Abstract
This paper considers a single-layer fourth order quasi-geostrophic equation in two-dimensional case. We prove the existence and uniqueness of global smooth solution to the Cauchy problem of this equation by using energy estimate. We also establish a new estimate for the nonlinear term and obtain decay estimates of the solution in \(L^{2}\).
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
A hierarchy of ocean models occur in the literature of wind-driven circulation, starting from the most complex model and ending with a very elementary model (see e.g. [1, 7]). One of them is the so-called quasi-geostrophic \(\beta\)-plane model, which is considered as a simplification of the shallow-water equations when the Rossby number is small and the magnitude of bottom topography variations is comparable to the Rossby number. This paper studies the homogeneous quasi-geostrophic model by ignoring the effect of the bottom friction and the wind-stress effect. In this case, the model takes the form
where \(\psi =\psi (x,y,t)\) is the geostrophic pressure (or the geostrophic stream function), and the nonlinear term J is defined by
The coefficients in equation (1.1) are: the rotational Froude number F, the Coriolis parameter \(\beta\) and the Reynolds number \(R_e\).
Equation (1.1) is a single-layer quasi-geostrophic model. A direct extension of this model is a two-layer quasi-geostrophic model, where the densities are constant in each layer and the motion of fluid in both layers is coupled through the continuity of pressure and vertical velocity (see [1]). Due to wide applications in meteorology and oceanography, these models have been intensively studied in the past years. In [8], the authors proved global existence of weak solutions for the fractional quasi-geostrophic equation with \(\Delta ^{2} \psi\) replaced by \((-\Delta )^{1+\alpha }\psi\) (\(\alpha \in (0,1]\)) and they also obtained long-time behavior of the solution when \(\alpha \in (0,\frac{1}{2}]\). The authors of [3] discussed the existence theory and decay estimates for two-layer quasi-geostrophic model with fractional dissipative term. Decay estimates were also studied in [2] for a type of two-layer quasi-geostrophic model with both viscosity and friction. Medjo in [6] investigated the existence of strong solutions and maximal attractor for the multi-layer quasi-geostrophic equations.
In this work, we study the existence and large time behavior of smooth solution for the initial-value problem equipped with the initial data
Throughout the paper, for \(1 \le p<+\infty\), we denote by \(L^{p}(\mathbb {R}^{2})\) the Lebesgue space equipped the norm
For \(s\in \mathbb {R}\), \(H^{s}(\mathbb {R}^{2})\) denotes the nonhomogeneous Sobolev space whose norm is defined by
where \(\widehat{u}(\xi )\) is the Fourier transform of u.
Now we state the main results of the paper.
Theorem 1.1
Assume that \(\psi _{0}\in H^{m}({\mathbb {R}^{2}})\) with \(m\ge 4\) be an integer, then system (1.1)–(1.2) admits a unique global solution \(\psi \in C(\mathbb {R}^+;H^m({\mathbb {R}^{2}}))\).
Theorem 1.2
Let \(\psi _{0}\in H^{m}({\mathbb {R}^{2}})\cap L^{1}({\mathbb {R}^{2}})\) with \(m\ge 4\) be an integer, and \(\psi\) is the solution obtained by Theorem 1.1. Then for any multi-index \(\alpha\) we have the decay estimates
and
Theorem 1.1 is proved via a-priori energy estimates, and the proof is given in the next section. In Section 3, we present the proof of Theorem 1.2. We remark that the decay estimates of system (1.1)–(1.2) are not obtained in the previous works due to the effect of the nonlinear term \(J(\psi ,\Delta \psi )\). In this work, a new estimate is established for this nonlinear term and we apply this estimate to get the large time behavior for all the derivatives of the solution.
