ON THE STRONG SOLUTIONS OF THE INHOMOGENEOUS INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN A THIN DOMAIN

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1 ON THE STRONG SOLUTIONS OF THE INHOMOGENEOUS INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN A THIN DOMAIN XIAN LIAO Abstract. In this work we will show the global existence of the strong solutions of the inhomogeneous incompressible Navier-Stokes equations in a thin three dimensional domain = R [, ɛ], with Dirichlet boundary condition on the top and bottom boundary: the global well posedness may hold for large initial data when the vertical size ɛ is sufficiently small. Furthermore, when ɛ the velocity tends to vanish away from the initial time. The analysis relies on the a priori H -estimate for the solutions (similar as in [4, 5, 1] and one pays attention to the dependence of the vertical size ɛ. 1. Introduction This paper is devoted to study the global existence of the strong solutions to the following evolutionary inhomogeneous incompressible Navier-Stokes system t ρ + div (ρu =, t (ρu + div (ρu u u + π =, div u =, on the thin three dimensional domain = R [, ɛ], ɛ (, 1. The above system (1 describes the motion of a viscous density-dependent incompressible fluid: the unknown ρ = ρ(t, x R + represents the density, u = u(t, x R 3 the velocity vector field and π = π(t, x R the pressure term respectively. The time variable t is non-negative and the space variable x = (x h, x 3 belongs to, with the horizontal variable x h = (x 1, x taking value on the whole plane R and the vertical variable x 3 on the thin interval [, ɛ]. We accompany this system with an initial condition ρ t= = ρ R +, u t= = u R 3, ( and Dirichlet boundary condition on the top and bottom boundary u(t, x h, x 3 =, t, if (x h, x 3 Γ := {(x h, x 3 x h R, x 3 = or x 3 = ɛ}. (3 In the present work we will simply assume the following regular initial data away from vacuum ρ [ρ, ρ ], < ρ < ρ < +, u H 1 ( H (, div u =. (4 We will show that the unique solution to the initial boundary value problem (1-(4 will exist globally in time, if we assume further the initial velocity vector field u to depend (precisely on the parameter ɛ in the following (critical way: ɛ 1 u L ( c, (5 with c some small enough positive constant depending only on ρ, ρ. Correspondingly, the global well-posedness result may follow for large initial data when the vertical size ɛ of the considered domain is sufficiently small. The proof consists of a priori H -estimates for the solutions. The mathematical analysis is similar as in Craig-Huang-Wang [5] and the authors there proved the global existence result under Mathematics Subject Classification. Primary 35Q3, Secondary 76D3. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 55 Zhongguancun East Road, 119, Beijing, P.R. China. (1 1

2 X. LIAO a smallness assumption (see Theorem 1. below. See also Paicu-Zhang-Zhang [1] for the global unique solvability under the less regular assumption u H 1 (R 3 : the authors there consider the time weighted norms, e.g. t 1/ u L t (L x. The smallness assumption (5 is quite similar (by virtue of (18 below as the one in [1]: u L (R 3 u L (R 3 η, with η small enough. Nevertheless, we pay much attention here to the dependence of the norms of the velocity on ɛ, and Lemma 1.6 below will be used constantly. Notice that for the homogeneous initial data ρ ρ >, any weak solution of (1 becomes a weak solution of the classical Navier-Stokes equations and the global existence result was well known in Leray [17]. As for our case of the viscous inhomogeneous incompressible flow, there are also rich literatures showing the global (in time existence result of weak solutions of Leray s type. See the books [, 18] and references therein. The analysis mainly relies on the following energy inequality: 1 ( T ρ u dx (T + u dxdt 1 ρ u dx, a.e. T (, +. (6 Let us consider the regularity issue of the above weak solutions. As stated in cf. [, 18], for the two-dimensional inhomogeneous incompressible flow, if we assume higher regularity on the initial velocity field u H 1 (R, then the weak solutions will have higher regularity u L ( [, T ]; H 1 (R L ( [, T ]; H (R, t u L ([, T ]; L (R. Indeed, if we test the momentum equation by t u, then we arrive at the following a priori estimate (by using Gagliardo-Nirenberg s inequality and Young s inequality d dt u L (R + ( tu, u L (R C(ρ, ρ u L (R u 4 L (R. Similarly, further regularity results can be deduced by routine arguments, e.g., one differentiates the velocity equation and then test it by its derivatives of higher order. However, such regularity results are not true for three dimensional domain due to the lack of the boundedness of the nonlinear convective term, and (hence the uniqueness of weak solutions remains an open problem. On the other side, one can show the local-in-time existence and uniqueness of strong solutions for the inhomogeneous incompressible Navier-Stokes equations (1, see cf. [1, 4, 6, 7, 8, 11, 16,, 1]. And in particular for our simple regular case away from vacuum, see Danchin & Mucha [9] (or Choe & Kim [4] 1 for the following local well-posedness result: Theorem 1.1. Suppose the initial data (ρ, u to satisfy the initial condition (4, then there exists a positive time T (, + such that the initial boundary value problem (1-(3 has a unique strong solution (ρ, u, π on the time interval [, T ], with the following property: ρ [ρ, ρ ], u L ( [, T ]; H 1 ( H (, u L ((, T ]; L ( W 1,6 (, (7 u t L ((, T ]; L ( L ([, T ]; H 1 (, π L ((, T ]; L ( L ((, T ]; L 6 (. We mention here that by use of Lagrangian approach, Danchin & Mucha [9] proved notably the uniqueness result of the solution (7, where the rough density case (e.g. piecewise-constant density patches is admissible. If we assume furthermore the smallness condition on the initial velocity field u, then Craig- Huang-Wang [5] showed the following global well-posedness result: 1 Notice that the embedding H 1 ( L 6 ( holds true for our unbounded domain and hence the proof in [4] works. Choe & Kim [4] showed only the existence result of the solution (7 while by view of the uniqueness result in Danchin & Mucha [9] we know that the solution is unique. In the original paper [5] there is smoothness hypothesis on the density, which is assumed only to ensure the uniqueness of the solution. Hence thanks to the uniqueness result in [9], the density could be assumed only to have upper and lower bounds.

