Startseite Mathematik Regularity of weak solutions to the 3D stationary tropical climate model
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Regularity of weak solutions to the 3D stationary tropical climate model

  • Huiyang Song , Qunyi Bie und Yanping Zhou EMAIL logo
Veröffentlicht/Copyright: 11. April 2025
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Open Mathematics
Aus der Zeitschrift Open Mathematics Band 23 Heft 1

Abstract

This article studies the regularity of weak solutions to the 3D stationary tropical climate model. We prove that when ( U , V , θ ) belongs to the homogeneous Morrey space M ˙ 2 , p ( R 3 ) with p > 3 , then ( U , V , θ ) ∈ C ∞ ( R 3 ) .

Keywords: weak solutions; tropical climate model; regularity
MSC 2010: 35B65; 35Q35

1 Introduction

In this article, we consider the following 3D stationary tropical climate model:

(1.1) − Δ U + div ( U ⊗ U ) + div ( V ⊗ V ) + ∇ P = 0 , − Δ V + div ( U ⊗ V ) + div ( V ⊗ U ) + ∇ θ = 0 , − Δ θ + U ⋅ ∇ θ + div V = 0 , div U = 0 ,

where U ( x ) = ( U 1 , U 2 , U 3 ) , V ( x ) = ( V 1 , V 2 , V 3 ) , θ = θ ( x ) , and P = P ( x ) denote the barotropic mode of the velocity field, the first baroclinic mode of the velocity field, the scalar temperature, and the scalar pressure, respectively. The ( i , j ) component of matrix U ⊗ V is U i V j with i , j = 1 , 2, 3.

System (1.1) becomes the standard stationary tropical climate model when the term div ( V ⊗ U ) in (1.1)2 is replaced by V ⋅ ∇ U . For the time-dependent version related to the standard stationary tropical climate model, i.e.,

(1.2) ∂ t u + μ ( − Δ ) α u + div ( u ⊗ u ) + div ( v ⊗ v ) + ∇ P = 0 , ∂ t v + ν ( − Δ ) β v + div ( u ⊗ v ) + v ⋅ ∇ u + ∇ θ = 0 , ∂ t θ + η ( − Δ ) γ θ + u ⋅ ∇ θ + div v = 0 , div u = 0 , ( u , v , θ ) ∣ t = 0 = ( u 0 , v 0 , θ 0 )

with the parameters μ , ν , η , α , β , γ ≥ 0 , there have been many studies on well-posedness and regularity. For the well-posedness results of system (1.2), when the viscosity depends on the temperature, Ye and Zhu [1] showed that there exists a global strong solution in R 2 for the case μ = ν = η = 0 . In this case, it is worth mentioning that system (1.2) reduces to the original tropical climate model derived by Frierson et al. [2]. For μ = ν = 1 , η = 0 , 0 ≤ α ≤ 1 , β ≥ 1 or α ≥ 1 , β ≥ 0 , Ma et al. [3] obtained the local well-posedness of strong solutions in R 2 . Li and Titi [4] showed the existence and uniqueness of strong solutions in R 2 with μ , ν > 0 , η = 0 , α = β = 1 . When μ , ν , η > 0 , α = β = 5 ⁄ 4 , γ = 0 , Li et al. [5] obtained the global well-posedness in R 3 . Zhang and Xu [6] established the global well-posedness of the classical solutions for system (1.2) with a full Laplacian term in R 3 . More 2D and 3D well-posedness results for system (1.2) were studied in [7–9].

For the regularity results of system (1.2), Ye [10] proved the global regularity in R 2 with μ , ν , η > 0 , α > 0 , β = γ = 1 . When μ , ν > 0 , η = 0 , 1 2 < α < 1 , 0 < β < 1 , 2 α + β = 2 , a regularity criterion obtained by Bisconti [11] in R 2 . Wang et al. [12] and Wu [13] proved the global regularity for system (1.2) with a full Laplacian term in R 3 , respectively. For more 2D and 3D regularity results related to system (1.2), one could see [14–16].

It is worth mentioning that when ∂ t u = ∂ t v = 0 , θ = 0 , and μ = ν = α = β = 1 , system (1.2) reduces to a system, which is similar to the standard stationary MHD system. Let us mention that Jarrín [17] obtained a criterion to improve the regularity of weak solutions to the standard stationary MHD system, provided that ( u , v ) ∈ M ˙ 2 , p ( R 3 ) .

To the best of our knowledge, with respect to the regularity of solutions to system (1.1), there are no corresponding results, which is our motivation in this article. Inspired by [17], we study the regularity of weak solutions to system (1.1) in the homogeneous Morrey space.

In this article, we write ‖ ( f 1 , f 2 , f 3 ) ‖ X ≔ ‖ f 1 ‖ X + ‖ f 2 ‖ X + ‖ f 3 ‖ X and use ‖ ⋅ ‖ X to denote ‖ ⋅ ‖ X ( R 3 ) . The notation D ′ ( R 3 ) is the dual space of D ( R 3 ) ≔ C 0 ∞ ( R 3 ) , where C 0 ∞ ( R 3 ) represents the set of infinitely differentiable functions with compact support in R 3 . The derivative of the multi-index α of the function f is ∂ α f = ∂ α 1 ∂ x 1 α 1 ∂ α 2 ∂ x 2 α 2 … ∂ α n ∂ x n α n f with α = ( α 1 , α 2 , … , α n ) and ∣ α ∣ = α 1 + α 2 + … + α n . The positive constant c may change from line to line.

