Global regularity for 2D SQG equation with variable viscosity term

Journal Pre-proof Global regularity for 2D SQG equation with variable viscosity term

Zhuan Ye

PII: DOI: Reference:

S0893-9659(19)30479-3 https://doi.org/10.1016/j.aml.2019.106153 AML 106153

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Applied Mathematics Letters

Received date : 7 October 2019 Revised date : 18 November 2019 Accepted date : 18 November 2019 Please cite this article as: Z. Ye, Global regularity for 2D SQG equation with variable viscosity term, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106153. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier Ltd. All rights reserved.

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Journal Pre-proof

Global regularity for 2D SQG equation with variable viscosity term Zhuan Ye Department of Mathematics and Statistics, Jiangsu Normal University, 101 Shanghai Road, Xuzhou 221116, Jiangsu, PR China E-mail: [email protected] Abstract: In this paper we establish the global regularity of the 2D SQG equation with variable viscosity term in non-divergence form. AMS Subject Classification 2010: 35Q35; 35B65; 76D03. Keywords: SQG equation; Global regularity; Variable viscosity. 1. Introduction

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In this paper, we consider the two-dimensional (2D) surface quasi-geostrophic (abbr. SQG) equation with variable viscosity term in non-divergence form  ∂ θ + (u · ∇)θ − κ(θ)∆θ = 0,   t u = R⊥ θ ≡ (−R2 θ, R1 θ), (1.1)   θ(x, 0) = θ0 (x), where θ is a scalar real-valued function and R1 , R2 are the standard 2D Riesz transforms. We shall assume κ(x) is a smooth function satisfying κ(x) ≥ α,

∀ x ≥ 0,

(1.2)

where α > 0 is some fixed positive constant. In the context of geophysical fluid dynamics, the SQG equation arises from the geostrophic study of the highly rotating flow [5] and has applications in both meteorological and oceanic flows [13]. Starting with the initial work [5], the global regularity results of the SQG equation have recently been studied very extensively and important progress has been made (see for example [7, 2, 11, 10, 6, 4, 9, 12, 3, 16]). The goal of this paper is to establish the global regularity for (1.1) under the assumption (1.2). More precisely, our main result is stated as follows. Theorem 1.1. Let θ0 ∈ H s (R2 ) with s > 1, then the SQG equation (1.1) has a unique global solution θ satisfying for any given T > 0

2. The proof of Theorem 1.1

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θ ∈ C([0, T ]; H s (R2 )) ∩ L2 (0, T ; H s+1 (R2 )).

This section is devoted to proving Theorem 1.1. In this paper, all constants will be denoted by C that is a generic constant. We shall write C(σ1 , σ2 , · · ·, σk ) as the constant C depends on the quantities σ1 , σ2 , · · ·, σk . As the local wellposedness result for (1.1) is standard, it suffices to establish the a priori estimates. We first show the L∞ -estimate and H 1 -estimate of θ. 1

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Lemma 2.1. Let θ be a smooth solution of (1.1), then there holds ∥θ(t)∥L∞ ≤ ∥θ0 ∥L∞ , ∥θ(t)∥2H 1

+

∫

(2.1)

t 0

∥θ(τ )∥2H 2 dτ ≤ C(θ0 , α, t).

(2.2)

Proof. Since θ(·, t) is a smooth solution for any t > 0, there exists a point xt ∈ R2 where |θ| attains its maximum value, with no loss of generality we let θ(xt , t) = ∥θ(t)∥L∞ .

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It should be pointed out that ∇x θ(xt , t) = 0 and ∆x θ(xt , t) ≤ 0. This along with (1.2) allows us to get d ∥θ(t)∥L∞ ≤ ∂t θ(xt , t) ≤ 0, for all t > 0. dt Integrating in time yields the desired estimate (2.1). We next show the H˙ 1 -estimate of u and θ. To this end, multiplying (1.1)2 by −∆θ and integrating it over the whole space imply ∫ ∫ 1d 2 ∥∇θ(t)∥L2 + κ(θ)∆θ∆θ dx = (u · ∇)θ ∆θ dx. 2 dt R2 R2 According to (1.2), we have ∫ κ(θ)∆θ∆θ dx ≥ α∥∆θ∥2L2 . R2

Noticing the fact ∇ · u = 0 and integrating by parts, one derives ∫ ∫ (u · ∇)θ ∆θ dx = − ∂j ui ∂i θ∂j θ dx R2