3 INHOMOGENEOUS INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN A THIN DOMAIN 3 Theorem 1.. Assume the hypothesis in Theorem 1.1. Then there exists a positive constant η depending on ρ such that if u H 1 ( η, then the unique strong solution constructed in Theorem 1.1 is defined for all positive time. All the above analysis leads us to consider the problem on the thin three dimensional domain = R [, ɛ] (viewed as a perturbation of the plane case, expecting better global existenceuniqueness result when the vertical size ɛ is sufficiently small. Indeed, we will show the a priori estimates for U, V, W defined as follows: by use of their initial data: U(t := u L t L ( + u L t L (, V (t := u L t L ( + ( t u, u L t L (, W (t := ( t u, u, π L t L ( + t u L t L (, U := u L (, V := u L (, W := u L (. Notice that we can apply Poincaré s inequality in the vertical direction (see (15 below on u H 1 (, to arrive at (keeping in mind (5 the smallness assumption on U : Our main result reads as follows: U = u L ( Cɛ 3 u L ( CɛV Cc ɛ 1. Theorem 1.3. Let the initial data (ρ, u satisfy (4 and (5. Then the initial-boundary value problem (1-(3 has a unique global-in-time strong solution (ρ, u, π such that, for any T (, +, ρ(t, x [ρ, ρ ], a.e. t [, T ], x, U(T MU, V (T MV, W (T MW, u L T (L ( MU e T/ɛ, u L T (L ( MV e T/ɛ, u L T (L ( MW e T/ɛ, with the positive constant M depending only on ρ, ρ, c. In particular, we see from (1 that for any τ (, + there holds lim u = uniformly in C([τ, + ]; ɛ H (. Remark 1.4. We give here some remarks about Theorem 1.3: Here we consider the regular data away from vacuum (4 just for the simplicity of the presentation. Indeed, the result can be generalized to the case including the vacuum ρ by following the same lines as in e.g. [4, 5], although the smoothness assumption on the density should be added for the uniqueness consideration. We can also consider the less regular case u H 1 as in [1], by use of the time weighted norms e.g. t 1 u L t (L x. Under the Dirichlet boundary condition we see in Theorem 1.3 that the velocity tends to vanish away from the initial time when the vertical size ɛ. While under the periodic boundary condition on the thin domain = [, 1] [, 1] [, ɛ], we expect that the vertical average of the solution (ρ, u (see the definition (11 below converges to the solution of the two dimensional inhomogeneous Navier-Stokes equations. (8 (9 (1