2 Global Existence of the Solution
In this section, we give the proof of Theorem 1.1. Indeed, the proof consists of two crucial steps. The first step is to obtain local existence of the solution to system (1.1)–(1.2), and the second step is to extend the local solution globally in time by establishing the a-priori estimates. For the first step, we can apply the regularized strategy of [5, Chapter 3] to study the approximated system
where \(\mathcal {J}_\epsilon f\) denotes the mollification of function f defined by
with \(\rho\) be a positive and radial \(C_0^\infty\) function whose mass is equal to one. By a limiting argument for the regularized system (2.1)–(2.2), it is not hard to obtain local existence and uniqueness of solution to system (1.1)–(1.2). Moreover, if \(T^*<+\infty\) is the maximal existence time of the solution, then there holds
Since the argument for the local existence part is standard, we omit further details. Hence, in order to complete the proof of Theorem 1.1, it is sufficient to establish the following three propositions which give the a-priori estimates.
Proposition 2.1
Let \(\psi\) be a sufficiently smooth solution to system (1.1)–(1.2). Assume \(\psi\) and its derivatives decay at infinity, then there hold
where \(C_1\), \(C_2\) depend only on \(\Vert \psi _0\Vert _{H^1},\ \Vert \nabla \psi _0\Vert _{H^1}\), respectively.
Proof
We multiply equation (1.1) with \(2\psi\) to get
Since
integrating (2.5) in time gives
Hence, the bound (2.3) follows.
To obtain (2.4), we multiply Eq. (1.1) with \(2\Delta \psi\) to get
Note that
thus the bound (2.4) follows immediately. \(\square\)
Proposition 2.2
With the same assumptions as Proposition 2.1, we have
where \(C_3\), \(C_4\) depend on \(\Vert \nabla \psi _0\Vert _{H^3}\) and t.
Proof
From Eq. (1.1), we can get the following energy identity
Note that
the nonlinear integral term of the above identity is estimated by
where we have used Young’s inequality and the fact \(\Vert \Delta \psi \Vert _{L^2}\le C\) in the last step. Hence, the bound (2.6) follows by choosing \(\delta =\frac{1}{R_{e}}\).
Similarly, taking energy estimate at the level of fourth order derivative gives
For the nonlinear term, we have
Then choosing \(\delta =\frac{1}{R_{e}}\) yields the desired bound (2.7). \(\square\)
Proposition 2.3
With the same assumptions as Proposition 2.2, there exists \(C_m>0\) depending on \(\Vert \psi _0\Vert _{H^m}\) and t such that
This proposition can be proved with an induction on m. We omit further details for simplicity. With these propositions, Theorem 1.1 thus follows.
3 Decay Estimates of the Solution
In this section we study the large time behaviour of solution to the Cauchy problem for the nonlinear quasi-geostrophic model (1.1)–(1.2). We first derive the integral identity of the solution. Applying Fourier transform to Eq. (1.1), we get
which implies that
In the succeeding arguments, we need to estimate the nonlinear term in (3.2) which is presented in Lemma 3.1 below.
Lemma 3.1
For any \(\psi \in H^{3}(\mathbb {R}^{2})\), there holds that
Proof
Recall that
we use integration by parts to rewrite \(\hat{J}(\psi , \Delta \psi )\) as
Thus there holds
\(\square\)
We remark that for the nonlinear estimate \(\hat{J}(\psi , \Delta \psi )\), in the works [2, 8], the authors used the following bound (due to [4])
However, we observe that it is not sufficient to prove Theorem 1.2 by using this bound. Therefore, the decay argument in [2, 8] can not cover our fourth-order quasi-geostrophic equation. Hence, the bound (3.3) is new and crucial in the following decay estimates. In particular, the step of establishing logarithmic decay bound is not needed in our proof by using this new bound (3.3). Now we can prove Theorem 1.2 in the framework of Fourier splitting method which is originally due to Schonbek [9, 10] and improved by Zhang [11].