R2

≤ C∥∇u∥L2 ∥∇θ∥2L4

−1 ∥∇θ∥ ˙ 1 ≤ C∥∇u∥L2 ∥∇θ∥B˙ ∞,∞ B2,2

≤ C∥∇u∥L2 ∥θ∥B˙ ∞,∞ ∥∆θ∥L2 0

≤ C∥∇u∥L2 ∥θ∥L∞ ∥∆θ∥L2 α ≤ ∥∆θ∥2L2 + C∥θ∥2L∞ ∥∇u∥2L2 . 2 where we have used the sharp interpolation inequality (see [1, Theorem 2.42])

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p−q

q

∥f ∥Lp ≤ C∥f ∥B˙ p−γ ∥f ∥ p γ( p −1) ∞,∞

B˙ q,qq

with γ > 0 and 1 ≤ q < p < ∞. Collecting all the above estimates yields

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d ∥∇θ(t)∥2L2 + α∥∆θ∥2L2 ≤ C∥θ∥2L∞ ∥∇u∥2L2 . dt Thanks to (2.1) and the Gronwall inequality, we deduce from (2.3) that ∫ t 2 ∥∆θ(τ )∥2L2 dτ ≤ C(θ0 , α, t). ∥∇θ(t)∥L2 + 0

(2.3)

(2.4)

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Now multiplying (1.1)1 by θ and integrating it over R2 , we derive ∫ 1d 2 κ(θ)∆θθ dx, ∥θ(t)∥L2 ≤ 2 dt R2 ≤ C∥θ∥L2 ∥∆θ∥L2 ≤ C∥θ∥2L2 + C∥∆θ∥2L2 .

(2.5)

Noting (2.4), we get by applying the Gronwall inequality to (2.5) that ∥θ(t)∥L2 ≤ C(θ0 , α, t), which along with (2.4) immediately yields the desired estimate (2.2). This ends the proof of Lemma 2.1.  With (2.2) in hand, we are ready to prove Theorem 1.1. 1

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Proof of Theorem 1.1. Let J := (I − ∆) 2 and [A, f ]g = A(f g) − f Ag is the classical commutator. We obtain by applying J s with s > 1 to (1.1)2 and taking L2 inner product with J s θ that ∫ ∫ ( ) s 1d 2 s ∥θ(t)∥H s − J κ(θ)∆θ J θ dx = − [J s , u · ∇]θJ s θ dx. (2.6) 2 dt 2 2 R R Direct computations show ∫ ∫ ∫ ( ) s ( ) s ( ) s s − J κ(θ)∆θ J θ dx = − J ∂i κ(θ)∂i θ J θ dx + J s ∂i κ(θ)∂i θ J s θ dx 2 2 R2 ∫ R ∫ R ( ) ( ) = J s κ(θ)∂i θ J s ∂i θ dx + J s ∂i κ(θ)∂i θ J s θ dx 2 2 ∫R ∫ R = κ(θ)J s ∂i θJ s ∂i θ dx + [J s , κ(θ)]∂i θJ s ∂i θ dx 2 2 R R ∫ ( ) + J s ∂i κ(θ)∂i θ J s θ dx 2 R ∫ 2 ≥ α∥∇θ∥H s + [J s , κ(θ)]∂i θJ s ∂i θ dx 2 R ∫ ( ) + J s ∂i κ(θ)∂i θ J s θ dx. (2.7) R2

Inserting (2.7) into (2.6) leads to

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1d ∥θ(t)∥2H s + α∥∇θ∥2H s = − 2 dt −

∫

R2

∫

R2

s

s

[J , u · ∇]θJ θ dx −

∫

( ) J s ∂i κ(θ)∂i θ J s θ dx.

R2

[J s , κ(θ)]∂i θJ s ∂i θ dx

Recall the following classical estimates (see [8]) ( ) ∥[J s , f ]g∥Lp ≤ C ∥J s−1 g∥Lp1 ∥∇f ∥Lp2 + ∥J s f ∥Lp3 ∥g∥Lp4 , ∥J s (f g)∥Lp ≤ C (∥J s g∥Lp1 ∥f ∥Lp2 + ∥J s f ∥Lp3 ∥g∥Lp4 ) ,

(2.8)

(2.9) (2.10)

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where s > 0, p, p1 , p3 ∈ (1, ∞) and p2 , p4 ∈ [1, ∞] satisfy p1 = to (2.9), one derives ∫ s s ≤C∥[J s , u · ∇]θ∥L2 ∥J s θ∥L2 [J , u · ∇]θJ θ dx 2 R

1 + p12 p1

=

1 + p14 . p3

According

≤C(∥∇u∥L∞ ∥J s θ∥L2 + ∥∇θ∥L∞ ∥J s u∥L2 )∥J s θ∥L2 ≤C(∥∇u∥L∞ + ∥∇θ∥L∞ )(∥J s u∥2L2 + ∥J s θ∥2L2 ) ≤C(∥∇u∥L∞ + ∥∇θ∥L∞ )∥θ∥2H s .

