The lifespan of solutions to the inviscid 3D Boussinesq system

Accepted Manuscript The lifespan of solutions to the inviscid 3D Boussinesq system Xiaojing Xu, Zhuan Ye PII: DOI: Reference:

S0893-9659(13)00090-6 http://dx.doi.org/10.1016/j.aml.2013.03.009 AML 4351

To appear in:

Applied Mathematics Letters

Received date: 8 February 2013 Revised date: 22 March 2013 Accepted date: 22 March 2013 Please cite this article as: X. Xu, Z. Ye, The lifespan of solutions to the inviscid 3D Boussinesq system, Appl. Math. Lett. (2013), http://dx.doi.org/10.1016/j.aml.2013.03.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.

The lifespan of solutions to the inviscid 3D Boussinesq system Xiaojing Xu,

Zhuan Ye∗

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, The People’s Republic of China. ( [email protected]; [email protected] )

Abstract: In this paper, we consider the inviscid 3D Boussinesq system in Besov spaces and obtain the lower bound for the lifespan of solutions. AMS Subject Classification: 35Q35, 35B30, 35D05, 76D03, 76D05. Keywords: Lifespan, inviscid Boussinesq system, Littlewood-Paley decomposition, Besov spaces.

1

Introduction and motivation

In this paper, we devote to the following inviscid Boussinesq system:

with the initial data

   θt + u · ∇θ = 0,   ut + u · ∇u + ∇P = θed ,     div u = 0, θ(0, x) = θ0 (x),

(1.1)

u(0, x) = u0 (x).

Here, u : R+ × Rd → Rd is a vector field denoting the velocity, θ : R+ × Rd → R is a scalar

function denoting the temperature in the content of thermal convection and the density in the modeling of geophysical fluids, P the scalar pressure and ed is the unit vector in the xd direction. u, θ and P are all unknown functions. The Boussinesq system is extensively used

in the atmospheric sciences and oceanographic turbulence in which rotation and stratification are important (see for example [18, 22] and references therein). Also it is well known that the 2D Boussinesq equations is one of the most commonly used models because it shares a similar ∗

Corresponding author.

1

vortex stretching effect as that in the 3D incompressible flow, see [16] for more details. Thus, it has lately received significant attention in mathematical fluid dynamics (One can refer for example [3, 5, 10] and references therein). An obvious consideration shows that in the case of zero initial temperature the system (1.1) is reduced to the incompressible Euler equations. Over the past few years, (1.1) has been studied extensively theoretically see [6–8, 13–15, 17] and references therein. Constructing global unique solutions for some nonconstant θ0 is a challenging open problem (even in the two-dimensional case) which has many similarities with the global existence problem for the three-dimensional incompressible Euler equations, thus the global well-posedness of (1.1) is still a standing open problem. In the past, local existence and blow-up criteria have been established for the inviscid Boussinesq system and related models (see [1, 6–8, 11, 15, 17] and references therein). In the absence of global well-posedness theory for large initial data, the lifespan is of major importance for both theoretical and practical purposes. So it is interesting to estimate the maximum existence time of solutions, namely lifespan. In this paper, we are aiming at estimating the lower bound lifespan of solutions.

2

Notations, preliminaries and the main results

Notation 2.1 Throughout the paper, C stands for some real positive constants which may be different in each occurrence and independent on the initial data. We shall sometimes use the natation A . B which stands for A ≤ CB. Before we state the main results, we first explain the notations and conventions used throughout this paper. In this preparatory section, we recall the so-called Littlewood-Paley operators and their elementary properties which allow us to define the Besov spaces. It will be also convenient to introduce some function spaces and review some important lemma that will be used constantly in the following pages. Related materials can be found in several books and many papers (see for example [2, 4, 19, 23]). 8 d d 3 Let ϕ ∈ C∞ 0 (R ) be supported in the annulus C , {ξ ∈ R , 4 ≤ |ξ| ≤ 3 } and meet

X j∈Z

ϕ(2−j ξ) = 1 f or ξ 6= 0.

