LOGARITHMICALLY IMPROVED REGULARITY CRITERIA FOR ...

Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 137, pp. 1–7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

LOGARITHMICALLY IMPROVED REGULARITY CRITERIA FOR SUPERCRITICAL QUASI-GEOSTROPHIC EQUATIONS IN ORLICZ-MORREY SPACES SADEK GALA, MARIA ALESSANDRA RAGUSA

Abstract. This article provides a regularity criterion for the surface quasigeostrophic equation with supercritical dissipation. This criterion is in terms of the norm of the solution in a Orlicz-Morrey space. The result shows that, if a weak solutions θ satisfies α

T

Z 0

k∇θ(·, s)k α−r 2/r

M 2 L logP L

1 + ln(e + k∇⊥ θ(·, s)kL2/r )

ds < ∞,

for some 0 < r < α and 0 < α < 1, then θ is regular at t = T . In view of the 2/r embedding L2/r ⊂ Mp 2/r ⊂ ML2 logP L with 2 < p < 2/r and P > 1, our result extends the results due to Xiang [29] and Jia-Dong [15].

1. Introduction The surface quasi-geostrophic equation is ∂t θ + u · ∇θ + Λα θ = 0, θ(x, 0) = θ0 (x),

(1.1)

where θ = θ(x, t) is a scalar real-valued function of (x, t) ∈ R2 ×R+ and u = (u1 , u2 ) is the associated incompressible velocity field of the fluid with ∇ · u = 0, and determined from θ by u := (−∂2 Λ−1 θ, ∂ 1 Λ−1 θ) = R⊥ θ = (−R2 θ, R1 θ), where Λ = (−∆)1/2 is the Zygmund operator and Ri , i = 1, 2 are the Riesz transforms. The surface quasi-geostrophic equation with subcritical (1 < α ≤ 2) or critical dissipation (α = 1) have been shown to possess global classical solutions whenever the initial data is sufficiently smooth. However, the global regularity problem remains open for the supercritical case (0 < α < 1). Various regularity (or blow-up) criteria have been produced to shed light on this difficult global regularity problem (see e.g. [1, 2, 4, 5, 7, 8, 16, 17, 28, 30] and the references therein). The difficulties in understanding this problem are similar to those in solving the three-dimensional Navier-Stokes equations. 2010 Mathematics Subject Classification. 35Q35, 76D03. Key words and phrases. Quasi-geostrophic equations; logarithmical regularity criterion; Orlicz-Morrey space. c

2016 Texas State University. Submitted September 28, 2015. Published June 8, 2016. 1

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S. GALA, M. A. RAGUSA

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For the study of the regularity criterion to (1.1) for the critical and supercritical case, Constantin, Majda and Tabak [6] obtained the following conditions Z T lim sup kθ(t)kH m < ∞ if and only if k∇⊥ θ(·, t)kL∞ dt < ∞, (1.2) t%T

0

with m ≥ 3 and ∇⊥ = (−∂x2 , ∂x1 ). Later on, Chae [2] (see also [3]) generalizes (1.2) to obtain the regularity criterion of the supercritical quasi-geostrophic equation (1.1) under the assumption Z T 2 2 α k∇⊥ θ(·, t)krLp dt < ∞ with + ≤ α and < p < ∞. p r α 0 Recently, Xiang [29] improved the Chae’s result and obtained another logarithmically improved regularity criterion in terms of the Lebesgue space subject to the assumption Z T k∇θ(·, t)krLp 2 α 2 dt < ∞ with + ≤ α and < p < ∞. (1.3) ∞ 1 + ln(e + k∇θ(·, t)k ) p r α L 0 Very recently, Jia and Dong [15] improves the above regularity criterion (1.3) from Lebesgue space framework to Morrey-Campanato space framework. More precisely, they show the regularity of weak solution when the temperature function θ satisfies the growth condition Z T k∇θ(·, t)krMpq 2 α 2 dt < ∞ with + = α and < p < ∞. (1.4) p 1 + ln(e + k∇θ(·, t)k ) p r α L 0 The regularity criterion presented in this article states that, if a weak solution of (1.1) satisfies α

Z 0

T

k∇θ(·, s)k α−r 2/r M

1 + ln(e +

L2 logP L

k∇⊥ θ(·, s)kL2/r )

ds < ∞ with 0 < r < α,

for some 0 < r < α and 0 < α < 1, then θ is actually regular in H 2 on [0, T ], 2/r where ML2 logP L denotes the Orlicz-Morrey space. Since the embedding relation 2/r