(2.11)

∥G(g)∥H s ≤ C(1 + ∥g∥L∞ )[s]+1 ∥g∥H s ,

(2.12)

Now we state the following estimate [14]: Let s > 0, g ∈ H s (R2 ) ∩ L∞ (R2 ) and G(.) is a smooth function on R with G(0) = 0, then it holds

R

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where the constant C depends on supk≤[s]+2, |a|≤∥g∥L∞ ∥G(k) (a)∥L∞ . Using (2.9) and (2.12), we infer ∫ s s ≤C∥[J s , κ(θ)]∂i θ∥L2 ∥J s ∇θ∥L2 − [J , κ(θ)]∂ θJ ∂ θ dx i i 2 ≤C(∥∇κ(θ)∥L∞ ∥J s θ∥L2 + ∥∇θ∥L∞ ∥J s κ(θ)∥L2 )∥∇θ∥H s ≤C∥∇θ∥L∞ ∥J s θ∥L2 ∥∇θ∥H s

+ CF (∥θ∥L∞ )∥∇θ∥L∞ ∥J s θ∥L2 ∥∇θ∥H s

≤C∥∇θ∥L∞ ∥J s θ∥L2 ∥∇θ∥H s

+ CF (∥θ0 ∥L∞ )∥∇θ∥L∞ ∥J s θ∥L2 ∥∇θ∥H s α (2.13) ≤ ∥∇θ∥2H s + C∥∇θ∥2L∞ ∥θ∥2H s , 4 where F (z) is a nondecreasing function on z ≥ 0. Thanks to (2.10) and (2.12), it is not difficult to check ∫ ( ) ( ) s s − ≤C∥J s ∂i κ(θ)∂i θ ∥L2 ∥J s θ∥L2 J ∂ κ(θ)∂ θ J θ dx i i 2 R

≤C(∥∇κ(θ)∥L∞ ∥J s ∇θ∥L2 + ∥∇θ∥L∞ ∥J s ∇κ(θ)∥L2 )∥J s θ∥L2 ≤C∥∇θ∥L∞ ∥J s θ∥L2 ∥∇θ∥H s

+ C∥∇θ∥L∞ ∥κ(θ)∥H s+1 ∥J s θ∥L2

≤C∥∇θ∥L∞ ∥J s θ∥L2 ∥∇θ∥H s

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+ CF (∥θ∥L∞ )∥∇θ∥L∞ ∥θ∥H s+1 ∥J s θ∥L2 α ≤ ∥∇θ∥2H s + C∥∇θ∥2L∞ ∥θ∥2H s . 4 Putting (2.11), (2.13) and (2.14) into (2.8) implies

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d ∥θ(t)∥2H s + α∥∇θ∥2H s ≤ C(1 + ∥∇u∥2L∞ + ∥∇θ∥2L∞ )∥θ∥2H s . dt Let us recall the following logarithmic interpolation inequality ( ) √ ∥∇θ∥L∞ ≤ C 1 + ∥∇θ∥L2 + ∥∆θ∥L2 ln(e + ∥∇θ∥H s ) , s > 1.

(2.14)

(2.15)

(2.16)

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By the relation u = R⊥ θ, we may follow the proof of (2.16) to deduce ( ) √ ∥∇u∥L∞ ≤ C 1 + ∥∇θ∥L2 + ∥∆θ∥L2 ln(e + ∥∇θ∥H s ) , s > 1.

(2.17)

Inserting (2.16) and (2.17) into (2.15), it yields

d ∥θ(t)∥2H s + α∥∇θ∥2H s ≤C(1 + ∥∇θ∥2L2 )∥θ∥2H s dt + C∥∆θ∥2L2 ln(e + ∥∇θ∥2H s )∥θ∥2H s .

