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Global Sobolev regular solution for Boussinesq system

  • Xiaofeng Zhao , Weijia Li EMAIL logo and Weiping Yan EMAIL logo
Published/Copyright: March 27, 2023
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Advances in Nonlinear Analysis
From the journal Advances in Nonlinear Analysis Volume 12 Issue 1

Abstract

This article is concerned with the study of the initial value problem for the three-dimensional viscous Boussinesq system in the thin domain Ω ≔ R 2 × ( 0 , R ) . We construct a global finite energy Sobolev regularity solution ( v , ρ ) ∈ H s ( Ω ) × H s ( Ω ) with the small initial data in the Sobolev space H s + 2 ( Ω ) × H s + 2 ( Ω ) . Some features of this article are the following: (i) we do not require the initial data to be axisymmetric; (ii) the Sobolev exponent s can be an arbitrary big positive integer; and (iii) the explicit asymptotic expansion formulas of Sobolev regular solution is given. The key point of the proof depends on the structure of the perturbation system by means of a suitable initial approximation function of the Nash-Moser iteration scheme.

Keywords: Boussinesq system; global Sobolev solution; Nash-Moser iteration scheme
MSC 2010: 35Q31; 35A01

1 Introduction and main results

In this article, we consider the three-dimensional incompressible Boussinesq system:

(1.1) ∂ t v + v ⋅ ∇ v + ∇ P − ν Δ v = ρ e z , ∂ t ρ + v ⋅ ∇ ρ − μ ∂ x 3 2 ρ = 0 , ∇ ⋅ v = 0 ,

with the initial data

(1.2) v ( 0 , x ) = v 0 ( x ) , ρ ( 0 , x ) = ρ 0 ( x ) ,

where for all ( t , x ) ∈ R + × Ω with x = ( x 1 , x 2 , x 3 ) ∈ Ω ≔ R 2 × ( 0 , R ) , and R is a positive constant. v denotes the 3D velocity field of the fluid, P stands for the pressure in the fluid, and the scalar function ρ denotes the density or the temperature. The constants ν and μ are the viscous coefficients and diffusivity coefficients, respectively. The term ρ e z , where e z = ( 0 , 0 , 1 ) T , takes into account the influence of the gravity and the stratification on the motion of the fluid. The pressure satisfies the form

(1.3) P ( t , x ) = − Δ − 1 ∑ i , j = 1 3 ∂ v i ∂ x j ∂ v j ∂ x i − ∂ ρ ∂ x 3 .

We impose the boundary condition

(1.4) v ( t , x ) ∣ x ∈ ∂ Ω = 0 , ρ ( t , x ) ∣ x ∈ ∂ Ω = 0 ,

that is, for i = 1 , 2 , 3 , it holds

lim ∣ x j ∣ → + ∞ v i ( t , x ) = 0 , lim ∣ x j ∣ → + ∞ ρ ( t , x ) = 0 , j = 1 , 2 , v i ( t , x ) ∣ x 3 = 0 = 0 , v i ( t , x ) ∣ x 3 = R = 0 , ρ ( t , x ) ∣ x 3 = 0 = 0 , ρ ( t , x ) ∣ x 3 = R = 0 ,

and we assume the initial data (1.2) satisfies the following conditions:

(1.5) ∇ ⋅ v 0 ( x ) = 0 , v 0 ( x ) ∣ x ∈ ∂ Ω = 0 , ρ 0 ( x ) ∣ x ∈ ∂ Ω = 0 .

It is easy to check that Boussinesq equations (1.1) admits the solution of scaling invariant property. More precisely, if ( v , ρ , P ) is an arbitrary solution of problem (1.1), then for any constant λ > 0 , the functions

v λ ( t , x ) = λ v ( λ 2 t , λ x ) , ρ λ ( t , x ) = λ 3 ρ ( λ 2 t , λ x ) , P λ ( t , x ) = λ 2 P ( λ 2 t , λ x )

are also solutions of Boussinesq equations (1.1). Here, the initial data ( v 0 ( x ) , ρ 0 ( x ) ) are changed into ( λ v 0 ( λ x ) , λ 3 ρ 0 ( λ x ) ) .

The Boussinesq equations form a fundamental block in many geophysical models, and it is relevant in the study of atmospheric and oceanographic turbulence, as well as other astrophysical situations in which rotation and stratification play a dominant role such as the Rayleigh-Benard convection, see, example, Majda [34,35], Pedlosky [42], and Vallis [51]. The full three-dimensional viscous Boussinesq equations takes the form

∂ t v − ν Δ v + v ⋅ ∇ v + ∇ P = ρ e z , ∂ t ρ − μ Δ ρ + v ⋅ ∇ ρ = 0 , ∇ ⋅ v = 0 .

In particular, when the initial density ρ 0 is identically constant, then the three-dimensional Boussinesq equations can reduce to the classical three-dimensional incompressible Navier equation. The Navier-Stokes equations are locally well-posed for smooth enough initial data as long as one imposes appropriate boundary conditions on the pressure at ∞ (see [31] for more details). The question of whether a solution of the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data with finite energy is one of the Millennium Prize problems [20]. In 1934, Leray [27] showed the 3D incompressible Navier-Stokes equations (1.1) admit global-forward-in-time weak solution of the initial value problem. Caffarelli et al. [7] established a ε -regularity criterion for equations (1.1). After that, Lin [29] gave a new and simpler proofs for the result of Caffarelli-Kohn-Nirenberg. Koch and Tataru [32] proved the global well-posedness for equations (1.1) in a space of arbitrary dimension with small initial data in BMO − 1 space. Kozono and Shimizu [33] presented the existence and uniqueness result of local strong solutions to the Navier-Stokes equations with arbitrary initial data and external forces in the homogeneous Besov space. Under some assumptions, Chemin et al. [12] obtained the global regularity for some classes of large solutions to the Navier-Stokes equations. Recently, Buckmaster and Vicol [6] proved that the Leray weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy. We refer the readers to [2,4,5,13,26,28,40,45,47,48,53] for more related results on this equations.

However, when all viscosities and diffusivity coefficients are zero, the global well-posedness is still an open problem for three- or two-dimensional Boussinesq equations, see [35]. Danchin and Paicu [14] showed the global existence of weak solution for L 2 -data and the global well-posedness for small initial data. Meanwhile, Danchin and Paicu [15] proved an existence and uniqueness result for system (1.1) with a small initial data belonging to some critical Lorentz spaces. Hmidi and Rousset [22] obtained the global well-posedness for the three-dimensional axisymmetric Navier-Stokes-Boussinesq system without swirl. According to the assumptions of partial viscosities and diffusivity coefficients, there are important contributions on the local and global well-posedness for this problem. Cao and Wu [8] proved that the two-dimensional Boussinesq equations with only vertical dissipation admit a unique global classical solution with the initial data in H 2 ( R 2 ) . Li and Titi [30] generalized the result of [8] by the assumption of the initial data in the space X ≔ { f ∈ L 2 ( R 2 ) ∣ ∂ x f ∈ L 2 ( R 2 ) } . We refer the readers to [1,9–11,16–18,21,22,36,52] for more results on this equations. To the best of our knowledge, there are few results on the global well-posedness for this equations without axisymmetric.

It is meaningful to consider the well-posedness problem of partial differential equations in thin domains. Thin domains are widely studied in solid mechanics, fluid dynamics, and magnetohydrodynamics, and one can see [19,38,44] for more details. Raugel and Sell [43] showed the existence of global strong solutions ( H 2 space) and attractors of the three-dimensional Navier-Stokes equations with external force in a bounded thin domain with a periodic boundary condition. After that, there are several results [24,25,49] on the three-dimensional Navier-Stokes equations. Recently, Xu [50] obtained the global well-posedness result for the ideal magnetohydrodynamics in three-dimensional thin domains Ω δ ≔ R 2 × ( − δ , δ ) with the slip boundary conditions by the construction of a global solution with infinite energy near Alfvén waves, where the constant δ must be sufficient small. Moreover, Xu found that the 3D Alfvén waves converge to the 2D Alfvén waves in R 2 as the constant δ → 0 .

In this article, we will construct a global finite energy small Sobolev regularity solution for the three-dimensional incompressible Boussinesq equations (1.1) in the thin domain, and we require the small initial data in Sobolev Space H s + 2 ( Ω ) × H s + 2 ( Ω ) . Here, the index s of Sobolev space can be any large fixed integer, but s ≠ + ∞ .

We now state the main result in this article.

Theorem 1.1

Let the viscous constants ν , μ > 1 . For any fixed constant s ≥ 1 , if the initial data (1.2) of the 3D incompressible Boussinesq equations (1.1) satisfies the conditions (1.5), and there exists a small constant ε ∈ ( 0 , 1 ) such that

‖ v 0 ( x ) ‖ H s + 2 ( Ω ) + ‖ ρ 0 ( x ) ‖ H s + 2 ( Ω ) ≲ ε ,

then equations (1.1) with the boundary conditions (1.4) admit a global Sobolev regular solution with finite energy

( v ( t , x ) , ρ ( t , x ) ) ∈ C ( ( 0 , + ∞ ) ; H s ( Ω ) × H s ( Ω ) ) .

Moreover, it holds

sup t ∈ ( 0 , + ∞ ) ( ‖ v ( t , x ) ‖ H s ( Ω ) + ‖ ρ ( t , x ) ‖ H s ( Ω ) ) ≲ ε , sup t ∈ ( 0 , + ∞ ) ‖ P ( t , x ) ‖ H s ( Ω ) ≲ ε ,

for any ( t , x ) ∈ ( 0 , ∞ ) × Ω . Here, the pressure is given by (1.3).

In particular, we have the following explicit representation formulas.

Corollary 1.1

Let the parameter 0 < ε ≪ 1 . Assume that the small initial data (1.2) satisfy the conditions (1.5). The 3D incompressible Boussinesq equations (1.1) (in the thin domain) admit an explicit expansion of global Sobolev regular solution with finite energy written as follows:

v ∗ ( t , x ) = v ( 0 ) ( t , x ) + v 0 ( x ) + ℛ 1 ( t , x ) , ρ ∗ ( t , x ) = ρ ( 0 ) ( t , x ) + ρ 0 ( x ) + ℛ 2 ( t , x ) ,

where the fixed function

( v ( 0 ) ( t , x ) , ρ ( 0 ) ( t , x ) ) = ( v 1 ( 0 ) ( t , x ) , v 2 ( 0 ) ( t , x ) , v 3 ( 0 ) ( t , x ) , ρ ( 0 ) ( t , x ) )

satisfies the conditions

∇ ⋅ v ( 0 ) ( t , x ) = 0 , v ( 0 ) ( 0 , x ) = 0 , ‖ v ( 0 ) ‖ H s ( Ω ) ≲ ε , v ( 0 ) ( t , x ) ∣ x ∈ ∂ Ω = 0 , ρ ( 0 ) ( 0 , x ) = 0 , ‖ ρ ( 0 ) ‖ H s ( Ω ) ≲ ε , ρ ( 0 ) ( t , x ) ∣ x ∈ ∂ Ω = 0 ,

and

∑ k = 0 s ‖ ∂ x i k v j ( 0 ) ( t , x ) ‖ L ∞ ≲ ε 0 ∀ i , j = 1 , 2 , 3 , ∑ k = 0 s ‖ ∂ x i k ρ ( 0 ) ( t , x ) ‖ L ∞ ≲ ε 0 ,

and the remainder terms ℛ 1 ( t , x ) ∈ H s ( Ω ) and ℛ 2 ( t , x ) ∈ H s ( Ω ) satisfies

ℛ 1 ( 0 , x ) = 0 , ∇ ⋅ ℛ 1 ( t , x ) = 0 , ℛ 1 ( t , x ) ∣ x ∈ ∂ Ω = 0 , sup t ∈ ( 0 , + ∞ ) ‖ ℛ 1 ( t , x ) ‖ H s ( Ω ) ∼ O ( ε 2 ) ,

and

ℛ 2 ( 0 , x ) = 0 , ℛ 2 ( t , x ) ∣ x ∈ ∂ Ω = 0 , sup t ∈ ( 0 , + ∞ ) ‖ ℛ 2 ( t , x ) ‖ H s ( Ω ) ∼ O ( ε 2 ) .

Moreover, the pressure is determined by

Δ P ∗ ( t , x ) = − ∑ i , j = 1 3 ∂ v i ∗ ∂ x j ∂ v j ∗ ∂ x i − ∂ ρ ∗ ∂ x 3 .

