) = 0 and u is symmetric, that is, there is a vertical axis of symmetry x1 = 4 such that for all (xr, x2) E T, x1 > ii, u(xr ,x2) = u(2q - x,, x2). Furthermore, (cWdx,)(x,, x2) < 0 for x1 > fl; or
(ii) lim sup,, _ +m u(xi, x2) > 0, and u is monotone increasing in the x1 direction in the entire strip, that is (&/ax,)(x, , x2) > 0 for all (x1, x2) E T.
A similar result can be obtained for a negative associated flow u, also called wave of depression, assuming its existence. In this case, however, the monotonicity appears with opposite sign.
We use the Alexandroff method of moving planes to prove both symmetry and monotonicity stated in the theorem.
Before we proceed with the proof of theorem 2 we present a few more definitions and simplify some notation used previously.
Let ui denote the partial derivative of u with respect to xi, i = 1,2, similarly for second partial derivatives, U,j, i,j = 1, 2. For a density p: E C(I), A E R, and u E C2(F) define
F(u, u:, u;, Ufl) U&) 24,) = ( >
& /x3X2)fi(VUf) - A/J:(x,)u. (6) I
The function F(u, pi, qij), i,j = 1,2, is continuous in all arguments for (x, ,x2) E T since 1 + 24: > 0 in T,
B(u, u:, u;, u;, ui) = PmLf2Pu-> - au-1 - Pmu-2P+) - Au1 on x, = 0. (7)
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Hence, equations (2) and (5) are, respectively, equivalent to
F(u',u:,u~,u~~,u:,,u:,) = 0 in T'
B(uf, u:, u;, u;, 24,) = 0 on x, = 0.
We shall work in the following...