Research Article | Open Access
Well-Posedness of the Two-Dimensional Fractional Quasigeostrophic Equation
This paper is concerned with the fractional quasigeostrophic equation with modified dissipativity. We prove the local existence of solutions in Sobolev spaces for the general initial data and the global existence for the small initial data when .
This paper is concerned with the nonlocal quasigeostrophic -plane model with modified dissipativity [1, 2] where can be either the 2D torus or the whole space , , and with being the modified dissipative term. Let denote the Jacobian operator; (1) can be notationally simplified as In this model, is the geostrophic pressure, also called the geostrophic stream function, is the vertical component of the relative vorticity, is a zeroth-order balance in the momentum equation, and , and are the rotational Froude number, the Coriolis parameter, and the Reynolds number, respectively. Usually, is also called viscosity parameter. It has some features in common with the much studied two-dimensional surface quasigeostrophic equation (SQGE) (see [3–9] and references therein). However the quasi-geostrophic -plane model has a number of novel and distinctive features.
Recently, this equation has been intensively investigated because of both its mathematical importance and its potential applications in meteorology and oceanography. The quasi-geostrophic -plane model is a simplified model for the shallow water -plane model [2, 10, 11] when the Rossby number is small under several assumptions on the magnitude of the bottom topography variations, which is used to understand the atmospheric and oceanic circulation, the gulf stream, and the variability of this circulation on time scales from several months to several years. In this regime, quasi-geostrophic theory is an adequate approximation to describe the flow and is developed for the simulation of large-scale geophysical currents in the middle latitudes.
When , this is the standard quasi-geostrophic model studied in , which was put forward as a simplified model of the shallow water model (see also  for a review). In , the author studied a multilayer quasi-geostrophic model, which is a generalization of the single layer model in the case . The general fractional power was considered by Pu and Guo . The equation is In , they proved the global existence of weak solutions by employing the Galerkin approximation method for initial data belonging to the (inhomogeneous) Sobolev space . If the initial data is in the (homogeneous) Sobolev space , it is natural for us to ask whether (3) has regular solutions.
In this paper, we only consider the 2D torus with periodic boundary conditions. And we will prove the well-posedness results of (3) under certain condition on initial data which belong to the (homogeneous) Sobolev space . In Section 3, the local existence and uniqueness of the solutions of the problem are proved in when for . That is, for any initial data and , there exists such that (3) has a uniqueness solution on , satisfying However, we may not obtain the global existence of solutions from energy (34), if the initial data has large norm. The main reason is that in the energy estimate for (3), the integral for , where denotes the integral as usual. Thus, it is necessary to control it. To overcome this essential difficulty, we will make use of the properties of the product estimates (Proposition 2) as well as those of the Sobolev embedding inequality.
In Section 4, global existence and uniqueness for small initial data in are also proved when . More precisely, we just need the following condition: where .
For the cases, and , we also obtain the unique global solution in proved by where and .
We conclude this introduction by mentioning the global existence result of weak solutions obtained .
2. Notations and Preliminaries
We now review the notations used throughout the paper. Let us denote . The Fourier transform of a tempered distribution on is defined as Generally, for can be identified with the Fourier series denotes the space of the th-power integrable functions normed by For any tempered distribution on and , we define denotes the homogeneous Sobolev space of all for which is finite. The homogeneous counterparts of are denoted by .
Next, this section contains a few auxiliary results used in the paper. In particular, we recall, by now, the classical, product, and commutator estimates, as well as the Sobolev embedding inequalities. Proofs of these results can be found for instance, in [14–16].
Proposition 2 (product estimate). If , then, for all , one has the estimates where and , and . In particular
In the case of a commutator we have the following estimate.
Proposition 3 (commutator estimate). Suppose that and . If , then where , , and , and .
We will use as well the following Sobolev inequality.
Proposition 4 (Sobolev inequality). Suppose that , , and Suppose that ; then and there is a constant such that
Proposition 5. If , and are nonnegative with except that one requires when , then there is a constant such that for all .
3. Local Existence and Large Data
In , the authors studied and established the existence and uniqueness of local and global solutions to the two-dimensional SQGE. It is natural that (3) is more complex than SQGE. However, we also establish an analogue. In this section we will prove that (3) is locally well-posed in when for . Regarding arbitrarily large initial data, we obtain the following result.
Proof. First of all, multiplying (3) by , we get the following energy inequality:
Integration by parts gives us the following estimate:
Then we get the inequality
We estimate the first term on the right side by
To handle the second term, we proceed as follows.
First note that The estimate of the product term follows from Proposition 2. Hence, we have We now fix an arbitrary such that
Note that since and the range for is nonempty since . For , our choice of and Proposition 5 give where may be computed explicitly from .
In order to estimate in (27), we split it into two cases.
Case 1 (). From Proposition 5 and Sobolev inequality, we have where . In addition, since has zero mean and , from the Sobolev embedding we obtain Combining estimates (27)–(31) gives where and is as defined earlier. The second term on the right side of (32) is bounded using the -Young inequality as and we finally obtain the following estimate: Using Gronwall’s inequality, from estimate (34) we may deduce the existence of a positive time such that Note that we have , and hence we may not obtain the global existence of solutions from the energy (34), if the initial data has large norm. These a priori estimates can be made formal using a standard approximation procedure. We omit further details
Case 2 (). Using Proposition 5, we obtain where . From Sobolev embedding, we have Then, using the same method as in Case 1, we can complete Theorem 6.
