Long time existence of regular solutions to non-homogeneous Navier-Stokes equations

We consider the motion of incompressible viscous non-homogeneous fluid described by the Navier-Stokes equations in a bounded cylinder under boundary slip conditions. Assume that the third co-ordinate axis is the axis of the cylinder. Assuming that the derivatives of density, velocity, external force with respect to the third co-ordinate are sufficiently small in some norms we prove large time regular solutions without any restriction on the existence time. The proof is divided into two parts. First an a priori estimate is shown. Next the existence follows from the Leray-Schauder fixed point theorem.

Then there exists a unique solution to problem (1.1) such that v ∈ H 2+s,1+s/2 (Ω T ), v ,x 3 ∈ H 2,1 (Ω T ), ∇p ∈ H s,s/2 (Ω T ), ∇p ,x 3 ∈ L 2 (Ω T ) (1.5) v H 2+s,1+s/2 (Ω T ) + v ,x 3 H 2,1 (Ω T ) + ∇p H s,s/2 (Ω T ) where ϕ is an increasing positive function and N = f H s,s/2 (Ω T ) + f ,x 3 L 2 (Ω T ) + f L 2 (0,T ;W 1 6/5 (Ω)) The result formulated in Theorem A describes a long time existence of solutions to problem (1.1) because the smallness condition (1.4) contains at most time integral norms of f . The aim of this paper is to prove long time existence of regular solutions to problem (1.1) such that there is no restriction on the magnitudes of the external force, the initial velocity and the density. The aim is covered by the smallness restriction (1.4) because it contains derivatives of the initial density and derivatives with respect to x 3 of the initial velocity and the external force. This kind of restrictions suggests that our solution remains close to two-dimensional solutions of incompressible Navier-Stokes equations because the initial density is close to a constant but the initial velocity and the external force change a little in the x 3 -direction. In view of the result on long time existence of solutions to two-dimensional incompressible nonhomogeneous Navier-Stokes equations (see [AKM, Ch. 3]) we could expect that smallness of ̺ 0,x can be replaced by smallness of ̺ 0,x 3 only. However, up to now, we do not know how to do it.
One could expect that looking for solutions close to two-dimensional solutions is nothing to do comparing with [AKM,Ch. 3]. But it is totally not true because we need three-dimensional imbeddings, solvability of three-dimensional problems and apply the three-dimensional technique of Sobolev and Sobolev-Slobodetski spaces. Moreover, we have to mention that many techniques used in this paper were developed in [Z2,Z3,Z4,RZ] in the case of a constant density. The next step in our considerations will be a global existence result which can be proved by extending [Z4,NZ1] to the nonhomogeneous fluids.
Finally we expect an existence of global attractor by applying the technique of [NZ2].
Many results on existence and estimates of weak solutions to nonhomogeneous incompressible Navier-Stokes equations can be found in [P].

Notation
We use isotropic and anisotropic Lebesgue spaces ∞]; isotropic and anisotropic Sobolev spaces with the norms In the case p = 2 we use the notation . Next we introduce a space natural for examining weak solutions to the Navier-Stokes and parabolic equations In the case of noneven s spaces W s p (Q) and W s,s/2 p (Q T ) are defined as sets of functions with the finite norms, respectively, In the case where either s = [s] or s 2 = s 2 the corresponding fractional derivatives vanish. For Q = S the above norms are defined by applying a partition of unity.
Theorems of imbedding and interpolation for above spaces can be found in [BIN]. By • C α (Ω T ) we denote a space of functions with the finite seminorm where α ∈ (0, 1). By c we denote a generic constant which changes its value from formula to formula. In general c depends on constants of imbeddings, on functions describing the boundary, but it does not depend on data. By ϕ we denote a generic function which depends on data, changes its form from formula to formula and is always positive increasing function of its arguments. The dependence of ϕ on data will be always expressed explicitly.
To simplify presentation we use the notation Let us consider the Stokes system in Ω.