4 4 X. LIAO This is not the issue of the present work through. We mention [4] for the results pertaining to various boundary condition assumptions for the homogeneous Navier-Stokes equations. There are many works making study of the homogeneous Navier-Stokes equations (i.e. Equation (1 with ρ ρ > in a thin three dimensional domain. The pioneer works by Raugel & Sell [, 3] proved that the global existence of strong solutions holds for a large class of initial data when the vertical size ɛ is small. Then progresses have been made, see cf. [3, 1, 1, 13, 14, 15, 19, 4]: for the Dirichlet boundary condition case, the global existence may hold under the critical initial assumption ɛ 1 u L ( η (see [4] or u H 1 ( η (see [13]. As for the periodic boundary condition, there are also similar results, see [1, 14, 15, 19]. However, as far as we know, there isn t any global existence result as above for the densitydependent incompressible Navier-Stokes equations (1. We will consider here this problem in the unbounded thin domain = R [, ɛ], which can be viewed to be a perturbation of the plane case. This is a singular perturbation problem: as pointed out in [], the dilated solution R(t; x h, x 3 = ρ(t; x h, ɛx 3, U(t; x h, x 3 := u(t; x h, ɛx 3, t, x h R, x 3 [, 1], satisfies the following dilated equation (1 ɛ : t R + ɛ (RU =, t (RU + ɛ (RU U ɛ U + ɛ π =, ɛ u =, with ɛ := ( 1,, ɛ 1 3 and ɛ := ( 11,, ɛ 33. The terms ɛ 1 3, ɛ 33 will be highly oscillating when ɛ is small. The proof hence consists of deriving precise a priori estimates for the velocity field u with respect to ɛ, see Section. Before going to the proof part, we finish this introduction part by some basic inequalities for the functions defined in. We start with the estimates for general functions f defined on. Following the original idea in Raugel & Sell [, 3], for any function f defined on the thin domain, we can decompose it into two parts, namely f = f + f, f := 1 ɛ ɛ f(x 1, x, x 3 dx 3, f := f f. (11 Notice that the average part f = f(x 1, x is defined on the two dimensional space R while the remainder part f has zero average on the x 3 direction. We can consider these two parts seperately: we apply Gagliardo-Nirenberg s inequalities (see (13 below on the main part f while bound the perturbation part f precisely by use of ɛ (see (14 below. Finally, following the same lines as in Section.1, Temam & Ziane [4], one obtains the following basic lemma: Lemma 1.5. One has the following basic inequalities concerning the three functions f, f, f: It follows immediately the relations between f and f, f: f L (R ɛ 1 f L (, f L ( ɛ 1 f L (R + f L ( f L (, (1 f L ( ɛ 1 f L (R + f L (; One has the following Gagliardo-Nirenberg s inequality for f on the whole plane R : f L 4 (R C f 1 L (R f 1 L (R ; f L6 (R C f 3 L4 (R f 1 3 L (R C f 1 3 L (R f 3 L (R. (13 Some L p -norms of f can be easily controlled by its derivatives in the following way: f L ( ɛ 3 f L (, f L 6 ( C f L (. (14

5 INHOMOGENEOUS INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN A THIN DOMAIN 5 Proof. Let us just prove (14, following the analysis in [14, 4] for bounded domain case. Since f has zero average on the vertical direction, we can use Poincaré s inequality with respect to this thin direction x 3 [, ɛ] to get ( ɛ f (x h, x 3 dx 3 1 ɛ ( ɛ 3 f (x h, x 3 dx 3 1, x h R. (15 We then take the square of the above inequality and integrate it with respect to the horizontal variable x h R. Thus (14 1 follows immediately. In order to get the second inequality (14, we need the Gagliardo-Nirenberg inequality (13 on the whole plane R. One therefore has ɛ ɛ ( ( f 6 (x h, x 3 dx h dx 3 C f 4 dx h h f dx h dx 3 R R R ( ( ɛ C max f 4 dx h h f dx h dx 3. (16 x 3 [,ɛ] R R One has to control the first member on the righthand side above. Since for any function u H 1 ([, ɛ], one has the following equality by integration by parts ɛ u 4 (ζ + ɛ s ds = ɛu 4 (ζ + ɛ 4 s u 3 (ζ + ɛ s u (ζ + ɛ s ds, ζ [, ɛ], we can bound u 4 as follows: u 4 1 ɛ ( u 4 + 4ɛ u 3 u dx 3 ɛ ( ɛ 1 ( ɛ C u 6 ( 1 dx 3 ɛ u + u 1 dx 3. (17 Correspondingly, one applies the above inequality on f and integrates it with respect to x h R, to arrives at the following for f: f 4 dx h max x 3 [,ɛ] R ( ɛ 1 C f ( ɛ 6 ( 1 dx h dx 3 R R ( ɛ 1 C f 6 dx h dx 3 3 f L, by (14 1. R Recalling (16, the inequality (14 follows. ɛ f + 3 f dx h dx 3 1, by Hölder s inequality, Notice in our case that the solution u L ([, T ]; H 1 ( of (1-(3 itself satisfies Poincaré s inequality in the vertical direction (15: u(t; x h, L ([,ɛ] Cɛ ( 3 u(t; x h, L ([,ɛ], for a.e. t [, T ], x h R. Similarly, the Poincaré s inequality (15 also holds for u, t u, a.e. t [, T ], x h R. Indeed, on one side, it is easy to see that i u j, t u L ([, T ]; H 1 (, i = 1,, j = 1,, 3. On the other side, the quantities 3 u j, j = 1,, 3 have zero average in the vertical direction. We then can follow the same lines of the proof for (14 to get the following lemma, which will be used constantly in Section : Lemma 1.6. Let the initial velocity u H 1 ( H ( satisfy (5. Then U V = u L ( u L ( C(ɛ 3 u L ( u L ( Cc. (18 Let (ρ, u, π be the strong solution to (1-(3 satisfying (7, defined on the time interval [, T ], T (,. Then for almost every t [, T ], we have the a priori estimate (14 for u, u, t u.