Notations. Throughout this article, let Ω ≔ R 2 × ( 0 , R ) , we denote the usual norm of L 2 ( Ω ) and Sobolev space H s ( Ω ) by ‖ ⋅ ‖ L 2 and ‖ ⋅ ‖ H s , respectively. The norm of Sobolev space H s ( Ω ) ≔ ( H s ( Ω ) ) 3 is denoted by ‖ ⋅ ‖ H s . The symbol a ≲ b means that there exists a positive constant C such that a ≤ C b . ( a , b , c ) denotes the column vector in R 3 . The letter C with subscripts to denote dependencies stands for a positive constant that might change its value at each occurrence.

The organization of this article is as follows. In Section 2, we show how to choose a suitable initial approximation functions, which lead to the dissipative structure of linearized system. After that, we give the existence of global time-decay Sobolev solution for the linearized equations of first approximation step. In Section 3, we establish the general approximation step for the construction of Nash-Moser iteration scheme. Section 4 shows how to construct a global Sobolev solution for the three-dimensional incompressible Boussinesq equations (1.1) by the proof of convergence for the Nash-Moser iteration scheme. This method has been used in [54–57,59]. For the general Nash-Moser implicit function theorem, one can see the celebrated works of Nash [39], Moser [37], and Hörmander [23].

2 The first approximation step

For m = 1 , 2 , … , by setting N m = 2 m , we introduce a family of smoothing operators (see [46] for more details) Π N m : L 2 → C ∞ such that

(2.1) ‖ Π N m U ‖ H s 1 ≲ N m s 1 − s 2 ‖ U ‖ H s 2 ∀ s 1 ≥ s 2 ≥ 0 , ‖ Π N m U − U ‖ H s 1 ≲ N m s 1 − s 2 ‖ U ‖ H s 2 ∀ 0 ≤ s 1 ≤ s 2 .

We consider the approximation problem of Boussinesq system (1.1) as follows:

(2.2) ℒ 1 ( v , ρ ) ≔ ∂ t v − ν Δ v + Π N m ( v ⋅ ∇ v + ∇ P − ρ e z ) , ℒ 2 ( v , ρ ) ≔ ∂ t ρ − μ ∂ x 3 2 ρ + Π N m ( v ⋅ ∇ ρ ) ,

with the initial data (1.2), the boundary condition (1.4), and the incompressible condition

∇ ⋅ v = 0 .

2.1 The initial approximation function

Let s ≥ 1 be a fixed finite constant and 0 < ε 0 < ε 2 ≪ 1 . For any ( x 1 , x 2 ) ∈ R 2 and x 3 ∈ ( 0 , R ) , we choose the initial approximation functions:

v ( 0 ) ( t , x ) = ( v 1 ( 0 ) ( t , x ) , v 2 ( 0 ) ( t , x ) , v 3 ( 0 ) ( t , x ) ) T ∈ H s ( Ω ) , ρ ( 0 ) ( t , x ) ∈ H s ( Ω ) .

Meanwhile, we require

(2.3) ∇ ⋅ v ( 0 ) ( t , x ) = 0 , v ( 0 ) ( 0 , x ) = 0 , ‖ v ( 0 ) ‖ H s ≲ ε 0 , v ( 0 ) ( t , x ) ∣ x ∈ ∂ Ω = 0 ,

and

(2.4) ρ ( 0 ) ( 0 , x ) = 0 , ‖ ρ ( 0 ) ‖ H s ≲ ε 0 , ρ ( 0 ) ( t , x ) ∣ x ∈ ∂ Ω = 0 .

Moreover, for any ( t , x ) ∈ Ω and i , j = 1 , 2 , 3 , it also needs the condition

(2.5) ∑ k = 0 s ‖ ∂ x i k v j ( 0 ) ( t , x ) ‖ L ∞ ≲ ε 0 ∀ i , j = 1 , 2 , 3 , ∑ k = 0 s ‖ ∂ x i k ρ ( 0 ) ( t , x ) ‖ L ∞ ≲ ε 0 ,

and the initial error terms

(2.6) ‖ E ( 0 ) ‖ H s ≲ ε 0 , ‖ E ¯ ( 0 ) ‖ H s ≲ ε 0 ,

where E ( 0 ) and E ¯ ( 0 ) denote the error terms taking the form

E ( 0 ) ≔ ℒ 1 ( v ( 0 ) , ρ ( 0 ) ) , E ¯ ( 0 ) ≔ ℒ 2 ( v ( 0 ) , ρ ( 0 ) ) .

2.2 The Carleman-type estimation at the first approximation step

We now construct the first approximation solution denoted by ( v ( 1 ) ( t , x ) , ρ ( 1 ) ( t , x ) ) of (2.2). The first approximation step between the initial approximation function and first approximation solution is denoted by

h ( 1 ) ( t , x ) ≔ v ( 1 ) ( t , x ) − v ( 0 ) ( t , x ) , ϱ ( 1 ) ( t , x ) ≔ ρ ( 1 ) ( t , x ) − ρ ( 0 ) ( t , x ) ,

then we linearize nonlinear system (2.2) around U ( 0 ) to obtain the linearized operators as follows:

(2.7) J 1 [ v 0 , ρ ( 0 ) ] ( h ( 1 ) , ϱ ( 1 ) ) ≔ h t ( 1 ) − ν Δ h ( 1 ) + Π N 1 [ ( v ( 0 ) ⋅ ∇ ) h ( 1 ) + ( h ( 1 ) ⋅ ∇ ) v ( 0 ) + ∇ ( ( D v ( 0 ) P ) h ( 1 ) + ∇ ( D ρ ( 0 ) P ) ϱ ( 1 ) ) − ϱ ( 1 ) e z ] , J 2 [ v 0 , ρ ( 0 ) ] ( h ( 1 ) , ϱ ( 1 ) ) ≔ ϱ t ( 1 ) − μ ∂ x 3 2 ϱ ( 1 ) − Π N 1 [ ( v ( 0 ) ⋅ ∇ ) ϱ ( 1 ) + ( h ( 1 ) ⋅ ∇ ) ρ ( 0 ) ] ,

where D v ( 0 ) denotes the Fréchet derivatives on v ( 0 ) , and by (1.3), it takes the form

(2.8) ∇ ( ( D v ( 0 ) P ) h ( 1 ) + ( D ρ ( 0 ) P ) ϱ ( 1 ) ) ≔ − ∇ Δ − 1 ∑ i , j = 1 3 ∂ x j h i ( 1 ) ∂ x i v j ( 0 ) + ∂ x j v i ( 0 ) ∂ x i h j ( 1 ) − ∂ ϱ ( 1 ) ∂ x 3 .

We now consider the linear system

(2.9) J 1 [ v 0 , ρ ( 0 ) ] ( h ( 1 ) , ϱ ( 1 ) ) = Π N 1 E ( 0 ) , J 2 [ v 0 , ρ ( 0 ) ] ( h ( 1 ) , ϱ ( 1 ) ) = Π N 1 E ¯ ( 0 ) , ∇ ⋅ h ( 1 ) = 0 , h ( 1 ) ( 0 , x ) = h 0 ( 1 ) ( x ) , ϱ ( 1 ) ( 0 , x ) = ϱ 0 ( 1 ) ( x ) ,

and the boundary condition

(2.10) h ( 1 ) ( t , x ) ∣ x ∈ ∂ Ω = 0 , ϱ ( 1 ) ( t , x ) ∣ x ∈ ∂ Ω = 0 ,

from which, the solution gives the first approximation step of 3D incompressible Boussinesq system (1.1).

Before we carry out some priori estimates, for j = 1 , 2 , 3 , we rewrite (2.9) as a coupled system:

(2.11) ∂ t h j ( 1 ) − ν Δ h j ( 1 ) + Π N 1 ∑ i = 1 3 v i ( 0 ) ∂ x i h j ( 1 ) + Π N 1 ∑ i = 1 3 h i ( 1 ) ∂ x i v j ( 0 ) + Π N 1 ∂ x j ( ( D v ( 0 ) P ) h ( 1 ) + ( D ρ ( 0 ) P ) ϱ ( 1 ) ) − Π N 1 ϱ ( 1 ) e z = Π N 1 E j ( 0 ) , ∂ t ϱ ( 1 ) − μ ∂ x 3 2 ϱ ( 1 ) + Π N 1 [ ( v ( 0 ) ⋅ ∇ ) ϱ ( 1 ) + ( h ( 1 ) ⋅ ∇ ) ρ ( 0 ) ] = Π N 1 E ¯ ( 0 ) ,

with the initial data

(2.12) h j ( 1 ) ( 0 , x ) = h 0 j ( 1 ) ( x ) , ϱ ( 1 ) ( 0 , x ) = ϱ 0 ( 1 ) ( x ) ,

and the boundary condition (2.10).

In fact, the linear system (2.11) is a partial dissipative system under conditions (2.3)–(2.6). We now derive L 2 -estimate of solution for the linear system (2.11). We first introduce a weighted function. Let ψ ( x 3 ) be a function defined in ( 0 , R ) such that

(2.13) 0 < κ ≤ ψ ″ ( x 3 ) − ( ψ ′ ( x 3 ) ) 2 − ∣ ψ ′ ( x 3 ) ∣ < 1 4 ,

and e − ψ ( x 3 ) is bounded in ( 0 , R ) . Here, the constant κ belongs to ( 0 , 1 4 ) . The condition (2.13) implies ψ ″ ( x 3 ) > 1 4 > 0 . In fact, there are many functions that can satisfy the aforementioned conditions. Here, we give a simple example. Let

ψ ( x 3 ) = 2 a b x 3 − 4 b ln ( e a 2 ( x 3 + b ) − 1 ) ,

where the constants a ≔ ( 1 4 − κ ) − 1 2 and b > 1 .

Direct computations give that

ψ ′ ( x 3 ) = − 2 a b ( e a 2 ( x 3 + b ) − 1 ) − 1 ,

and thus, there exists a fixed constant b > 1 such that the ordinary differential equation inequality (2.13) holds. e − ψ ( x 3 ) is a bounded smooth function in ( 0 , R ) .

Lemma 2.1

Let the parameters ν , μ > 1 . Assume the initial approximation function ( v ( 0 ) , ρ ( 0 ) ) satisfying (2.3)–(2.6). Then the solution ( h ( 1 ) ( t , x ) , ϱ ( 1 ) ( t , x ) ) of the linear system (2.11) satisfies

(2.14) ∫ Ω ∑ j = 1 3 ( h j ( 1 ) ) 2 + ( ϱ ( 1 ) ) 2 d x ≤ e − C ν , λ , μ , ε t ∫ Ω ∑ j = 1 3 ( h 0 j ( 1 ) ) 2 + ( ϱ 0 ( 1 ) ) 2 d x + Π N 1 ∫ 0 + ∞ ∫ Ω ∑ j = 1 3 ( E j ( 0 ) ) 2 + ( E ¯ ( 0 ) ) 2 d x d t ∀ t > 0 ,

where C ν , μ , ε denotes a positive constant depending on ν , μ , ε .

Proof

Multiplying both sides of two equations in (2.11) by e − ψ ( x 3 ) h j ( 1 ) and e − ψ ( x 3 ) ϱ ( 1 ) , respectively, and then integrating over Ω , by noticing the boundary condition (2.10), for j = 1 , 2 , 3 , it holds

(2.15) 1 2 d d t ∫ Ω ( h j ( 1 ) ) 2 e − ψ ( x 3 ) d x + ν ∑ i = 1 3 ∫ Ω ( ∂ x i h j ( 1 ) ) 2 e − ψ ( x 3 ) d x + ν 2 ∫ Ω ( ψ ″ ( x 3 ) − ( ψ ′ ( x 3 ) ) 2 ) ( h j ( 1 ) ) 2 e − ψ ( x 3 ) d x + Π N 1 ∑ i = 1 3 ∫ Ω ( v i ( 0 ) ∂ x i h j ( 1 ) ) h j ( 1 ) e − ψ ( x 3 ) d x + Π N 1 ∑ i = 1 3 ∫ Ω ( h i ( 1 ) ∂ x i v j ( 0 ) ) h j ( 1 ) e − ψ ( x 3 ) d x + Π N 1 ∫ Ω ∂ x j ( ( D v ( 0 ) P ) h ( 1 ) + ( D ρ ( 0 ) P ) ϱ ( 1 ) ) h j ( 1 ) e − ψ ( x 3 ) d x − Π N 1 ∫ Ω ( ϱ ( 1 ) e z ) h j ( 1 ) e − ψ ( x 3 ) d x = Π N 1 ∫ Ω E j ( 0 ) h j ( 1 ) e − ψ ( x 3 ) d x ,

and

(2.16) 1 2 d d t ∫ Ω ( ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + μ ∫ Ω ( ∂ x 3 ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + μ 2 ∫ Ω ( ψ ″ ( x 3 ) − ( ψ ′ ( x 3 ) ) 2 ) ( ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + Π N 1 ∑ i = 1 3 ∫ Ω ( v i ( 0 ) ∂ x i ϱ ( 1 ) ) ϱ ( 1 ) e − ψ ( x 3 ) d x + Π N 1 ∑ i = 1 3 ∫ Ω ( h i ( 1 ) ∂ x i ρ ( 0 ) ) ϱ ( 1 ) e − ψ ( x 3 ) d x = Π N 1 ∫ Ω E ¯ ( 0 ) ϱ ( 1 ) e − ψ ( x 3 ) d x .