Our main result can be stated as follows. Here, the definitions of weak solutions, M ˙ 2 , p ( R 3 ) and W k + 2 , p ( R 3 ) will be given in the next section.

Theorem 1.1

Let ( U , V , θ ) be a weak solution to system (1.1). Assume that ( U , V , θ ) ∈ M ˙ 2 , p ( R 3 ) with p > 3 and P ∈ D ′ ( R 3 ) , then, for all k ≥ 0 we have

( U , V , θ , P ) ∈ W k + 2 , p ( R 3 )

and

( ∂ α U , ∂ α V , ∂ α θ , ∂ α P ) ∈ C 1 − 3 ⁄ p ( R 3 ) , ∣ α ∣ ≤ k + 1 .

Remark 1.1

In fact, the function ( U , V , θ , P ) belongs to C ∞ ( R 3 ) since the exponent k could be large enough.

In order to prove Theorem 1.1, we need to overcome a difficulty, which is the estimation associated with ∇ θ and div V in system (1.1). In fact, Jarrín gave a way to deal with ( div ( F ) , div ( G ) ) (see [17], pp. 6–7), i.e.,

sup 0 ≤ t ≤ T ∫ 0 t e ( t − s ) Δ ( div ( F ) , div ( G ) ) d s M ˙ 2 , p ≤ sup 0 ≤ t ≤ T ∫ 0 t ‖ e ( t − s ) Δ ( div ( F ) , div ( G ) ) ‖ M ˙ 2 , p d s ≤ c sup 0 ≤ t ≤ T ‖ ( div ( F ) , div ( G ) ) ‖ M ˙ 2 , p ∫ 0 t d s ≤ c T ‖ ( div ( F ) , div ( G ) ) ‖ M ˙ 2 , p

and

sup 0 ≤ t ≤ T t 3 2 p ∫ 0 t e ( t − s ) Δ ( div ( F ) , div ( G ) ) d s L ∞ ≤ sup 0 ≤ t ≤ T t 3 2 p ∫ 0 t ‖ e ( t − s ) Δ ( div ( F ) , div ( G ) ) ‖ L ∞ d s ≤ c sup 0 ≤ t ≤ T t 3 2 p ∫ 0 t ( t − s ) − 3 2 p ‖ ( div ( F ) , div ( G ) ) ‖ M ˙ 2 , p d s ≤ c sup 0 ≤ t ≤ T t 3 2 p ‖ ( div ( F ) , div ( G ) ) ‖ M ˙ 2 , p ∫ 0 t ( t − s ) − 3 2 p d s ≤ c T ‖ ( div ( F ) , div ( G ) ) ‖ M ˙ 2 , p ,

where ( F , G ) denote the external force and p > 3 . However, if we follow the above estimates, namely,

sup 0 ≤ t ≤ T ∫ 0 t e ( t − s ) Δ ( ∇ θ , div V ) d s M ˙ 2 , p ≤ c T ‖ ( ∇ θ , div V ) ‖ M ˙ 2 , p

and

sup 0 ≤ t ≤ T t 3 2 p ∫ 0 t e ( t − s ) Δ ( ∇ θ , div V ) d s L ∞ ≤ T ‖ ( ∇ θ , div V ) ‖ M ˙ 2 , p ,

then the condition ( ∇ θ , div V ) ∈ M ˙ 2 , p ( R 3 ) is required. To avoid this additional condition, in this article, we will choose another way to deal with these two terms, see the estimates of J 4 and K 3 below.

The rest of this article is divided into two sections. In Section 2, we will introduce some lemmas, definitions of weak solutions and function spaces. In Section 3, we will give the proof of Theorem 1.1 by three steps.

2 Preliminaries

In this section, we first recall the definitions of homogeneous Morrey space M ˙ r , p ( R 3 ) and weak solutions to system (1.1). Then, we introduce the Sobolev-Morrey space W k , p ( R 3 ) and some lemmas.

Definition 2.1

The function ( U , V , θ ) ∈ L loc 2 ( R 3 ) is called a weak solution to system (1.1) if it satisfies

∫ R 3 ( − U ⋅ Δ ϕ + ( U ⊗ U ) : ∇ ϕ + ( V ⊗ V ) : ∇ ϕ + P ( ∇ ⋅ ϕ ) ) d x = 0 , ∫ R 3 ( − V ⋅ Δ ϕ + ( U ⊗ V ) : ∇ ϕ + ( V ⊗ U ) : ∇ ϕ + θ ( ∇ ⋅ ϕ ) ) d x = 0 , ∫ R 3 ( − θ ⋅ Δ ϕ + ( U ⋅ ∇ θ ) : ∇ ϕ + V ⋅ ( ∇ ⋅ ϕ ) ) d x = 0 , div U = 0 ,

for any test function ϕ ∈ C 0 ∞ ( R 3 ) .

Definition 2.2

Let 1 < r < p and 1 < p < + ∞ . The homogeneous Morrey space M ˙ r , p ( R 3 ) is the set of all functions f ∈ L loc r ( R 3 ) such that

‖ f ‖ M ˙ r , p ( R 3 ) ≔ sup x 0 ∈ R 3 , R > 0 R 3 p 1 ∣ B ( x 0 , R ) ∣ ∫ B ( x 0 , R ) ∣ f ( x ) ∣ r d x 1 r < + ∞ ,

where B ( x 0 , R ) is a ball of radius R centered at x 0 .

