On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions

Transcription

1 Nečas Center for Mathematical Modeling On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions Donatella Donatelli and Eduard Feireisl and Antonín Novotný Preprint no. 9-3 Research Team Mathematical Institute of the Academy of Sciences of the Czech Republic Žitná 5, Praha 1

2 On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions Donatella Donatelli Eduard Feireisl Antonín Novotný Dipartimento di Matematica Pure ed Applicata Università degli Studi dell Aquila, 67 1 L Aquila, Italy Institute of Mathematics of the Academy of Sciences of the Czech Republic Žitná 5, Praha 1, Czech Republic Université du Sud Toulon-Var, BP 13, La Garde, France 1 Introduction A proper choice of boundary conditions plays an important role in fluid mechanics. This issue have been subjected to discussion for over two centuries by many distinguished scientists who developed the foundations of fluid mechanics, including Bernoulli, Coulomb, Navier, Couette, Poisson, Stokes, to name only a few. Consider a viscous fluid confined to a domain in the Euclidean physical space R 3, the boundary of which represents a solid wall. Under the hypothesis of perfect impermeability of the wall we have u n = on, 1.1 where u is the fluid velocity and the symbol n denotes the outer normal vector to. The commonly accepted hypothesis asserts that there is no relative motion between a viscous fluid and the solid wall, meaning, [u] τ = on, 1. where [u] τ stands for the tangential component of u, provided the wall is at rest. The so-called no-slip boundary condition 1.1, 1. turned out to be extremely successful in reproducing the velocity profiles for macroscopic flows, in particular for incompressible fluids liquids. On the other hand, Navier suggested to replace 1. by a slip hypothesis β[u] τ + [Sn] τ = on, 1.3 The work of D.D. was supported by the general research programme of the Academy of Sciences of the Czech Republic, Institutional Research Plan AVZ The work of E.F. was supported by Grant 1/8/315 of GA ČR as a part of the general research programme of the Academy of Sciences of the Czech Republic, Institutional Research Plan AVZ The work of A.N. was partially supported by the Nečas Center for Mathematical Modeling LC65 financed by MŠMT 1

3 where S is the deviatoric viscous stress tensor and β is a friction coefficient. Note that, at least formally, condition 1.3 becomes 1. provided β. In the presence of slip, the fluid motion is opposed by a force proportional to the relative velocity between the fluid and the solid wall. Hypothesis 1.3 may be viewed as a convenient alternative to 1. whenever the rate of flow is sufficiently strong turbulent regimes and the medium is a compressible gas of low viscosity, as in meteorological models see Priezjev nad Troian [1] for relevant discussion. Our aim is to study the low Mach number limit for the Navier-Stokes system describing the motion of a compressible viscous fluid subjected to the slip boundary conditions 1.3, with the friction coefficient β inversely proportional to a certain power of the Mach number. Motivated by possible applications in meteorology, we consider the problem on an unbounded exterior spatial domain R 3. The paper can be viewed as a counterpart of [7], where a similar problem was studied on a bounded domain with a regular boundary although the methods used are of fairly different nature. Ignoring the effect of external forces, we consider a scaled Navier-Stokes system of equations t ϱ + div x ϱu =, 1.4 t ϱu + div x ϱu u + 1 xpϱ = div x S x u, 1.5 where ϱ = ϱt, x is the density, u = ut, x the velocity, p = pϱ the pressure, and S denotes the viscous stress tensor determined through the standard Newton s rheological law S x u = µ x u + t xu 3 div xui + ηdiv x ui, with µ >, η. 1.6 The fluid velocity u satisfies Navier s slip boundary condition together with the boundary conditions at infinity u n =, [u] τ + α [Sn] τ = on, α >, 1.7 ux, ϱx ϱ as x, 1.8 where ϱ > is a given constant. Finally, we consider the so-called ill-prepared initial data in the form ϱ, = ϱ, = ϱ + ϱ 1,, u, = u,, 1.9 where the functions ϱ 1, are bounded uniformly for. System contains several terms depending on a small parameter that may be viewed as a result of a suitable scaling of the original physical variables. Our goal is to study the behavior of solutions to problem for. From the physical viewpoint, the situation corresponds to the singular limit of the Navier-Stokes system, where the Mach number as well as the slip coefficient in Navier s boundary condition are proportional to. Loosely speaking, the fluid is almost incompressible and driven to the rest at the boundary cf. Klein et al. [1]. The pressure p = pϱ satisfies p C 1 [, C,, p =, p ϱ > for all ϱ >, 1.1

