Local Existence for Boussinesq Equations with Slip Boundary Condition in a Bounded Domain

In this paper, we are interested in the local existence for the Boussinesq equations with the slip boundary conditions. Energy method and Gerlakin approach are employed in this paper to get the main result.


Introduction
The Boussinesq equations are as follows: where Ω is a bounded smooth domain of In this paper, the initial data is given by ( ) ( ) and the boundary condition is where n is the unit outward normal on ∂Ω , u τ is the tangential part of u , Su denotes the deformation tensor:

Su n Bu
( ) Now, what we need to do is the study for the local existence of the problem (1.1) -(1.3).Usually, the followed existences are considered.First, the smooth of the existence is so small [8] [9] [10] that the initial data and the existence are close to a constant in ( )

2
H Ω , however this existence does not have singularity.Second, the existence is the "large energy" proposed by Lions [11].This existence has regularity but the analysis of the characteristic are more difficult and the uniqueness and the continuous dependence can't be solved [12] [13].Last, according to documents [14] [15] [16], the initial data in ( ) L Ω is small and the initial density is positive and bounded.For example, when the initial data is piecewise smooth, the solutions shown in [17] satisfy the Rankine-Hugoniot conditions in a strict point wise sense.On the other hand, these solutions have so enough structure and regularity that the uniqueness and continuous dependence theory can be proved in [18].The well-posedness theory is nearly complete in the whole space, but for the Dirichlet condition: on ∂Ω , the station 0 u = may have problems.However, if considering the Navier condition de- scribed above, the global weak small-energy solutions can be proved to exist in a half space for initial data with small energy and bounded density.David Hoff [ where ( ) div ⋅ denotes divergence.Satisfying the bounded conditions: 0, , 0 , where ( ) is the unit outward normal on ∂Ω , ( ) is the projection onto the tangent plant to ∂Ω at x, then the existence of the solutions can be given.T.

2) For any times
[ ] with , W φ ϕ ∈ .Remark: the system parameters Ω and B will be assumed to satisfy the following conditions: C in a neighborhood of ∂Ω .Theorem 1.2.Assume the hypotheses hold, let ( ] 3, 6 q ∈ be fixed constant and the ( ) 0 0 , u θ , p and f satisfy the following conditions: , 0, ; 0, ; , 0, ; Then there is a small 0 T * > and a unique strong solution ( )

The Lame Operator and the Regularity of −∆ Operator
In this part, we introduce the regularity of Laplace and the −∆ .First we assume the condition (1.2) holds, Ω and B satisfy the above hypotheses.Considering the following problem: find ( ) where for all W ω ∈ .Furthermore, because the embedding L and the symmetry condition guarantees S β is self-adjoint.
The following lemmas are taken from document [21].
C boundary and that ( ) ∂Ω is a symmetric matrix.For β enough large, there is a compact self-adjoint operator :

S L L
β Ω → Ω , whose range is contained in W and for which (2.2) holds for ( ) , and which are eigenfunctions of S β × matrices, then there exists a constant ( ) For the operator −∆ , consider the following problem: find Ω to multiply by the above differential equation, and integration by parts, then we get ( ) where is the bilinear form: ( ) Similar to the Lame operator, there exist a bounded operator : for all

A Prior Estimates for Higher Regularity
In this part, we need the following prior estimates to prove the local existence of the solution.Assume that the following inequalities hold: where

Estimate for Temperature
Lemma 3.1.1.

∫
Proof: According to the second equation of (1.1), we have then use the elliptic regularity: Both sides of the above inequality multiply by itself, then And integrating over ( ) 0,t , we have ( ) Both sides of the above inequality multiply by itself, then And integrating over ( )

The Local Existence of the Solution of Boussinesq Equations
First, we consider the following linearized system: Let Ω be a bounded domain in 3 R with smooth boundary, when 3 6 q < ≤ , we have ( ) ( ) Then there is a unique strong solution ( ) , 0, ; 0, ; Proof: It follows from Theorem 4 in chapter 5.9 [22], then we obtain , , , , , , , We hope that the coefficients

The Proof of Theme 1.2
Similar to the prior estimates in part 3, we have where C has no connection with m , then we have , , in 0, ; It follows from Theorem 3 in chapter 7 [22], let m → ∞ then we obtain that    Above all, we complete the proof of the Theorem 1.2.
temperature, p is the pressure function, f is the ex- ternal force,

3 )
Boussinesq equations are the classical model of fluid mechanics.There are a lot of important applications in marine ecology and weather forecasting.There are a lot of related conclusions about 2-D Boussinesq equations.Ye Zhuan[1] , and another is the famous boundary condition proposed by Navier: studied a fractional diffusion Boussinesq model which couples a Navier-Stokes type equation with fractional diffusion for the velocity and a transport equation for the temperature by establishing global well-posedness results with rough initial data.In this paper, we consider the Boussinesq equations.Because the temperature and the fluid are coupled together, the study becomes more difficult.The key we solve the problem is how to deal with the temperature.

(
the second equation of (1.1) by θ and integrating over Ω , one has the second equation of (1.1) by t θ and integrating over Ω , one has ( ) to the first equation of (1.1), we have the elliptic regularity: to the first equation of (1.1), we have ( ) use the elliptic regularity:

(
approach is applied to prove the local existence of the solution of Equation (4eigenvectors of the operator −∆ and the operator L are smooth functions. satisfy the "projection" (4.4) of problem (4.1) onto the finite dimensional subspace spanned by { } Theorem 1 in chapter 7[22], then we obtain that for m the solution of (4.1).The proof of Lemma 4.1 is completed.Next, the iteration method is used to prove the local existence of the solution of Boussinesq equations.Construct approximate solutions of Boussinesq equations that meet the initial and boundary problems (1.2) -(1

W
where C has no connection with k .denotes the dual space of W .It follows from Theorem 3 in chapter 5.9[22], then we obtain to the proof of the continuity of t u , it is easy to know Differentiating the second equation of (1.1) with respect to t , multiplying the second equation of (1.1) by t θ and integrating over Ω , one has DOI: 10.4236/jamp.2017.5101651957 Journal of Applied Mathematics and Physics Proof: