Abstract
We study solutions of the initial value problem for the 2D regularized surface quasigeostrophic
(RSQG) equation. For
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
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 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
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
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
Lemma 2.1 Let
This lemma immediately yields the following corollary:
Corollary 2.2 Let
Now, we state and prove the global existence result for weak solutions for all time
if the initial condition θ_{0 }belongs to
Theorem 2.3 Let
Proof. Due to (1.3) and (2.2), we can have
Therefore, by Lemma 2.1, we know that
Taking
by the 2D GagliardoNirenbergLadyzhenskaya interpolation inequality, we can obtain
which implies that div(uθ) is bounded in
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
we have
On the other hand, using the facts that the functional operator
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
Let
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
where 1 < q < ∞.
By the Sobolev imbedding
Putting (2.11) into (2.8), then (2.7) becomes
In the above analysis, a is essentially arbitrary and we can choose
Putting (2.13) into (2.6), we have
where we have used the inequality
Thanks to the Gronwall's lemma, we obtain
This guarantees that the
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
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
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
The case m = 1 is proved in Theorem 2.3. Assume by induction that
Let
where
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
Moreover, if θ_{0}(x) ≥ 0, ∀x ∈Ω, then
Proof. Let θ(x, t) be the solution of (1.3), then
Denote u_{ }= max{u, 0}, if
Multiplying the above equation (3.2) by
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
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
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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