We sum up (2.15) and (2.16) from j = 1 to j = 3 , and then it holds

(2.17) 1 2 d d t ∫ Ω ∑ j = 1 3 ( h j ( 1 ) ) 2 + ( ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + ∫ Ω ν ∑ j = 1 3 ∑ i = 1 3 ( ∂ x i h j ( 1 ) ) 2 + μ ( ∂ x 3 ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + 1 2 ∑ j = 1 3 ∫ Ω ( ψ ″ ( x 3 ) − ( ψ ′ ( x 3 ) ) 2 ) ( ν ( h j ( 1 ) ) 2 + μ ( ϱ ( 1 ) ) 2 ) e − ψ ( x 3 ) d x + Π N 1 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( v i ( 0 ) ∂ x i h j ( 1 ) ) h j ( 1 ) e − ψ ( x 3 ) d x + Π N 1 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( h i ( 1 ) ∂ x i v j ( 0 ) ) h j ( 1 ) e − ψ ( x 3 ) d x + Π N 1 ∑ j = 1 3 ∫ Ω ∂ x j ( ( D v ( 0 ) P ) h ( 1 ) + ( D ρ ( 0 ) P ) ϱ ( 1 ) ) h j ( 1 ) e − ψ ( x 3 ) d x − Π N 1 ∫ Ω ϱ ( 1 ) h 3 ( 1 ) e − ψ ( x 3 ) d x + Π N 1 ∑ i = 1 3 ∫ Ω ( v i ( 0 ) ∂ x i ϱ ( 1 ) ) ϱ ( 1 ) e − ψ ( x 3 ) d x + Π N 1 ∑ i = 1 3 ∫ Ω ( h i ( 1 ) ∂ x i ρ ( 0 ) ) ϱ ( 1 ) e − ψ ( x 3 ) d x = Π N 1 ∫ Ω ∑ j = 1 3 E j ( 0 ) h j ( 1 ) + E ¯ ( 0 ) ϱ ( 1 ) e − ψ ( x 3 ) d x .

On the one hand, note that we have chosen the initial approximation function ( v ( 0 ) , ρ ( 0 ) ) satisfying (2.3)–(2.6). We integrate by parts to obtain

(2.18) ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( v i ( 0 ) ∂ x i h j ( 1 ) ) h j ( 1 ) e − ψ ( x 3 ) d x = − 1 2 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ∂ x i v i ( 0 ) ( h j ( 1 ) ) 2 e − ψ ( x 3 ) d x + 1 2 ∑ j = 1 3 ∫ Ω ψ ′ ( x 3 ) v 3 ( 0 ) ( h j ( 1 ) ) 2 e − ψ ( x 3 ) d x ,

since the initial approximation function v ( 0 ) satisfies ∇ ⋅ v ( 0 ) , inequality (2.18) is reduced into

(2.19) ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( v i ( 0 ) ∂ x i h j ( 1 ) ) h j ( 1 ) e − ψ ( x 3 ) d x = 1 2 ∑ j = 1 3 ∫ Ω ψ ′ ( x 3 ) v 3 ( 0 ) ( h j ( 1 ) ) 2 e − ψ ( x 3 ) d x ,

and direct computation gives that

(2.20) ∑ j = 1 3 ∑ i = 1 3 ∫ Ω h i ( 1 ) ∂ x i v j ( 0 ) h j ( 1 ) e − ψ ( x 3 ) d x = ∑ j = 1 3 ∫ Ω ∂ x j v j ( 0 ) ( h j ( 1 ) ) 2 e − ψ ( x 3 ) d x + ∑ j = 1 3 ∑ i ≠ j ∫ Ω h i ( 1 ) ∂ x i v j ( 0 ) h j ( 1 ) e − ψ ( x 3 ) d x ,

and noticing the incompressible condition

∇ ⋅ h ( 1 ) = 0 ,

by (2.8), it holds

(2.21) ∑ j = 1 3 ∫ Ω ∂ x j ( ( D v ( 0 ) P ) h ( 1 ) + ( D ρ ( 0 ) P ) ϱ ( 1 ) ) h j ( 1 ) e − ψ ( x 3 ) d x = − ∑ j = 1 3 ∫ Ω ( ( D v ( 0 ) P ) h ( 1 ) + ( D ρ ( 0 ) P ) ϱ ( 1 ) ) ∂ x j h j ( 1 ) e − ψ ( x 3 ) d x + 3 ∑ i = 1 3 ∫ Ω ψ ′ ( x 3 ) ( ∂ x 3 h i ( 1 ) ∂ x i v 3 ( 0 ) + ∂ x 3 v i ( 0 ) ∂ x i h 3 ( 1 ) + ∂ x 3 ϱ ( 1 ) e z ) h 3 ( 1 ) e − ψ ( x 3 ) d x = 3 ∑ i = 1 3 ∫ Ω ψ ′ ( x 3 ) ( ∂ x 3 h i ( 1 ) ∂ x i v 3 ( 0 ) + ∂ x 3 v i ( 0 ) ∂ x i h 3 ( 1 ) + ∂ x 3 ϱ ( 1 ) e z ) h 3 ( 1 ) e − ψ ( x 3 ) d x ,

and thus, using the standard Calderon-Zygmund theory and Young’s inequality, it holds

(2.22) ∑ j = 1 3 ∫ Ω ∂ x j ( ( D v ( 0 ) P ) h ( 1 ) + ( D ρ ( 0 ) P ) ϱ ( 1 ) ) h j ( 1 ) e − ψ ( x 3 ) d x ≲ 1 2 ∑ i = 1 3 ∑ j = 1 3 ∫ Ω ∣ ψ ′ ( x 3 ) ( ∂ x i v j ( 0 ) + ∂ x j v i ( 0 ) ) ∣ ( h 1 ( 1 ) ) 2 e − ψ ( x 3 ) d x + 1 2 ∫ Ω ∣ ψ ′ ( x 3 ) ∣ ( ∂ x 3 ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + 1 2 ∑ i = 1 3 ∑ j = 1 3 ∫ Ω ∣ ψ ′ ( x 3 ) ∣ ( ∣ ∂ x i v j ( 0 ) ∣ ( ∂ x j h i ( 1 ) ) 2 + ∣ ∂ x j v i ( 0 ) ∣ ( ∂ x i h j ( 1 ) ) 2 ) e − ψ ( x 3 ) d x .

On the other hand, by Young’s inequality, we derive

(2.23) ∑ j = 1 3 ∑ i ≠ j ∫ Ω h i ( 1 ) ∂ x i v j ( 0 ) h j ( 1 ) e − ψ ( x 3 ) d x ≤ 1 2 ∫ Ω ∣ ∂ x 2 v 1 ( 0 ) + ∂ x 1 v 2 ( 0 ) + ∂ x 3 v 1 ( 0 ) + ∂ x 1 v 3 ( 0 ) ∣ ( h 1 ( 1 ) ) 2 e − ψ ( x 3 ) d x + 1 2 ∫ Ω ∣ ∂ x 2 v 1 ( 0 ) + ∂ x 1 v 2 ( 0 ) + ∂ x 3 v 2 ( 0 ) + ∂ x 2 v 3 ( 0 ) ∣ ( h 2 ( 1 ) ) 2 e − ψ ( x 3 ) d x + 1 2 ∫ Ω ∣ ∂ x 2 v 3 ( 0 ) + ∂ x 3 v 2 ( 0 ) + ∂ x 3 v 1 ( 0 ) + ∂ x 1 v 3 ( 0 ) ∣ ( h 3 ( 1 ) ) 2 e − ψ ( x 3 ) d x ,

and

(2.24) ∑ j = 1 3 ∫ Ω ( ϱ ( 1 ) e z ) h j ( 1 ) e − ψ ( x 3 ) d x ≤ 1 2 ∫ Ω ( ( ϱ ( 1 ) ) 2 + ( h 3 ( 1 ) ) 2 ) e − ψ ( x 3 ) d x , ∑ i = 1 3 ∫ Ω ( v i ( 0 ) ∂ x i ϱ ( 1 ) ) ϱ ( 1 ) e − ψ ( x 3 ) d x = − ∑ i = 1 3 ∫ Ω ( ∂ x i v i ( 0 ) ) ( ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + ∫ Ω ψ ′ ( x 3 ) v 3 ( 0 ) ( ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x , ∑ i = 1 3 ∫ Ω ( h i ( 1 ) ∂ x i ρ ( 0 ) ) ϱ ( 1 ) e − ψ ( x 3 ) d x ≤ 1 2 ∑ i = 1 3 ∫ Ω ( ∂ x i ρ ( 0 ) ) ( ( h i ( 1 ) ) 2 + ( ϱ ( 1 ) ) 2 ) e − ψ ( x 3 ) d x , ∑ j = 1 3 ∫ Ω E j ( 0 ) h j ( 1 ) e − ψ ( x 3 ) d x ≤ 1 2 ∑ j = 1 3 ∫ Ω ( 4 ( E j ( 0 ) ) 2 + ∣ ψ ″ ( x 3 ) ∣ ( h j ( 1 ) ) 2 ) e − ψ ( x 3 ) d x , ∫ Ω E ¯ ( 0 ) ρ ( 1 ) e − ψ ( x 3 ) d x ≤ 1 2 ∫ Ω ( 4 ( E ¯ ( 0 ) ) 2 + ∣ ψ ″ ( x 3 ) ∣ ( ρ ( 1 ) ) 2 ) e − ψ ( x 3 ) d x .

Thus, by (2.18)–(2.24), it holds

(2.25) 1 2 d d t ∫ Ω ∑ j = 1 3 ( h j ( 1 ) ) 2 + ( ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + ∫ Ω μ − 1 2 ∣ ψ ′ ( x 3 ) ∣ ( ∂ x 3 ρ ( 1 ) ) 2 e − ψ ( x 3 ) d x + ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ν − 1 2 ∣ ψ ′ ( x 3 ) ∣ ( ∣ ∂ x i v j ( 0 ) ∣ + ∣ ∂ x j v i ( 0 ) ∣ ) ( ∂ x i h j ( 1 ) ) 2 e − ψ ( x 3 ) d x + 3 Π N 1 ∑ j = 1 3 ∫ Ω A j ( t , x ) ( h j ( 1 ) ) 2 e − ψ ( x 3 ) d x + ∫ Ω B ( t , x ) ( ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x ≲ Π N 1 ∑ j = 1 3 ∫ Ω ( ( E j ( 0 ) ) 2 + ( E ¯ ( 0 ) ) 2 ) e − ψ ( x 3 ) d x ,

where the coefficients

A 1 ( t , x ) ≔ ν 2 ( ψ ″ ( x 3 ) − ( ψ ′ ( x 3 ) ) 2 ) + 1 2 v 3 ( 0 ) ψ ′ ( x 3 ) + ∂ x 1 v 1 ( 0 ) − 1 2 ∑ i = 1 3 ∑ j = 1 3 ∣ ψ ′ ( x 3 ) ( ∂ x i v j ( 0 ) + ∂ x j v i ( 0 ) ) ∣ − 1 2 ∣ ∂ x 2 v 1 ( 0 ) + ∂ x 1 v 2 ( 0 ) + ∂ x 3 v 1 ( 0 ) + ∂ x 1 v 3 ( 0 ) ∣ − 1 2 ∣ ψ ″ ( x 3 ) ∣ − 1 2 ∣ ∂ x 1 ρ ( 0 ) ∣ , A 2 ( t , x ) ≔ ν 2 ( ψ ″ ( x 3 ) − ( ψ ′ ( x 3 ) ) 2 ) + 1 2 v 3 ( 0 ) ψ ′ ( x 3 ) + ∂ x 2 v 2 ( 0 ) − 1 2 ∣ ∂ x 2 v 1 ( 0 ) + ∂ x 1 v 2 ( 0 ) + ∂ x 3 v 2 ( 0 ) + ∂ x 2 v 3 ( 0 ) ∣ − 1 2 ∣ ψ ″ ( x 3 ) ∣ − 1 2 ∣ ∂ x 2 ρ ( 0 ) ∣ , A 3 ( t , x ) ≔ ν 2 ( ψ ″ ( x 3 ) − ( ψ ′ ( x 3 ) ) 2 ) + 1 2 v 3 ( 0 ) ψ ′ ( x 3 ) + ∂ x 3 v 3 ( 0 ) − ∣ ∂ x 2 v 3 ( 0 ) + ∂ x 3 v 2 ( 0 ) + ∂ x 3 v 1 ( 0 ) + ∂ x 1 v 3 ( 0 ) ∣ − 1 2 ∣ ψ ″ ( x 3 ) ∣ − 1 2 − 1 2 ∣ ∂ x 3 ρ ( 0 ) ∣ , B ( t , x ) ≔ μ 2 ( ψ ″ ( x 3 ) − ( ψ ′ ( x 3 ) ) 2 ) − 1 2 − ∑ i = 1 3 ∣ ∂ x i v i ( 0 ) ∣ − ∣ ψ ′ ( x 3 ) ∣ ∣ v 3 ( 0 ) ∣ − 1 2 ∑ i = 1 3 ∣ ∂ x i ρ ( 0 ) ∣ .