6 6 X. LIAO Remark 1.7. As we assume the Dirichlet boundary condition, we can consider the norms of the velocity u directly by use of Lemma 1.6, without resorting to the norms of u and ũ separately. If we assume other boundary conditions, say, periodic boundary condition, then we can use the Raugel-Sell decomposition to consider the two dimensional part u and the perturbed part ũ separately. Finally we can get the estimates for u as follows: u L 4 ( ɛ 1 4 u L 4 (R + ũ L 4 ( Cɛ 1 4 u 1 L ( u 1 L ( + C u 1 4 L ( u 3 4 L (, (19 u L 6 ( Cɛ 1 3 u 1 3 L ( u 3 L ( + C u L (. In the following context, the L p -norm will always denote the L p -norm on the domain where the functions are defined, and A B always means A C B for some universal positive constant C (may depend on ρ, ρ, unless otherwise specified.. Proof This section is devoted to the proof of Theorem 1.3. It consists of two a priori estimates (see Propositions.1 and. below, for the strong solution (ρ, u, π of the initial boundary value problem (1-(3, defined on [, T ], T (, +. The proof is similar as that in [4, 5, 1], except that we will take use of Lemma 1.6 constantly which pays attention on the precise dependence of the vertical size ɛ..1. H 1 Estimate. In this subsection, we will prove the following fact: Proposition.1. Assume the initial data to satisfy (5. There exists a positive constant C 1 (depending on ρ, ρ, c such that for any strong solution (ρ, u, π (satisfying (7 defined on [, T ], one has ρ(t, x [ρ, ρ ], a.e. t [, T ], x, U(t C 1 U, u L t (L ( C 1 U e t/ɛ, t [, T ], V (t C 1 V, u L t (L ( C 1 V e t/ɛ, t [, T ], with U(t, V (t and U, V defined in (8 and (9 respectively. Proof. First of all, thanks to the incompressibility of the flow, the evolutionary density retains the same lower and upper bounds as the initial data ρ(t, x [ρ, ρ ]. One has the energy equality for the strong solution (ρ, u: 1 d ρ u + u =. (1 dt Indeed, we follow the classical argument: we test the velocity equation (1 by u itself to get ρ t u u + ρu u u u u + π u = ; we integrates by parts to arrive at ρ t u u = ρ 1 t u = 1 d ρ u 1 t ρ u, dt ρu u u = 1 ρu u = 1 div (ρu u, u u = u, π u = ; (

7 INHOMOGENEOUS INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN A THIN DOMAIN 7 finally the mass conservation law (1 1 entails the energy equality. Thus it follows U(T = u L ([,T ];L ( + u L ([,T ];L ( C(ρ, ρ u L (. ( On the other side, by Lemma 1.6 we know that from which and (1 we infer that Hence we derive that u L ( x3 u L ( C 1 ɛ 1 u L (, 1 d dt ρu L ( + C 1 ɛ u L (. (3 t u(t L ( C u L ( C 1 ɛ u L ( dt. We multiply both sides by large enough constant C 1 and Gronwall s inequality then implies (. Notice that for the Stokes operator one has the classic estimate u + π = ρ t u ρu u, ( u, π L t u L + u u L. (4 Following the lines to get (1, taking the L (-scalar product between the equation of u and t u entails the following quasi-energy inequality for u: d u + t u + u u u. (5 dt By virtue of Lemma 1.6, we bound the righthand side as follows: u u u L 6 ( u L 6 ( u L ( Young s inequality hence entails (by choosing δ small enough u 3 L ( u L (. u 3 L ( u L ( C δ u 6 L ( + δ u L (, t t u(t L ( + ( t u, u L ( u L ( + C u 6 L (. Recalling the definition (8 for V (t and keeping in mind (, one has ( ( V (t C u L ( + u L t L ( u 4 L t L ( C V + U V 4. (6 Hence, if c is small enough, then by virtue of (18 it follows that (by bootstrap argument V (t C 1 V, for some C 1 depending on ρ, ρ, c. Finally, similar as to get (3, we get from (5 that d dt u L ( + ɛ u L ( C u 6 L (,