The space M ˙ r , p ( R 3 ) is a homogeneous space of degree − 3 p and satisfies the embedding relation L p ( R 3 ) ⊂ L p , q ( R 3 ) ⊂ M ˙ r , p ( R 3 ) . Here, L p , q ( R 3 ) is the Lorentz space and for more details of this space, one could refer to [18].

Definition 2.3

Let p > 2 , k ≥ 0 . The Sobolev-Morrey space W k , p ( R 3 ) is defined by

W k , p ( R 3 ) ≔ { ∂ α f ∈ M ˙ 2 , p ( R 3 ) for all ∣ α ∣ ≤ k } .

Lemma 2.1

[17] Let f ∈ M ˙ r , p ( R 3 ) ∩ L ∞ ( R 3 ) . For all 1 ≤ σ < + ∞ , there holds

‖ f ‖ M ˙ r σ , p σ ≤ c ‖ f ‖ M ˙ r , p 1 σ ‖ f ‖ L ∞ 1 − 1 σ .

Lemma 2.2

[19] The space M ˙ r , p ( R 3 ) is stable under convolution with functions in the space L 1 ( R 3 ) , and we have

‖ f ⁎ g ‖ M ˙ r , p ≤ c ‖ f ‖ M ˙ r , p ‖ g ‖ L 1 .

Lemma 2.3

[19] Let t > 0 and h t be the heat kernel: h t ( x , y ) = ( 4 π t ) − d 2 e − ∣ x − y ∣ 2 4 t , where ( x , y ) ∈ R d with d = 1 , 2 , … , n . Then, there holds

t 3 2 p ‖ h t ⁎ f ‖ L ∞ ≤ c ‖ f ‖ M ˙ r , p .

Lemma 2.4

[20] For i = 1 , 2, 3, R i ≔ ∂ i − △ denotes the ith Riesz transform. Then,

‖ R i R j ( f ) ‖ M ˙ r , p ≤ c ‖ f ‖ M ˙ r , p .

Lemma 2.5

[21] Let f ∈ S ′ such that ∇ f ∈ M ˙ 1 , p ( R 3 ) with p > 3 , where S ′ is the collection of all continuous linear functional defined on S , i.e., the tempered distributions space. There exists a constant c > 0 such that for all x , y ∈ R 3 , we have ∣ f ( x ) − f ( y ) ∣ ≤ c ‖ ∇ f ‖ M ˙ 1 , p ∣ x − y ∣ 1 − 3 ⁄ p . Here, M ˙ 1 , p ( R 3 ) is defined as the space of locally finite Borel measures d μ such that

sup x 0 ∈ R 3 , R > 0 R 3 p 1 ∣ B ( x 0 , R ) ∣ ∫ B ( x 0 , R ) d ∣ μ ∣ ( x ) < + ∞ .

Now, we introduce the following lemma, which is a simple result of the Banach fixed point theorem.

Lemma 2.6

[22,23] Let Y be a Banach space. Given a bilinear from H : Y × Y → Y such that ‖ H ( x 1 , x 2 ) ‖ Y ≤ c 0 ‖ x 1 ‖ Y ‖ x 2 ‖ Y for some constant c 0 > 0 and for all x 1 , x 2 ∈ Y , we have the following assertions for the equation:

(2.1) u = z + H ( u , u ) .

Suppose that z ∈ B ε ( 0 ) ≔ { f ∈ Y : ‖ f ‖ Y < ε } for some ε ∈ ( 0 , 1 4 c 0 ) , then (2.1) has a solution u ∈ B 2 ε ( 0 ) ≔ { f ∈ Y : ‖ f ‖ Y < 2 ε } . It means that the unique solution lies in the ball B 2 ε ( 0 ) ¯ . Moreover, the solution u depends continuously on z, that is, when z ¯ ∈ B ε ( 0 ) , u ¯ ∈ B 2 ε ( 0 ) , and u ¯ = z ¯ + H ( u ¯ , u ¯ ) , it follows that

(2.2) ‖ u − u ¯ ‖ Y ≤ 1 1 − 4 ε c 0 ‖ z − z ¯ ‖ Y .

We can conclude the local well-posedness of the solution from (2.2) when c 0 = c T a for some a > 0 .

3 The proof of main result

Proof of Theorem 1.1

For the sake of clarity, the proof is divided into three steps. In step 1, we prove the local existence and uniqueness of solution ( u , v , θ ) to system (3.1), i.e., Proposition 3.1. In step 2, we focus on system (1.1) and obtain the boundedness of ( U , V , θ ) in R 3 , i.e., Proposition 3.2. In step 3, we first study the regularity of the solution ( U , V , θ ) to system (1.1), then from the relation between ( U , V ) and pressure P (see (3.17)), we are able to obtain the regularity of P .

Step 1. The local existence and uniqueness of solution ( u , v , θ ) to system (3.1).

