1.2 The 3D quasi–geostrophic model
1.2.2 Rotating patches around the vertical axis
The main contribution of this thesis to the field of quasi–geostrophic motion is to analyze the existence of 3D patches uniformly rotating around the vertical axis motivated by the V–states
described in Section 1.1.2. From now on we assume that the buoyancy frequency N2is constant and N2= 1. Hence, we aim to study the following system
8>
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:
@tq + u@1q + v@2q = 0, (t, x)2 [0, +1) ⇥ R3, = q,
u = @2 , v = @1 , q(t = 0, x) = q0(x).
(1.2.12)
The stream function is given by
(t, x) = 1 4⇡
ˆ
R3
q(t, y)
|x y|dA(y),
where dA denotes the usual Lebesgue measure. Then, the velocity field takes the form
(u, v)(t, x) = 1 4⇡
ˆ
R3
(x1 y1, x2 y2)?
|x y|3 q(t, y)dA(y).
From the evolution equation we recover that the potential vorticity is transported along the trajectories. Considering an initial datum in the patch form q0 = 1D, where D is a bounded domain in R3, then
q(t,·) = 1Dt,
where Dtis the image of D by the flow. Similarly to the 2D case, as for Euler equations, we can implement the contour dynamics equations. Namely, for any sufficiently smooth parametriza-tion tof the boundary @Dtone has that
(@t t U (t, t))· n( t) = 0,
where U = (u1, u2, 0)and n( t)is a unit normal vector to the boundary at the point t.
Compared to the 2D equations that we have seen before, the quasi–geostrophic equations enjoy richer structures and offer several new perspectives. Actually, at the level of stationary solutions, those in the patch form are more abundant here than in the planar case. Indeed, any domain with a revolution shape about the z axis generates a stationary solution. The analogous to Kirchhoff ellipses still surprisingly survive in the 3D case. In [108], MEACHAM
showed that a standing ellipsoid of arbitrary semi–axis lengths a, b and c rotates uniformly about the z axis with the angular velocity
⌦ = µ
1RD(µ2, , 1) RD(µ2, 1, )
3( 1 ) ,
where = ab is the horizontal aspect ratio, µ := pc
ab the vertical aspect ratio and RD denotes the elliptic integral of second order
RD(x, y, z) := 3 2
ˆ +1 0
p dt
(t + x)(t + y)(t + z)·
For more details about the stability of those ellipsoids we refer to works of DRISTCHEL, SCOTT, RENAUD, MCKIVER[54, 55, 57].
The main objective of Chapter 4 is to show the existence of non trivial rotating patches by suitable perturbation of stationary solutions given by generic revolution shapes around the vertical axis. Indeed, we look for solutions of the type
q(t, x) = q0(e i⌦txh, x3), q0= 1D, xh= (x1, x2). (1.2.13)
1.2. THE3DQUASI–GEOSTROPHIC MODEL
The initial domain is chosen so that it can be parametrized in the following way D =n
(rei✓, cos( )) : 0 r r( , ✓), 0 ✓ 2⇡, 0 ⇡o
, (1.2.14)
where the shape is sufficiently close to a revolution shape domain, meaning that r( , ✓) = r0( ) + f ( , ✓),
for a small non–axisymmetric perturbation f( , ✓) of the generatrix curve r0( ). Moreover, we assume the following Dirichlet boundary conditions
r0(0) = r0(⇡) = f (0, ✓) = f (⇡, ✓) = 0,
so that the poles (0, 0, ±1) remain unchanged. Inserting the ansatz (1.2.13) in the contour dy-namics equation and using the parametrization of the domain given by (1.2.14), we get the following equivalent formulation
F (⌦, f )( , ✓) := 0(r( , ✓)ei✓, cos( )) ⌦
2r2( , ✓) m(⌦, f )( ) = 0, (1.2.15) for any ( , ✓) 2 [0, ⇡] ⇥ [0, 2⇡], with
m(⌦, f )( ) := 1 2⇡
ˆ
2⇡0
⇢
0(r( , ✓)ei✓, cos( )) ⌦
2r2( , ✓) d✓,
where 0stands for the stream function associated to q0. Moreover, from the structure of the stream function 0we achieve that F (⌦, 0) = 0 for any angular velocity ⌦ 2 R.
The main tool to find periodic solutions around the stationary shapes is the bifurcation theory, and in particular, finding nontrivial solutions (⌦, f) to (1.2.15). However when imple-menting such program several hard difficulties emerge at the linear and nonlinear levels. In particular, the spectral problem here is very delicate and strongly depends on the shape of the initial stationary solutions (that is, r0). Surprisingly, in this case we are finally able to bifurcate from a quite large family of initial domains with mild regularity conditions. Consequently, we do not need to restrict to a specific initial shape, as opposed to the non uniform rotating patches in the Euler equations in Section 1.1.3, where we had to restrict to specific initial vorticity con-figurations, namely, quadratic profiles.
