Existence and Decay Estimates of Solution for a Fourth Order Quasi-Geostrophic Equation

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 L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}$$\end{document}.


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 -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 = (x, y, t) is the geostrophic pressure (or the geostrophic stream function), and the nonlinear term J is defined by Throughout the paper, for 1 ≤ p < +∞ , we denote by L p (ℝ 2 ) the Lebesgue space equipped the norm For s ∈ ℝ , H s (ℝ 2 ) denotes the nonhomogeneous Sobolev space whose norm is defined by where û( ) is the Fourier transform of u.Now we state the main results of the paper.
Theorem 1.2 Let 0 ∈ H m (ℝ 2 ) ∩ L 1 (ℝ 2 ) with m ≥ 4 be an integer, and is the solution obtained by Theorem 1.1.Then for any multi-index 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( , Δ ) .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.

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 J f denotes the mollification of function f defined by with be a positive and radial C ∞ 0 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 * < +∞ 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.
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 ‖Δ ‖ L 2 ≤ C in the last step.Hence, the bound (2.6) follows by choosing = 1 R e .
Similarly, taking energy estimate at the level of fourth order derivative gives For the nonlinear term, we have This proposition can be proved with an induction on m.We omit further details for simplicity.With these propositions, Theorem 1.1 thus follows.

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 ∈ H 3 (ℝ 2 ) , there holds that Proof Recall that we use integration by parts to rewrite Ĵ(, Δ) as Thus there holds Δ y e −i(x 1 +y 2 ) dxdy x e −i(x 1 +y 2 ) dxdy �� 2 e −i(x 1 +y 2 ) dxdy yy e −i(x 1 +y 2 ) dxdy.

◻
We remark that for the nonlinear estimate Ĵ(, Δ) , 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 quasigeostrophic 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].
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.