We consider the following time-dependent version of system (1.1):

(3.1) ∂ t u − Δ u + div ( u ⊗ u ) + div ( v ⊗ v ) + ∇ P = 0 , ∂ t v − Δ v + div ( u ⊗ v ) + div ( v ⊗ u ) + ∇ θ = 0 , ∂ t θ − Δ θ + u ⋅ ∇ θ + div v = 0 , div u = 0 , ( u , v , θ ) ∣ t = 0 = ( u 0 , v 0 , θ 0 ) .

Now, we give the following proposition:

Proposition 3.1

Let ( u 0 , v 0 , θ 0 ) ∈ M ˙ 2 , p ( R 3 ) with p > 3 . There exists a unique solution ( u , v , θ ) ∈ C * ( [ 0 , T 0 ] , M ˙ 2 , p ( R 3 ) ) to system (3.1), where T 0 is a positive time, depending on ‖ ( u 0 , v 0 , θ 0 ) ‖ M ˙ 2 , p such that

sup 0 ≤ t ≤ T 0 t 3 2 p ‖ ( u , v , θ ) ( t , ⋅ ) ‖ L ∞ < + ∞ .

Here, C * ( [ 0 , T ] , M ˙ 2 , p ( R 3 ) ) denotes the functional space of bounded and weak-⁎continuous functions on [ 0 , T ] with values in the homogeneous Morrey space M ˙ 2 , p ( R 3 ) .

Proof

Applying Leray’s projector P = I + ∇ ( − Δ ) − 1 div to (3.1)1, then by Duhamel’s formulae, we obtain

(3.2) u ( t , ⋅ ) = e t Δ u 0 + ∫ 0 t e ( t − s ) Δ P ( div ( u ⊗ u ) ) ( s , ⋅ ) d s + ∫ 0 t e ( t − s ) Δ P ( div ( v ⊗ v ) ) ( s , ⋅ ) d s ≔ I 1 + I 2 + I 3 ,

(3.3) v ( t , ⋅ ) = e t Δ v 0 + ∫ 0 t e ( t − s ) Δ ( div ( u ⊗ v ) ) ( s , ⋅ ) d s + ∫ 0 t e ( t − s ) Δ ( div ( v ⊗ u ) ) ( s , ⋅ ) d s + ∫ 0 t e ( t − s ) Δ ∇ θ ( s , ⋅ ) d s ≔ J 1 + J 2 + J 3 + J 4 ,

and

(3.4) θ ( t , ⋅ ) = e t Δ θ 0 + ∫ 0 t e ( t − s ) Δ ( u ⋅ ∇ θ ) ( s , ⋅ ) d s + ∫ 0 t e ( t − s ) Δ div v ( s , ⋅ ) d s ≔ K 1 + K 2 + K 3 ,

where ( I 1 , J 1 , K 1 ) is the linear term and ( I 2 , I 3 , J 2 , J 3 , J 4 , K 2 , K 3 ) is the bilinear term.

Exploiting Picard’s fixed point argument, we introduce the following Banach space:

E T ≔ f ∈ C * ( [ 0 , T ] , M ˙ 2 , p ( R 3 ) ) : sup 0 ≤ t ≤ T t 3 2 p ‖ f ( t , ⋅ ) ‖ L ∞ < + ∞ ,

with the norm

‖ f ‖ E T ≔ sup 0 ≤ t ≤ T ‖ f ( t , ⋅ ) ‖ M ˙ 2 , p + sup 0 ≤ t ≤ T t 3 2 p ‖ f ( t , ⋅ ) ‖ L ∞ .

We will estimate ( I 1 , I 2 , I 3 , J 1 , J 2 , J 3 , J 4 , K 1 , K 2 , K 3 ) in the space E T . As ( u 0 , v 0 , θ 0 ) ∈ M ˙ 2 , p ( R 3 ) , by Lemma 2.2 and Lemma 2.3, it is easy to obtain

(3.5) ‖ ( I 1 , J 1 , K 1 ) ‖ E T ≤ c ‖ ( u 0 , v 0 , θ 0 ) ‖ M ˙ 2 , p .

Next, we will show that

(3.6) ‖ ( I 2 , I 3 , J 2 , J 3 , J 4 , K 2 , K 3 ) ‖ E T ≤ c T 1 2 − 3 2 p ‖ ( u , v , θ ) ‖ E T 2 + c T 1 2 ‖ ( v , θ ) ‖ E T ,

where p > 3 .

We first give the following estimate, which will be used frequently in subsequent estimates:

∫ 0 t 1 ( t − s ) 1 2 s q d s = ∫ 0 t ⁄ 2 1 ( t − s ) 1 2 s q d s + ∫ t ⁄ 2 t 1 ( t − s ) 1 2 s q d s ≤ c t − 1 2 ∫ 0 t ⁄ 2 1 s q d s + t − q ∫ t ⁄ 2 t 1 ( t − s ) 1 2 d s ≤ c t 1 2 − q ,

where 0 ≤ q < 1 . For I 2 , recalling that P is continuous in M ˙ 2 , p ( R 3 ) , then by Lemma 2.2 and the well-known estimate on the heat kernel ‖ ∇ h ( t − s ) ( ⋅ ) ‖ L 1 ≤ c ( t − s ) 1 2 , we obtain

sup 0 ≤ t ≤ T ‖ I 2 ‖ M ˙ 2 , p = sup 0 ≤ t ≤ T ∫ 0 t e ( t − s ) Δ P ( div ( u ⊗ u ) ) ( s , ⋅ ) d s M ˙ 2 , p ≤ c sup 0 ≤ t ≤ T ∫ 0 t ∥ e ( t − s ) Δ ( div ( u ⊗ u ) ) ( s , ⋅ ) ∥ M ˙ 2 , p d s ≤ c sup 0 ≤ t ≤ T ∫ 0 t 1 ( t − s ) 1 2 s 3 2 p s 3 2 p ‖ u ( s , ⋅ ) ‖ L ∞ ‖ u ( s , ⋅ ) ‖ M ˙ 2 , p d s ≤ c T 1 2 − 3 2 p ‖ u ‖ E T 2 .