(2.18)

Noticing the simple interpolation

2− 2

2

2

∥∆θ∥2L2 ≤ C∥∇θ∥L2 s ∥∇θ∥Hs s ≤ C∥∇θ∥Hs s

with 2s < 2 due to s > 1, we derive by applying a refined logarithmic Gronwall inequality [15, Lemma 2.7] to (2.18) that ∫ t 2 ∥∇θ(τ )∥2H s dτ ≤ C(θ0 , α, t). (2.19) ∥θ(t)∥H s + 0

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which implies the desired global H s -estimate of Theorem 1.1. Moreover, the uniqueness can be deduced by (2.19) with s > 1. In fact, let θb and θe be two solutions of (1.1). e it is obvious to check Denoting δθ := θb − θ, ( )  b θb − κ(θ)∆ e θe = −(δu · ∇)θ, e  ∂ δθ + (b u · ∇)δθ − κ( θ)∆  t  (2.20) δu = R⊥ δθ,    δθ(x, 0) = 0, e Taking the L2 -inner product of (2.20) with δθ yields where δu := u b−u e ≡ R⊥ θb − R⊥ θ. 1 ∫ ( ∫ ) 1d b θb − κ(θ)∆ e θe δθ dx = − e dx ∥δθ(t)∥2L2 − κ(θ)∆ (δu · ∇)θδθ 2 dt R2 R2 e L∞ ∥δu∥L2 ∥δθ∥L2 ≤ C∥∇θ∥ e H s ∥δθ∥2 2 . ≤ C∥∇θ∥ L

(2.21)

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Moreover, we can show that ∫ ( ∫ ( ) ( ) ) b b e e b b − κ(θ) e ∆θe δθ dx − κ(θ)∆θ − κ(θ)∆θ δθ dx = − κ(θ)∆δθ + κ(θ) 2 R2 ∫R ∫ ( ) b b − κ(θ) e ∆θδθ e dx =− κ(θ)∆δθδθ dx − κ(θ) 2 2 ∫ R ∫R b i δθ∂i δθ dx + b i δθδθ dx = κ(θ)∂ ∂i κ(θ)∂ 2 2 R R ∫ ( ) b − κ(θ) e ∆θδθ e dx − κ(θ) 2 R ∫ 2 b i δθδθ dx ≥ α∥∇δθ∥L2 + ∂i κ(θ)∂ R2 ∫ ( ) b e e dx − κ(θ) − κ(θ) ∆θδθ (2.22) R2

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Putting (2.22) into (2.21) shows

∫ 1d 2 2 2 b i δθδθ dx e s ∂i κ(θ)∂ ∥δθ(t)∥L2 + α∥∇δθ∥L2 ≤C∥∇θ∥H ∥δθ∥L2 − 2 dt R2 ∫ ( ) b − κ(θ) e ∆θδθ e dx. + κ(θ)

(2.23)

R2

The Young inequality allows us to derive ∫ ∫ b − = − ∂ κ( θ)∂ δθδθ dx i i 2 R

R2

b b κ (θ)∂i θ∂i δθδθ dx ′

b L∞ ∥∇δθ∥L2 ∥δθ∥L2 ≤ C∥∇θ∥ α b 2 s ∥δθ∥2 2 . ≤ ∥∇δθ∥2L2 + C∥∇θ∥ H L 4

By the mean value theorem, we get ∫ ( ∫ ) b e e − κ(θ) − κ(θ) ∆θδθ dx = R2

R2

(2.24)

e dx κ′ (ζ)δθ∆θδθ

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e L2 ∥δθ∥2 4 ≤ C∥∆θ∥ L e ≤ C∥∆θ∥L2 ∥δθ∥L2 ∥∇δθ∥L2 α e 2 s ∥δθ∥2 2 . ≤ ∥∇δθ∥2L2 + C∥∇θ∥ H L 4

(2.25)

e Inserting (2.25) and (2.24) into (2.23) implies that where ζ lies between θb and θ. d e H s + ∥∇θ∥ b 2 s + ∥∇θ∥ e 2 s )∥δθ∥2 2 . ∥δθ(t)∥2L2 + α∥∇δθ∥2L2 ≤ C(∥∇θ∥ H H L dt Recalling (2.19) and the Gronwall inequality, we obtain δθ = 0,

which is the desired uniqueness. Therefore, the proof of Theorem 1.1 is completed.



Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 11701232) and the Natural Science Foundation of Jiangsu Province (No. BK20170224).

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