On the Incompressible Navier-Stokes Equations with Damping *

We consider dynamics system with damping, which are obtained by some transformations from the system of incompressible Navier-Stokes equations. These have similar properties to original Navier-Stokes equations the scaling invariance. Due to the presence of the damping term, conclusions are different with proving the origin of the incompressible Navier-Stokes equations and get some new conclusions. For one form of dynamics system with damping we prove the existence of solution, and get the existence of the attractors. Moreover, we discuss with limit-behavior the deformations of the Navier-Stokes equation.


Introduction
Concerned with the perturbed Navier-Stokes equations: where is a smooth bounded domain with boundary , and p, u is the velocity vector, is the pressure at x at time t, and  is the kinematic viscosity, and f represents volume forces that are applied to the fluid, and , where 1  is the first eigenvalue of A (see Remark 4).The Equation (1.1) is Navier-Stokes equations, as 0   , which show the existence of absorbing sets and the existence of a maximal attractor, the universal attractor, attractor in unbounded domain (see [1][2][3][4][5]).ACTA Mathematical Application Sinica.In [6], where they some interesting results, as 0   .In [7,8] Babin, Vishik and Abergel consider maximal attractors of semigroups corresponding to evolution differential equations, existence and finite dimensionality of global attractor for evolution equations on unbounded domains.In [9,10] A. Pazy consider Semigroups of linear operator and application to partial differential equation.
We need the following preliminaries: Equations (1.1) are supplemented with a boundary condition.Two cases will be considered: The nonslip boundary condition.The boundary is solid and at rest; thus The space-periodic case.Here     Remark 2. That is u and p take the same values at corresponding points of  .
Furthermore, we assume in this case that the average flow vanishes (1.4) When an initial-value problem is considered we supplement these equations with For the mathematical setting of this problem we consider a Hilbert space H (see [8]) which is a close subspace of and in the periodic case We refer the reader to R. Temam [2] for more details on these spaces and, in particular, a trace theorem showing that the trace of on  exists and belong to The space H is endowed with the scalar product and the norm of denoted by  and   Remark 3. and are the faces in the nonslip case and , in the space-periodic case, ( where , , . We denoted ea by A the lin r unbounded operator in H which is associated with V, H and the scalar product

 
D A can be fully characte by using the regularity theory of linear elliptic systems (see [1,3]).
and in the nonslip and periodic The evolution ical system is described by a family of operators  , 0 S t t  , that map H into itself and enjoy the usual sem perties (see [8]): igroup pro The operator   S t are that for ev uniformly compact for t large.By this we mean ery bounded set X there exists 0 t which may depend on X such that for every bound Of course, if H is Banach space, any family of operators satisfying (1.14) also satisfies (1.15) with 2 0 S  .Theorem 1.2.(see [4]) We assume that H tr is a me ic space and that the operators   S t are given and satisfy (1.12), (1.13) and either (1.14 1.15).We also assume that there exists an open set , is a compact attractor which attracts the f  .It is the maximal bounded attractor in  (for the incl ion relation).Furthermore, if H is a nach space, if U is convex 2 , and the mapping such that Section 2 contains a sketch of existence and uniqueness of solution of the equations; in Section 3 we show the existence of absorbing set and the existence of a maximal attractor; in Section 4 contain the proof of existence and uniqueness of solution of the equations, in Section 5 discussed the perturbation coefficients  .

Th
Navier-Stokes equations due to J.
the Equations e weak from of the Leray [1-3] involves only u, as 0   .It is obtained by multiply (1.1) by a test function V and integrating over  .Using the Green formula (1.1) and the boundary co dition, we find that the term involving p disappears and there remains , , d , whenever the integrals make sense.Actually, the from b is trilinear continuous on and in particular on V. We have the f alities giving various continuity properties of b: .
where is an appropriate constant using the op ive from of (2.1) can be en erator A and the bilinear operator we also set and we easily see that (2.1) is equivalent t the equation while (1.5) can be rewritten We assume that f is in depen dynam dent of t so that the ical system associated with (2.5) is autonomous Theorem 1.3 will be giv

Abs ng orbi Sets and Attractor
similar to the The part proof about global attractor is Temam's book, but the exists of perturbation term is different from the Temam's book, so we reprove it for integrality.
Theorem 1.4.The dynamical system associated with the tow-dimensional modified Navier-Stokes equations, supplemented by boundary (1.2) or (1.3), (1.4) possesses an attractor  that is compact, connected,and maximal in H.  attracts the bounded sets of H and  is also maximal among the functional invariant set bounded in H.
Proof.We first prove the existence of an absorbing set in H.A first energy-type equality is obtained by taking the scalar product of (2.5) with u .Hence We see that We know that where 1  is the first eigenvalue of A .Hence, we can majorize the right-hand side of (3.1) by Hence we obtain Using the classical Gronwall Lemma, we obtain We the infer from (3.3), after integration in t, that


With the use of (3.6) we conclude that d , 0 , and if and 2 3 1 1) Absorbing set in V An continue and show the existence of a set in V.For that purpose we obtain another energy-type eq alar product of (2.5) with and using the second inequality (2.3) , .
We a priori estimate of easily from (3.14) by the cla ing the previous estimates o terested in an estimate valid for large belo ssical Gronwall lem n u.We are more in t.Assuming that f H and that follows ma, us- as in (3.7), we apply the uniform Gronwall lemma to (3.14) with , , g h y replaced by Thanks to 4), (2.18) we (2.1 estimate the quantities in Lemma 1.1 by All the assumption of T em 1.1 are satisfied and we deduce from this theorem the existence of a maximal for modified Navier-Stokes equations. 

Proof of Theorem 1.3
The existence of a solution 2.4) (2.5) that belong to , is first obtain by the Fa ]) m 0  ethod.ap edo-Gakerkin (see [3 proximation procedure with the function j w representing the eigenvalues of A (see Remark 4).For each m we look for an approximate solution m u of the form where is projector in H (or V) on e space spanned .Since A and commute, the relation uivalent to 0 , We prove on

 
, .This is sufficient to pass to the limit in (4.1)-(4.3)and we find (2.4), (2.5) at the limit.For (2.5) we simply observe that (4.7) implies that wea  by multiply (5.4) by a function an ating over Using the second inequality (2.3) and is trilinear continuous: (5.4) where is an appropriate consta Hence These operator enjoy the semigro p properties (1.12) an u d the are continuous from H into itself and even from H into   D A .