The main result in Chapter 4 reads as Theorem 1.2.1. Assume that r0satisfies (H1) r02 C2([0, ⇡]).
(H2) There exists C > 0 such that
8 2 [0, ⇡], C 1sin r0( ) C sin( ).
(H3) r0is symmetric with respect to = ⇡2, i.e., r0 ⇡
2 = r0 ⇡
2 + , for any 2 [0,⇡2].
Then for any m 2, there exists a curve of non trivial rotating solutions with m-fold symmetry to the equation (1.2.12) bifurcating from the trivial revolution shape associated to r0at some angular velocity
⌦m.
We precise by m–fold shape symmetric shape of R3 we mean a surface whose horizontal sections are m–fold symmetric as for the 2D case.
There are many difficulties when trying to apply bifurcation theory to (1.2.15), and the main one is related to the choice of the function spaces. First, notice the following simplification in (1.2.15):
r2( , ✓) 1 2⇡
ˆ 2⇡
0
r2( , ✓)d✓ =(r0( ) + f ( , ✓))2 1 2⇡
ˆ 2⇡
0
(r0( ) + f ( , ✓)2d✓
=2r0( )f ( , ✓) + f ( , ✓)2 1 2⇡
ˆ 2⇡
0
(2r0( )f ( , ✓) + f ( , ✓)2)d✓.
Then, we observe that the nonlinear functional (1.2.15) can be written as F = ⌦ r0Id + F , for an appropriate functional F . Note that r0 is vanishing at the boundary {0, ⇡}, and thus we need to include this degeneracy in the function spaces. Specifically, in order to avoid this problem we can rescale F as follows
F (⌦, f ) :=˜ F (⌦, f )
r0 , (1.2.16)
and work with ˜F instead of F . By doing so we can write ˜F = ⌦Id + ˜F , that is a better suited operator. Our bifucation method then is divided into two main parts: the spectral study and the regularity study, that we briefly discuss in the sequel.
On the one hand, from the point of view of the spectral study, we shall look at the linearized operator and show that it is a Fredholm operator of zero index. Take h( , ✓) =P
n 1hn( ) cos(n✓), then
@fF (⌦, 0)h( , ✓) =˜ X
n 1
cos(n✓) [{F1(1)( ) ⌦} hn( ) Fn(hn)( )] , (1.2.17) where
Fn(hn)( ) := 1 4⇡r0( )
ˆ ⇡ 0
ˆ 2⇡
0 Hn( , ', ⌘)h(')d⌘d', (1.2.18)
Hn( , ', ⌘) := sin(')r0(') cos(n⌘)
((r0( ) r0('))2+ 2r0( )r0(')(1 cos(⌘)) + (cos( ) cos('))2)12. (1.2.19) Observe then that the linearized operator contains both a local and nonlocal part. Moreover, the function ⌫⌦defined as
⌫⌦( ) :=F1(1)( ) ⌦,
is crucial in order to have a Fredholm operator. Defining := inf 2(0,⇡)F1(1)( ), we get that
⌫⌦is strictly positive when ⌦ 2 ( 1, ) and thus we will restrict to angular velocities lying in such interval.
Furthermore, after computing explicitly the integral in ⌘, we achieve that the linearized operator can be expressed in terms of Gauss Hypergeometric functions as follows:
@fF (⌦, 0)h( , ✓) = ⌫˜ ⌦( )X
n 1
cos(n✓)Ln(hn)( ),
where
Ln(hn)( ) :=hn( ) K⌦n(hn)( ),
1.2. THE3DQUASI–GEOSTROPHIC MODEL
K⌦n(h) :=
ˆ ⇡ 0
Hn( , ')
sin(')⌫⌦( )⌫⌦(')r2(')h(')dµ⌦('), Hn( , ') :=22n 1 12 2n
(2n)!
sin(')rn 10 ( )rn+10 (') [R( , ')]n+12 Fn
✓4r0( )r0(') R( , ')
◆ . Moreover ⌫⌦can be recovered as
⌫⌦( ) = ˆ ⇡
0
H1( , ')d' ⌦, and we set
R( , ') := (r0( ) + r0('))2+ (cos( ) cos('))2. Here Fndenotes the Gauss Hypergeometric function
Fn(x) := F
✓ n +1
2, n +1
2, 2n + 1, x
◆
, x2 [0, 1),
and we refer to Appendix C for more information about these special functions. In addition, the measure dµ⌦has the following expression
dµ⌦(') := sin(')r20(')⌫⌦(')d'.