Auxiliary results
This section is devoted to obtain some a priori estimates for solutions to problem (1.1). Therefore, we assume existence of such solutions to (1.1) that the derived estimates can be satisfied.
Hence (3.6) holds. For p < 1, p = 0 we assume additionally that ̺, ̺ 0 are different from zero. Hence, performing the same considerations as in the case p ≥ 1, we obtain the same equality as above. Finally, (3.6) also holds for p < 1, p = 0. This concludes the proof.
To prove the existence with large data we follow the ideas developed in [RZ,Z2,Z4]. therefore we introduce the quantities Lemma 3.6. Let v, ̺ be given. Then (h, q) is a solution to the problem in Ω.
To formulate problem for χ we introduce Lemma 3.7. Let ̺, v, h be given. Then χ is a solution to the problem in Ω, where the summation convention over the repeated indices is assumed.
Proof. (3.19) 1 follows from applying two-dimensional rot operator to the first two equations of (1.1) 1 . The boundary condition (3.19) 2 was proved in [Z2]. This ends the proof.
To apply the energy type method to problem ( in Ω.
Then the function χ ′ = χ −χ satisfies Let us consider problem (3.5). Then we have Then the following a priori estimate is valid Proof. For solutions to problem (3.5) we have for any p > 1.
Integrating with respect to time yields Let us consider the expression for s > 5 2 , we obtain from (3.25)-(3.27) estimate (3.23). This concludes the proof.
Lemma 3.10. Let ̺ and v be given and sufficiently regular. Assume also that vectorsn,τ α , α = 1, 2, are defined in a neighbourhood of S and a αβ , where the last equation was added to have uniqueness of solutions to (3.29).
Integrating with respect to time is the L 2 -norm and using the Hölder inequality we obtain (3.41) Using (3.10) we get (3.42). This concludes the proof.

Estimates
First we obtain an estimate for solutions to problem (3.19).
Using he above estimates in (4.7), assuming that ε is sufficiently small, using Lemma 3.5 and integrating with respect to time we obtain (4.5) in the case (4.8) and (4.6) for (4.9). Let us mention that the time integral of the first term in I is deleted. This concludes the proof.
Since ̺ x , ̺ t ∈ L ∞ (Ω T ) and since we are interested in the case s < 1 we have To estimate the last term in (4.33) it is sufficient to examine the highest order terms. First we use the splitting (4.34) v · ∇v H s,s/2 (Ω ) = v · ∇v L 2 (0,T ;H s (Ω)) + v · ∇v L 2 (Ω;H s/2 (0,T )) .
It is sufficient to examine only one norm. Therefore, we consider Hence we examine only the first two norms. By the Hölder inequality we have where κ 1 = 3 2 +s 2+s < 1 and there is no restrictions on s ∈ (1/2, 1). Similarly, we have Finally, we examine the term with pressure. We have (4.38) p H s,s/2 (Ω T ) = p L 2 (Ω;H s/2 (0,T )) + p L 2 (0,T ;H s (Ω)) .
In view of the above considerations we express (4.42) in the form where (4.45) q ≤ 3 3 2 − s .
Remark 4.5. In formulas (3.25), (3.26), (4.41) and (4.44) we have the expression In view of the imbedding v L 2 (0,T ;W 1 ∞ (Ω)) ≤ cV s (T ), s > 1 2 we obtain the estimate which is not convenient because factor t 1/2 appears under the exponent functions. The difficulty can be cancelled in virtue of the assumption The above inequality is not restrictive because the case v W 1 ∞ (Ω) ≤ 1 implies in view of Lemma 3.5 the following estimate for solutions to prob- Hence regularity H s+2,s/2+1 (Ω T ) follows immediately. In this case the constant in the above estimate depends on time but this does not imply any restrictions on magnitudes of the data. Now we pass to problem (3.17). Then we have Lemma 4.6. Assume that 1 2 < σ ≤ s < 1 and σ can be chosen as very close to s. Let us take Remark 4.5 under account and let (4.54) B = g L 2 (0,T ;L 6/5 (Ω)) + f 3 L 2 (0,T ;L 4/3 (S 2 )) , v ∈ H 2+σ,1+σ/2 (Ω T ) and let (v, ̺) be the weak solution to (1.1) described by Lemma 3.5. Then solutions to (3.17) satisfy the inequality where X 1 is introduced in (3.22) and (4.56) Proof. For solutions to (3.17) we have (4.57) We need the inequalities In view of (4.6) we have Employing ( Then for sufficiently small X the estimate holds (4.69) V s (T ) ≤ ϕ(̺ * , ̺ * , G, K, K 0 , X, L), where K 0 is introduced in (4.56), K in(4.31), G in (4.18) and (4.70) L = f 3 L 2 (0,T ;H 1/2 (S)) + f H s,s/2 (Ω T ) .
Using (4.55) with σ = s in (4.71) and applying again a fixed point argument for sufficiently small X we obtain (4.69). This concludes the proof.