Regularity criteria via horizontal component of velocity for the Boussinesq equations in anisotropic Lorentz spaces

Definition 1.1

(Weak solution) Given θ 0 ∈ L 1 ( R 3 ) ∩ L ∞ ( R 3 ) and u 0 ∈ L 2 ( R 3 ) with ∇ ⋅ u 0 = 0 , a pair of functions { u ( t ) , θ ( t ) } defined for t ≥ 0 is called a weak solution of the initial value Problem (1.1) if the following statements are valid:

1. (1.2) ( u , θ ) ∈ L ∞ ( 0 , T ; L 2 ( R 3 ) ) ∩ L 2 ( 0 , T ; H 1 ( R 3 ) ) , θ ∈ L ∞ ( 0 , T ; L 1 ( R 3 ) ∩ L ∞ ( R 3 ) ) ,

   for all T > 0 .

2. (1.1) 1 , 2 , 3 are satisfied in the sense of distributions;

3. the energy inequality

   ∥ u ( ⋅ , t ) ∥ L 2 2 + 2 ∫ 0 t ∥ ∇ u ( ⋅ , τ ) ∥ L 2 2 d τ ≤ ∥ u 0 ∥ L 2 2 + 2 ∫ 0 t ∫ R 3 θ u 3 d x d τ , ∥ θ ( ⋅ , t ) ∥ L 2 2 + 2 ∫ 0 t ∥ ∇ θ ( ⋅ , τ ) ∥ L 2 2 d τ ≤ ∥ θ 0 ∥ L 2 2 ,

   for all 0 ≤ t ≤ T .

Remark 1.1

A weak solution of (1.1) is regular on time interval I if the Sobolev norm ∥ ( u , θ ) ∥ H s is continuous for s > 1 2 on I . One can apply a standard bootstrap argument to show if a solution is regular, then u and θ are smooth.

Remark 1.2

System (1.1) has scaling property that if ( u , θ , π ) is a solution of System (1.1), then for any λ > 0 , the functions

are also solutions of (1.1), with the corresponding initial data

This motivates the study of (1.1) in function spaces that are left invariant by the aforementioned scaling.

Theorem 1.2

Denote r ˜ = ( r 1 , r 2 , r 3 ) > 2 . Let ( u 0 , θ 0 ) ∈ H 1 ( R 3 ) with ∇ ⋅ u 0 = 0 . Assume that ( u , θ ) is a weak solution of (1.1) on some interval [ 0 , T ) with 0 < T < ∞ . If the tangential components of the velocity u ˜ satisfy one of the following two conditions:

1. (1.3) u ˜ ∈ L 2 1 − ( 1 r 1 + 1 r 2 + 1 r 3 ) ( 0 , T ; L r ˜ , ∞ ( R 3 ) ) with ∑ i = 1 3 1 r i < 1 .

2. Assume that ∑ i = 1 3 1 r i = 1 . Then, there exists δ > 0 such that if

   (1.4) u ˜ ∈ L ∞ ( [ 0 , T ] ; L r ˜ , ∞ ( R 3 ) ) and ∥ u ˜ ∥ L ∞ ( [ 0 , T ] ; L r ˜ , ∞ ( R 3 ) ) ≤ δ ,

   then the solution ( u , θ ) ∈ C ∞ ( ( 0 , T ) × R 3 ) .

Theorem 1.3

Denote r ˜ = ( r 1 , r 2 , r 3 ) > 1 . Let ( u 0 , θ 0 ) ∈ H 1 ( R 3 ) with ∇ ⋅ u 0 = 0 . Assume that ( u , θ ) is a weak solution of (1.1) on some interval [ 0 , T ) with 0 < T < ∞ . If

then the solution ( u , θ ) ∈ C ∞ ( ( 0 , T ) × R 3 ) .

Remark 1.3

Our results improve almost all known regularity criteria involving Lorentz spaces or two components.

Remark 1.4

Since the blow-up criteria in Theorems 1.2 and 1.3 are only on the velocity field u and there is no any additional condition on the temperature field θ , on the one hand, the velocity vector field plays a very important role than the temperature field in the regularity criterion for the Boussinesq equations. On the other hand, thanks to the fact that Boussinesq equations (1.1) with θ = 0 reduces the Navier-Stokes equations, Theorems 1.2 and 1.3 cover the previous results [33] on Navier-Stokes equations in the largest anisotropic Lorentz spaces.