Proof of Theorem 1.2
We first show
From the basic energy estimate (2.5), namely,
we have
Applying Plancherel’s theorem (that is, \(\Vert f\Vert _{L^2}=\Vert \hat{f}\Vert _{L^2}\) for any \(f\in L^2\)) to (3.5), we see
Define
then
Inserting (3.8) into (3.7), we obtain
From (3.2), Lemma 3.1 and (3.6), we can get
Using this estimate, we have
Now it follows from (3.10) and (3.11) that
then we have
Integrating the above inequality over interval [0, t], we get
Thus, we get the decay bound (3.4).
Next, we want to prove
From the proof of Proposition 2.1, we get the energy estimate
which can be rewritten in the Fourier space as
Define
We now treat the dissipative term as
and we use (3.10) to get
Inserting the above two estimates into (3.13), we can get
and integrating this inequality gives (3.12).
Then we will prove
From the proof of Proposition 2.2, we actually obtain the estimate
and by (3.12), we have
Denote
then there holds
where
Combining these estimates yields that
so we obtain (3.14) as desired.
Finally, applying the same treatment as above, we can get
or
As the idea of the proof is similar to (3.14), so it is omitted here. By the bounds (3.4), (3.12), (3.14), (3.15) and (3.16), we thus obtain the decay bounds (1.3) and (1.4) in Theorem 1.2. \(\square\)
Availability of data and materials
Not applicable.
References
Dijkstra, H.A.: Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and el niño. Springer, New York (2005)
Guo, B., Huang, D., Zhang, J.: Decay of solutions to a two-layer quasi-geostrophic model. Anal. Appl. 15(4), 595–606 (2017)
Guo, B., Han, Y., Huang, D., Bian, D., Zhang, L.: Global smooth solution of a two-dimensional nonlinear singular system of differential equations arising from geostrophics. J. Differ. Equ. 262(7), 3980–4020 (2017)
Guo, B., Zhang, L.: Decay of solutions to magnetohydrodynamics equations in two space dimensions. Proc. R. Soc. Lond.Ser. A 449(1935), 79–91 (1995)
Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2002)
Medjo, T.T.: On strong solutions of the multi-layer quasi-geostrophic equations of the ocean. Nonlinear Anal. 68(11), 3550–3564 (2008)
Pedlosky, J.: Geophysical Fluid Dynamics. Springer, New York (1987)
Pu, X., Guo, B.: Existence and decay of solutions to the two-dimensional fractional quasigeostrophic equation. J. Math. Phys. 51, 083101 (2010)
Schonbek, M.E.: L\(^2\) decay for weak solutions of the nonlinear Navier–Stokes equations. Arch. Ration. Mech. Anal. 88, 209–222 (1985)
Schonbek, M.E.: Large time behaviour to the Navier–Stokes equations. Commun. Partial Differ. Equ. 11(7), 733–763 (1986)
Zhang, L.: Sharp rate of decay of solutions to 2-dimensional Navier–Stokes equations. Commun. Partial Differ. Equ. 20(1–2), 119–127 (1995)
Acknowledgements
The authors would like to thank the anonymous referee for comments and suggestions.
Funding
J. Li is supported by Hunan Provincial Natural Science Foundation of China grant 2021JJ30697, and by Scientific Research Project of the Hunan Provincial Office of Education grant 20A022. J. Zhang is supported by the NSFC grants 11771183, 11971503 and Zhejiang Provincial Natural Science Foundation of China (Grant No. LY23A010006).
Author information
Authors and Affiliations
Contributions
HL focus on the decay estimates of the solution and the writing of the manuscript. JL completed the global existence results of the QG model. JZ conceived the study and the manuscript design as well as the revision of the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no competing interests.
Ethics approval and consent to participate
The authors approve and consent to participate.
Consent for publication
The authors agree to publication.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, 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/.
About this article
Cite this article
Li, H., Li, J. & Zhang, J. Existence and Decay Estimates of Solution for a Fourth Order Quasi-Geostrophic Equation. J Nonlinear Math Phys 30, 1282–1294 (2023). https://doi.org/10.1007/s44198-023-00130-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s44198-023-00130-8