L2/r ⊂ ML2 logP L with P > 1 holds, our regularity criterion can be understood as an extension of the regularity results of Xiang [29] and Jia-Dong [15]. Main tools used in this paper are a weighted norm inequality for the Riesz potential and the Gagliardo-Nirenberg inequality. 2. Orlicz-Morrey spaces and statement of the main result Before stating our result, let us recall some definitions and properties of the spaces that we are going to use (see e.g. [10, 11, 12, 13, 26] and references therein). Definition 2.1. For P ∈ R and 1 < w < v < ∞, the Orlicz-Morrey space MvLu logP L is defined by  kf kvMLu logP L := sup r2/v kf kB(x,r),Lu logP L : x ∈ R2 , r > 0 , (2.1) where kf kB(x,r),Lu logP L denotes the tw logP (3 + t) average given by kf kB(x,R),Lu logP L

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LOGARITHMICALLY IMPROVED REGULARITY

n := inf λ > 0 :

1 |B(x, R)|

 |f (x)| w

Z

λ

B(x,R)

3

 o |f (x)| P log 3 + dx ≤ 1 . λ

Our main result now reads as follows. Theorem 2.2. Let θ be a Leray-Hopf weak solutions of (1.1) with 0 < α < 1, namely θ ∈ L∞ (0, T ; L2 (R2 )) ∩ L2 (0, T ; H˙ α (R2 )). and satisfies the condition α

Z 0

T

k∇θ(·, s)k α−r 2/r M

L2 logP L

1 + ln(e +

k∇⊥ θ(·, s)k

L2/r )

ds < ∞

with 0 < r < α.

(2.2)

Then, the solution θ(x, t) is regular on (0, T ]. Remark 2.3. This criterion is in terms of the norm of the solution in a OrliczMorrey space. It is clear that Theorem 2.2 gives a logarithmic improvement of Xiang’s regularity criteria (1.3) (see also 1.4). As a consequence, this result extends several previous works. Meanwhile, the definition of classical Morrey-Campanato spaces is as follows (see e.g. [18]): Definition 2.4. For 1 < p ≤ q ≤ +∞, the Morrey-Campanato space is defined by n o Mpq = f ∈ Lploc (R2 ) : kf kMpq = sup sup |B|1/q−1/p kf kLp (B(x,R)) < ∞ , (2.3) x∈R2 R>0

where B(x, R) denotes the closed ball in R2 with center x and radius R. In view of (2.1) and (2.3), the definition (2.1) covers (2.3) as a special case when P = 0. Here and below we write (log a)P =: logP a. Recall the following crucial result established in [10, 13, 12] (see also [22, 23, 24, 25]). Theorem 2.5. Let 0 < α < 1 and fractional integral operator Iα be defined by Z f (y) Iα f (x) = dy. (2.4) |x − y|2−α R2 If P > 1, then kg · Iα f kL2 ≤ C kgkM3/α

L2 logP L

kf kL2 .

(2.5)

Additionally, we have the following embeddings: for P > 0 and 0 < u < u ˜ < v, Lv ,→ Lv,∞ ,→ Mvu˜ ,→ MvLu logP L ,→ Mvu

(2.6)

in the sense of continuous embedding and the inclusion is proper, where Lp,∞ denotes the usual Lorentz (weak-Lp ) space. For more details see [10, 13, 12]. We shall use as well the following useful Sobolev inequality. Lemma 2.6. Suppose that s > 1 and p ∈ [2, +∞]. Then, there is a constant C ≥ 0 such that kf kLp (R2 ) ≤ Ckf kH s (R2 ) . In particular, kf kL2/r (R2 ) ≤ Ckf kH 2 (R2 ) with 0 ≤ r ≤ 1.

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The above lemma can be proved using the well-known boundedness property of the Riesz potential operator (see, e.g., Stein [27]). In the proof of the main result, we employ the following Gagliardo-Nirenberg inequality having fractional derivatives contained in [14]. Lemma 2.7. Let 1 < p, p0 , p1 ≤ ∞, s, γ ∈ R+ , 0 ≤ β ≤ 1. Then, there exists a constant C such that β kf kH˙ s ≤ Ckf k1−β ˙γ , Lp kf kH p

where

0

p1

1 s γ
1−β 1 − = +β 1 − , p 2 p0 p 2

s ≤ βγ.

In particular, 1− s

s/γ

kf kH˙ s ≤ Ckf kL2 γ kf kH˙ γ .

