Solutions and Drift Homogenization for a Class of Viscous Lake Equations in ) ( 2 

address:1760724097@qq.com Abstract In this paper we study solutions and drift homogenization for a class of viscous lake equations by using the method of semigroups of bounded operators. Suppose that the initial value , ) , ( 0 t u = for some Hölder continuous function  T , 0  ) ( x b ) . (   C Then the initial value problem (2) for viscous lake equations has a unique smooth local strong solution. Using this result we study the drift homogenization for three-dimensional stationary Stokes equation in

(1)Introduction The viscous lake equations considered in this paper have the following equations (see the formula (1.3)in [17]) , a bounded domain with smooth boundary   of class , 3 C µ > 0 represents the eddy viscosity coefficient, ), , is the deformation tensor.
The equation (1) shows that the system does not describe incompressible flow, it is a constraint that plays a role similar to that played by the incompressibility condition for the incompressible Navier-Stokes system. We do not have an existence result concerning the solution of viscous shallow water equations with viscosity term given by ) (i until now.
We consider in this paper the well posedness of system (1)-) (ii with initial and boundary conditions which is given as follows The existence，uniqueness and regularity properties of solutions for the viscous lake equations are extensively studied. There is an extensive literature on the solvability of the initial value problem for viscous lake equations. The terms and symbols in this paper are the same as [18]. For some narratives and background, please refer to [18]. In case when ( )

 =  DL bDL
Similarly to p.270 in [4] we can prove that if (see lemma 1) We will see in lemma 7 that if (see lemma 2) Let P be the orthogonal projection from ) By applying P to the first equation of (3) and taking account of the other equations , we are let the following abstract initial value problem, We take a rectangular in [11] implies that  is the infinitesimal generator of an analytic semigroup of contractions on . Hence  is also the infinitesimal generator of an analytic semigroup of contraction on H respectively. We will prove that  is also the infinitesimal generator of an analytic semigroup of contraction on ).
Since  − is a strongly elliptic operator of order 2 on  . From Theorem 7.3.2 in [11] it follows that there exist constant C ˃0 , 0  r and 0 < < 2  such that Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 8 November 2021 on a Banach space X and assume that the restricted semigroup (subspace ).
be the restriction of the analytic semigroup generated by  on )  [11] it follows that can also be extended to an analytic semigroup on ).
From the formula (10) and Theorem 2.5.2(c) in [11] we have [11] implies that can also be extended to an analytic semigroup on X From the results of section 2.6 in [11] we can define the fraction powers Problem 8 in [7]), therefore there exists 0 L ˃0 such that for any ) ) (( From the Lemma 8 and the formulas (11) (12) we see that if take The function  (14) has a unique local solution In what follows we will need Banach lattice (see [1] ). A real vector space G which is ordered by some order relation  is called a vector lattice (or Riesz space) if any two elements G g f  , have a least upper bound, denoted by g f  , and a greatest lower bound , denoted by g f  , and the following properties are satisfied: A Banach lattice is a real Banach space G endowed with an ordering  such that ( )  , G is a vector lattice and the norm is a lattice norm, that is is the absolute value of f and • is the norm in . G In a Banach lattice In what follows we will need the above formula .

(3)The solutions of lake equations
Now we study the viscous lake equations (2). In the following proof of Theorem In the following we will use these facts. The bilinear     ( We used lemma 9 in the above third step . For any We used the Lemma 9 in the above third step and the formula (16) (16), (17) and (18) Changing the value of u on   to zero we get a unique local strong solution for (2).
Using a similar induction way as Theorem 3.9 in [9] or as Theorem 5.1 in [15] we can prove that the solution We can also prove directly that ) , ( x t u is smooth. In fact, the solution (19) of (15) is also the solution of (6). The Theorem 3.4 in [9] mean that as long as the solution of (6) exists , this solution is smooth. From Theorem 3.4 in [9] we have the is the unique local strong solution of the initial value problem (2) for viscous lake equations. The Theorem 1 has the following corollary.

) Drift homogenization
A composite is a material containing two or more finely mixed components. Composite materials are widely used nowadays in any kind of industries. How to determine the properties of a composite material, for example thermal, electrical or linear elastic properties of materials ? It can be solved by the homogenization of a set of partial differential equations. The homogenization theory allows to describe the asymptotic behaviour as 0 →  of partial differential equations of many types. A classical problem is the elliptic Dirichlet problem x u The prescription of traditional Chinese medicine consists of a variety of Chinese herbal medicines. Its efficacy can be obtained from the homogenization of a group of partial differential equations controlled by each drug. The actual speed of a ship can be obtained from the homogenization of a set of Lake equations disturbed by the ship's dynamic speed ( drift) where the oscillations are produced by the sequence of vector-value functions  v which strongly converges to some v in 2 1 ) Here v is the scheduled ship speed and  v is the actual approximate speed, In what follows all   v u , are velocity fields.

Theorem 2. Suppose that the smooth initial value
are Hölder continuous in   Similarly to Theorem 1. from (16), (17) and (18) for any    Proof. For each fixed  ˃0 let ) (t y  be the fixed point in the proof of Theorem 6.3.1 in [11]. Then is bounded by Theorem 2.16.13(c) in [11], hence the below of (23) has a limit when 0 →  by Theorem 7.16 in [13].
Therefor the above of (23) also has a limit and so there exists ).
It is clear that