Since the weighted function ψ ( x 3 ) satisfying (2.13), the main term of A i ( t , x ) ( i = 1 , 2 , 3 ) is

ν 2 ( ψ ″ ( x 3 ) − ( ψ ′ ( x 3 ) ) 2 ) − 1 2 ∣ ψ ″ ( x 3 ) ∣ .

Thus, there are suitable big constants ν , μ > 1 such that

A 1 ( t , x ) ≥ ν 2 ( ψ ″ ( x 3 ) − ( ψ ′ ( x 3 ) ) 2 ) − 1 2 ∣ ψ ″ ( x 3 ) ∣ − 1 2 ∣ ψ ′ ( x 3 ) ∣ ‖ v 3 ( 0 ) ‖ L ∞ ( Ω ) − ‖ ∂ x 1 v 1 ( 0 ) ‖ L ∞ ( Ω ) − 1 2 ∑ i = 1 3 ∑ j = 1 3 ∣ ψ ′ ( x 3 ) ∣ ( ‖ ∂ x i v j ( 0 ) ‖ L ∞ + ‖ ∂ x j v i ( 0 ) ‖ L ∞ ) − 1 2 ( ‖ ∂ x 2 v 1 ( 0 ) ‖ L ∞ + ‖ ∂ x 1 v 2 ( 0 ) ‖ L ∞ + ‖ ∂ x 3 v 1 ( 0 ) ‖ L ∞ + ‖ ∂ x 1 v 3 ( 0 ) ‖ L ∞ ) − 1 2 ‖ ∂ x 1 ρ ( 0 ) ‖ L ∞ ≳ ν 2 ( ψ ″ ( x 3 ) − ( ψ ′ ( x 3 ) ) 2 ) − 1 2 ∣ ψ ″ ( x 3 ) ∣ − ε ∣ ψ ′ ( x 3 ) ∣ − ε ,

from which and (2.13), one can see that there exists a positive constant C ν , ε depending on ν , ε such that

A 1 ( t , x ) ≳ ν 2 ( ψ ″ ( x 3 ) − ( ψ ′ ( x 3 ) ) 2 − ∣ ψ ′ ( x 3 ) ∣ ) − ε ≥ ν κ 2 − ε ≥ C ν , ε > 0 ,

where κ ∈ ( 0 , 1 4 ) . Similarly, it holds

A 2 ( t , x ) , A 3 ( t , x ) ≳ C ν , ε ,

and

ν − 1 2 ∣ ψ ′ ( x 3 ) ∣ ( ∣ ∂ x i v j ( 0 ) ∣ + ∣ ∂ x j v i ( 0 ) ∣ ) ≳ C ν , μ − 1 2 ∣ ψ ′ ( x 3 ) ∣ ≳ C μ .

We integrate (2.25) over ( 0 , t ) to obtain

(2.26) 1 2 d d t ∫ Ω ∑ j = 1 3 ( h j ( 1 ) ) 2 + ( ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + C ν ∑ j = 1 3 ∑ i = 1 3 ∫ 0 t ∫ Ω ( ∂ x i h j ( 1 ) ) 2 e − ψ ( x 3 ) d x d t + C μ ∫ 0 t ∫ Ω ( ∂ x 3 ρ ( 1 ) ) 2 e − ψ ( x 3 ) d x d t + C μ , ν , ε Π N 1 ∫ 0 t ∫ Ω ∑ j = 1 3 ( h j ( 1 ) ) 2 + ( ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x d t ≲ Π N 1 ∫ 0 t ∫ Ω ∑ j = 1 3 ( E j ( 0 ) ) 2 + ( E ¯ ( 0 ) ) 2 e − ψ ( x 3 ) d x d t ,

which gives the following inequality:

(2.27) ∑ j = 1 3 ∑ i = 1 3 ∫ 0 t ∫ Ω ( ∂ x i h j ( 1 ) ) 2 e − ψ ( x 3 ) d x d t + ∫ 0 t ∫ Ω ( ∂ x 3 ρ ( 1 ) ) 2 e − ψ ( x 3 ) d x d t ≲ Π N 1 ∫ 0 t ∫ Ω ∑ j = 1 3 ( E j ( 0 ) ) 2 + ( E ¯ ( 0 ) ) 2 e − ψ ( x 3 ) d x d t .

Therefore, we apply Gronwall’s inequality to inequality (2.26) to obtain

∫ Ω ∑ j = 1 3 ( h j ( 1 ) ) 2 + ( ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x ≤ e − C ν , μ , ε t ∫ Ω ∑ j = 1 3 ( h 0 j ( 1 ) ) 2 + ( ϱ 0 ( 1 ) ) 2 e − ψ ( x 3 ) d x + Π N 1 ∫ 0 t ∫ Ω ∑ j = 1 3 ( E j ( 0 ) ) 2 + ( E ¯ ( 0 ) ) 2 e − ψ ( x 3 ) d x d t ,

which combines with the function e − ψ ( x 3 ) being bounded in ( 0 , R ) , and it gives (2.14). This completes the proof.□

Furthermore, we derive the higher order derivative estimates. For a fixed constant s ≥ 1 , we apply D i s ≔ ∂ x i s ( ∀ i = 1 , 2 , 3 ) to both sides of (2.11), and it holds

(2.28) ∂ t D i s h j ( 1 ) − ν Δ D i s h j ( 1 ) + Π N 1 ∑ i = 1 3 v i ( 0 ) ∂ x i D i s h j ( 1 ) + Π N 1 ∑ i = 1 3 D i s h i ( 1 ) ∂ x i v j ( 0 ) + Π N 1 ∂ x j D i s ( ( D v ( 0 ) P ) h ( 1 ) + ( D ρ ( 0 ) P ) ϱ ( 1 ) ) − Π N 1 ( D i s ϱ ( 1 ) ) e z = F j , for j = 1 , 2 , 3 ,

and

(2.29) ∂ t D i s ϱ ( 1 ) − μ ∂ x 3 2 D i s ϱ ( 1 ) + Π N 1 ∑ i = 1 3 ( v i ( 0 ) ( ∂ x i D i s ϱ ( 1 ) ) + ( D i s h i ( 1 ) ) ( ∂ x i ρ ( 0 ) ) ) = F ¯ ,

with the boundary condition

(2.30) D i l h j ( 1 ) ( t , x ) ∣ x ∈ ∂ Ω = 0 , D i l ϱ ( 1 ) ( t , x ) ∣ x ∈ ∂ Ω = 0 ,

where the constant 1 ≤ l ≤ s , and

F j ≔ Π N 1 D i s E j ( 0 ) − Π N 1 ∑ s 1 + s 2 = s , 0 ≤ s 2 ≤ s − 1 ∑ i = 1 3 D i s 1 v i ( 0 ) ∂ x i D i s 2 h j ( 1 ) − Π N 1 ∑ s 1 + s 2 = s , 0 ≤ s 2 ≤ s − 1 ∑ i = 1 3 ( D i s 2 h i ( 1 ) ) ( D i s 1 ∂ x i v j ( 0 ) ) , F ¯ ≔ Π N 1 D i s E ¯ ( 0 ) − Π N 1 ∑ s 1 + s 2 = s , 0 ≤ s 2 ≤ s − 1 ∑ i = 1 3 D i s 1 v i ( 0 ) ∂ x i D i s 2 ϱ ( 1 ) − Π N 1 ∑ s 1 + s 2 = s , 0 ≤ s 2 ≤ s − 1 ∑ i = 1 3 ( D i s 2 h i ( 1 ) ) ( D i s 1 ∂ x i ρ ( 0 ) ) .

Next we derive higher derivative estimate of solution for (2.11).

Lemma 2.2

Let the parameters ν , μ > 1 . Assume the initial approximation function ( v ( 0 ) , ρ ( 0 ) ) satisfying (2.3)–(2.6). Then the solution ( h ( 1 ) ( t , x ) , ϱ ( 1 ) ( t , x ) ) of the linear system (2.11) satisfies

(2.31) ∑ i = 1 3 ∫ Ω ∑ j = 1 3 ( D i s h j ( 1 ) ) 2 + ( D i s ϱ ( 1 ) ) 2 d x ≲ e − C ν , μ , ε t Π N 1 ∑ i = 1 3 ∑ l 0 = 0 s ∫ Ω ∑ j = 1 3 ( D i l 0 h 0 j ( 1 ) ) 2 + ( D i l 0 ϱ 0 ( 1 ) ) 2 d x + ∫ 0 ∞ ∫ Ω ∑ j = 1 3 ( D i l 0 E j ( 0 ) ) 2 + ( D i l 0 E ¯ ( 0 ) ) 2 d x d t ,

where C ν , μ , ε denotes a positive constant depending on ν , μ , ε .

Proof

This proof is based on the induction. Let s = 1 , by (2.28), it holds

(2.32) ∂ t D i 1 h j ( 1 ) − ν Δ D i 1 h j ( 1 ) + Π N 1 ∑ i = 1 3 v i ( 0 ) ∂ x i D i 1 h j ( 1 ) + Π N 1 ∑ i = 1 3 D i 1 h i ( 1 ) ∂ x i v j ( 0 ) + Π N 1 ∂ x j D i 1 ( ( D v ( 0 ) P ) h ( 1 ) + ( D ρ ( 0 ) P ) ϱ ( 1 ) ) + Π N 1 ∑ i = 1 3 D i 1 v i ( 0 ) ∂ x i h j ( 1 ) + Π N 1 ∑ i = 1 3 h i ( 1 ) D i 1 ∂ x i v j ( 0 ) − Π N 1 ( D i 1 ϱ ( 1 ) ) e z = Π N 1 D i 1 E j ( 0 ) , for j = 1 , 2 , 3 ,

and

(2.33) ∂ t D i 1 ϱ ( 1 ) − μ ∂ x 3 2 D i 1 ϱ ( 1 ) + Π N 1 ∑ i = 1 3 ( v i ( 0 ) ( ∂ x i D i 1 ϱ ( 1 ) ) + ( D i 1 v i ( 0 ) ) ( ∂ x i ϱ ( 1 ) ) + ( D i 1 h i ( 1 ) ) ( ∂ x i ρ ( 0 ) ) + h i ( 1 ) ( ∂ x i D i 1 ρ ( 0 ) ) ) = Π N 1 D i 1 E ¯ ( 0 ) ,

with the boundary condition

(2.34) D i 1 h j ( 1 ) ( t , x ) ∣ x ∈ ∂ Ω = 0 , D i 1 ϱ ( 1 ) ( t , x ) ∣ x ∈ ∂ Ω = 0 .