4 and p ϱ lim ϱ ϱ γ 1 = p > for a certain γ In view of the existence theory developed by Lions [16] and in [9], problem admits a weak solution {ϱ, u } for any fixed > provided γ > 3/. The necessary modifications to accommodate the exterior domains and the conditions at infinity 1.8 are available in [, Theorem 7.15]. The asymptotic limit for in a bounded periodic slab was studied in [7]. It was shown that the limit problem can be identified with the standard incompressible Navier-Stokes system, more specifically, where ϱ ϱ in L, T ; L γ, u U weakly in L, T ; W 1, ; R 3, 1.1 div x U =, 1.13 t ϱu + div x ϱu U + x Π = div x S x U, 1.14 and the velocity field U satisfies the no-slip boundary condition U = Note that the slip boundary condition 1.7 describes an acoustically hard boundary. Accordingly, if R 3 is a bounded domain, the gradient part of the velocity field associated to acoustic waves may exhibit fast oscillations in time, meaning the convergence of {u } > claimed in 1.1 is really weak with respect to the time variable. However, if α 1/, meaning if the friction coefficient in 1.7 is large enough, a boundary layer may appear due to the presence of viscosity and specific geometrical properties of the boundary. As a result, we obtain a strong decay of the acoustic waves, specifically, u U strongly in L, T ; R cf. Desjardins et al. [3], and [7]. The main objective of the present paper is to show a similar result, in particular the strong convergence of the velocity field claimed in 1.16, at least locally over the compact subsets of, in the case when the underlying spatial domain is unbounded, with a compact boundary. In sharp contrast with [7], such a result may be viewed as a consequence of the local energy decay of acoustic waves in the asymptotic limit due to the dispersive phenomena. Note that the local decay of the acoustic energy expressed in terms of the Strichartz estimates was exploited by Desjardins and Grenier [] who showed strong convergence of the velocity field in the low Mach number limit for = R 3. For a general exterior domain, however, the Strichartz estimates become much more delicate and require severe restrictions to be imposed on the shape of see Metcalfe [19]. In this paper, we use a new approach, developed in [6], which is based on weighted space-time estimates for abstract wave equations due to Kato [11]. We essentially exploit the fact that relation 1.1 provides compactness of the velocity field with respect to the space variable. Accordingly, it is sufficient to apply Kato s result only to a fixed range of frequencies of the acoustic waves. In such a way, the problem may be reduced to verification of the so-called limiting absorption principle for the wave operator that is relatively easy to establish for a large class of spatial domains. 3

5 The paper is organized as follows. In Section, we introduce the concept of weak solution for both the primitive and the target system and state our main result. Uniform bounds independent of are derived in Section 3. The heart of the paper is the analysis of acoustic waves, the motion of which is governed by Lighthill s equation derived in Section 4. In Section 4.1, we show compactness of the solenoidal part of the velocity field. Finally, a detailed analysis of propagation of acoustic waves and the proof of the main result are given in Section 5. Main result Let R 3 be an unbounded exterior domain with a compact regular boundary. We shall say that ϱ, u is a weak solution to the Navier-Stokes system if ϱ, ϱ ϱ L, T ; L γ + L u L, T ; W 1, ; R 3, u n =,.1 and the following holds : ϱ u L, T ; L 1, equation of continuity 1.4 holds in the sense of renormalized solutions introduced by DiPerna and Lions [5], specifically, bϱ t ϕ + bϱu x ϕ + bϱ b ϱϱ div x u ϕ dx dt = bϱ, ϕ, dx. for any b BC[,, b zz BC[,, and any ϕ C c [, ; equation 1.5, together with the boundary conditions 1.7, are satisfied in the sense of distributions, more precisely, the integral identity ϱu t ϕ + ϱu u : x ϕ + 1 pϱdiv xϕ dx dt.3 = S x u : x ϕ dx dt 1 α holds for any ϕ C c [, T ; R 3, ϕ n = ; u ϕ ds x dt ϱ, u, ϕ dx the energy inequality t ϱ u + P ϱ ϱ P ϱϱ ϱ P ϱ t, dx + S x u : x u dx dt.4 + α u ds x dt E, ϱ, u, + P ϱ, ϱ P ϱϱ, ϱ P ϱ dx holds for a.a. t, T, where we have set P ϱ = ϱ ϱ 1 pz z dz..5 4

