Abstract
We study solutions of the initial value problem for the 2D regularized surface quasigeostrophic (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 quasigeostrophic equation.
Mathematics Subject Classifications
35Q35; 76U05; 86A10.
Keywords:
regularized surface quasigeostrophic equation; initial value problem; existence; uniqueness; regularization1 Introduction
The quasigeostrophic equation (QG) with periodic boundary conditions on a basic period box Ω = [0, 2π]^{2 }⊂ ℝ^{2 }is
where θ(x, t) is a realvalued function of x and t, which represents the potential temperature, and u represents the incompressible horizontal velocity at the surface. The advective velocity u in these equations is determined from θ by a stream function φ via the auxiliary relations
and the relationship (1.1)_{2}. The equality relating u to θ in (1.2) and (1.1)_{2 }can be reformulated in terms of periodic Riesz transforms
where , denotes the Riesz transforms defined by Fourier transform: , is the pseudodifferential operator defined in the Fourier space by , here is the horizontal Laplacian operator.
We also write down the regularized surface quasigeostrophic equation
where 0 ≤ α ≤ 1 and κ > 0 are real numbers. This model comes from [1]. The quasigeostrophic 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., [27] for further references), but there are few results on the surface quasigeostrophic 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 socalled subcritical case, the initial value problem with smooth periodic initial data θ_{0 }has a global smooth solution [1,8].
The quasigeostrophic 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 [3]). 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 NavierStokes equations when α = 1. In recent years, the 2D quasigeostrophic equations with and without the dissipative term have attracted significant attention. For the system (1.1), in the previous works of Wu [1,912], the wellposedness results for initial data θ_{0 }in Lebesgue space L^{p}, homogeneous Sobolev space , Morry space M_{p,λ }and Hölder space C^{r }have been studied. Chae [7] obtained these results for the initial data θ_{0 }which belongs to the TriebelLizorkin space with and p, q ∈(1, ∞). For the critical case, that is , this problem was first dealt with by Constantin et al. [13] who showed the global existence in Sobolev space H^{1 }under smallness assumption of the L^{∞}norm of the initial temperature θ, but the uniqueness is proved for initial data H^{2}. A. Córdoba and D. Córdoba [14] proved that the viscosity solutions are smooth on the interval t ≤ T_{1 }and t ≥ T_{2}. A. Kiselev et al. [5] proved global existence of large smooth solutions. Later, in [1], Wu reformulates the problem as an integral equation and applies the Banach contraction mapping principle to prove local existence with initial value θ_{0 }∈H^{s}(s > 1). Another recent progress on the critical dissipative QG equation was given in the work by Caffarelli and Vasseur [15]. The supercritical case is open. For more results, see [1,9,1618] and references therein. However, all these results mainly concentrate on the general quasigeostrophic equation with dissipative term κ(Δ)^{α}θ.
Recently, Khouider and Titi [6] study the following regularized model of surface quasigeostrophic equation:
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 θ_{0 }∈H^{s}(s > 1) in [1]. 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 quasigeostrophic equation to classical surface quasigeostrophic equation.
2 Global existence for the regularized surface quasigeostrophic equation
In this section, we establish existence and uniqueness of global weak solutions of regularized model (1.3).
Firstly, we rewrite the equation (1.3)_{1 }as a functional differential equation in the form
or
Noticing that if θ is in the Sobolev space H^{1}(Ω), then belongs to the Sobolev space H^{2}(Ω). This implies that u = ∇^{⊥}ψ is in H^{1}(Ω). Furthermore, we have the following lemma, which is proved in [6].
Lemma 2.1 Let and then , where is the dual space of . Moreover, for fixed, θ → div(θu) is a linear continuous operator from to .
This lemma immediately yields the following corollary:
Corollary 2.2 Let satisfying 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 θ_{0 }belongs to . More precisely, we have the following theorem.
Theorem 2.3 Let and , 2 < q < ∞, then the initial value problem (1.3) has a global unique solution .
Proof. Due to (1.3) and (2.2), we can have
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.
by the 2D GagliardoNirenbergLadyzhenskaya interpolation inequality, we can obtain
which implies that div(uθ) is bounded in by using lemma 2.1.