We also choose the weighted function satisfying (2.13). Multiplying both sides of (2.32)–(2.33) by D i 1 h j ( 1 ) e − ψ ( x 3 ) and D i 1 ϱ ( 1 ) e − ψ ( x 3 ) , respectively, then integrating over Ω by noticing (2.34), and summing up those equalities from i , j = 1 to i , j = 3 , it holds

(2.35) 1 2 d d t ∑ i = 1 3 ∫ Ω ∑ j = 1 3 ( D i 1 h j ( 1 ) ) 2 + ( D i 1 ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + ν ∑ k = 1 3 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( ∂ x k D i 1 h j ( 1 ) ) 2 e − ψ ( x 3 ) d x + μ ∑ i = 1 3 ∫ Ω ( ∂ x 3 D i 1 ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + ν 2 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( ψ ″ ( x 3 ) − ( ψ ′ ( x 3 ) ) 2 ) ( D i 1 h j ( 1 ) ) 2 e − ψ ( x 3 ) d x + μ 2 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( ψ ″ ( x 3 ) − ( ψ ′ ( x 3 ) ) 2 ) ( D i 1 ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + Π N 1 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω v i ( 0 ) ( ∂ x i D i 1 h j ( 1 ) ) ( D i 1 h j ( 1 ) ) e − ψ ( x 3 ) d x + Π N 1 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( D i 1 h i ( 1 ) ) ( ∂ x i v j ( 0 ) ) ( D i 1 h j ( 1 ) ) e − ψ ( x 3 ) d x + Π N 1 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ∂ x j D i 1 ( ( D v ( 0 ) P ) h ( 1 ) + ( D ρ ( 0 ) P ) ϱ ( 1 ) ) D i 1 h j ( 1 ) e − ψ ( x 3 ) d x − Π N 1 ∑ i = 1 3 ∫ Ω ( D i 1 ϱ ( 1 ) ) ( D i 1 h 3 ( 1 ) ) e − ψ ( x 3 ) d x + Π N 1 ∑ i = 1 3 ∫ Ω v i ( 0 ) ( ∂ x i D i 1 ϱ ( 1 ) ) ( D i 1 ϱ ( 1 ) ) e − ψ ( x 3 ) d x + Π N 1 ∑ i = 1 3 ∫ Ω ( D i 1 h i ( 1 ) ) ( ∂ x i ρ ( 0 ) ) ( D i 1 ϱ ( 1 ) ) e − ψ ( x 3 ) d x + Π N 1 ∑ a = 1 5 I a = 0 ,

where

I 1 ≔ ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( D i 1 v i ( 0 ) ) ( ∂ x i h j ( 1 ) ) ( D i 1 h j ( 1 ) ) e − ψ ( x 3 ) d x , I 2 ≔ ∑ j = 1 3 ∑ i = 1 3 ∫ Ω h i ( 1 ) ( D i 1 ∂ x i v j ( 0 ) ) ( D i 1 h j ( 1 ) ) e − ψ ( x 3 ) d x , I 3 ≔ ∑ i = 1 3 ∫ Ω ( D i 1 v i ( 0 ) ) ( ∂ x i ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x , I 4 ≔ ∑ i = 1 3 ∫ Ω h i ( 1 ) ( ∂ x i D i 1 ρ ( 0 ) ) ( D i 1 ϱ ( 1 ) ) e − ψ ( x 3 ) d x , I 5 ≔ ∑ i = 1 3 ∫ Ω ∑ j = 1 3 ( D i 1 E j ( 0 ) ) ( D i 1 h j ( 1 ) ) + ( D i 1 E ¯ ( 0 ) ) ( D i 1 ϱ ( 1 ) ) e − ψ ( x 3 ) d x .

We now estimate each terms in (2.35). On the one hand, since we have chosen the initial approximation function ( v ( 0 ) , ρ ( 0 ) ) satisfying (2.3)–(2.6), using the similar method of getting (2.18)–(2.24), we obtain

(2.36) ∑ j = 1 3 ∑ i = 1 3 ∫ Ω v i ( 0 ) ( ∂ x i D i 1 h j ( 1 ) ) ( D i 1 h j ( 1 ) ) e − ψ ( x 3 ) d x = 1 2 ∑ j = 1 3 ∫ Ω ψ ′ ( x 3 ) v 3 ( 0 ) ( D i 1 h j ( 1 ) ) 2 e − ψ ( x 3 ) d x ,

(2.37) ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( D i 1 h i ( 1 ) ) ( ∂ x i v j ( 0 ) ) ( D i 1 h j ( 1 ) ) e − ψ ( x 3 ) d x = ∑ j = 1 3 ∫ Ω ∂ x j v j ( 0 ) ( D j 1 h j ( 1 ) ) 2 e − ψ ( x 3 ) d x + ∑ j = 1 3 ∑ i ≠ j ∫ Ω ∂ x i v j ( 0 ) ( D i 1 h i ( 1 ) ) ( D i 1 h j ( 1 ) ) e − ψ ( x 3 ) d x ,

and by the incompressible condition ∇ ⋅ h ( 1 ) = 0 , we integrate by parts to obtain

(2.38) ∑ j = 1 3 ∫ Ω ∂ x j D i 1 ( ( D v ( 0 ) P ) h ( 1 ) + ( D ρ ( 0 ) P ) ϱ ( 1 ) ) D i 1 h j ( 1 ) e − ψ ( x 3 ) d x = ∫ Ω ψ ′ ( x 3 ) D i 1 ( ( D v ( 0 ) P ) h ( 1 ) + ( D ρ ( 0 ) P ) ϱ ( 1 ) ) ( D i 1 h 3 ( 1 ) ) e − ψ ( x 3 ) d x ,

from which, we use the standard Calderon-Zygmund theory and Young’s inequality to derive

(2.39) ∫ Ω ψ ′ ( x 3 ) D i 1 ( ( D v ( 0 ) P ) h ( 1 ) + ( D ρ ( 0 ) P ) ϱ ( 1 ) ) ( D i 1 h 3 ( 1 ) ) e − ψ ( x 3 ) d x = ∑ i = 1 3 ∑ j = 1 3 ∫ Ω ψ ′ ( x 3 ) D i 1 ( ∂ x j h i ( 1 ) ∂ x i v j ( 0 ) + ∂ x j v i ( 0 ) ∂ x i h j ( 1 ) + ∂ x 3 ϱ ( 1 ) e z ) ( D i 1 h 1 ( 1 ) ) e − ψ ( x 3 ) d x ≲ ε 2 ∑ i = 1 3 ∫ Ω ∣ ψ ′ ( x 3 ) ∣ ( D i 1 h 1 ( 1 ) ) 2 + ∑ j = 1 3 ( ( ∂ x j D i 1 h i ( 1 ) ) 2 + ( ∂ x i D i 1 h j ( 1 ) ) 2 ) e − ψ ( x 3 ) d x + ε 2 ∑ i = 1 3 ∑ j = 1 3 ∫ Ω ∣ ψ ′ ( x 3 ) ∣ ( ( ∂ x j h i ( 1 ) ) 2 + ( ∂ x i h j ( 1 ) ) 2 ) e − ψ ( x 3 ) d x + 1 2 ∑ i = 1 3 ∫ Ω ∣ ψ ′ ( x 3 ) ∣ ( ( ∂ x 3 D i 1 ϱ ( 1 ) ) 2 + ( D i 1 h 1 ( 1 ) ) 2 ) e − ψ ( x 3 ) d x ,

and

∑ j = 1 3 ∑ i ≠ j ∫ Ω ∂ x i v j ( 0 ) ( D i 1 h i ( 1 ) ) ( D i 1 h j ( 1 ) ) e − ψ ( x 3 ) d x ≲ 3 ε 2 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( D i 1 h j ( 1 ) ) 2 e − ψ ( x 3 ) d x , ∑ i = 1 3 ∫ Ω ( D i 1 ϱ ( 1 ) ) ( D i 1 h 3 ( 1 ) ) e − ψ ( x 3 ) d x ≤ 1 2 ∑ i = 1 3 ∫ Ω ( ( D i 1 ϱ ( 1 ) ) 2 + ( D i 1 h 3 ( 1 ) ) 2 ) e − ψ ( x 3 ) d x , ∑ i = 1 3 ∫ Ω v i ( 0 ) ( ∂ x i D i 1 ϱ ( 1 ) ) ( D i 1 ϱ ( 1 ) ) e − ψ ( x 3 ) d x ≲ ε 2 ∑ i = 1 3 ∫ Ω ( D i 1 ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x , ∑ i = 1 3 ∫ Ω ( D i 1 h i ( 1 ) ) ( ∂ x i ρ ( 0 ) ) ( D i 1 ϱ ( 1 ) ) e − ψ ( x 3 ) d x ≲ ε 2 ∑ i = 1 3 ∫ Ω ( ( D i 1 h i ( 1 ) ) 2 + ( D i 1 ϱ ( 1 ) ) 2 ) e − ψ ( x 3 ) d x .

On the other hand, it holds

(2.40) I 1 ≲ ε ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( D i 1 h j ( 1 ) ) 2 e − ψ ( x 3 ) d x , I 2 ≲ ε 2 ∑ j = 1 3 ∫ Ω ( h j ( 1 ) ) 2 + ∑ i = 1 3 ( D i 1 h j ( 1 ) ) 2 e − ψ ( x 3 ) d x , I 3 ≲ ε ∑ i = 1 3 ∫ Ω ( ∂ x i ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x , I 4 ≲ ε ∑ i = 1 3 ∫ Ω ( ( h i ( 1 ) ) 2 + ( D i 1 ϱ ( 1 ) ) 2 ) e − ψ ( x 3 ) d x , I 5 ≲ ∑ i = 1 3 ∫ Ω ∑ j = 1 3 ( ( D i 1 E j ( 0 ) ) 2 + ∣ ψ ″ ( x 3 ) ∣ ( D i 1 h j ( 1 ) ) 2 ) + ( D i 1 E ¯ ( 0 ) ) 2 + ∣ ψ ″ ( x 3 ) ∣ ( D i 1 ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x .

Thus, we use (2.36)–(2.40) to derive

(2.41) d d t ∑ i = 1 3 ∫ Ω ∑ j = 1 3 ( D i 1 h j ( 1 ) ) 2 + ( D i 1 ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + ( ν − ε ) ∑ k = 1 3 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( ∂ x k D i 1 h j ( 1 ) ) 2 e − ψ ( x 3 ) d x + ( μ − ε ) ∑ i = 1 3 ∫ Ω ( ∂ x 3 D i 1 ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + ∑ i = 1 3 ∫ Ω ∑ j = 1 3 A ¯ 1 ( x 3 ) ( D i 1 h j ( 1 ) ) 2 + A ¯ 2 ( x 3 ) ( D i 1 ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x ≲ Π N 1 ∑ i = 1 3 ∫ Ω ∑ j = 1 3 ( D i 1 E j ( 0 ) ) 2 + ( D i 1 E ¯ ( 0 ) ) 2 e − ψ ( x 3 ) d x + ε ∑ j = 1 3 ∫ Ω ( h j ( 1 ) ) 2 e − ψ ( x 3 ) d x ,

where

A ¯ 1 ( x 3 ) ≔ ν 2 ( ψ ″ ( x 3 ) − ( ψ ′ ( x 3 ) ) 2 − ∣ ψ ′ ( x 3 ) ∣ − ε ) − c 0 , A ¯ 2 ( x 3 ) ≔ μ 2 ( ψ ″ ( x 3 ) − ( ψ ′ ( x 3 ) ) 2 − ∣ ψ ′ ( x 3 ) ∣ − ε ) − c 0 ,

with c 0 denotes a positive constant.

Note that ψ ′ ( x 3 ) satisfying inequality (2.13). There exist sufficient big positive constants ν , μ such that

ν − ε > C ν , ε > 0 , μ − ε > C μ , ε > 0 , max { A ¯ 1 ( x 3 ) , A ¯ 1 ( x 3 ) } > C ν , μ , ε > 0 .

Thus, we integrate inequality (2.41) over ( 0 , t ) to obtain

(2.42) ∑ i = 1 3 ∫ 0 t ∫ Ω ∑ j = 1 3 ( D i 1 h j ( 1 ) ) 2 + ( D i 1 ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x d t + C ν , ε ∑ k = 1 3 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( ∂ x k D i 1 h j ( 1 ) ) 2 e − ψ ( x 3 ) d x + C μ , ε ∑ i = 1 3 ∫ Ω ( ∂ x 3 D i 1 ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + C ν , μ , ε ∑ i = 1 3 ∫ Ω ∑ j = 1 3 ( D i 1 h j ( 1 ) ) 2 + ( D i 1 ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x ≲ Π N 1 ∑ i = 1 3 ∫ Ω ∑ j = 1 3 ( D i 1 E j ( 0 ) ) 2 + ( D i 1 E ¯ ( 0 ) ) 2 e − ψ ( x 3 ) d x + ε ∑ j = 1 3 ∫ Ω ( h j ( 1 ) ) 2 e − ψ ( x 3 ) d x .