(2.7)

Now we are in a position to prove our regularity criterion. Proof of Theorem 2.2. It suffices to prove that (2.2) ensures the a priori estimate Z T k∇⊥ θ(·, t)kL∞ dt < ∞, 0

hence guaranteeing the desired regularity until T by (1.2). For this, applying Λ2 to (1.1) and taking the L2 inner product of the resulting equation with Λ2 θ and integrating by parts, we obtain α d 2 kΛ θ(·, t)k2L2 + 2kΛ2+ 2 θ(·, t)k2L2 dt Z

(u · ∇θ)Λ4 θ dx

= −2 R2

Z = −2

Λ2 (u · ∇θ)Λ2 θ dx

R2

Z = −2

(Λ2 u · ∇θ)Λ2 θ dx − 2

Z

R2

Z = −2

(Λu · ∇Λθ)Λ2 θ dx − 2

R2 2

2

Z

(Λ u · ∇θ)Λ θ dx − 2 R2

Z

(u · ∇Λ2 θ)Λ2 θ dx

R2

(Λu · ∇Λθ)Λ2 θ dx,

R2

where we have used the following cancelation property Z Z Z 1 1 2 2 2 2 u · ∇|Λ θ| dx = − (∇ · u).|Λ2 θ|2 dx = 0. (u · ∇Λ θ)Λ θ dx = 2 R2 2 R2 R2 Notice that Λs and ∇ commute. Hence, by H¨older inequality, first we estimate J. r By using the Schwarz inequality, the fact that B˙ 2.2 = H˙ r and the interpolation inequality, we have α d 2 kΛ θ(·, t)k2L2 + 2kΛ2+ 2 θ(·, t)k2L2 dt Z

= −2

r

(Λ2 u · ∇θ)Ir (−∆) 2 Λ2 θ dx − 2

R2 ⊥ 2

Z R2

r 2

2

≤ 2kR Λ θ(·, t)kL2 k(∇θIr (−∆) Λ θ)(·, t)kL2 r

r

(Λu · ∇Λθ)Ir (−∆) 2 Λ2 θ dx

+ 2k∇Λθ(·, t)kL2 k(R⊥ ΛθIr (−∆) 2 Λ2 θ)(·, t)kL2 .

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LOGARITHMICALLY IMPROVED REGULARITY

5

If we invoke Theorem 2.5 and (2.7), then we have by Young inequality and the 2/r boundedness of R⊥ in the space L2 and ML2 logP L α d 2 kΛ θ(·, t)k2L2 + 2kΛ2+ 2 θ(·, t)k2L2 dt 2/r ≤ CkΛ2 θ(·, t)kL2 kΛ2 θ(·, t)kH˙ r k∇θ(·, t)kM 2 L

logP L

⊥

2

2

+ CkΛ θ(·, t)kL2 kΛ θ(·, t)kH˙ r kR Λθ(·, t)k 2− 2r

2r

2/r

≤ CkΛ2 θ(·, t)kL2 α kΛ2 θ(·, t)k α˙ α2 k∇θ(·, t)kM 2 P L log L H 1− αr  r/α  α α−r 2 2 = CkΛ θ(·, t)kL2 k∇θ(·, t)k 2/r kΛ2 θ(·, t)k2H˙ α2 M

L2 logP L

α 1 ≤ kΛ2 θ(·, t)k2H˙ α/2 + CkΛ2 θ(·, t)k2L2 k∇θ(·, t)k α−r . 2/r M 2 2 L logP L

Consequently, by absorbing the diffusion term into the left hand side, we obtain α d 2 kΛ θ(·, t)k2L2 + kΛ 2 +2 θ(·, t)k2L2 dt α

kΛ2 θ(·, t)k2L2

≤ Ck∇θ(·, t)k α−r 2/r M

L2 logP L α α−r 2/r M 2 L logP L

k∇θ(·, t)k ≤C

1 + ln(e + k∇⊥ θ(·, t)kL2/r )

[1 + ln(e + k∇⊥ θ(·, t)k

2

Lr

)]kΛ2 θ(·, t)k2L2

α

k∇θ(·, t)k α−r 2/r ≤C

M

L2 logP L

1 + ln(e + k∇⊥ θ(·, t)kL2/r )

[1 + ln(e + kΛ2 θ(·, t)kL2 )]kΛ2 θ(·, t)k2L2 ,

where we have used the Sobolev embedding (see Lemma 2.6) k∇⊥ θ(·, t)kL2/r ≤ CkΛ2 θ(·, t)kL2

for 0 < r < 1.