4. Global Existence and Small Data
The main result of this section concerns global well-posedness in case of small initial data.
Theorem 7 (global existence). Let , and let have zero mean on , where . There exists a small enough constant depending on , such that if where , then the unique smooth solution of the Cauchy problem (3)-(4) is global in time; that is, .
Proof. We proceed as in the proof of Theorem 6. The product term in (27) is now estimated by
where , so that
Similarly, in order to estimate in (40), we split it into two cases.
Case 3 (). From Sobolev imbedding, we have
Case 4 (). Using Sobolev imbedding, we have
So, we can always obtain the following estimate
With this choice of and the above embedding, the product estimate gives us Combining (24) with (45) and proceeding as in (34) we obtain which in turn implies
Observe that where . Therefore, if estimate (47) combined with Sobolev imbedding inequality shows that and hence By Sobolev imbedding, we have Combining (51) and (52), we get Note that taking the -product of (3) with gives for any Thus, there exists some constant (dependent on ) such that which gives us a basic uniform estimate of in .
Hence, from (53) and (55) we obtain that condition (49) is satisfied for all as long as we have where is sufficiently small, thereby concluding the proof of the theorem.
Note also that the proof of Theorem 7 fails for the value . Thus, indeed is the limit of the local well-posedness theory. Nonetheless, we still can prove that the considered system is globally well-posed for small data.
Theorem 8 (global existence for small data). Let and assume that the initial data and have zero mean on . There exists a sufficiently small constant depending on , such that if where and , then the unique smooth solution of the Cauchy problem (3)-(4) is global in time.
Proof. We proceed as in the proof of Theorem 7 and obtain the energy estimate
The second term on the right side is estimated using the product estimate in Proposition 2. Thus we obtain, similar to (45), By interpolation inequality, we have where and . Combining estimates (61) gives
We obtain the desired result as in the proof of Theorem 7.
Remark 9. When , the result of Theorem 8 is still open.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
This work is supported by the Doctoral Starting-up Foundation of Minnan Normal University, China-NSAF (Grant no. L21228).
- H. A. Dijkstra, Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Cirrculation and El Nino, Springer, Berlin, Germany, 2nd edition, 2005.
- J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, NY, USA, 1987.
- L. C. Berselli, “Vanishing viscosity limit and long-time behavior for 2D quasi-geostrophic equations,” Indiana University Mathematics Journal, vol. 51, no. 4, pp. 905–930, 2002.
- P. Constantin and J. Wu, “Behavior of solutions of 2D quasi-geostrophic equations,” SIAM Journal on Mathematical Analysis, vol. 30, no. 5, pp. 937–948, 1999.
- P. Constantin, D. Córdoba, and J. Wu, “On the critical dissipative quasi-geostrophic equation,” Indiana University Mathematics Journal, vol. 50, pp. 97–107, 2001.
- A. Córdoba and D. Córdoba, “A maximum principle applied to quasi-geostrophic equations,” Communications in Mathematical Physics, vol. 249, no. 3, pp. 511–528, 2004.
- N. Ju, “Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space,” Communications in Mathematical Physics, vol. 251, no. 2, pp. 365–376, 2004.
- A. Kiselev, F. Nazarov, and A. Volberg, “Global well-posedness for the critical 2D dissipative quasi-geostrophic equation,” Inventiones Mathematicae, vol. 167, no. 3, pp. 445–453, 2007.
- J. Wu, “Inviscid limits and regularity estimates for the solutions of the 2-D dissipative quasi-geostrophic equations,” Indiana University Mathematics Journal, vol. 46, no. 4, pp. 1113–1124, 1997.
- R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, Berlin, Germany, 1988.
- Y. L. Zhou, B. L. Guo, and L. H. Zhang, “Periodic boundary problem and Cauchy problem for the fluid dynamic equation in geophysics,” Journal of Partial Differential Equations, vol. 6, no. 2, pp. 173–192, 1993.
- T. T. Medjo, “On strong solutions of the multi-layer quasi-geostrophic equations of the ocean,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 11, pp. 3550–3564, 2008.
- X. Pu and B. Guo, “Existence and decay of solutions to the two-dimensional fractional quasigeostrophic equation,” Journal of Mathematical Physics, vol. 51, no. 8, Article ID 083101, 15 pages, 2010.
- C. E. Kenig, G. Ponce, and L. Vega, “Well-posedness of the initial value problem for the Korteweg-de Vries equation,” Journal of the American Mathematical Society, vol. 4, no. 2, pp. 323–347, 1991.
- E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1993.
- T. Tao, Nonlinear Dispersive Equations, vol. 106 of CBMS Regional Conference Series in Mathematics, Local and Global Analysis, The Conference Board of the Mathematical Sciences, Washington, DC, USA, 2006.
- D. B. Henry, “How to remember the Sobolev inequalities,” in Differential Equations, Sao Paulo 1981, vol. 957 of Lecture Notes in Mathematics, pp. 97–109, Springer, Berlin, Germany, 1982.
- H. Triebel, Theory of Function Spaces I, vol. 78 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 1983.
- H. Triebel, Theory of Function Spaces II, vol. 84 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 1992.
Copyright © 2013 Yongqiang Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.