Furthermore, one can see that the last term on the right-hand side of (2.42) can be controlled by using (2.14), and we apply Gronwall’s inequality to (2.42) to derive

(2.43) ∑ i = 1 3 ∫ Ω ∑ j = 1 3 ( D i 1 h j ( 1 ) ) 2 + ( D i 1 ϱ ( 1 ) ) 2 d x ≲ e − C ν , μ , ε t Π N 1 ∑ i = 1 3 ∑ l 0 = 0 1 ∫ Ω ∑ j = 1 3 ( D i l 0 h 0 j ( 1 ) ) 2 + ( D i l 0 ϱ 0 ( 1 ) ) 2 d x + ∫ 0 ∞ ∫ Ω ∑ j = 1 3 ( D i l 0 E j ( 0 ) ) 2 + ( D i l 0 E ¯ ( 0 ) ) 2 d x d t .

Assume that the 2 ≤ l ≤ s − 1 derivative case holds, i.e.,

(2.44) ∑ i = 1 3 ∫ Ω ∑ j = 1 3 ( D i l h j ( 1 ) ) 2 + ( D i l ϱ ( 1 ) ) 2 d x ≲ e − C ν , μ , ε t Π N 1 ∑ i = 1 3 ∑ l 0 = 0 l ∫ Ω ∑ j = 1 3 ( D i l 0 h 0 j ( 1 ) ) 2 + ( D i l 0 ϱ 0 ( 1 ) ) 2 d x + ∫ 0 ∞ ∫ Ω ∑ j = 1 3 ( D i l 0 E j ( 0 ) ) 2 + ( D i l 0 E ¯ ( 0 ) ) 2 d x d t .

We now prove the s th derivative case holds. Multiplying both sides of equations (2.28) and (2.29) by D i s h j ( 1 ) e − ψ ( x 3 ) and D i s ϱ ( 1 ) e − ψ ( x 3 ) , respectively, then integrating over Ω by using the boundary condition (2.30), and summing up those equalities from j = 1 to j = 3 , it holds

(2.45) 1 2 d d t ∑ i = 1 3 ∫ Ω ∑ j = 1 3 ( D i s h j ( 1 ) ) 2 + ( D i s ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + ν ∑ k = 1 3 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( ∂ x k D i s h j ( 1 ) ) 2 e − ψ ( x 3 ) d x + μ ∑ i = 1 3 ∫ Ω ( ∂ x 3 D i s ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + ν 2 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( ψ ″ ( x 3 ) − ( ψ ′ ( x 3 ) ) 2 ) ( D i s h j ( 1 ) ) 2 e − ψ ( x 3 ) d x + μ 2 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( ψ ″ ( x 3 ) − ( ψ ′ ( x 3 ) ) 2 ) ( D i s ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + Π N 1 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω v i ( 0 ) ( ∂ x i D i s h j ( 1 ) ) ( D i s h j ( 1 ) ) e − ψ ( x 3 ) d x + Π N 1 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( D i s h i ( 1 ) ) ( ∂ x i v j ( 0 ) ) ( D i s h j ( 1 ) ) e − ψ ( x 3 ) d x + Π N 1 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ∂ x j D i s ( ( D v ( 0 ) P ) h ( 1 ) + ( D ρ ( 0 ) P ) ϱ ( 1 ) ) D i s h j ( 1 ) e − ψ ( x 3 ) d x − Π N 1 ∑ i = 1 3 ∫ Ω ( D i s ϱ ( 1 ) ) ( D i s h 3 ( 1 ) ) e − ψ ( x 3 ) d x + Π N 1 ∑ i = 1 3 ∫ Ω v i ( 0 ) ( ∂ x i D i s ϱ ( 1 ) ) ( D i s ϱ ( 1 ) ) e − ψ ( x 3 ) d x + Π N 1 ∑ i = 1 3 ∫ Ω ( D i s h i ( 1 ) ) ( ∂ x i ρ ( 0 ) ) ( D i s ϱ ( 1 ) ) e − ψ ( x 3 ) d x = ∑ j = 1 3 ∫ Ω ( F j D i s h j ( 1 ) + F ¯ D i s ϱ ( 1 ) ) e − ψ ( x 3 ) d x .

We notice that

(2.46) ∫ Ω ψ ′ ( x 3 ) D i s ( ( D v ( 0 ) P ) h ( 1 ) + ( D ρ ( 0 ) P ) ϱ ( 1 ) ) ( D i s h 1 ( 1 ) ) e − ψ ( x 3 ) d x = ∑ i = 1 3 ∑ j = 1 3 ∫ Ω ψ ′ ( x 3 ) D i s ( ∂ x j h i ( 1 ) ∂ x i v j ( 0 ) + ∂ x j v i ( 0 ) ∂ x i h j ( 1 ) + ∂ x 3 ϱ ( 1 ) e z ) ( D i s h 1 ( 1 ) ) e − ψ ( x 3 ) d x = ∑ i = 1 3 ∑ j = 1 3 ∫ Ω ψ ′ ( x 3 ) ∑ j 1 + j 2 = s , 0 ≤ j 1 , j 2 ≤ s ( D i j 2 ∂ x j h i ( 1 ) D i j 1 ∂ x i v j ( 0 ) + D i j 1 ∂ x j v i ( 0 ) D i j 2 ∂ x i h j ( 1 ) + ∂ x 3 D i s ϱ ( 1 ) e z ) ( D i s h 1 ( 1 ) ) e − ψ ( x 3 ) d x ≲ ε 2 ∑ i = 1 3 ∑ j = 1 3 ∫ Ω ∣ ψ ′ ( x 3 ) ∣ ( D i s h 1 ( 1 ) ) 2 e − ψ ( x 3 ) d x + ε 2 ∑ i = 1 3 ∫ Ω ( ∂ x 3 D i s ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + ε 2 ∑ i = 1 3 ∑ j = 1 3 ∑ j 1 = 0 s − 1 ∫ Ω ∣ ψ ′ ( x 3 ) ∣ ( ( ∂ x j D i j 1 h i ( 1 ) ) 2 + ( ∂ x i D i j 1 h j ( 1 ) ) 2 ) e − ψ ( x 3 ) d x + ε 2 ∑ i = 1 3 ∑ j = 1 3 ∫ Ω ∣ ψ ′ ( x 3 ) ∣ ( ( ∂ x j D i s h i ( 1 ) ) 2 + ( ∂ x i D i s h j ( 1 ) ) 2 ) e − ψ ( x 3 ) d x ,

and thus, similar to obtain (2.42), we can obtain

(2.47) ∑ i = 1 3 ∫ 0 t ∫ Ω ∑ j = 1 3 ( D i s h j ( 1 ) ) 2 + ( D i s ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x d t + C ν , ε ∑ k = 1 3 ∑ j = 1 3 ∑ i = 1 3 ∫ Ω ( ∂ x k D i s h j ( 1 ) ) 2 e − ψ ( x 3 ) d x + C μ , ε ∑ i = 1 3 ∫ Ω ( ∂ x 3 D i s ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x + C ν , μ , ε ∑ i = 1 3 ∫ Ω ∑ j = 1 3 ( D i s h j ( 1 ) ) 2 + ( D i s ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x ≲ Π N 1 ∑ i = 1 3 ∫ Ω ∑ j = 1 3 ( D i s E j ( 0 ) ) 2 + ( D i s E ¯ ( 0 ) ) 2 e − ψ ( x 3 ) d x + ε 2 ∑ i = 1 3 ∑ j = 1 3 ∑ j 1 = 0 s − 1 ∫ Ω ∣ ψ ′ ( x 3 ) ∣ ( ∂ x j D i j 1 h i ( 1 ) ) 2 e − ψ ( x 3 ) d x ,

and then by (2.44), we apply Gronwall’s inequality to (2.47), it holds

(2.48) ∑ i = 1 3 ∫ Ω ∑ j = 1 3 ( D i s h j ( 1 ) ) 2 + ( D i s ϱ ( 1 ) ) 2 e − ψ ( x 3 ) d x ≲ e − C ν , μ , ε t Π N 1 ∑ i = 1 3 ∑ l 0 = 0 s ∫ Ω ∑ j = 1 3 ( D i l 0 h 0 j ( 1 ) ) 2 + ( D i l 0 ϱ 0 ( 1 ) ) 2 e − ψ ( x 3 ) d x + ∫ 0 ∞ ∫ Ω ∑ j = 1 3 ( D i l 0 E j ( 0 ) ) 2 + ( D i l 0 E ¯ ( 0 ) ) 2 e − ψ ( x 3 ) d x d t .

Note that e − ψ ( x 3 ) to be a bounded function in ( 0 , R ) , it follows from inequality (2.48) that

∑ i = 1 3 ∫ Ω ∑ j = 1 3 ( D i s h j ( 1 ) ) 2 + ( D i s ϱ ( 1 ) ) 2 d x ≲ e − C ν , μ , ε t Π N 1 ∑ i = 1 3 ∑ l 0 = 0 s ∫ Ω ∑ j = 1 3 ( D i l 0 h 0 j ( 1 ) ) 2 + ( D i l 0 ϱ 0 ( 1 ) ) 2 d x + ∫ 0 ∞ ∫ Ω ∑ j = 1 3 ( D i l 0 E j ( 0 ) ) 2 + ( D i l 0 E ¯ ( 0 ) ) 2 d x d t .□

2.3 The existence of first approximation step

On the basis of the aforementioned priori estimates, we are ready to prove the existence of first approximation step by the semigroup theory [3].

Proposition 2.1

Let the parameters ν , μ > 1 . Assume the initial approximation function ( v ( 0 ) , ρ ( 0 ) ) satisfying (2.3)–(2.6). Then the linearized system (2.9) with the boundary condition (2.10) admits a global solution

( h ( 1 ) ( t , x ) , ϱ ( 1 ) ( t , x ) ) T ∈ C ( ( 0 , + ∞ ) ; H s ( Ω ) × H s ( Ω ) ) .

Moreover, this global solution satisfies

(2.49) ‖ h ( 1 ) ‖ H s 2 + ‖ ϱ ( 1 ) ‖ H s 2 ≲ ‖ h 0 ( 1 ) ‖ H s 2 + ‖ ϱ 0 ( 1 ) ‖ H s 2 + ‖ Π N 1 E ( 0 ) ‖ H s 2 + ‖ Π N 1 E ¯ ( 0 ) ‖ H s 2 ∀ t > 0 .

Proof

Let P be the Leray projector onto the space of divergence free functions. We apply the Leray projector to equations (2.9), hence

(2.50) h t ( 1 ) − ν Δ h ( 1 ) + N 1 ( h ( 1 ) , ϱ ( 1 ) ) = P E 0 , ϱ t ( 1 ) − μ ∂ x 3 2 ϱ ( 1 ) + N 2 ( h ( 1 ) , ϱ ( 1 ) ) = P E ¯ 0 ,

where

N 1 ( h ( 1 ) , ϱ ( 1 ) ) ≔ P Π N 1 [ ( v ( 0 ) ⋅ ∇ ) h ( 1 ) + ( h ( 1 ) ⋅ ∇ ) v ( 0 ) + ϱ ( 1 ) e z ] , N 2 ( h ( 1 ) , ϱ ( 1 ) ) ≔ P Π N 1 [ ( v ( 0 ) ⋅ ∇ ) ϱ ( 1 ) + ( h ( 1 ) ⋅ ∇ ) ρ ( 0 ) ] .

For convenience, we rewrite equation (2.50) as an abstract evolution equation in the following form:

(2.51) ∂ ∂ t h ( 1 ) ϱ ( 1 ) = Z h ( 1 ) ϱ ( 1 ) + P E 0 P E ¯ 0 ,

where

Z h ( 1 ) ϱ ( 1 ) ≔ − − ν Δ h ( 1 ) + N 1 ( h ( 1 ) , ϱ ( 1 ) ) − μ ∂ x 3 2 ϱ ( 1 ) + N 2 ( h ( 1 ) , ϱ ( 1 ) ) .

We follow the idea of [3,58] to show that the linear operator Z generates a strongly continuous semigroup e Z τ in Sobolev space H s ( Ω ) × H s ( Ω ) . To see this, by the same process of getting (2.31), we obtain

∑ k = 1 3 ∑ j = 1 3 ∫ Ω ( ∂ k s h j ( 1 ) , ∂ k s ϱ ( 1 ) ) ⋅ ∂ k s Z h ( 1 ) ϱ ( 1 ) d x ≤ 0 .

We deduce that the linear operator Z is a linear dissipative operator in H s ( Ω ) × H s ( Ω ) . Moreover, if we set

Z h ( 1 ) ϱ ( 1 ) = 0 ,

we know that the linear operator Z is injective. Furthermore, we can verify that this linear operator is surjective by using the standard theory of elliptic-type equations of the general order. Thus, by the Lumer-Phillips theorem [41], the linear operator Z can generate a strongly continuous semigroup S 0 ( τ ) ≔ e Z τ in the Sobolev space H s ( Ω ) × H s ( Ω ) . Therefore, the linear system (2.50) has a global solution in H s ( Ω ) × H s ( Ω ) . Furthermore, it follows from Lemmas 2.1 and 2.2 that (2.49) holds.□

3 The m th approximation step

Let ε ∈ ( 0 , 1 ) be a fixed constant. We define

(3.1) ℬ ε ≔ { ( v ( k ) , ϱ ( k ) ) : ‖ v ( k ) ‖ H s + ‖ ϱ ( k ) ‖ H s ≲ ε < 1 }

with the integers 2 ≤ k ≤ m − 1 and s ≥ 1 .

Assume that the m th approximation solutions of (2.2) is denoted by ( h ( m ) ( t , x ) , ϱ ( m ) ( t , x ) ) T with m = 2 , 3 , … . Let

h ( m ) ( t , x ) ≔ v ( m ) ( t , x ) − v ( m − 1 ) ( t , x ) , ϱ ( m ) ( t , x ) ≔ ρ ( m ) ( t , x ) − ρ ( m − 1 ) ( t , x ) .

It follows that

v ( m ) ( t , x ) = v ( 0 ) ( t , x ) + h ( 1 ) ( t , x ) + ∑ i = 2 m h ( i ) ( t , x ) , ρ ( m ) ( t , x ) = ρ ( 0 ) ( t , x ) + ϱ ( 1 ) ( t , x ) + ∑ i = 2 m ϕ ( i ) ( t , x ) .

We linearize the nonlinear system (2.2) around ( v ( m − 1 ) ( t , x ) , ρ ( m − 1 ) ( t , x ) ) T to obtain the following initial value problem:

(3.2) J 1 [ v ( m − 1 ) , ρ ( m − 1 ) ] ( h ( m ) , ϱ ( m ) ) = Π N m E ( m − 1 ) , J 2 [ v ( m − 1 ) , ρ ( m − 1 ) ] ( h ( m ) , ϱ ( m ) ) = Π N m E ¯ ( m − 1 ) , ∇ ⋅ h ( m ) = 0 , h ( m ) ( 0 , x ) = h 0 ( m ) ( x ) , ϱ ( m ) ( 0 , x ) = ϱ 0 ( m ) ( x ) ,

with the boundary conditions

(3.3) h ( m ) ( t , x ) ∣ x ∈ ∂ Ω = 0 , ρ ( m ) ( t , x ) ∣ x ∈ ∂ Ω = 0 ,

where the error term is

(3.4) E ( m − 1 ) ≔ ℒ 1 [ v ( m − 1 ) ( t , x ) , ρ ( m − 1 ) ( t , x ) ] = ℛ 1 ( h ( m ) ( t , x ) , ϱ ( m ) ( t , x ) ) , E ¯ ( m − 1 ) ≔ ℒ 2 [ v ( m − 1 ) ( t , x ) , ρ ( m − 1 ) ( t , x ) ] = ℛ 2 ( h ( m ) ( t , x ) , ϱ ( m ) ( t , x ) ) ,

and

(3.5) ℛ 1 ( h ( m ) ( t , x ) , ϱ ( m ) ( t , x ) ) ≔ ℒ 1 ( v ( m − 1 ) + h ( m ) , ρ ( m − 1 ) + ϱ ( m ) ) − ℒ 1 ( v ( m − 1 ) , ρ ( m − 1 ) ) − ℒ 1 [ v ( m − 1 ) , ρ ( m − 1 ) ] ( h ( m ) , ϱ ( m ) ) = Π N m ( h ( m ) ⋅ ∇ h ( m ) + ∇ P ( m ) ) , ℛ 2 ( h ( m ) ( t , x ) , ϱ ( m ) ( t , x ) ) ≔ ℒ 2 ( v ( m − 1 ) + h ( m ) , ρ ( m − 1 ) + ϱ ( m ) ) − ℒ 2 ( v ( m − 1 ) , ρ ( m − 1 ) ) − ℒ 2 [ v ( m − 1 ) , ρ ( m − 1 ) ] ( h ( m ) , ϱ ( m ) ) = Π N m ( h ( m ) ⋅ ∇ ϱ ( m ) ) ,

where

P ( m ) ( t , x ) = − Δ − 1 ∑ i , j = 1 3 ∂ x j h i ( m ) ∂ x i h j ( m ) .

Here, we notice that this term − ∂ x 3 ϱ ( m ) e z has been concluded in the linearized operator.

The following result establishes how to construct the m th approximation solution.

Proposition 3.1

Let the parameters ν , μ > 1 . Assume the initial approximation function ( v ( 0 ) , ρ ( 0 ) ) satisfying (2.3)–(2.6), ( v ( m − 1 ) ( t , x ) , ρ ( m − 1 ) ( t , x ) ) T ∈ ℬ ε and

∑ i = 1 m − 1 ( ‖ h ( i ) ‖ H s 2 + ‖ ϱ ( i ) ‖ H s 2 ) ≲ ε 2 .

Then the linearized problem (3.2) with the boundary condition (3.3) admits a global solution

( h ( m ) ( t , x ) , ϱ ( m ) ( t , x ) ) ∈ C ( ( 0 , + ∞ ) ; H s ( Ω ) × H s ( Ω ) ) ,

which satisfies

(3.6) ‖ h ( m ) ‖ H s 2 + ‖ ϱ ( m ) ‖ H s 2 ≲ ‖ h 0 ( m ) ‖ H s 2 + ‖ ϱ 0 ( m ) ‖ H s 2 + ‖ Π N m E ( m − 1 ) ‖ H s 2 + ‖ Π N m E ¯ ( m − 1 ) ‖ H s 2 ∀ t > 0 ,

where the error term satisfies

(3.7) ‖ E ( m ) ‖ H s + ‖ E ¯ ( m ) ‖ H s ≲ N m 2 ( ‖ h ( m ) ‖ H s 2 + ‖ ϱ ( m ) ‖ H s 2 ) .

Proof

By direct computation, we find

(3.8) ∂ x i v j ( m − 1 ) ( t , x ) = ∂ x i v j ( 0 ) ( t , x ) + ∂ x i h ( 1 ) ( t , x ) + ∑ i = 2 m − 1 ∂ x i h ( i ) ( t , x ) , ∂ x i ϱ ( m − 1 ) ( t , x ) = ∂ x i ρ ( 0 ) ( t , x ) + ∂ x i ϱ ( 1 ) ( t , x ) + ∑ i = 2 m − 1 ∂ x i ϱ ( i ) ( t , x ) .

By the assumption ∑ i = 1 m − 1 ( ‖ h ( i ) ‖ H s 2 + ‖ ϱ ( i ) ‖ H s 2 ) ≲ ε 2 , it follows that

∂ x i v j ( m − 1 ) ( t , x ) ∼ ∂ x i v j ( 0 ) ( t , x ) + O ( ε 2 ) , ∂ x i ρ ( m − 1 ) ( t , x ) ∼ ∂ x i ρ ( 0 ) ( t , x ) + O ( ε 2 ) .

Thus, noticing that ( v ( 0 ) ( t , x ) , ρ ( 0 ) ( t , x ) ) satisfies (2.3)–(2.6), by small modification of ∂ x i v j ( 0 ) ( t , x ) and ∂ x i ρ ( 0 ) ( t , x ) , it follows that

(3.9) ∑ k = 0 s ‖ Π N m ∂ x i k h j ( m − 1 ) ( t , x ) ‖ L ∞ ≲ ε 2 ∀ i , j = 1 , 2 , 3 , ∑ k = 0 s ‖ Π N m ∂ x i k ϱ ( m − 1 ) ( t , x ) ‖ L ∞ ≲ ε 2 .

Moreover, we notice that the ( m − 1 )th approximation solution is

v ( m − 1 ) ( t , x ) = v ( 0 ) ( t , x ) + h ( 1 ) ( t , x ) + ∑ i = 2 m − 1 h ( i ) ( t , x ) , ρ ( m − 1 ) ( t , x ) = ρ ( 0 ) ( t , x ) + ϱ ( 1 ) ( t , x ) + ∑ i = 2 m − 1 ϱ ( i ) ( t , x ) ,

and

∇ ⋅ h ( m − 1 ) = 0 .

We obtain

∇ ⋅ v ( m − 1 ) ( t , x ) = 0 , v ( m − 1 ) ( 0 , x ) = 0 , ‖ v ( m − 1 ) ‖ H s ≲ ε , v ( m − 1 ) ( t , x ) ∣ x ∈ ∂ Ω = 0 ,

and

ϱ ( m − 1 ) ( 0 , x ) = 0 , ‖ ϱ ( m − 1 ) ‖ H s ≲ ε , ϱ ( m − 1 ) ( t , x ) ∣ x ∈ ∂ Ω = 0 .

Then we will find the m th ( m ≥ 2 ) approximation solution ( v ( m ) ( t , x ) , ρ ( m ) ( t , x ) ) , which is equivalent to find ( h ( m ) ( t , x ) , ϱ ( m ) ( t , x ) ) such that

(3.10) v ( m ) ( t , x ) = v ( m − 1 ) ( t , x ) + h ( m ) ( t , x ) , ρ ( m ) ( t , x ) = ρ ( m − 1 ) ( t , x ) + ϱ ( m ) ( t , x ) .

By substituting (3.10) into (2.2), we have

ℒ 1 ( v ( m ) , ρ ( m ) ) = ℒ 1 ( v ( m − 1 ) , ρ ( m − 1 ) ) + ℒ 1 [ ( v ( m − 1 ) , ρ ( m − 1 ) ) ] ( h ( m ) , ϱ ( m ) ) + ℛ 1 ( h ( m ) , ϱ ( m ) ) , ℒ 2 ( v ( m ) , ρ ( m ) ) = ℒ 2 ( v ( m − 1 ) , ρ ( m − 1 ) ) + ℒ 2 [ ( v ( m − 1 ) , ρ ( m − 1 ) ) ] ( h ( m ) , ϱ ( m ) ) + ℛ 2 ( h ( m ) , ϱ ( m ) ) .

Set

ℒ 1 [ ( v ( m − 1 ) , ρ ( m − 1 ) ) ] ( h ( m ) , ϱ ( m ) ) = − ℒ 1 ( v ( m − 1 ) , ρ ( m − 1 ) ) = − E ( m − 1 ) , ℒ 2 [ ( v ( m − 1 ) , ρ ( m − 1 ) ) ] ( h ( m ) , ϱ ( m ) ) = − ℒ 2 ( v ( m − 1 ) , ρ ( m − 1 ) ) = − E ¯ ( m − 1 ) .

We supplement this definition with the boundary conditions (3.3).

Since we assume ( v ( m − 1 ) ( t , x ) , ρ ( m − 1 ) ( t , x ) ) T ∈ ℬ ε , there is the same structure between the linear system (2.9) and the linear system of m th approximation solutions. Thus, by means of the same proof process as in Proposition 2.1, we can show that the aforementioned problem admits nonuniqueness solutions ( h ( m ) ( t , x ) , ϱ ( m ) ( t , x ) ) T ∈ H s ( Ω ) × H s ( Ω ) . Here, we should use (2.1). Furthermore, similar to (2.49), we can use (3.8) and (3.9) to derive

‖ h ( m ) ‖ H s 2 + ‖ ϱ ( m ) ‖ H s 2 ≲ ‖ h 0 ( m ) ‖ H s 2 + ‖ ϱ 0 ( m ) ‖ H s 2 + ‖ E ( m − 1 ) ‖ H s 2 + ‖ E ¯ ( m − 1 ) ‖ H s 2 ∀ t > 0 ,

where one can take the ( m − 1 ) th error term ( E ( m − 1 ) , E ¯ ( m − 1 ) ) T such that

E ( m − 1 ) ≔ ℒ 1 ( v ( m − 1 ) , ρ ( m − 1 ) ) = ℛ 1 ( h ( m ) , ϱ ( m ) ) , E ¯ ( m − 1 ) ≔ ℒ 2 ( v ( m − 1 ) , ρ ( m − 1 ) ) = ℛ 2 ( h ( m ) , ϱ ( m ) ) .

