The vortex patch problem for the surface quasi-geostrophic equation
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Communicated by Charles L. Fefferman, Princeton University, Princeton, NJ, December 15, 2003 (received for review October 13, 2003)
Abstract
In this article, the analog of the Euler vortex patch problem for the surface quasi-geostrophic equation is considered.
The surface quasi-geostrophic (QG) equation was originally introduced as a model for atmospheric turbulence. It represents
the evolution of the temperature on the 2D boundary of a half space with small Rossby and Ekman numbers. For a more detailed
analysis of the geophysical properties of QG see ref. 1. The QG equation is given by 
where 
and 
For simplicity we are considering fronts on the cylinder, i.e., we take (x, y) in 
. In this setting we define -Δ-½ that comes from inverting the third equation by convolution with the kernel.†
where 
in |x - y| ≤ r and suppχ is contained in {|x - y| ≤ R} with 0 < r < R < ½. Also 
.
The mathematical interest lies in the strong connections with the 3D Euler equation. Constantin, Majda, and Tabak (2, 3) noticed that the vorticity ω of the Euler equation satisfies the same equation as ▿⊥θ, precisely 
There are many other analogies that have been analyzed in refs. 2 and 3. Other analyses of the equation are found in refs. 4–11.‡
The question we are going to explore here is an analog of the vortex patch problem for 2D Euler equation, i.e. the evolution of a solution that consists of two regions, where the function θ takes the values 0 and 1, separated by a smooth curve (see Fig. 1). For 2D Euler the global regularity of the vortex patches has been proved by Chemin (12). A simpler proof has been obtained by Bertozzi and Constantin (13) and Majda and Bertozzi (14). See also refs. 15 and 16.§
Sharp front.
We notice that the question we are addressing here is the local existence of a solution for the QG-vortex patch problem. Deciding whether there is global regularity of the solutions is a very interesting open problem.
I refer the reader to ref. 17 for a discussion of a possible relationship between the problem studied here and a problem arising from 3D Euler.
We need the following definition:
Definition. A bounded function θ is a weak solution of QG if for any 
we have 
where u is determined by Eqs. 2 and 3.
In the case of a sharp front we have the following form for the scalar 
For a sharp front for the surface QS equation we have the following results.
Theorem 1.
If θ is a weak solution of the surface QS (see Definition) of the form described in
Eq. 5, then the function ϕ satisfies the equation,
Theorem 2. Given any periodic, smooth function ϕ0(x) the initial value problem determined by Eq. 6 with initial data ϕ(x, 0) = ϕ0(x) has a unique smooth solution for a small time, determined by the initial data ϕ. Moreover the function θ defined by Eq. 5 is a weak solution of the QG equation.
To prove Theorem 1 we substitute the above expression for θ in Eq. 4 and try to obtain an equation for the evolution of the curve ϕ.
Notice that by using Eqs. 2 and 3 we obtain u(x, y, t) = K *θ(x, y, t), where K looks locally like the orthogonal of the Riesz transform. More details about the Riesz transform can be found in refs. 18 and 19. 
we obtain that u is in BMO (see refs. 19 and 20 for more details) and hence the integrals below will make perfect sense. 
As for the other term in Eq. 4 (considering only the space integration) 
Now we look more closely to the integrand of the above expression. We have 
The precise expression of K is given by 
where χ(u, v) is supported in |u - v| ≤ ¼ and η is compactly supported.
And so¶
Hence we have 
Putting these two equalities together we have 
From that equality we obtain the equation we were looking for, 
This completes the proof of Theorem 1.
Acknowledgments
This work was partially supported by Spanish Ministry of Science and Technology Grant BFM2002-02042.
Footnotes
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↵ * E-mail: jrodrigo{at}math.princeton.edu.
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Abbreviation: QC, quasi-geostrophic.
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↵ † To avoid irrelevant considerations at ∞, we will take η to be compactly supported.
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↵ ‡ Constantin, P., Proceedings of the Workshop on Earth Climate as a Dynamical System, September 25–26, 1992, Argonne, IL; ANL/MCSTM-170.
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↵ § Krasny, R., Proceedings of the International Congress of Mathematics, August 21–29, 1990, Kyoto, Japan.
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↵ ¶ We move the ⊥ that appears in K to the factor (∂ϕ/∂x, - 1).
- Copyright © 2004, The National Academy of Sciences