It follows that α

k∇θ(·, t)k α−r 2/r

M 2 d L logP L ln(e + kΛ2 θ(·, t)k2L2 ) ≤ C [1 + ln(e + kΛ2 θ(·, t)k2L2 )] ⊥ dt 1 + ln(e + k∇ θ(·, t)kL2/r )

and thus by Gronwall’s inequality, ln(e + kΛ2 θ(·, t)k2L2 ) α

 Z ≤ ln(e + kΛ2 θ0 (·, t)k2L2 ) exp C 0

T

k∇θ(·, t)k α−r 2/r M

L2 logP L

1 + ln(e + k∇⊥ θ(·, t)kL2/r )

 dt .

This gives the uniform boundedness of kΛ2 θ(·, t)k2L2 in the time interval [0, T ]. Recall that α d 2 kΛ θ(·, t)k2L2 + kΛ 2 +2 θ(·, t)k2L2 dt α (2.8) k∇θ(·, t)k α−r ≤C

2/r L2 logP L

M

1 + ln(e + k∇⊥ θ(·, t)kL2/r )

[1 + ln(e + kΛ2 θ(·, t)kL2 )]kΛ2 θ(·, t)k2L2 .

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S. GALA, M. A. RAGUSA

EJDE-2016/137

Integrating (2.8) over [0, T ], we have Z T α kΛ2 θ(·, t)k2L2 + kΛ 2 +2 θ(·, t)k2L2 dt 0 α

Z

T

k∇θ(·, t)k α−r 2/r M

2

P

L log L dt ⊥ θ(·, t)k 2/r ) 1 + ln(e + k∇ 0 L  sup [1 + ln(e + kΛ2 θ(·, t)kL2 )]kΛ2 θ(·, t)k2L2 + kΛ2 θ0 k2L2 ,

≤C

0≤t≤T

which implies Z

T

α

kΛ 2 +2 θ(·, t)k2L2 dt < ∞.

0

On the other hand, by the Gagliardo-Nirenberg inequality in R2 , it follows that 4

α

α+4 ⊥ k∇⊥ θkL∞ ≤ CkθkLα+4 2 k∇ θk 1+ α ˙ 2

H

≤ Ckθk

α α+4 L2

kΛ

α 2 +2

4

θkLα+4 2 .

Noting that kθkL2 ≤ kθ0 kL2 , implies Z T Z T 4 α α α+4 2 +2 θ(·, t)k k∇⊥ θ(·, t)kL∞ dt ≤ C kθ(·, t)kLα+4 2 kΛ L2 dt 0

0

α

≤ Ckθ0 kLα+4 2 α α+4 L2

≤ Ckθ0 k

T

Z 0

T

4

α

kΛ 2 +2 θ(·, t)kLα+4 2 dt

α+2 α+4

Z (

T

α

2

kΛ 2 +2 θ(·, t)k2L2 dt) α+4 < ∞.

0

By the blow-up criterion (2.2) of smooth solutions to (1.1), we complete the proof.  Acknowledgments. This work was done, while the first author was visiting University of Catania. He thanks the Department of Mathematics at the University of Catania for its support and hospitality. The authors want to express their sincere thanks to the editor and the referees for their invaluable comments and suggestions for improving this paper. References [1] L. Caffarelli, A. Vasseur; Drift diffusion equations with fractional diffusion and the quasigeostrophic equation, Ann. Math. 171 (2010), 1903-1930. [2] D. Chae; On the regularity conditions for the dissipative quasi-geostrophic equations, SIAM J. Math. Anal. 37 (2006), 1649-1656. [3] D. Chae, J. Lee; Global well-posedness in the supercritical dissipative quasi-geostrophic equations, Commun Math. Phys, 233 (2003), 297-311. [4] A. Cordoba, D. Cordoba; A maximum principle applied to quasi-geostrophic equations, Commum. Math. Phys. 249 (2004), 511-528. [5] P. Constantin, D. Cordoba, J. Wu; On the critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J. 50 (2001), 97-107. [6] P. Constantin, A. Majda, E. Tabak; Formation of strong fronts in the 2D quasi-geostrophic thermal active scalar, Nonlinearity 7 (1994), 1495-1533. [7] P. Constantin, J. Wu; Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal. 30 (1999), 937-948. [8] P. Constantin, J. Wu; Regularity of H¨ older continuous solutions of the super-critical quasigeostrophic equation, Ann. Inst. H. Poincar´ e Anal. NonLineaire 25 (2008), 1103-1110.