Furthermore, by (3.5) and the standard Calderon-Zygmund theory, we conclude that

‖ E ( m ) ‖ H s + ‖ E ¯ ( m ) ‖ H s = ‖ Π N m ( h ( m ) ⋅ ∇ h ( m ) ) ‖ H s + ‖ Π N m ∇ P ( m ) ‖ H s + ‖ Π N m ( h ( m ) ⋅ ∇ ϱ ( m ) ) ‖ H s ≲ N m 2 ( ‖ h ( m ) ‖ H s 2 + ‖ ϱ ( m ) ‖ H s 2 ) .

The proof is now completed.□

4 Convergence of the approximation scheme

Our target is to prove that ( v ( ∞ ) ( t , x ) , ρ ∞ ( t , x ) ) T is a global solution of nonlinear equations (1.1). This is equivalent to show that the series ∑ i = 1 m h ( i ) ( t , x ) and ∑ i = 1 m ϱ ( i ) ( t , x ) are convergent.

For a fixed integer s ≥ 1 , let 1 ≤ s = k ¯ < k 0 ≤ k and

k m ≔ k ¯ + k − k ¯ 2 m , k + ∞ = k ¯ , α m + 1 ≔ k m − k m + 1 = k − k ¯ 2 m + 1 .

It follows that

(4.1) k 0 > k 1 > … > k m > k m + 1 > … .

Proposition 4.1

Let the parameters ν , μ > 1 . Assume the initial approximation function ( v ( 0 ) , ρ ( 0 ) ) satisfying (2.3)–(2.6). Then the Boussinesq equation (1.1) in the thin domain with the small initial data (1.2) and the boundary condition (1.4) admits a global Sobolev solution

v ( ∞ ) ( t , x ) = v ( 0 ) ( t , x ) + ∑ m = 1 ∞ h ( m ) ( t , x ) + v 0 ( x ) ∈ H s ( Ω ) , ρ ( ∞ ) ( t , x ) = ρ ( 0 ) ( t , x ) + ∑ m = 1 ∞ ϱ ( m ) ( t , x ) + ρ 0 ( x ) ∈ H s ( Ω ) .

Moreover, it holds

‖ v ( ∞ ) ‖ H s + ‖ ρ ( ∞ ) ‖ H s ≲ ε .

Proof

The proof is based on induction arguments. We first deal with the case of zero initial data, that is, ( v 0 ( x ) , ρ 0 ( x ) ) T = ( 0 , 0 , 0 , 0 ) T . Next, we discuss the case v 0 ( x ) ≢ 0 and ρ 0 ( x ) ≢ 0 . Note that N m = N 0 m with N 0 > 1 . For all m = 1 , 2 , … , we claim that there exists a sufficient small positive constant ε such that

(4.2) ‖ h ( m ) ‖ H k m − 1 + ‖ ϱ ( m ) ‖ H k m − 1 < ε 2 m − 1 , ‖ E ( m ) ‖ H k m − 1 + ‖ E ¯ ( m ) ‖ H k m − 1 < ε 2 m , ( v ( m ) , ρ ( m ) ) T ∈ ℬ ε .

For the case m = 1 , we recall the assumptions (2.3)–(2.6) on the initial approximation function ( v ( 0 ) ( t , x ) , ρ ( 0 ) ( t , x ) ) T . By (2.49), let 0 < ε 0 < N 0 − ( 8 + k − k ¯ ) ε 2 < ε ≪ 1 , hence,

‖ h ( 1 ) ‖ H k 0 + ‖ ϱ ( 1 ) ‖ H k 0 ≲ ‖ E ( 0 ) ‖ H k 0 + ‖ E ¯ ( 0 ) ‖ H k 0 < ε .

Moreover, by (3.7) and the aforementioned estimate, we obtain

‖ E ( 1 ) ‖ H k 0 + ‖ E ¯ ( 1 ) ‖ H k 0 ≲ ‖ ℛ 1 ( h ( 1 ) , ϱ ( 1 ) ) ‖ H k 0 + ‖ ℛ 2 ( h ( 1 ) , ϱ ( 1 ) ) ‖ H k 0 ≲ N 1 2 ( ‖ h ( 1 ) ‖ H k 0 2 + ‖ ϱ ( 1 ) ‖ H k 0 2 ) ≲ ε 0 N 1 2 < ε 2 ,

and

‖ v ( 1 ) ‖ H k 0 + ‖ ρ ( 1 ) ‖ H k 0 ≲ ‖ v ( 0 ) ‖ H k 0 + ‖ ρ ( 0 ) ‖ H k 0 + ‖ h ( 1 ) ‖ H k 0 + ‖ ϱ ( 1 ) ‖ H k 0 ≲ ε ,

which means that ( v ( 1 ) , ρ ( 1 ) ) T ∈ ℬ ε .

Assume that our assertion holds for m − 1 , that is,

(4.3) ‖ h ( m − 1 ) ‖ H k m + ‖ ϱ ( m − 1 ) ‖ H k m < ε 2 m − 2 , ‖ E ( m − 1 ) ‖ H k m + ‖ E ¯ ( m − 1 ) ‖ H k m < ε 2 m − 1 , ( v ( m − 1 ) , ρ ( m − 1 ) ) T ∈ ℬ ε ,

then we prove the case of m holds. By using (2.1), (3.6), and the second inequality of (4.3), we derive

(4.4) ‖ h ( m ) ‖ H k m − 1 + ‖ ϱ ( m ) ‖ H k m − 1 ≲ ‖ Π N m E ( m − 1 ) ‖ H k m − 1 + ‖ Π N m E ¯ ( m − 1 ) ‖ H k m − 1 ≲ N m α m ( ‖ E ( m − 1 ) ‖ H k m + ‖ E ¯ ( m − 1 ) ‖ H k m ) < ε 2 m − 2 .

Next, by combining with (2.1), (3.7), and (4.1), we obtain

(4.5) ‖ E ( m ) ‖ H k m + ‖ E ¯ ( m ) ‖ H k m ≲ N m 2 ( ‖ h ( m ) ‖ H k m 2 + ‖ ϱ ( m ) ‖ H k m 2 ) ≲ N m 2 + α m + 1 ( ‖ E ( m − 1 ) ‖ H k m + 1 + ‖ E ¯ ( m − 1 ) ‖ H k m + 1 ) 2 ≲ N 0 ( 2 + α m + 1 ) m + 2 ( 2 + α m + 2 ) ( m − 1 ) ( ‖ E ( m − 2 ) ‖ H k m + 2 + ‖ E ¯ ( m − 2 ) ‖ H k m + 2 ) 2 2 ≲ … ≲ [ N 0 8 + k − k ¯ ( ‖ E ( 0 ) ‖ H k 2 m + ‖ E ¯ ( 0 ) ‖ H k 2 m ) ] 2 m .

We choose a sufficient small positive constant ε 0 such that

0 < N 0 8 + k − k ¯ ( ‖ E ( 0 ) ‖ H k ¯ + ‖ E ¯ ( 0 ) ‖ H k ¯ ) < 2 N 0 4 ε 0 < ε 2 ,

and thus, (4.5) implies that

‖ E ( m ) ‖ H k m + ‖ E ¯ ( m ) ‖ H k m < ε 2 m ,

and

0 ≤ lim m → + ∞ ( ‖ E ( m ) ‖ H k m + ‖ E ¯ ( m ) ‖ H k m ) ≲ [ N 0 8 + k − k ¯ ( ‖ E ( 0 ) ‖ H k + ∞ + ‖ E ¯ ( 0 ) ‖ H k + ∞ ) ] 2 + ∞ ⟶ 0 .

So the error term goes to 0 as m → ∞ , i.e.,

lim m → ∞ ( ‖ E ( m ) ‖ H k m + ‖ E ¯ ( m ) ‖ H k m ) = 0 .

On the other hand, note that N m = N 0 m . By (4.3)–(4.4), we have

‖ v ( m ) ‖ H k m + ‖ ρ ( m ) ‖ H k m ≲ ‖ v ( m − 1 ) ‖ H k m + ‖ ρ ( m − 1 ) ‖ H k m + ‖ h ( m ) ‖ H k m + ‖ ϱ ( m ) ‖ H k m ≲ ε + N m 3 ε 2 m ≲ ε .

This means that ( v ( m ) , ρ ( m ) ) T ∈ ℬ ε . Hence, we conclude that (4.2) holds.

Therefore, the Boussinesq equations (1.1) with the zero initial data admit a global solution

v ( ∞ ) ( t , x ) = v ( 0 ) ( t , x ) + ∑ m = 1 ∞ h ( m ) ( t , x ) = v ( 0 ) ( t , x ) + O ( ε ) , ρ ( ∞ ) ( t , x ) = ρ ( 0 ) ( t , x ) + ∑ m = 1 ∞ ϱ ( m ) ( t , x ) = ρ ( 0 ) ( t , x ) + O ( ε ) .

Next, we discuss the case of small nonzero initial data. We introduce an auxiliary function

v ¯ ( t , x ) = v ( t , x ) − v 0 ( x ) , x ∈ Ω , ρ ¯ ( t , x ) = ρ ( t , x ) − ρ 0 ( x ) , x ∈ Ω .

Then the initial data are reduced to

v ¯ ( 0 , x ) = ( 0 , 0 , 0 ) T , ρ ¯ ( 0 , x ) = 0 ,

and equations (1.1) are transformed into equations of ( v ¯ , ρ ¯ ) T . Thus, we can follow the aforementioned iteration scheme to construct a global Sobolev solution ( v ¯ , ρ ¯ ) T . Furthermore, the global Sobolev solution of equations (1.1) with a small nonzero initial data of the form

( v ¯ ( t , x ) + v 0 , ρ ¯ ( t , x ) + ρ 0 ) T .

Finally, we use (1.3) and apply the Calderon-Zygmund theory. Thus, for the Riesz operator ℛ , there exists ‖ ℛ w ‖ L s 0 ≤ ‖ w ‖ L s 0 with 1 < s 0 < ∞ , such that

‖ P ‖ H s ≲ ε .

This completes the proof.□

Acknowledgment

Weiping Yan was supported by Guangxi Natural Science Foundation (No. 2021JJG110002) and NSFC (No. 12161006).

  1. Conflict of interest: The authors state no conflict of interest.

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Received: 2022-09-08
Revised: 2022-11-14
Accepted: 2023-01-24
Published Online: 2023-03-27

© 2023 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/anona-2022-0298
Keywords for this article
Boussinesq system; global Sobolev solution; Nash-Moser iteration scheme
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Articles in the same Issue

  1. Regular Articles
  2. On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth
  3. On the critical Choquard-Kirchhoff problem on the Heisenberg group
  4. On the local behavior of local weak solutions to some singular anisotropic elliptic equations
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  44. On the dynamics of grounded shallow ice sheets: Modeling and analysis
  45. A survey on some vanishing viscosity limit results
  46. Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions
  47. Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation
  48. Front propagation in a double degenerate equation with delay
  49. Positive solutions for a class of singular (p, q)-equations
  50. Higher integrability for anisotropic parabolic systems of p-Laplace type
  51. The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor
  52. On a system of multi-component Ginzburg-Landau vortices
  53. Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities
  54. Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
  55. On double phase Kirchhoff problems with singular nonlinearity
  56. Estimates for eigenvalues of the Neumann and Steklov problems
  57. Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
  58. Dirichlet problems involving the Hardy-Leray operators with multiple polars
  59. Incompressible limit for compressible viscoelastic flows with large velocity
  60. Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces
  61. Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings
  62. Noncoercive parabolic obstacle problems
  63. Touchdown solutions in general MEMS models
  64. Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
  65. Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth
  66. Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
  67. Symmetries of Ricci flows
  68. Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
  69. On the topological gradient method for an inverse problem resolution
  70. Supersolutions to nonautonomous Choquard equations in general domains
  71. Uniform complex time heat Kernel estimates without Gaussian bounds
  72. Global existence for time-dependent damped wave equations with nonlinear memory
  73. Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
  74. Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
  75. Lamé system with weak damping and nonlinear time-varying delay
  76. Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds
  77. Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
  78. Boundary regularity results for minimisers of convex functionals with (p, q)-growth
  79. Parametric singular double phase Dirichlet problems
  80. Special Issue on Nonlinear analysis: Perspectives and synergies
  81. Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”
  82. Identification of discontinuous parameters in double phase obstacle problems
  83. Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
  84. On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
  85. On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
  86. The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
  87. Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data
  88. On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth
  89. Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
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