8 8 X. LIAO from which and the estimates for U(t, V (t we infer t u(t L ( u L ( ɛ u L ( dt + C u 4 L t (L u L t (L t V ɛ u L ( dt + CU (tv 4 (t t V ɛ u L ( dt + C(U V V. Therefore noticing the third term on the righthand side bounded by CV by (18, one deduces ( 3 by Gronwall s inequality... H Estimate. In this subsection we will consider the H estimate for the velocity field u. We will firstly get the energy estimate for its time derivative t u and then the bound for u L T L ( follows. ( In order to consider the initial data lim t + t u (t, let us test the momentum equation (1 by the divergence free vector field t u L ([, T ]; H 1 ( to obtain ρ t u = ( ρu u + u t u. Thus one has the following bound by use of Lemma 1.6 ( lim ρ t u (t C lim ( u t + t + u + u dx ( C lim u(t L u(t L 6 u(t t L + 6 u(t L + C( u L u L u L + u L C ( U V u L + u L C u L. (7 And we are going to prove the following H -Estimate: Proposition.. Assume the same hypothesis as in Proposition.1. Then there exists a positive constants C (depending on ρ, ρ, c such that W (t C W, ( t u, u, π L t (L ( C W e t/ɛ, t [, T ], (8 with W (t and W defined in (8 and (9 respectively. Proof. In this proof, we will denote t u by w for notational simplicity. Let us take the derivative t on the equation for u to get the evolutionary equation for w = t u: { ρ t w + ρu w w + t π = t ρ w t ρ u u ρw u, (9 div w =. Testing the equation for w by itself entails 1 d ( dt ρ w L + w L = t ρ w + ρw u w := I 1 (t + I (t. t ρ u u w Thanks to the density equation t ρ = div (ρu,

9 INHOMOGENEOUS INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN A THIN DOMAIN 9 by integration by parts, we can rewrite I i, i = 1, separately: I 1 = ρu w w + ρw u w, I = ρu u u w + ρu u : ( u w + ρu u : ( u ( w T. By use of Lemma 1.6, one has w L 6 w L. Noticing that it is hence easy to bound I 1 as follows: w L 3 w 1 L w 1 L 6 w 1 L w 1 L, I 1 w L u L 6 w L 3 + u L w L 6 w L 3 u L w 1 L w 3 L δ w L + C δ u 4 L w L. Similarly, by use of Lemma 1.6 on u, u, one has the estimate for I : I u L 6 u L 6 u L w L 6 + u L 6 u L w L 6 + w L u L 6 u L 6 δ w L + C δ u 4 L u L. To conclude, by choosing δ sufficiently small and by use of Gronwall s inequality, one finally has the following energy inequality for w (keeping in mind (7: t t t w L + w L {C C exp u 4 L }(lim t w(t L + u 4 L u L } C exp {C u L u t L L t L ( u L + u L u t L L t L u L. t L By virtue of (18 and Proposition.1, one has w L t (L + w L t (L C exp{cu V }( W + U V W (t C(ρ, ρ, c (W + c 4 W (t. On the other side, by virtue of (4, it rests to control u u L t (L ( so as to bound ( u, π L t (L (. Similar as the above analysis, one has u u L t L ( u L t L 3 ( u L t L 6 ( C u 1 L t L ( u 1 L t L ( u L t L ( C(U V 1 u L t L (. Thanks to (18: U V Cc with c small enough, (4 entails then from which and (3 we infer (3 ( u, π L t L ( C w L t L (, (31 W (t C(W + c 4 W (t. Hence the smallness of c entails W (t C W. Finally, since 3 w L ( C 1 ɛ 1 w L (, we deduce from (3 and Gronwall s inequality that w L t (L C W e t/ɛ. Hence by virtue of (31, (8 follows. To conclude, Theorem 1.3 follows from Theorem 1.1 and Propositions.1 and..

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