The goal of such new measure is to symmetrize the linearized operator by working over the new Hilbert spaces determined by the weighted Lebesgue spaces: L2µ⌦(0, ⇡). We emphasize that our hypothesis that ⌦ 2 ( 1, ) guarantees that µ⌦is a signed measure and the weighted space is well defined.
Bearing all the above notation in mind, the kernel equation reduces to study whether 1 is an eigenvalue of K⌦n. In fact, we prove that Kn⌦: L2µ⌦ ! L2µ⌦ is a compact self–adjoint integral operator, and more precisely, it is of Hilbert–Schmidt type. A careful spectral study allows us to determine that that the largest eigenvalue n(⌦)is simple and monotone. Moreover, there exists sequence ⌦n ! such that n(⌦n) = 1, which is crucial in order to have that the kernel of the linearized operator is one dimensional.
On the other hand, from the point of view of regularity, at this moment the preceding weighted spaces are too weak to get the persistence and regularity of the nonlinear functional ˜F on those spaces. In order to solve that, our candidate will be the H¨older space C1,↵with Dirich-let boundary conditions, and ↵ 2 (0, 1). First, we need to check that the above eigenfunctions of K⌦n belong to this new space, which a priori is not trivial and is equivalent to show such regularity for h satisfying Fn(h) = ⌫⌦h. By using a bootstrap argument starting at h 2 L2µ⌦, we are able to achieve that h 2 C1,↵and fulfills the Dirichlet conditions, for any n 2. Here it appears the restriction m 2in Theorem 1.2.1. Note also that the integral kernel of Fn, that is (1.2.19), is singular inside the interval (0, 2⇡) but also on the boundary since r0is vanishing and thus one strongly needs the Dirichlet conditions for h. In particular, this creates some problems in order to guarantee that ⌫⌦=Fn(1) ⌦belongs to C1,↵since the singularity at the boundary can not be compensated with any function inside the integral. This will be solved with a more delicate analysis using the Gauss Hypergeometric functions. Secondly, the persistence of the nonlinear functional can not be achieved by standard potential theory arguments. Note that the Euclidean kernel of the stream function 0 is deformated by the cylindrical coordinates amounting to new singularities at the boundary and a more refined analysis is needed.
Finally, let us mention that the prototypes of domains satisfying the hypotheses (H1)–
(H3) are the sphere, agreeing with r0 = sin( ), or ellipsoids with same x and y axes, that is,
r0 = A sin( ). For these particular shapes, the associated stream function is very well–known and thus some of the previous computations can be simplified, we refer to Section 4.6.1 for more details. Indeed, hypothesis (H2) means that the initial domain can be located between two ellipsoids and thus gives us some regularity on the domain. However, this is a technical assumption needed for the persistence of the nonlinear functional but not for the spectral study.
Hence, the bifurcation of singular shapes not verifying (H2) may occur using some appropriate weaker arguments. This will be discussed in Section 5.4.
Chapter 2
Non uniform rotating vortices and
periodic orbits for the two–dimensional Euler equations
This chapter is the subject of the following publication:
C. Garc´ıa, T. Hmidi, J. Soler, Non uniform rotating vortices and periodic orbits for the two–dimensional Euler equations, arXiv:1807.10017.
Accepted for publication in Archive for Rational Mechanics and Analysis.
Contents
2.1 Introduction . . . 30 2.2 Preliminaries and statement of the problem . . . 36 2.2.1 Equation for rotating vortices . . . 36 2.2.2 Function spaces . . . 38 2.3 Boundary equation . . . 40 2.4 Density equation . . . 47 2.4.1 Reformulation of the density equation . . . 47 2.4.2 Functional regularity . . . 51 2.4.3 Radial solutions . . . 52 2.5 Linearized operator for the density equation . . . 58 2.5.1 General formula and Fredholm index of the linearized operator . . . 58 2.5.2 Kernel structure and negative results . . . 62 2.5.3 Range structure . . . 87 2.6 Spectral study . . . 89 2.6.1 Reformulations of the dispersion equation . . . 89 2.6.2 Qualitative properties of hypergeometric functions . . . 91 2.6.3 Eigenvalues . . . 99 2.6.4 Asymptotic expansion of the eigenvalues . . . 112 2.6.5 Separation of the singular and dispersion sets . . . 115 2.6.6 Transversal property . . . 117 2.7 Existence of non–radial time–dependent rotating solutions . . . 126