X j∈Z

ϕ(2−j ξ) = 1, ∀ξ 6= 0.

2

We denote a function χ(ξ) = 1 −

P

j∈N

ϕ(2−j ξ). For every u ∈ S 0 (tempered distributions) we

define the non-homogeneous Littlewood-Paley operators ∆j u = 0 j ≤ −2;

∆−1 u = χ(D)u;

∆j u = ϕ(2−j D)u

∀j ∈ N,

and

Sj u =

X

∆k u.

−1≤k≤j−1

Let us recall the definition of Besov spaces through the dyadic decomposition. s is defined as a space of Let s ∈ R, (p, r) ∈ [1, +∞]2 . The inhomogeneous Besov space Bp,r

f ∈ S 0 such that

s , kf kBp,r

 X

j≥−1

2jrs k∆j f krLp

1 r

and

s , sup 2js k∆j f kLp . kf kBp,∞

j≥−1

s as follows Thus we can define the homogeneous Besov space B˙ p,r

kf kB˙ s , p,r

X j∈Z

˙ j f kr p 2jrs k∆ L

1 r

and

kf kB˙ s

p,∞

˙ j f kLp . , sup 2js k∆ j∈Z

s as For s > 0, (p, r) ∈ [1, +∞]2 , we define the inhomogeneous Besov space norm Bp,r s kf kBp,r = kf kLp + kf kB˙ s . p,r

(2.1)

The following laws of product can be found in the reference [2] thus we omit its proof here. Lemma 2.1 For all s > 0, (p, r) ∈ [1, +∞]2 , the following inequalities hold s kuvkBp,r ≤

C s+1 s s ), (kukL∞ kvkBp,r + kvkL∞ kukBp,r s

kuvkB˙ s ≤

C s+1 (kukL∞ kvkB˙ s + kvkL∞ kukB˙ s ). p,r p,r s

and p,r

Next, we recall the well-known Calderon-Zygmund operators, which will be used to get the control between the gradient of velocity and the vorticity (see reference [4]). Lemma 2.2 There exists a universally positive constant C such that for every p ∈ (1, ∞), holding

k∇ukLp ≤ C where ω is the vorticity, namely ω , ∇ × u.

p2 kωkLp , p−1

In this paper, we need some simple commutator estimates as follows, the detailed proofs have been stated by many references, see for example [2, 11, 12]. 3

Lemma 2.3 Let s > −1, (p, r) ∈ [1, +∞]2 and u is a divergence-free vector field, namely div u = 0, we have the following inequality

s + k∇vkL∞ kukBp,r k2js k[∆j , u · ∇]vkLp kljr ≤ C(k∇ukL∞ kvkBp,r s−1 ),

(2.2)

where [∆j , u · ∇]v = ∆j (u · ∇v) − u · ∇(∆j v). If we set v = ω , ∇ × u, then (2.2) reduces to s , ∀s > −1. k2js k[∆j , u · ∇]ωkLp kljr ≤ Ck∇ukL∞ kωkBp,r

(2.3)

Lemma 2.4 (see [20, 21].) Let j ∈ Z be an integer and 1 ≤ p ≤ ∞. Then ˙ j , u · ∇]vkLp . k∇ukL∞ k∆ ˙ j vkLp + 2j kvkL∞ k∆ ˙ j ukLp . k[∆

(2.4)

Finally, let us now state our main results as follows s × B s for every 1 < p < ∞ and all s with s = Theorem 2.1 Suppose that (ω0 , ∇θ0 ) ∈ Bp,r p,r

if r = 1 and s >

d p

d p

if r ∈ (1, ∞], then the lifespan T ∗ of system (1.1) satisfies T∗ ≥

s kω0 kBp,r

where C is universal constant.