6 Similarly, we introduce a concept of weak solution of the target system : U L, T ; L, R 3 L, T ; W 1,, R 3 ;.6 div x U = for a.a. t, x, T ;.7 the integral identity ϱu t ϕ + ϱu U : x ϕ dx dt = is satisfied for any test function ϕ C c [, T ; R 3, div x ϕ =. S x U : x ϕ dx dt ϱu ϕ, dx.8 Note that the integral identity.8 implicitly includes the satisfaction of the initial condition U, = H[U ] where H denotes the standard Helmholtz projection onto the space of solenoidal divergenceless functions, namely v = H[v] + H [v], H = x Ψ.9 where Ψ D 1, is the unique solution of the problem x Ψ x ϕ dx = v x ϕ dx for all ϕ D 1,..1 Here, the symbol D 1. denotes the completion of C c with respect to the norm x L ;R 3. Our main result reads as follows. Theorem.1 Let R 3 be an unbounded domain with a compact boundary of class C. Assume that the pressure p satisfies 1.1, 1.11, with γ > 3/. Let ϱ, u be a weak solution to the Navier-Stokes system. -.4 emanating from the initial data 1.9, where < α < 1,.11 and {ϱ 1, } > is bounded in L 1 L, {u, } > is bounded in L L ; R 3..1 Then, passing to subsequences if necessary, we have and ess sup t,t u, U weakly in L ; R 3, ϱ ϱt, L +L q c, q = min{, γ}, u U weakly in L, T ; W 1, ; R 3, u U strongly in L, T K; R 3 for any compact K, where U L, T ; W 1, ; R 3 L, T ; L ; R 3 is a weak solution of problem.6 -.8, with the initial datum U, = H[U ]. 5

7 The rest of the paper is devoted to the proof of Theorem.1. As already pointed out, the principal issue is to establish strong a.a. pointwise convergence of the velocity fields. Note that weak convergence of {u } > can be shown by means of the local method developed by Lions and Masmoudi [17], [18]. 3 Uniform bounds Proof of Theorem.1 leans on uniform bounds for {ϱ } >, {u } >. Following [8] we introduce the notation: h = [h ] ess + [h ] res, where χ is a fixed function such that [h] ess = χϱ h, [h] res = 1 χϱ h χ C c,, χ 1, χr = 1 for all r [ϱ/, ϱ]. 3.1 Energy estimates It follows from hypotheses 1.1, 1.11 that the function is strictly convex in [,, ϱ P ϱ ϱ P ϱϱ ϱ P ϱ P ϱ ϱ = 1 pϱ ϱ ϱ, 3.1 attaining its global minimum on [, at ϱ. Consequently, by virtue of hypotheses 1.9,.1, the expression on the right-hand side of the energy inequality.4 divided on remains bounded uniformly for. Thus we may infer that and ess ess sup t,t sup t,t ess sup t,t As P is strictly convex, relation 3.5 yields [ ] ess sup ϱ ϱ ϱ u dx c, 3. S x u : x u dxdt c, 3.3 u ds x dt α c, 3.4 [ ] P ϱ ϱ P ϱϱ ϱ P ϱ dx c, 3.5 ess [ ] P ϱ ϱ P ϱϱ ϱ P ϱ dx c. 3.6 res t,t 6 ess L c, 3.7

