Global existence for the regularized surface Quasi-geostrophic equation and its inviscid limits
© Li and Wang; licensee Springer. 2011
Received: 23 May 2011
Accepted: 27 October 2011
Published: 27 October 2011
We study solutions of the initial value problem for the 2D regularized surface quasi-geostrophic (RSQG) equation. For initial data, we prove the global existence and uniqueness of weak solution for RSQG equation with subcritical powers. For RSQG equation, we establish some regularization results and prove the inviscid limit of the RSQG equation to the classical quasi-geostrophic equation.
Mathematics Subject Classifications
35Q35; 76U05; 86A10.
where , denotes the Riesz transforms defined by Fourier transform: , is the pseudo-differential operator defined in the Fourier space by , here is the horizontal Laplacian operator.
where 0 ≤ α ≤ 1 and κ > 0 are real numbers. This model comes from . The quasi-geostrophic equation with dissipative term κ(-Δ) α θ has received an extensive amount of attentions and has many results in theory and numerical analysis (see e.g., [2–7] for further references), but there are few results on the surface quasi-geostrophic equation with regularized term κ(-Δ) α θ t . Therefore, in this paper, we mainly pay more attention to the regularized equation (1.3) to obtain the global existence, regularity for the solution and the inviscid limit of (1.3). The key issue is still whether weak solutions are regular for all the time. It is well known that for , the so-called subcritical case, the initial value problem with smooth periodic initial data θ0 has a global smooth solution [1, 8].
The quasi-geostrophic equation (1.1) is an important model in geophysical fluid dynamics. It is derived in the special case of constant potential vorticity and buoyancy frequency. Indeed, Equation (1.1) is an important example of a 2D active scalar with a specific structure most closely related to the 3D Euler equation (see ). The regularized version of (1.1), (1.1) with the dissipative term κ(-Δ) α θ or the regularized term κ(-Δ) α θ t , is the dimensionally correct analogue of the 3D incompressible Navier-Stokes equations when α = 1. In recent years, the 2D quasi-geostrophic equations with and without the dissipative term have attracted significant attention. For the system (1.1), in the previous works of Wu [1, 9–12], the well-posedness results for initial data θ0 in Lebesgue space L p , homogeneous Sobolev space , Morry space M p,λ and Hölder space C r have been studied. Chae  obtained these results for the initial data θ0 which belongs to the Triebel-Lizorkin space with and p, q ∈(1, ∞). For the critical case, that is , this problem was first dealt with by Constantin et al.  who showed the global existence in Sobolev space H1 under smallness assumption of the L∞-norm of the initial temperature θ, but the uniqueness is proved for initial data H2. A. Córdoba and D. Córdoba  proved that the viscosity solutions are smooth on the interval t ≤ T1 and t ≥ T2. A. Kiselev et al.  proved global existence of large smooth solutions. Later, in , Wu reformulates the problem as an integral equation and applies the Banach contraction mapping principle to prove local existence with initial value θ0 ∈H s (s > 1). Another recent progress on the critical dissipative QG equation was given in the work by Caffarelli and Vasseur . The supercritical case is open. For more results, see [1, 9, 16–18] and references therein. However, all these results mainly concentrate on the general quasi-geostrophic equation with dissipative term κ(-Δ) α θ.
They showed that the model (1.4) admits a maximum principle and obtained a necessary and sufficient condition that the solution of the regularized QG equations (1.4) develops a singularity in finite time and proves that, if the initial condition is smooth, then the regularized solution remains as smooth as the initial data for all times.
In the present paper, we will study the model (1.3) in the subcritical and critical case . On the one hand, we will establish the solution with lower regularity for initial data , and this improved the global existence for initial data θ0 ∈H s (s > 1) in . On the other hand, we generalize the results for (1.4) to the generally regularized models (1.3).
The rest of this article is organized as follows. In Section 2, we present the global existence for the regularized Equation (1.3). In Section 3, we give the regularization results for the regularized model (1.3) and obtain the maximum principle. Section 4 is devoted to the inviscid limit from the regularized surface quasi-geostrophic equation to classical surface quasi-geostrophic equation.
2 Global existence for the regularized surface quasi-geostrophic equation
In this section, we establish existence and uniqueness of global weak solutions of regularized model (1.3).
Noticing that if θ is in the Sobolev space H1(Ω), then belongs to the Sobolev space H2(Ω). This implies that u = ∇⊥ψ is in H1(Ω). Furthermore, we have the following lemma, which is proved in .
Lemma 2.1 Letandthen, whereis the dual space of. Moreover, forfixed, θ → div(θu) is a linear continuous operator fromto.
This lemma immediately yields the following corollary:
Corollary 2.2 Letsatisfying divu = 0 and, then < div(uθ), θ > = 0.
Now, we state and prove the global existence result for weak solutions for all time if the initial condition θ0 belongs to . More precisely, we have the following theorem.
Theorem 2.3 Letand, 2 < q < ∞, then the initial value problem (1.3) has a global unique solution.
Therefore, by Lemma 2.1, we know that (i.e., ). We first prove local existence and uniqueness. For this, it is enough to establish that the function div(uθ) is locally Lipshitz as a map from into . Before proving the locally Lipshitz condition, we give some important inequality we will use.
which implies that div(uθ) is bounded in by using lemma 2.1.
Thus, by the fixed point theory, we have the short time existence and uniqueness of solution for the functional differential equation (2.3).
Suppose that [0, T*] is the maximal interval of existence of the solution .
Now, we show the global existence for (1.3). To do this, it suffices to prove that the norm stays bounded on the maximal interval of existence for the solution θ of the regularized surface quasi-geostrophic equation (1.3) in the subcritical case .
where 1 < q < ∞.
This guarantees that the norm of θ is bounded. Therefore, the local solution can be extended uniquely to [0, 2T0] and the global solution is obtained by repeating this procedure. This completes the proof of Theorem 2.1.
3 Regularity results
In this section, we investigate the higher regularity and prove the maximum principle for the regularized system (1.3).
Theorem 3.1 (Regularity) Let, m ≥ 1, α = 1, then the solution for the regularized problem of quasi-geostrophic equation exists a solution.
Proof. The case m = 1 follows from Theorem 2.1. The case m > 1 had been obtained by Wu in . For completeness, here we give a different proof by following the proof. We need to proceed the steps by induction.
This proves the local existence and uniqueness of smooth solutions for the (1.3) in .
Next, we will show that the θ(t) is bounded in . To prove the global existence, i.e., it suffices to prove that remains bounded in any finite interval of time.
Thanks to that ψ(t) is bounded by the induction assumption, we get ϕ(t) is bounded on any finite interval of time. This completes the proof of Theorem 3.1.
Hence (3.1) holds.
Owing to θ0 ≥ 0, we have θ-(t) ≡ 0, ∀t > 0. Therefore, θ(x, t) ≥ 0. This completes the proof of Theorem 3.2.
4 Inviscid limit
In this section, we investigate the convergence of the solution of the regularized surface quasi-geostrophic equation (1.3) to a solution of the classical surface quasi-geostrophic equation (1.1) as κ tends to zero. We have the following result
where C is a constant depending only T* and
where the constant C does not depend on κ. By the Gronwall's Lemma, we get the desired result.
This work is supported by NSFC (Grant No. 10771009), BSFC (Grant No. 1085001) of China, Funding Project for Academic Human Resources Development in Institutions of Higher Leading Under the Jurisdiction of Beijing Municipality(PHR-IHLB 200906103) and Beijing Education Committee Funds.
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