Local and global solvability for the Boussinesq system in Besov spaces

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:

Here u ( x , t ) = ( u 1 ( x , t ) , u 2 ( x , t ) , … , u n ( x , t ) ) , θ ( x , t ) , and π ( x , t ) represent the velocity, temperature (or salinity), and inner pressure of the fluid at the point x ∈ R n ( n ≥ 3 ) and the time t > 0 , respectively. However, u 0 ( x ) and θ 0 ( x ) 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 κ θ e n arises, where κ > 0 is the proportion coefficient, and e n = ( 0 , … , 0 , 1 ) 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 ( θ 0 , u 0 ) ∈ L 2 ( R n ) × L σ 2 ( R n ) , where L σ 2 ( R n ) denotes the collection of all solenoidal 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 0 ∈ L 1 . Danchin and Paicu [5] readdressed existence and uniqueness of the global weak solution by assuming that ( θ 0 , u 0 ) ∈ ( L n ∕ 3 , ∞ ∩ L p 0 , ∞ ) × L n , ∞ for some p 0 > 1 , 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 θ 0 ∈ B ˙ n , 1 0 ∩ L n ∕ 3 , and u 0 ∈ B ˙ p , 1 n ∕ p − 1 ∩ L n , ∞ for some p ≥ n , and condition (1.2), where ‖ θ 0 ‖ L n ∕ 3 , ∞ is replaced by ‖ θ 0 ‖ L n ∕ 3 , [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 3 D Boussinesq system (1.1) with full viscosity has a unique global strong solution ( θ , u ) in the class

Here the initial spaces L n ∕ 3 , L ∞ ( R 3 , ∣ x ∣ 3 d x ) , and L n , B ˙ p , 1 n ∕ p − 1 , L ∞ ( R 3 , ∣ x ∣ d x ) are critical, in other words, they are invariant under the following scaling transformation:

So [1,3] can be viewed as partial extensions of [7–9], where the critical space 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 Δ θ , the first equation of (1.1) exhibits different mechanical characteristics to the transport equation. This means that we could not make a time-independent estimate for the function θ under the single assumption ∇ u ∈ L 1 ( ( 0 , T ) ; L ∞ ) 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 1 < p < ∞ and − min { n ∕ p , 2 − n ∕ p } < s ≤ n ∕ p , then for any ( θ 0 , u 0 ) ∈ B ˙ p , 1 s − 1 × B ˙ p , 1 n ∕ p − 1 , system (1.1) possesses a unique strong solution ( θ , u , ∇ π ) on a short interval [ 0 , T ] in the class

where the existing time T is uniform on a neighbourhood of ( θ 0 , u 0 ) in B ˙ p , 1 s − 1 × B ˙ p , 1 n ∕ p − 1 (see Theorem 2.4 and Remark 3.1).

We also pursuit global existence and uniqueness of the strong solution under the assumption n ≤ p < ∞ and − n ∕ p < s ≤ n ∕ p , or especially n ≤ p < 2 n and − n ∕ p < s < n ∕ p − 1 . By making investigations on the global boundedness and convergence of the approximate solutions in L p − spaces, we will prove that if ( θ 0 , u 0 ) ∈ ( B ˙ p , 1 s − 1 ∩ L n ∕ 3 ) × ( B ˙ p , 1 n ∕ p − 1 ∩ L n ) and

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 ‖ θ 0 ‖ B ˙ p , 1 s − 1 + ‖ θ 0 ‖ L n ∕ 3 (or ‖ θ 0 ‖ B ˙ p , 1 s − 1 singly) and ‖ u 0 ‖ B ˙ p , 1 n ∕ p − 1 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 − n ∕ p < s ≤ n ∕ p , another function space L t ∞ ( B ˙ p , ∞ σ − 1 ) ∩ L ˜ t 1 ( B ˙ p , ∞ σ + 1 ) with lower index σ ∈ ( − 1 , min { s , n ∕ p − 1 } ) is applied. We think here the value of σ is taken reasonable. In fact, it fits the treatment of ‖ θ u ‖ L ˜ t 1 ( B ˙ p , 1 σ ) through Bony’s decomposition well. Since L n ∕ 3 × L n is a pair of critical spaces for the initial data, and L n ∕ 3 = L 1 for n = 3 , initial assumption employed here is optimal.