8 while, in accordance with 1.11, 3.1, ess sup t,t [1 + ϱ γ ] res t, dx c. 3.8 Relation 3.3 may be used to deduce uniform estimates on the velocity gradient. To this end, we need the following version of Korn s inequality: Proposition 3.1 [see [8, Theorem 1.17]] Let B be a bounded Lipschitz domain. Assume that V B such that V V >. Then v W 1, B cv, B [ x v + txv 3 div xvi LB;R3 3 + V ] 1/ v dx for any v W 1, B; R 3. As a direct consequence of 3., we get which, combined 3.3, 3.8, and Proposition 3.1, yields Finally, it follows from 3., 3.8 that {[ϱ u ] ess } > bounded in L, T ; L, 3.9 {u } > is bounded in L, T ; W 1, ; R [ϱ u ] res in L, T ; L s as for any 1 s γ/γ Lighthill s acoustic equation Motivated by Lighthill [14], [15], we rewrite the original Navier-Stokes system in the form t ϱ ϱ + div x ϱ u =, 4.1 ϱ ϱ t ϱ u + p x 4. = div x S x u div x ϱ u u 1 x pϱ p ϱ ϱ pϱ, p = ϱ pϱ >, supplemented with the boundary conditions 1.7, 1.8. Equations 4.1, 4., together with 1.7, are understood in a weak sense, specifically, the integral identity r t ϕ + p V x ϕ dx dt = 4.3 7

9 holds for any ϕ Cc, T, while V t ϕ + r div x ϕ dx dt = F 1 : x ϕ dx dt F ϕ ds x dt 4.4 for any ϕ C c, T ; R 3, ϕ n =, where we have set and Writing we use 3.7, 3.8 to obtain { [ ]} ϱ ϱ r p > ϱ ϱ r = p, V = ϱ u, F 1 = S x u ϱ u u 1 pϱ p ϱ ϱ pϱ I, F = 1 α u. [ ] [ ] ϱ ϱ ϱ ϱ r = p + p, V = [ϱ u ] ess + [ϱ u ] res ess res is bounded in L, T ; L + L q, q = min{γ; }, 4.5 and, by virtue of 3.9, 3.11, { } V [ϱ u ] is bounded in L, T ; L + L s ; R 3 for any s [1, γ/γ + 1]. 4.6 with and > Moreover, combining 3., 3.7, 3.8, and 3.1, we get Finally, in accordance with 3.4, F 1 = F 1,1 + F 1,, {F 1,1 } > bounded in L, T ; L 1 ; R 3 3, 4.7 {F 1, } > bounded in L, T ; L ; R F t, L ;R 3 dt c α

10 4.1 Compactness of the solenoidal component of the velocity We conclude this section by a simple observation that the solenoidal part of the vector field V is weakly compact in time. Indeed relations imply that ϱ ϱ in L, T ; L + L γ, 4.1 and u U weakly in L, T ; W 1,, 4.11 V [ϱ u ] V weakly-* in L, T ; L + L γ/γ+1 ; R 3, with V L, T ; L ; R 3, where, by virtue of 4.4 and the uniform bounds established in 4.7, 4.8 [ ] [ ] t V ϕ dx t V ϕ dx in C[, T ] 4.1 for any ϕ C c ; R 3, div x ϕ =. Moreover, it follows from 3.4, 4.1, 4.11 and equation of continuity. that V = ϱu L, T ; L ; R 3 L, T ; W 1, ; R 3, Compactness of the gradient part div x U = a.a. in, T In view of 4.1, it remains to establish compactness of the gradient component of the vector field V. Our goal is to apply the Helmholtz projection to equation 4.4, specifically, we consider test functions in the form ϕ = x Ψ, with to obtain where we have set Ψ = χ C c [, T, x Ψ n =, Ψt, D 1, 5.1 = Φ, χ, dx Φ t χ + r χ dx dt 5. F 1,1 + F 1, : xψ dx dt + F x Ψ, x Φ = H [V ], x Φ, = H [ϱ, u, ]. 5.3 Note that, as a consequence of the bounds obtained in 3.9, 3.11 and the basic properties of the Helmholtz projection see Galdi [1], we have Φ = Φ 1 + Φ, with {Φ 1 } > bounded in L, T ; D 1,, 5.4 Φ in L, T ; D 1,s for any s [1, γ/γ + 1]