C q , s + 4e k∇θ0 kBp,r + kω0 k2Bp,r s

s × B s , p = 1 or p = ∞ and all s with s = d , if Theorem 2.2 Suppose that (ω0 , ∇θ0 ) ∈ Bp,r p,r p

r = 1 and s >

d p

if r ∈ (1, ∞], we need additional condition on initial data that (ω0 , ∇θ0 ) ∈

Lr × Lr for some 1 < r < ∞, then the lifespan T ∗ of system (1.1) satisfies T∗ ≥

s ∩Lr kω0 kBp,r

C q . 4 s ∩Lr + kω0 k2Bp,r s ∩Lr + e k∇θ0 kBp,r

Remark 2.1 In fact, the above theorems is also true when the dimension d is strictly larger than 3. Without loss of generality, we only consider the case that the dimension d is three. In dimension d = 2 the vorticity equation is simpler than in the general case due to the absence of the stretching term. Therefore, the two-dimensional case is in a certain sense better. The author in [11] obtained a logarithmic form lifespan in Besov space with null regularity index. However, the presence of vortex stretching term in the high dimension (d ≥ 3) adds more difficulties

because there is no good estimate on vortex stretching term in Besov space with null regularity index. 4

Remark 2.2 As θ ≡ 0, we get analogous result in [9] for the standard incompressible Euler equations. Obviously, Theorems 2.1 and 2.2 are also true for Boussinesq system with viscosity and partial viscosity.

3

The proof of the Theorem 2.1

In this section we will prove Theorem 2.1. The following inequalities are some easy consequences of Besov space imbedding properties and Lemma 2.2. s s (1 < p < ∞). . kωkBp,r k∇ukL∞ . k∇ukBp,r

(3.1)

We differentiate the temperature equation of system (1.1) in x, then we get ∂t ∇θ + u · ∇(∇θ) + (∇u) · ∇θ = 0.

(3.2)

˙ j operator to above equality, we have Hence applying homogeneous blocks ∆ ˙ j ∇θ + u · ∆ ˙ j ∇(∇θ) = −∆ ˙ j (∇u · ∇θ) − [∆ ˙ j , u · ∇]∇θ. ∂t ∆ Which, together with Lp -norm in x, the H¨ older inequality and div u = 0, directly gives d ˙ ˙ j (∇u · ∇θ)kLp + k[∆ ˙ j , u · ∇]∇θkLp . k∆j ∇θkLp . k∆ dt This fact combining with simple computations, Lemmas 2.1, 2.4 and (3.1) gives (s > 0) d k∇θkB˙ s p,r dt

. k∇ukL∞ k∇θkB˙ s + k∇θkL∞ kukB˙ p,r s+1 + k∇θ · ∇uk ˙ s B

p,r

p,r

. k∇ukL∞ k∇θkB˙ s + k∇θkL∞ kukB˙ p,r + k∇θkL∞ k∇ukB˙ s s+1 + k∇ukL∞ k∇θk ˙ s B p,r

p,r

p,r

. k∇ukL∞ k∇θkB˙ s + k∇θkL∞ k∇ukB˙ s p,r

p,r

s k∇θkB s . . kωkBp,r p,r

(3.3)

Where we have used the two facts: kukB˙ p,r in the homogeneous Besov spaces and s+1 ' k∇uk ˙ s B p,r

kf kB˙ s . kf k p,r

s Bp,r

for every s > 0.

Multiplying (3.2) by |∇θ|p−2 ∇θ and integrating over R3 with respect to variable x, we have

immediately

d k∇θkLp dt

. k∇ukL∞ k∇θkLp s k∇θkLp . . k∇ukBp,r

5

(3.4)

Combining (3.3) with (3.4), we can conclude that d s k∇θkB s . s . kωkBp,r k∇θkBp,r p,r dt

(3.5)

