Local and global solvability for the Boussinesq system in Besov spaces

: This article focuses on local and global existence and uniqueness for the strong solution to the Boussinesq system in n (cid:2) ( ≥ n 3 ) with full viscosity in Besov spaces. Under the hypotheses < < ∞ p 1 and − ∕ − ∕ < ≤ ∕ n p n p s n p min , 2 { } , and the initial condition ∈ × − ∕ − θ u B B , ˙ ˙ ps pn p

Abstract: This article focuses on local and global existence and uniqueness for the strong solution to the Boussinesq system in n ( )with sufficiently small norms 1 Introduction Motion of the ocean or the atmosphere can be simulated by the Boussinesq system.By Boussinesq approximation, we can neglect variation of the density of the fluid and derive a simplified model as follows: θ ν θ u θ t x u μ u u u π κθe t x u t x θ x θ x u x u x x Δ 0 , 0 , ; Δ , 0 , ; 0, 0, ; 0, , 0, , .
( ) ( ( ) ( ) ( )), θ x t , ( ), and π x t , ( ) represent the velocity, temperature (or sali- nity), and inner pressure of the fluid at the point ∈ x n ( ≥ n 3) and the time > t 0, respectively.However, u x 0 ( ) and θ x 0 ( ) represent initial distributions of the velocity and temperature (or salinity) in the fluid.Because of the variation of the temperature (or salinity), a vortex buoyancy force κθe n arises, where > κ 0 is the proportion coefficient, and = e 0, …,0, 1 n ( )is the unit vertical vector.For the sake of simplicity, by re-scaling of the unknowns, we always assume that = κ 1.Moreover, the viscosity coefficients ν μ , appearing in the diffusion terms are both assumed to be larger than 0.
Danchin and Paicu [1] and Cannon and DiBenedetto [2] addressed global existence of the weak solution of (1.1) under the initial condition where L σ n 2 ( ) denotes the collection of all sole- noidal vector fields with L 2 components.Brandolese and Schonbek [3] and Han [4] investigated time-decaying property of the weak solutions in energy space under the additional assumption ∈ θ L readdressed existence and uniqueness of the global weak solution by assuming that , , 0 for some > p 1 0 , and for sufficiently small number > c 0. Here ∞ L r, denotes the Lorentz space, which is equivalent to the weak L r space for > r 1 (refer to [6, §7.24]).If = n 3, then ∕ ∞ L n 3, is replaced by L 1 .We are much concerned with strong solvability of the Boussinesq system.Under the initial assumption , for some ≥ p n, and condition (1.2), where [1] proved that system (1.1) with partial viscosity ( = ν 0) admits a unique global strong solution θ u , ( ) in the class Under the initial assumption for some sufficiently small number > ε 0, [3] proved that D 3 Boussinesq system (1.1) with full viscosity has a unique global strong solution θ u , ( ) in the class Here the initial spaces So [1,3] can be viewed as partial extensions of [7][8][9], where the critical space Δ , the first equation of (1.1) exhibits different mechanical characteristics to the transport equation.This means that we could not make a timeindependent estimate for the function θ under the single assumption ∇ ∈ ∞ u L T L 0, ; 1 (( ) ) anymore.In fact, the two equations in (1.1) together comprise a mixed evolution system, where asymptotic actions of θ and u are influenced by the convective terms ⋅∇ u θ and ⋅∇ u u simultaneously.By reducing (1.1) into a sequence of approximate systems and making estimates for the approximate solutions, we will prove that if where the existing time T is uniform on a neighbourhood of θ u , 0 0 for some sufficiently small number > ε 0, then the local strong solution of (1.1) can be extended to the whole interval ∞ 0, [ ), and the norms can be controlled by the initial data ‖ ‖ uniformly on any bounded interval (see Theorem 2.5).This is a beneficial attempt to deal with strong solvability of the Boussinesq system (1.1) with full viscosity in Besov spaces.Compared with [1,5], treatment of the first equation of (1.1) here is much different.In order to make suitable estimates for θ from the transport-diffusion equation in general case ≤ < ∞ n p and ) is applied.We think here the value of σ is taken reasonable.In fact, it fits the treatment of is a pair of critical spaces for the initial data, and 1 for = n 3, initial assumption employed here is optimal.

Preliminaries and main results
We first make a brief review on the theory of Littlewood-Paley decomposition and homogeneous Besov spaces, for the detailed discussions, please refer to [10, §2.3].Let ∞ C n 0 ( ) be the collection of all smooth functions with compact supports and ′ n ( ) be the set of all temperate distributions.For each ≤ ≤ ∞ p 1 , let L p n ( ) be the common Lebesgue space, scalar or vector-valued type, with the norm denoted by 1 fo r a ll 0. where , where ≾ A B means that ≤ A CB for some > C 0. Define By this definition, it is easy to check that ], define the homogeneous Besov space B ˙p r s , as follows: for ≤ r β, and Hereinafter, > C 0 denotes a universal constant, it may even change from line to line, but does not depend on the time t and involved functions.Now we can state main results of the article.
. Then there exists > T 0 such that on the interval T 0, [ ], Boussinesq system (1.1) has a uniqueness strong solution Then there exists a small number > ε 0, such that under the initial condition )with the restriction and obeying the following estimates: (2.8) (2.9) and ) 0, ; ˙0, ; ˙0, ; ˙0, ; ˙0, ; 0, ; ṗs and verifies the following estimates: (2.11) and ) where the norms are defined by Step )}, where = θ u ¯, ¯0, 0 0 0 ( ) ( ), and ( )is the integral solution of the following parabolic equation system: where is the Leray projection.According to Lemmas 2.1 and 2.2, with the method of interpolation, we can deduce that Similarly, by taking ) and take > T 0 so small that ‖ ‖ ( ) .Under this setting and the assumptions: , by means of iteration, we obtain the uniform boundedness of θ u Step 2. Convergence of the approximate solutions.
. Evidently, ( )is the integral solution of the following problem: Similar to (3.2) and (3.3), we can derive that and .