Using the Hölder inequality, the facts that e ( t − s ) Δ P ( div ( ⋅ ) ) can be written as a matrix of convolution operators and ‖ K i , j ( t − s , ⋅ ) ‖ L 1 ≤ c ( t − s ) 1 2 , with K i , j being the kernel of e ( t − s ) Δ P ( div ( ⋅ ) ) (see Proposition 11.1 of [23]), we obtain

sup 0 ≤ t ≤ T t 3 2 p ‖ I 2 ‖ L ∞ = sup 0 ≤ t ≤ T t 3 2 p ∫ 0 t e ( t − s ) Δ P ( div ( u ⊗ u ) ) ( s , ⋅ ) d s L ∞ ≤ sup 0 ≤ t ≤ T t 3 2 p ∫ 0 t ∥ e ( t − s ) Δ P ( div ( u ⊗ u ) ) ( s , ⋅ ) ∥ L ∞ d s ≤ c sup 0 ≤ t ≤ T t 3 2 p ∫ 0 t 1 ( t − s ) 1 2 s 3 p s 3 2 p ‖ u ( s , ⋅ ) ‖ L ∞ 2 d s ≤ c T 1 2 − 3 2 p ‖ u ‖ E T 2 ,

which together with the estimate of ‖ I 2 ‖ M ˙ 2 , p yields ‖ I 2 ‖ E T ≤ c T 1 2 − 3 2 p ‖ u ‖ E T 2 . Observing that U ⋅ ∇ θ = div ( U θ ) in K 2 , then similar to the estimate of I 2 , we have

‖ ( I 3 , J 2 , J 3 , K 2 ) ‖ E T ≤ c T 1 2 − 3 2 p ‖ ( u , v , θ ) ‖ E T 2 .

As stated above, the estimation of J 4 cannot follow directly from the estimate given by Jarrín (see pp. 6–7 in [17]), and we deal with J 4 as

sup 0 ≤ t ≤ T ‖ J 4 ‖ M ˙ 2 , p = sup 0 ≤ t ≤ T ∫ 0 t e ( t − s ) Δ ∇ θ ( s , ⋅ ) d s M ˙ 2 , p ≤ sup 0 ≤ t ≤ T ∫ 0 t ∥ e ( t − s ) Δ ∇ θ ( s , ⋅ ) ∥ M ˙ 2 , p d s ≤ c sup 0 ≤ t ≤ T ∫ 0 t 1 ( t − s ) 1 2 d s ‖ θ ‖ E T ≤ c T 1 2 ‖ θ ‖ E T

and

sup 0 ≤ t ≤ T t 3 2 p ‖ J 4 ‖ L ∞ = sup 0 ≤ t ≤ T t 3 2 p ∫ 0 t e ( t − s ) Δ ∇ θ ( s , ⋅ ) d s L ∞ ≤ c sup 0 ≤ t ≤ T t 3 2 p ∫ 0 t 1 ( t − s ) 1 2 s 3 2 p s 3 2 p ‖ θ ( s , ⋅ ) ‖ L ∞ d s ≤ c sup 0 ≤ t ≤ T t 3 2 p ∫ 0 t 1 ( t − s ) 1 2 s 3 2 p d s ‖ θ ‖ E T ≤ c T 1 2 ‖ θ ‖ E T ,

which yields that ‖ J 4 ‖ E T ≤ c T 1 2 ‖ θ ‖ E T . Similarly, we directly obtain ‖ K 3 ‖ E T ≤ c T 1 2 ‖ v ‖ E T .

The above estimates show that (3.6) holds. This together with (3.5) and Lemma 2.6 gives the local existence and uniqueness of solution ( u , v , θ ) to system (3.1) for a time 0 < T 0 ( u 0 , v 0 , θ 0 ) < + ∞ small enough, which completes the proof of Proposition 3.1.□

Step 2. The boundness of ( U , V , θ ).

Proposition 3.2

Let ( U , V , θ ) be a weak solution to system (1.1). If ( U , V , θ ) ∈ M ˙ 2 , p ( R 3 ) with p > 3 , then we have ( U , V , θ ) ∈ L ∞ ( R 3 ) .

Proof

First, applying Leray’s projector P to system (1.1)1, we obtain

(3.7) ∂ t U − Δ U + P ( div ( U ⊗ U ) ) + P ( div ( V ⊗ V ) ) = 0 , ∂ t V − Δ V + div ( U ⊗ V ) + div ( V ⊗ U ) + ∇ θ = 0 , ∂ t θ − Δ θ + U ⋅ ∇ θ + div V = 0 , div U = 0 ,

where ∂ t U = ∂ t V = 0 and ∂ t θ = 0 due to ( U , V , θ ) being the time-independent function.