11 In particular, as a direct consequence of Sobolev s embedding D 1,6/5 L we have Accordingly, equation 4.3 can be rewritten as r t ϕ + p x Φ x ϕ dx dt = for any ϕ C c [, T. 5.1 Abstract variational formulation Φ in L, T ; L. 5.6 Our aim is to rewrite system 5., 5.7 in terms of an abstract differential operator with N, N [v] = p v, x v n =, vx as x, D N = {w L w W,, x w n = }. sup λ C,<α Re[λ] β<, Im[λ] r, ϕ, dx 5.7 It can be shown, see e.g. Leis [13], that N is a self-adjoint, non-negative operator in L, with an absolutely continuous spectrum [,. Moreover, N satisfies the limiting absorption principle V λ 1 V c L[L ;L ] α,β, 5.8 where Vx = 1 + x s, s > 1. We denote by {P λ } λ [, the spectral resolution associated to the operator N in the Hilbert space L. Accordingly, for any continuous function G on the spectrum of N, we can define G N [v] by setting < G N [v]; ϕ >=< v; G N [ϕ] > = Gλd P λ [ϕ] v dx for ϕ Cc as long as the integral on the right-hand side converges see Dunford and Schwartz [4, Chapter XII] for details about the spectral families of self-adjoint operators and about the operator calculus related to these families which will be used in the sequel. Next, we claim that the mapping χ F 1,1 + F 1, : xψ dx + F x Ψ ds x, 5.9 where Ψ, χ are interrelated through 5.1, represents a bounded linear form on the Hilbert space 1 D N D, 5.1 the norm of which can be estimated in terms of F 1,1 L 1 ;R 3 3, F1, L ;R 3 3, F L ;R 3. To this end, it is enough to observe that for χ belonging to 5.1, the function Ψ defined through 5.1 meets the following properties: 1

12 i χ dx = N χ 1/ N χ dx and N χ dx = N χ χ dx so that 1 L + N L is an equivalent norm on D N D 1 ; ii xχ dx = N χ L ; iii xψ L L ;R 3 3 c xψ W,,R c N χ L + χ L, where we have used the Sobolev embedings, elliptic regularity to problem 5.1 and observations i, ii; iv x Ψ L ;R 3 c xψ W 1,,R 3 1 c N χ L + χ L, where we have used the trace theorem and observations i, ii. Thus we may apply to 5.9 the Riesz representation theorem on the Hilbert space 5.1 to get that the functional 5.9 can by written as h 1 N [χ] + h 1 [χ] dx with where {h 1 } >, {h } > bounded in L, T Accordingly, equation 5. can be written in the form = Φ, χ, dx + 1 α χ C c Φ t χ + r χ dx dt 5.1 h 1 N [χ] + h 1 [χ] dx dt, 1 [, T ; D N D. 11

13 Similarly, equation 5.7 reads for all χ C c r t + N [Φ ] N [χ] dx dt 5.13 = r, χ, dx [, T ; D N. Note that, since Φ is given by 5.3, we have [Φ ] N [χ] dx = p V x χ dx. Thus, at least formally, system 5.1, 5.13 can be written as t Φ + r = 1 α N [h 1 1 ] + [h ], 5.14 with the initial conditions t r + N [Φ ] =, 5.15 r, = r,, Φ, = Φ, Duhamel s formula Solutions to can be expressed in terms of the data by means of Duhamel s formula as Φ t, = exp i t [ ] 1 Φ i, + [r, ] 5.17 N + exp i t [ ] 1 Φ i, [r, ] N meaning, in particular, that for any G C c t t s + α/ cos = [ G N exp [ N [h 1 s, ] + i t 1 [h s, ] Φ t, G N [χ] dx 5.18 ] ds, + exp i t ] [χ]φ, dx ig N [exp i t exp i t ] [χ]r, dx t t s + α/ N G N cos [χ]h 1 s, dx ds t + α/ G N, and any χ L. t s cos [χ]h s, dx ds 1