Definition 1.4

Let multi-indexes p ˜ = ( p 1 , p 2 , p 3 ) and q ˜ = ( q 1 , q 2 , q 3 ) with 0 < p i < ∞ , 0 < q i ≤ ∞ for all i ∈ { 1 , 2 , 3 } , the anisotropic Lorentz space L p ˜ , q ˜ ( R 3 ) is a set of measurable functions h ( x 1 , x 2 , x 3 ) on R 3 such that

Lemma 1.5

Let p ˜ = ( p 1 , p 2 , p 3 ) and q ˜ = ( q 1 , q 2 , q 3 ) with 1 ≤ p i < ∞ , 0 < q i ≤ ∞ for all i ∈ { 1 , 2 , 3 } .

1. For any 0 < κ < ∞ , if ∣ h ∣ κ ∈ L p ˜ , q ˜ ( R 3 ) , then h ∈ L κ p ˜ , κ q ˜ ( R 3 ) and

   ∥ ∣ h ∣ κ ∥ L p ˜ , q ˜ ( R 3 ) = ∥ h ∥ L κ p ˜ , κ q ˜ ( R 3 ) κ ,

   where κ p ˜ = ( κ p 1 , κ p 2 , κ p 3 ) and κ q ˜ = ( κ q 1 , κ q 2 , κ q 3 ) .

2. If f , g ∈ L p ˜ , q ˜ ( R 3 ) , then f + g ∈ L p ˜ , q ˜ ( R 3 ) and

   ∥ f + g ∥ L p ˜ , q ˜ ( R 3 ) ≤ C ( ∥ f ∥ L p ˜ , q ˜ ( R 3 ) + ∥ g ∥ L p ˜ , q ˜ ( R 3 ) ) ,

   where C = C ( p ˜ , q ˜ ) is a positive constant.

Lemma 1.6

Let p ˜ = ( p 1 , p 2 , p 3 ) , q ˜ = ( q 1 , q 2 , q 3 ) , and r ˜ = ( r 1 , r 2 , r 3 ) be the vector-valued indices. If 1 < p i < ∞ , 1 ≤ q i ≤ r i ≤ ∞ for all i ∈ { 1 , 2 , 3 } , then we have the following sequence of continuous embeddings:

with

where C = C ( p ˜ , q ˜ , r ˜ ) is a positive constant.

Lemma 1.7

Let 1 < p i , s i , α i < ∞ and 1 ≤ q i , r i , β i ≤ ∞ for all i ∈ { 1 , 2 , 3 } . Then, if for each i = 1 , 2 , 3 , it holds the following relations:

then for any f ∈ L α ˜ , β ˜ ( R 3 ) and g ∈ L s ˜ , r ˜ ( R 3 ) , we have f g ∈ L p ˜ , q ˜ ( R 3 ) with the estimate

Lemma 1.8

Let us assume that 2 ≤ q 1 , q 2 , q 3 < ∞ and

Then, for f ∈ C 0 ∞ ( R 3 ) , we have the following estimate:

and C is a constant independent of f.

Lemma 1.9

Let η ( ⋅ ) be a nonnegative, absolutely continuous function on [ 0 , T ] , which satisfies for a.e. t the inequality

where φ ( t ) and ψ ( t ) are nonnegative, summable functions on [ 0 , T ] . Then,

Proof

Let T > 0 be any given fixed time. From the second equation of (1.1), one may easily show that for any s ∈ [ 2 , ∞ ] ,

where we have used the divergence-free condition ∇ ⋅ u = 0 , so

Next, taking the L 2 -inner product of the first equation and the second equation in (1.1) with ( − Δ u ) and ( − Δ θ ) , respectively, and integrating by parts, we obtain

where

For ℐ 3 , it follows from the Cauchy-Schwarz inequality that

By the Sobolev inequality ∥ f ∥ L 6 ≤ C ∥ ∇ f ∥ L 2 , the Hölder inequality, and Young inequality, we obtain

where we have used the following interpolation inequality with (2.1):

due to the interpolation inequality and the energy inequality in Definition 1.1.

Now, we work on the first term on the right-hand side ℐ 1 . Using integration by parts, we obtain

Case 1. Let

Applying for the Hölder inequality in anisotropic Lorentz spaces and Lemma 1.8, we estimate the first integral ℐ 11 as:

Similarly, the second integral ℐ 12 can be estimated as:

Using integration by parts twice and divergence-free condition, we infer that

By a slight modification of the proof of ℐ 11 , we find that

Substituting ℐ 11 , ℐ 12 , and ℐ 13 into ℐ 1 , we obtain

Inserting the aforementioned estimates into (2.3), we have