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LOGARITHMICALLY IMPROVED REGULARITY

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[9] S. Gala; Regularity criteria for the 3D magneto-micropolar fluid equations in the MorreyCampanato space, Nonlinear Differential Equations and Applications 17 (2010), 181-194. [10] S. Gala, Y. Sawano, H. Tanaka; A new Beale-Kato-Majda criteria for the 3D magnetomicropolar fluid equations in Orlicz-Morrey spaces, Math. Meth. Appl. Sci. 35 (2012), 13211334. [11] S. Gala, Y. Sawano, H. Tanaka; A remark on two generalized Orlicz–Morrey spaces, J. Approx. Theory 198 (2015), 1-9. [12] S. Gala, Y. Sawano, H. Tanaka; On the uniqueness of weak solutions of the 3D MHD equations in the Orlicz-Morrey space, Applicable Analysis 92 (2013), 776-783. [13] S. Gala, Y. Sawano, H. Tanaka, M. A. Ragusa; On the uniqueness of weak solutions of the 3D MHD equations in the Orlicz–Morrey space, Applicable Analysis 93 (2014), 356-368. [14] D. Gilbarg, N. S. Trudinger; Elliptic Partial Difrential Equations of Second Order, SpringerVerlag, Berlin, New York, 1983. [15] Y. Jia, B.-Dong; Remarks on the logarithmical regularity criterion of the supercritical surface quasi-geostrophic equation in Morrey spaces, Appl. Math. Lett. 43 (2015), 80-84. [16] N. Ju; Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equation in the Sobolev space, Comm. Math. Phys. 251 (2004), 365-376. [17] A. Kiselev, F. Nazarov, A. Volberg; Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent.Math. 167 (2007), 445-453. [18] P. G. Lemari´ e-Rieusset; Recent Developments in the Navier-Stokes Problem, London: Chapman & Hall/CRC, 2002. [19] F. Marchand; Weak-strong uniqueness criteria for the critical quasi-geostrophic equation, Physica D 237 (2008), 1346-1351. [20] J. Pedlosky; Geophysical Fluid Dynamics, Springer, New York (1987). [21] C. P´ erez; Sharp Lp-weighted Sobolev inequalities, Ann. Inst. Fourier (Grenoble) 45 (1995), 809-824. [22] Y. Sawano, S. Sugano, H. Tanaka; Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces, Trans. Amer. Math. Soc. 363 (2011), 6481–6503. [23] Y. Sawano, S. Sugano, H. Tanaka; A note on Generalized Fractional Integral Operators on Generalized Morrey Space, Boundary Value Problems, 2009 (2009), Article ID 835865, 18pp. [24] Y. Sawano, S. Sugano, H. Tanaka; Orlicz-Morrey spaces and fractional operators, Potential Analysis 36 (2012), 517-556. [25] Y. Sawano, S. Sugano, H. Tanaka; Olsen’s inequality and its applications to Schr¨ odinger equations. Harmonic analysis and nonlinear partial differential equations, 51–80, RIMS Kˆ okyˆ uroku Bessatsu, B26, Res. Inst. Math. Sci. (x RIMS), Kyoto, 2011. [26] Y. Sawano, S. Sugano, H. Tanaka, S. Gala; Olsen’s inequality and its applications to the MHD equations (Regularity and Singularity for Geometric Partial Differential Equations and Conservation Laws), RIMS Kokyuroku 1845 (2013), 121-144. [27] E. M. Stein; Harmonic Analysis. Princeton University Press, Princeton, 1993. [28] J. Wu; The 2D dissipative quasi-geostrophic equation, Appl. Math. Lett. 15 (2002), 925-930. [29] Z. Xiang; A regularity criterion for the critical and supercritical dissipative quasi-geostrophic equations, Appl. Math. Lett. 23 (2010), 1286-1290. [30] B. Yuan; Regularity condition of solutions to the quasi-geostrophic equations in Besov spaces with negative indices, Acta Mathematicae Applicatae Sinica, English Series 26 (2010), 381386. Sadek Gala Department of Mathematics, University of Mostaganem, Box 227, Mostaganem 27000, Algeria. ` di Catania, Viale Andrea Doria, Dipartimento di Matematica e Informatica, Universita 6 95125 Catania, Italy E-mail address: [email protected] Maria Alessandra Ragusa ` di Catania, Viale Andrea Doria, Dipartimento di Matematica e Informatica, Universita 6 95125 Catania, Italy E-mail address: [email protected]