Now, in the dimension 3 the vorticity equation can be read as follows ∂t ω + u · ∇ω = ω · ∇u + ∇ × (θe3 ). Applying inhomogeneous blocks ∆j operator to above equality, some calculations yields ∂t ∆j ω + u · ∇∆j ω = ∆j (ω · ∇u) + ∆j (∇ × (θe3 )) − [∆j , u · ∇]ω. Taking Lp -norm in x, using the H¨older inequality together with divergence-free condition, we can conclude the following inequality d k∆j ωkLp . k∆j (ω · ∇u)kLp + k∆j (∇θ)kLp + k[∆j , u · ∇]ωkLp . dt Together with Lemma 2.1, (2.3) and (3.1) and above equality leads to d s kωkBp,r dt

s s s . k∇ukL∞ kωkBp,r + k∇θkBp,r + kω · ∇ukBp,r s s s s . k∇ukL∞ kωkBp,r + k∇θkBp,r + kωkL∞ k∇ukBp,r + k∇ukL∞ kωkBp,r s . . kωk2Bp,r + k∇θkBp,r s

(3.6)

Applying Gronwall’s inequality to (3.5), we can obtain s s e k∇θkBp,r . k∇θ0 kBp,r

Rt 0

s dτ kω(τ )kBp,r

.

(3.7)

Substituting the inequality (3.7) into the (3.6), it follows Rt d s 0 kω(τ )kBp,r dτ . s s kω(t)kBp,r . kωk2Bp,r + k∇θ k e s 0 Bp,r dt

(3.8)

Now we denote s , X(t) , kωkBp,r

s . Ω0 , k∇θ0 kBp,r

(3.8) can be rewritten as follows Rt d X(t) . X 2 (t) + Ω0 e 0 X(τ ) dτ . dt

We can easily find the simple fact that  Rt d X(t)e− 0 X(τ ) dτ . Ω0 . dt 6

(3.9)

Hence, we conclude that Rt

X(t) . (Ω0 t + X(0))e

0

X(τ ) dτ

.

(3.10)

We first temporarily assume that following inequality holds X(t) . AX(0),

(3.11)

where A stands for some positive constant depending on initial data to be fixed hereafter. From the inequality (3.10), we infer that X(t) . (Ω0 t + X(0))eAX(0)t . We see that for every t ≤ T1 ,

B−X(0) Ω0

(3.12)

> 0,

Ω0 t + X(0) ≤ B,

(3.13)

where B > X(0) is a constant depending only on the initial data to be fixed later. Combining the inequalities (3.11), (3.12) and (3.13), we have X(t) . BeAX(0)t . AX(0),

(3.14)

provided ln( AX(0) B ) t ≤ T2 , , (AX(0) > B). AX(0) Now we define a new function F (y) ,

(B − X(0))X(0) yX(0) y − ln( ). Ω0 B

It is easy to see that F (y) ≥ F (A0 ), where A0 =

Ω0 (B−X(0))X(0) .

(In fact, A0 is the minimum point of F (y) for y > 0 and thus

F (y) > F (A0 )). Hence, we set A= Thus, we choose B such that

Ω0 . (B − X(0))X(0) Ω0 ≤ e. B(B − X(0)) 7

(3.15)

So it implies T2 ≤ T1 . Together AX(0) > B with above inequality, we finally choose B such

that

1<

Ω0 ≤ e. B(B − X(0))

Now we denote B as follows B=

X(0) +

q X 2 (0) +

4Ω0 λ

2

,

1 < λ ≤ e,

(3.16)

and T ∗ ≥ T2 ,

ln( AX(0) B ) . AX(0)

(3.17)

We can get the lifespan T ∗ easily from inequalities (3.15)-(3.17). 2 ln λ q T∗ ≥  λ X(0) + X 2 (0) +

4Ω0 λ



1 < λ ≤ e.

As the right side of the above expression is increasing as the variable λ, thus we let λ = e, then the proof of Theorem 2.1 is completed.

4

The proof of Theorem 2.2

For the case p = 1 or p = ∞, we need the following estimate. Applying the standard Lr -estimate to the vorticity and temperature equations yields the following two elementary estimates kωkLr ≤ kω0 kLr +

Z t 0

 kωkLr k∇ukL∞ + k∇θkLr dτ , Rt

k∇θkLr ≤ k∇θ0 kLr e

0

k∇ukL∞ dτ

,

(4.1)

(4.2)

where the Gronwall’s inequality and some basic computations have been used. We only make minor modification for the proof of the case 1 < p < ∞. The difference is

that the Calderon-Zygmund operator is not bounded on Lp (R3 ) when p = 1 or ∞. To overcome s ∩ Lr . Hence we have this difficulty we need to change the functional setting to the spaces Bp,r

completed the proof of Theorem 2.2.