͠
Summing up, we have Consequently, θ u , ( ) is an integral solution of the system Local and global solvability for the Boussinesq system  7 ( ), we have which combined with the a.e.differentiability of θ t L ( ) and u t L ( ) leads to the a.e.differentiability of θ t ( ) and u t ( ) and the validity of (3.5).As a straight consequence, we have ∂ ∈ ).We should mention that under the additional condition: , and θ u , ( ) exists globally.This can be proved by contraction as follows.
Assume that < < +∞ T 0 * , then similar to step 3 of the previous proof, we find that where the constants > C c , 0 are independent of t q , , and θ 1 , it follows that For arbitrary > ε 0, take ∈ k so large that . Thus, there exists > T 0 0 such that for all < < T T 0 0 and all ) can be eventually extended beyond T *.This however contradicts the definition of the maximal interval of existence.
Proof of Theorem 2.5.Assume that T 0, * [ )is the maximal interval of existence for the strong solution θ u , ( ) to system (1.1).By Remark 3.1, to show = ∞ T * , it suffices to verify Condition (3.6).For this purpose, we make estimates for the approximate solutions at first.
Given a Banach space X , for each > α 0 and < < T T 0 *, define Evidently, endowed with the norm ) is also a Banach space.Using the estimates for heat semigroups, one can easily check that, under the initial assumption , and Local and global solvability for the Boussinesq system  9 where the constants ), by using the following estimates (refer to [3]): where ≤ ≤ < ∞ p q 1 , f and F are vector and tensor fields, respectively, we can deduce that , ; 0, ; 1 1 2 0, ; where the constant > C 0 1 depends only on n.Meanwhile, continuity of respectively, can be checked directly.
)) with the norm Then putting (3.7), (3.8), (3.9), and (3.10) together, we have for all ∈ k .Making the same arguments on (3.4), we can derive the following estimates: …, which means that θ u , , we first consider Bony's decom- , and , qσ q θ L L q σ n p q q q q q L L q L L q σ n p L L q q q q q n p q L L L B L L q q q q q q σ n p Putting them together, we have Using this estimate, with application of Lemma 2.2 to the first equation of (1.1), we can deduce from the assumption < ≤ + , and Local and global solvability for the Boussinesq system  11 ) .Moreover, by performing the operators P and = − Q I P, respectively, on both sides of the second equation of (1.1), we obtain and Applying Lemma 2.2 to (3.18), we have ) and use the boundedness of P, Q on 1 , with application of logarithmic interpolation inequality (cf.[14]), then we can deduce that Here two terms remain untreated.One is ‖ ‖ ( ) ͠ . From (3.17), it follows that and    ).In the remaining part of the proof, we will derive (2.11) 2 , qs q θ L qs q q q q q L q L qs q q q q q n p q L q n p q L B L q q q q s q q q q s n p q qs q u L qs q q q L q L B B q q q q s q 4 1 ˙˙4 qs q L q s q q q L q L B L q q q q s q 1

3 (
are critical, in other words, they are invariant under the following scaling transformation:

∕ H ˙1 2
or L n was employed to deal with the global and strong solvability for the classical Navier-Stokes equations.This article focuses on the local and global strong solvability of the Boussinesq system (1.1) with full viscosity in Besov spaces.Because of the existence of the diffusion term −ν θ plugging (3.20) and (3.21) into (3.19),we can derive that that estimates (3.22) and (3.23) both hold on T 0, * ( ).Hence, if we assume < ∞ T * , then from (3.22) and (3.23), we obtain (3.6) easily, which just leads to the contradiction as shown in Remark 3.1.This proves global existence of the strong solution.Besides, from (3.17), (3.23), and (3.22), we reach the desired estimates (2.8), (2.9), and (2.10), respectively.Finally, uniqueness of the global strong solution comes from uniqueness of the local strong solution and connectedness of the interval ∞ 0, [ and applying Lemma 2.2 to the first equation of (1.1), we have 27) into (3.19),we obtainLocal and global solvability for the Boussinesq system  13 1. Construction of the approximate solutions.
by the closedness of Δ and analyticity of e νtΔ and Pe μtΔ in neighbourhood with small radius.We only deal with the former limit, and the latter one can be treated in the same way.Assume that − < a Cauchy sequence in W T .Consequently, it converges to θ u Let us return to the first equation of (1.1).By invoking Lemma 2.3, we can deduce the following estimate: and (2.12) in the special case ≤ <