14 5.3 An abstract result of Kato In order to deduce the desired space-time decay estimates for the group expit N, we report a result of Kato [11], see also Burq et al. [1]. Theorem 5.1 [ Reed and Simon [, Theorem XIII.5 and Corollary] ] Let A be a closed densely defined linear operator and H a self-adjoint densely defined linear operator in a Hilbert space X. For λ / R, let R H [λ] = H λid 1 denote the resolvent of H. Suppose that Γ = sup A R H [λ] A [v] X < λ/ R, v DA, v X =1 Then π sup A exp ith[w] X dt Γ. w X, w X =1 where We intend to apply Theorem 5.1 to X = L, H = N, A[v] = ϕg N [v], v X, To begin, we have to verify hypothesis Since G C c,, ϕ C c are given functions. A R H [λ] A 1 = ϕg N λ G Nϕ, 5. it is enough to consider the values of the parameter λ belonging to a bounded set Q of the complex plane, namely λ Q = {z C Re[z] [a, b], < Im[z] < d}, where < a < b <, supp[g] a, b and d >. Indeed the operator 5. is bounded in the operator norm on L uniformly with respect to λ C \ Q in terms of the parameters a, b, d. Continuing the investigation of 5. with λ Q, we observe that with A R H [λ] A = ϕ M N, λ N λ ϕ, M N, λ = G N N + λg N - a bounded linear operator in L as soon as λ Q. Finally, using the standard operator calculus cf. Dunford, Schwartz [4, Chapter XII], we may write ϕ M N, λ N λ ϕ 5.1 = sup v L, w L 1 L[L ;L ] ϕ M N, λ [ϕv]w dx N λ 13

15 M N, λ = sup v L, w L 1 λ [ϕv] 1 λ [ϕw] dx 1 c λ ϕ = c sup v L 1 L[L ;L ] 1 λ [ϕv] 1 λ [ϕv] dx 1 = c sup ϕ [ϕv]v dx v L 1 N λ = c ϕ 1 N λ ϕ. L[L ;L ] Therefore hypothesis 5.19 is satisfied as a direct consequence of the limiting absorption principle stated in 5.8. Going back to formula 5.17 we can apply Theorem 5.1 to obtain ϕg N exp ±i t [Φ, ] dt 5. L c 1 ϕg N exp ±it N [Φ, ] dt c x Φ, L L Analogously, we deduce ϕg N and c 3 V, L. exp ±i t Finally, by means of similar arguments, t α ϕ N G N exp α t 1 α 1 α c 1 ϕg N N ϕ N G N exp ±i t s exp ±i s [h 1 s, ] exp ±i t s [r, ] L dt c r, L. 5.3 [h 1 s, ] ±i t s L [h s, ] L L [h 1 s, ] ds = 1 α c 1 L ds dt 1 α c 1 ds dt 5.4 h 1 L dt ds ds; h L ds. 5.5 Combining relations with the uniform bounds established in 5.11 we may infer from 5.17 that G N [Φ ] L,T K 1 α ck, G 5.6 for any compact K, and any G C c,. 14

16 5.4 Compactness of the gradient part, conclusion Consider a family [ ] t V w dx, with w Cc ; R 3, where, in accordance with 5.3, V w dx = H[V ] w dx Φ div x w dx. Writing H[V ] w dx = V H[w] dx and using the density of Cc ; R 3 in the space of solenoidal functions, we deduce from 4.1 that [ ] [ ] t H[V ] w dx t H[V] w dx in C[, T ] 5.7 for any fixed w Cc ; R 3. On the other hand, Φ div x w dx = = G N [Φ ]div x w dx + G N [Φ ]div x w dx + [Id G N ][Φ ]div x w dx Φ [Id G N ][div x w] dx, where, by virtue of 5.6, the former integral on the right-had side tends to zero in L, T as for any fixed w C c ; R 3, G C c,. Furthermore, by means of relations , we have Φ = Φ 1 + Φ, Φ 1 in L, T ; L, while { N [Φ ]} > is bounded in L, T ; L. [ ] Finally, it is easy to check that for G 1, the quantity Id G [div x w] will be small in L. In view of the previous estimates, we conclude that [ ] [ ] t V t, w dx t Vt, w dx in L, T for any w Cc ; R Convergence Relation 5.8, together with 4.11, and compactness of the embedding W 1, L K, yields the desired conclusion u U strongly in L, T K for any compact K. 5.9 With 4.1, 5.9 at hand, it is not difficult to pass to the limit in momentum equation. to recover.8, which completes the proof of Theorem.1. 15

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