8

Acknowlegements Xu was partially supported by NSFC (No.11171026), BNSF (No.2112023) and the Fundamental Research Funds for the Central Universities of China. The authors would like to thank the anonymous referees for his or her careful reading of the manuscript, useful comments and suggestions for its improvement.

References [1] H. Abidi, T. Hmidi, On the global well-posedness for Boussinesq system, J. Differential Equations 233 (1) (2007) , 199–220. [2] H. Bahouri, J.-Y. Chemin and R. Danchin: Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, Springer (2011). [3] C. Cao, J. Wu, Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation, Arch. Rational Mech, 2013, in press. [4] J.-Y. Chemin: Perfect incompressible Fluids, Oxford University Press. [5] D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Advances in Math., 203 (2006) ,497–513. [6] X. Cui, C. Dou and Q. Jiu, Local well-posedness and blow up criterion for the inviscid Boussinesq system in H¨older spaces, J. Partial Differential Equations, 25 (2012), 220–238. [7] D. Chae, S. K. Kim, H. S. Nam, Local existence and blow up criterion of H¨ older continuous solutions of the Boussinesq equations, Nagoya Math. J. 155 (1999), 55–80. [8] D. Chae, H. S. Nam, Local existence and blow-up criterion for the boussinesq equations, Proceedings of the Royal Society of Edinburgh, Section A, 127(5)1997, 935-946. [9] D. Chae, Local existence and blow-up criterion for the Euler equations in the Besov spaces, Asymptotic Analysis, 38 (2004) 339–358. [10] D. Chae, J. Wu, The 2D Boussinesq equations with logarithmically supercritical velocities, Advances in Math. 230 (2012), 1618–1645. [11] R. Danchin, Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics, Proc. AMS, 121(6)(2013), 1979–1993. [12] R. Danchin, Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients, Revista Matem´ atica Iberoamericana 21 (2005), 861–886. [13] R. Danchin, M. Paicu, Global well-posedness issues for the inviscid Boussinesq system with Yudovich’s type data, Comm. Math. Phys. 290 (1) (2009), 1-14. [14] R. Danchin, M. Paicu, Global existence results for the anisotropic Boussinesq system in dimension two, Math. Models Methods Appl. Sci. 21 (2011), 421-457. [15] T. Hmidi, S. Keraani, On the global well-posedness of the Boussinesq system with zero viscosity, Indiana Univ. Math. J. 58 (4) (2009), 1591–1618. [16] T. Y. Hou, C. Li, Global well-posedness of the viscous Boussinesq equations, Discrete and Continuous Dynalical Systems A, 12(1)(2005), 1–12. [17] X. Liu, M. Wang, and Z. Zhang, Local well-posedness and blowup criterion of the Boussinesq equations in critical Besov spaces, Journal of Mathematical Fluid Mechanics, vol. 12(2)(2010), 280–292. [18] A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, in: Courant Lecture Notes in Mathematics, vol. 9, AMS/CIMS, 2003. [19] C. Miao, J. Wu and Z. Zhang, Littlewood-Paley Theory and its Applications in Partial Differential Equations of Fluid Dynamics, Science Press, Beijing, China, 2012 (in Chinese). [20] J. Wu, The Generalized Incompressible Navier-Stokes Equations in Besov Spaces. Dynamics of PDE, 1(4)(2004), 381–400. 9

[21] J. Wu, Solutions of the 2-D quasi-geostrophic equation in H¨ older spaces, Nonlinear Anal, 62 (2005), 579–594. [22] J. Pedlosky, Geophysical fluid dynamics, New York, Springer-Verlag, 1987. [23] H. Triebel, Theory of Function Spaces II, Birkhauser Verlag, 1992.

10