2 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 0 ∞ ( R n ) be the collection of all smooth functions with compact supports and S ′ ( R n ) be the set of all temperate distributions. For each 1 ≤ p ≤ ∞ , let L p ( R n ) be the common Lebesgue space, scalar or vector-valued type, with the norm denoted by ‖ ⋅ ‖ L p .

Suppose that φ ∈ C 0 ∞ ( R n ) with 0 ≤ φ ≤ 1 and supp φ ⊆ { ξ ∈ R n : 3 ∕ 4 ≤ ∣ ξ ∣ ≤ 8 ∕ 3 } such that

For each q ∈ Z and f ∈ S ′ ( R n ) , define

where χ ( ξ ) = 1 − ∑ q ≥ 0 φ ( 2 − q ξ ) for all ξ ∈ R n . It is easy to check that ‖ Δ ˙ q f ‖ L p ≾ ‖ f ‖ L p and ‖ S ˙ q f ‖ L p ≾ ‖ f ‖ L p for all 1 ≤ p ≤ ∞ , where A ≾ B means that A ≤ C B for some C > 0 .

Define

By this definition, it is easy to check that

for all f ∈ S h ′ ( R n ) .

For each triple of exponents ( s , p , r ) ∈ R × [ 1 , ∞ ] × [ 1 , ∞ ] , define the homogeneous Besov space B ˙ p , r s as follows:

where ‖ ( a q ) ‖ l r = ( ∑ q ∈ Z ∣ a q ∣ r ) 1 ∕ r if 1 ≤ r < ∞ and ‖ ( a q ) ‖ l r = sup q ∈ Z ∣ a q ∣ if r = ∞ . Evidently, endowed with the norm ‖ ⋅ ‖ B ˙ p , r s defined above, B ˙ p , r s is a normed space. Additionally, if s < n ∕ p or s = n ∕ p and r = 1 , then B ˙ p , r s turns to be a Banach space. In this case, f ∈ S h ′ ( R n ) for all f ∈ S ′ ( R n ) verifying ‖ f ‖ B ˙ p , r s < ∞ . There are some elementary properties of Besov spaces. If 1 ≤ p 1 ≤ p 2 ≤ ∞ and 1 ≤ r 1 ≤ r 2 ≤ ∞ , then B ˙ p 1 , r 1 s ↪ B ˙ p 2 , r 2 s − n ( 1 ∕ p 1 − 1 ∕ p 2 ) and L p 1 s ↪ B ˙ p 2 , ∞ − n ( 1 ∕ p 1 − 1 ∕ p 2 ) . For any f ∈ S h ′ ( R n ) , it holds that ∇ f ∈ S h ′ ( R n ) . Moreover, f ∈ B ˙ p , r s + 1 if and only if ∇ f ∈ B ˙ p , r s , and the norms ‖ f ‖ B ˙ p , r s + 1 and ‖ ∇ f ‖ B ˙ p , r s are equivalent.

Suppose that s ∈ R , p , r , β ∈ [ 1 , ∞ ] , and 0 < T ≤ ∞ , suppose also f ∈ S ′ ( ( 0 , T ) × R n ) such that f ( t ) ∈ S h ′ ( R n ) for all 0 < t < T . We say f ∈ L ˜ β ( ( 0 , T ) ; B ˙ p , r s ) (or L ˜ T β ( B ˙ p , r s ) if T < ∞ ), if

Direct calculation shows that

for r ≤ β , and

for β ≤ r .

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.