Next, operating (3.7)1, (3.7)2, and (3.7)3 by e ( t − s ) Δ and integrating over [ 0 , t ] with respect to s , we obtain

(3.8) U = e t Δ U + ∫ 0 t e ( t − s ) Δ P ( div ( U ⊗ U ) ) d s + ∫ 0 t e ( t − s ) Δ P ( div ( V ⊗ V ) ) d s ,

(3.9) V = e t Δ V + ∫ 0 t e ( t − s ) Δ div ( U ⊗ V ) d s + ∫ 0 t e ( t − s ) Δ div ( V ⊗ U ) d s + ∫ 0 t e ( t − s ) Δ ∇ θ d s ,

and

(3.10) θ = e t Δ θ + ∫ 0 t e ( t − s ) Δ ( U ⋅ ∇ θ ) d s + ∫ 0 t e ( t − s ) Δ div V d s .

Since (3.8), (3.9), and (3.10) are structurally similar to (3.2), (3.3), and (3.4), respectively, we could obtain the existence and uniqueness of solution ( U , V , θ ) to system (3.7) by the same method as in Step 1. Then, by Proposition 3.1, we have

sup 0 ≤ t ≤ T 0 t 3 2 p ‖ ( U , V , θ ) ‖ L ∞ < + ∞ .

As ( U , V , θ ) does not depend on the time variable, hence ( U , V , θ ) ∈ L ∞ ( R 3 ) .□

Step 3. The regularity of weak solution ( U , V , θ ) and pressure P.

Applying Leray’s projector P to system (1.1)1, then we can write (1.1) 1 , (1.1) 2 , and (1.1) 3 as, respectively,

∂ α U = − 1 − Δ P ( ∂ α div ( U ⊗ U ) ) − 1 − Δ P ( ∂ α div ( V ⊗ V ) ) ≔ A 1 + A 2 , ∂ α V = − 1 − Δ ( ∂ α div ( U ⊗ V ) ) − 1 − Δ ( ∂ α div ( V ⊗ U ) ) − 1 − Δ ( ∂ α ∇ θ ) ≔ B 1 + B 2 + B 3 ,

and

∂ α θ = − 1 − Δ ( ∂ α ( U ⋅ ∇ θ ) ) − 1 − Δ ( ∂ α div V ) ≔ D 1 + D 2 .

Claim: The following estimates hold:

(3.11) ∥ ∂ α U ∥ M ˙ 2 σ , p σ < + ∞ ,

(3.12) ∥ ∂ α V ∥ M ˙ 2 σ , p σ < + ∞ ,

and

(3.13) ∥ ∂ α θ ∥ M ˙ 2 σ , p σ < + ∞ ,

where 1 ≤ σ < + ∞ and 1 ≤ ∣ α ∣ ≤ k + 2 with k ≥ 0 .

Proof of Claim

The proof is divided into four cases: ∣ α ∣ = 1 , 2, k + 1 and k + 2 with k ≥ 0 . We first consider the cases ∣ α ∣ = 1 and 2, and then we complete the proofs of the cases ∣ α ∣ = k + 1 and k + 2 with the aid of a recurrence hypothesis that ( ∂ α U , ∂ α V , ∂ α θ ) ∈ M ˙ 2 σ , p σ ( R 3 ) for all 1 ≤ σ < + ∞ and 1 ≤ ∣ α ∣ ≤ k .

In the case of ∣ α ∣ = 1 . For A 1 , by Lemmas 2.1, 2.4, Hölder inequalities, Proposition 3.2, and the fact that P is continuous in M ˙ r , p ( R 3 ) , we obtain

(3.14) ∥ A 1 ∥ M ˙ 2 σ , p σ ≤ c ‖ U ⊗ U ‖ M ˙ 2 σ , p σ ≤ c ‖ U ‖ M ˙ 2 , p 1 σ ‖ U ‖ L ∞ 2 − 1 σ < + ∞ .

Similar to (3.14), we infer

∥ A 2 ∥ M ˙ 2 σ , p σ ≤ c ‖ V ⊗ V ‖ M ˙ 2 σ , p σ ≤ c ‖ V ‖ M ˙ 2 , p 1 σ ‖ V ‖ L ∞ 2 − 1 σ < + ∞ .

Regarding B 1 , we conclude from Lemmas 2.1, 2.4, Hölder inequalities, and Proposition 3.2 that

(3.15) ‖ B 1 ‖ M ˙ 2 σ , p σ = ‖ − 1 − Δ ( ∂ α div ( U ⊗ V ) ) ‖ M ˙ 2 σ , p σ ≤ c ‖ U ⊗ V ‖ M ˙ 2 σ , p σ ≤ c ‖ U ‖ M ˙ 2 σ , p σ ‖ V ‖ L ∞ ≤ c ‖ U ‖ M ˙ 2 , p 1 σ ‖ U ‖ L ∞ 1 − 1 σ ‖ V ‖ L ∞ < + ∞ .

By a similar argument to (3.15), B 2 and B 3 are handled as

∥ ( B 2 , B 3 ) ∥ M ˙ 2 σ , p σ ≤ c ‖ U ‖ M ˙ 2 , p 1 σ ‖ U ‖ L ∞ 1 − 1 σ ‖ V ‖ L ∞ + ‖ θ ‖ M ˙ 2 , p 1 σ ‖ θ ‖ L ∞ 1 − 1 σ < + ∞ .