Next, we use the inequality (2.4) to show the div(uθ) is locally Lipschitz
where we have used the boundedness of Riesz transforms in space. Then, using the Poincaré inequality
we have
On the other hand, using the facts that the functional operator is an isomorphism from into and θ → (1 + κΛ^{2α})^{1}θ_{t }is a bounded operator from into , and using the Poincaré inequality (2.5), we know that the following norm is equivalent
Therefore,
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 quasigeostrophic equation (1.3) in the subcritical case .
Let be the solution of the initial value problem (1.3). Take the inner produce of Λ^{22α}θ with the first Equation in (1.3)
For the righthand side of (2.6), we have
where a ≤ α remains to be determined. By the calculus inequality for the CalderonZygmund type singular integral, we obtain
where and 2 < q < ∞. Considering the second equation in (1.3), by the CalderonZygmund inequalities, we obtain
and
where 1 < q < ∞.
By the Sobolev imbedding , then (2.10) becomes
Putting (2.11) into (2.8), then (2.7) becomes
In the above analysis, a is essentially arbitrary and we can choose without loss of generality so that . Therefore, we get
Putting (2.13) into (2.6), we have
where we have used the inequality , 2 < q < ∞, it is easy to prove for the regularized surface quasigeostrophic equation. By the inequality (2.14), we obtain
Thanks to the Gronwall's lemma, we obtain
This guarantees that the norm of θ is bounded. Therefore, the local solution can be extended uniquely to [0, 2T_{0}] 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 quasigeostrophic equation exists a solution .
Proof. The case m = 1 follows from Theorem 2.1. The case m > 1 had been obtained by Wu in [1]. For completeness, here we give a different proof by following the proof. We need to proceed the steps by induction.
It is obvious that if m ≥ 2 then equivalent to , which hints that . By applying the GagliardoNirenbergLadyzhenskaya interpolation inequality, as in Theorem 2.1, we have
In addition, similar steps as in the proof of Theorem 2.1 yield
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.
The case m = 1 is proved in Theorem 2.3. Assume by induction that . If , then , thus and we can obtain
where . Thus, by the Gronwall's lemma we obtain,
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.
Then, we will extend the maximum principle in [14,17] to the regularized surface quasigeostrophic equation by using the method of Khouider and Titi [6].
Theorem 3.2 Let and α = 1, κ is positive number, then the solution θ of the regularized problem (1.3) satisfies
Moreover, if θ_{0}(x) ≥ 0, ∀x ∈Ω, then
Proof. Let θ(x, t) be the solution of (1.3), then satisfies
Denote u_{ }= max{u, 0}, if then , we have
Multiplying the above equation (3.2) by , we have
which yields
It is obvious that the righthand side of (3.4) is zero, by the fact
Therefore,
which implies
Similarly, we have
Hence (3.1) holds.
Next, we show the rest of Theorem 3.2. Assume θ_{0}(x) ≥ 0, x ∈Ω. Multiplying the evolution equation for θ by θ_{ }= max{θ, 0} and integrating over the domain in the similar way as above, we have
i.e.,
Owing to θ_{0 }≥ 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 quasigeostrophic equation (1.3) to a solution of the classical surface quasigeostrophic equation (1.1) as κ tends to zero. We have the following result
Theorem 4.1 Let θ_{1 }and θ_{2 }be the smooth solutions of the RSQG equations (1.3) and the classical QG equations (1.1) with the same initial data , defined on the maximal time interval of existence [0, T*], then for any t < T*,
where C is a constant depending only T* and
Proof. Let u_{1 }and u_{2 }be the velocity field corresponding θ_{1 }and θ_{2}, respectively. Then, the difference θ(x, t) = θ_{1}(x, t)  θ_{2}(x, t) solves the equation
where u = u_{1 } u_{2}. Multiplying (4.1) by θ(x, t) and integrating over Ω, we get
where the two terms on the righthand side of (4.2) may be estimated as follows,
Since u_{2 }≤ C θ_{2}, it follows that,
To proceed, we need the calculus inequality for the CalderonZygmund type singular integral
where γ > 0, 1 < γ ≤ p ≤ ∞ and . Noticing the fact that (θ_{2})_{t }+ u_{2}·∇θ_{2 }= 0 and applying the inequality (4.3), we can bound the second term of righthand side of (4.2) by
Therefore,
where the constant C does not depend on κ. By the Gronwall's Lemma, we get the desired result.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All the authors typed, read, and approved the final manuscript.
Acknowledgements
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(PHRIHLB 200906103) and Beijing Education Committee Funds.
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