With respect to D 1 , applying Lemmas 2.1, 2.4, Hölder inequalities, and Proposition 3.2, one deduces

‖ D 1 ‖ M ˙ 2 σ , p σ = ‖ − 1 − Δ ∂ α div ( U θ ) ‖ M ˙ 2 σ , p σ ≤ c ‖ U ‖ M ˙ 2 , p 1 σ ‖ U ‖ L ∞ 1 − 1 σ ‖ θ ‖ L ∞ < + ∞ .

Similarly, it holds that

∥ D 2 ∥ M ˙ 2 σ , p σ ≤ c ‖ V ‖ M ˙ 2 σ , p σ ≤ c ‖ V ‖ M ˙ 2 , p 1 σ ‖ V ‖ L ∞ 1 − 1 σ < + ∞ .

In the case of ∣ α ∣ = 2 . We split α = α 1 + α 2 with ∣ α 1 ∣ = 1 , and ∣ α 2 ∣ = 1 . For A 1 , B 1 , and D 1 , we write

(3.16) A 1 = − 1 − Δ P ( ∂ α 1 div ∂ α 2 ( U ⊗ U ) ) , B 1 = − 1 − Δ ( ∂ α 1 div ∂ α 2 ( U ⊗ V ) ) , D 1 = − 1 − Δ ( ∂ α 1 div ∂ α 2 ( U θ ) ) .

Recalling the Leibinz rule, for i , j = 1 , 2, 3, we obtain ∂ α 2 ( U i V j ) = V j ∂ α 2 U i + U i ∂ α 2 V j . Using Lemmas 2.1, 2.4, Hölder inequalities, and Proposition 3.2, it follows that

∥ ( A 1 , B 1 , D 1 ) ∥ M ˙ 2 σ , p σ ≤ c ‖ ( U ∂ α 2 U , U ∂ α 2 V , V ∂ α 2 U , U ∂ α 2 θ , θ ∂ α 2 U ) ‖ M ˙ 2 σ , p σ ≤ c ‖ ∂ α 2 U ‖ M ˙ 2 σ , p σ ‖ ∂ α 2 V ‖ M ˙ 2 σ , p σ ‖ ∂ α 2 θ ‖ M ˙ 2 σ , p σ ‖ U ‖ L ∞ ‖ V ‖ L ∞ ‖ θ ‖ L ∞ < + ∞ .

Similarly, the following estimate holds:

∥ ( A 2 , B 2 , B 3 , D 2 ) ∥ M ˙ 2 σ , p σ ≤ c ‖ ( V ∂ α 2 V , U ∂ α 2 V , V ∂ α 2 U , ∂ α 2 θ , ∂ α 2 V ) ‖ M ˙ 2 σ , p σ ≤ c ‖ ∂ α 2 U ‖ M ˙ 2 σ , p σ ‖ ∂ α 2 V ‖ M ˙ 2 σ , p σ ‖ ∂ α 2 θ ‖ M ˙ 2 σ , p σ ‖ U ‖ L ∞ ‖ V ‖ L ∞ < + ∞ .

In the case of ∣ α ∣ = k + 1 . We split α = α 1 + α 2 with ∣ α 1 ∣ = 1 and ∣ α 2 ∣ = k . For A 1 , B 1 , and D 1 , we deduce

A 1 = − 1 − Δ P ( ∂ α 1 div ∂ α 2 ( U ⊗ U ) ) , B 1 = − 1 − Δ ( ∂ α 1 div ∂ α 2 ( U ⊗ V ) ) , D 1 = − 1 − Δ ( ∂ α 1 div ∂ α 2 ( U θ ) ) .

Applying the Leibinz rule, one has

∂ α 2 ( U i U j ) = ∑ ∣ β ∣ ≤ k c α 2 , β ∂ β U i ∂ α 2 − β U j , ∂ α 2 ( U i V j ) = ∑ ∣ β ∣ ≤ k c α 2 , β ∂ β U i ∂ α 2 − β V j , ∂ α 2 ( U i θ ) = ∑ ∣ β ∣ ≤ k c α 2 , β ∂ β U i ∂ α 2 − β θ ,

where i , j = 1 , 2, 3 and c α 2 , β > 0 . Making use of the recurrence hypothesis, Lemmas 2.1, 2.4, Hölder inequalities, and Proposition 3.2, we obtain

∥ ( A 1 , B 1 , D 1 ) ∥ M ˙ 2 σ , p σ ≤ c ∑ ∣ β ∣ ≤ k ( ∂ β U i ∂ α 2 − β U j , ∂ β U i ∂ α 2 − β V j , ∂ β U i ∂ α 2 − β θ ) M ˙ 2 σ , p σ ≤ c ∑ 1 ≤ ∣ β ∣ ≤ k − 1 ( ‖ ∂ β U ‖ M ˙ 2 σ , p σ ‖ ∂ α 2 − β U ‖ M ˙ 2 σ , p σ ‖ ∂ α 2 − β V ‖ M ˙ 2 σ , p σ × ‖ ∂ α 2 − β θ ‖ M ˙ 2 σ , p σ + ‖ ( ∂ k U , ∂ k V , ∂ k θ ) ‖ M ˙ 2 σ , p σ ‖ U ‖ L ∞ ‖ V ‖ L ∞ ‖ θ ‖ L ∞ ) < + ∞ .

Similarly, it holds that

∥ ( A 2 , B 2 , B 3 , D 2 ) ∥ M ˙ 2 σ , p σ ≤ c ∑ ∣ β ∣ ≤ k ( ∂ β V i ∂ α 2 − β V j , ∂ β V i ∂ α 2 − β U j , ∂ α 2 θ , ∂ α 2 V ) M ˙ 2 σ , p σ ≤ c ∑ 1 ≤ ∣ β ∣ ≤ k − 1 ( ‖ ∂ β V ‖ M ˙ 2 σ , p σ ‖ ∂ α 2 − β U ‖ M ˙ 2 σ , p σ ‖ ∂ α 2 − β V ‖ M ˙ 2 σ , p σ + ‖ ( ∂ k U , ∂ k V ) ‖ M ˙ 2 σ , p σ ‖ U ‖ L ∞ ‖ V ‖ L ∞ + ‖ ( ∂ α 2 V , ∂ α 2 θ ) ‖ M ˙ 2 σ , p σ ) < + ∞ .

In the case of ∣ α ∣ = k + 2 . We repeat again the above estimates to obtain ( ∂ α U , ∂ α V , ∂ α θ ) ∈ M ˙ 2 σ , p σ ( R 3 ) for ∣ α ∣ = k + 2 . Therefore, the claim holds.□

Next, we study the pressure term in system (1.1). Applying the operator ∂ α div to system (1.1)1, it holds that

(3.17) ∂ α P = 1 − Δ ( div ∂ α div ( U ⊗ U ) ) + 1 − Δ ( div ∂ α div ( V ⊗ V ) ) ,

where ∣ α ∣ ≤ k + 2 with k ≥ 0 . By (3.11), (3.12), the Leibinz rule, Lemmas 2.1 and 2.4, Hölder inequalities, and Proposition 3.2, we obtain ∂ α P ∈ M ˙ 2 σ , p σ ( R 3 ) for all 1 ≤ σ < + ∞ .

The above estimates show that ( U , V , θ , P ) ∈ W k + 2 , p ( R 3 ) with p > 3 , k ≥ 0 . Moreover, we can conclude from the embedding M ˙ 2 , p ( R 3 ) ⊂ M ˙ 1 , p ( R 3 ) and Lemma 2.5 that ( ∂ α U , ∂ α V , ∂ α θ , ∂ α P ) ∈ C 1 − 3 ⁄ p ( R 3 ) for all p > 3 and ∣ α ∣ ≤ k + 1 with k ≥ 0 . The proof of Theorem 1.1 is completed.□

Acknowledgements

The authors are grateful to the anonymous referees for their very helpful comments and suggestions that greatly improve the quality and presentation of the original manuscript.

  1. Funding information: This work was supported by research supported by the Natural Science Foundation of China (Grant No. 11901346).

  2. Author contributions: All authors contributed equally to this work. All authors read and approved the final manuscript.

  3. Conflict of interest: The authors state no conflicts of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2024-07-03
Revised: 2024-12-11
Accepted: 2025-02-05
Published Online: 2025-04-11

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2025-0137
Schlagwörter für diesen Artikel
weak solutions; tropical climate model; regularity
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Artikel in diesem Heft

  1. Special Issue on Contemporary Developments in Graphs Defined on Algebraic Structures
  2. Forbidden subgraphs of TI-power graphs of finite groups
  3. Finite group with some c#-normal and S-quasinormally embedded subgroups
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  96. Number Theory
  97. Fibonacci vector and matrix p-norms
  98. Recurrence for probabilistic extension of Dowling polynomials
  99. Carmichael numbers composed of Piatetski-Shapiro primes in Beatty sequences
  100. The number of rational points of some classes of algebraic varieties over finite fields
  101. Classification and irreducibility of a class of integer polynomials
  102. Decompositions of the extended Selberg class functions
  103. Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals
  104. Fibonacci Cartan and Lucas Cartan numbers
  105. Recurrence relations satisfied by some arithmetic groups
  106. Numerical Methods
  107. A modified predictor–corrector scheme with graded mesh for numerical solutions of nonlinear Ψ-caputo fractional-order systems
  108. A kind of univariate improved Shepard-Euler operators
  109. Probability and Statistics
  110. Statistical inference and data analysis of the record-based transmuted Burr X model
  111. Multiple G-Stratonovich integral in G-expectation space
  112. p-variation and Chung's LIL of sub-bifractional Brownian motion and applications
  113. Real Analysis
  114. Chebyshev polynomials of the first kind and the univariate Lommel function: Integral representations
  115. Multiple solutions for a class of fourth-order elliptic equations with critical growth
  116. Majorization-type inequalities for (m, M, ψ)-convex functions with applications
  117. The evaluation of a definite integral by the method of brackets illustrating its flexibility
  118. Some new Fejér type inequalities for (h, g; α - m)-convex functions
  119. Some new Hermite-Hadamard type inequalities for product of strongly h-convex functions on ellipsoids and balls
  120. Topology
  121. Unraveling chaos: A topological analysis of simplicial homology groups and their foldings
  122. A generalized fixed-point theorem for set-valued mappings in b-metric spaces
  123. On SI2-convergence in T0-spaces
  124. Generalized quandle polynomials and their applications to stuquandles, stuck links, and RNA folding
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Heruntergeladen am 17.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/math-2025-0137/html
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