A review of basic theoretical results concerning the Navier-Stokes and other similar equations

Bol. Soc. Esp. Mat. Apl. no32(2005), 45–73 A review of basic theoretical results concerning the Navier-Stokes and other similar equations Enrique Fern´ andez-Cara Dpto. E.D.A.N., Universidad de Sevilla Aptdo. 1180, 41080 Sevilla, Spain [email protected] Abstract These notes are devoted to provide an introductory approach to the Navier-Stokes and some other related equations. Most concepts and arguments recalled below are very general and we believe that this presentation can be of help for the theoretical analysis of many PDE’s arising from Sciences and Engineering. First, we recall the Navier-Stokes equations, we explain the meaning of the variables and data and we state some technical results needed for our study. Then, we state and give the proofs of some basic existence, uniqueness and regularity results. In the proof of existence, we apply usual compactness arguments to a family of Galerkin approximations. We also discuss briefly some of the main open problems arising in the three-dimensional case. In a final section, we review briefly the state of the art for other similar equations and we indicate some related open questions. Key words: Navier-Stokes equations; nonlinear PDE’s in fluid mechanics; compactness methods for nonlinear PDE’s; Galerkin’s approximations; existence, uniqueness and regularity results. AMS subject classifications: 35Q30, 35Q35, 35K55, 65M60 1 Introduction. Formulation of the problem and main results In these notes, Ω ⊂RNis a bounded connected open set at least of class C0,1 (N= 2 or N= 3) and we have 0 < T ≤+∞. We will use the notation Q= Ω ×(0, T) and Σ = ∂Ω×(0, T) and we will denote by Ca generic positive constant, usually depending on Ω, Tand maybe other data. 45 46 E. Fern´ andez-Cara We will first be concerned with the nonlinear problem    ut+ (u·∇)u−ν∆u+∇p=f, ∇·u= 0 in Q, u= 0 on Σ, u(x, 0) = u0(x) in Ω, (1) where ν > 0, f=f(x, t) and u0=u0(x) are given. This serves to model the behavior of a Newtonian viscous incompressible fluid whose particles fill the spatial domain Ω during the time interval (0, T). In (1), the unknowns are the RN-valued function u=u(x, t) (the velocity field) and the real-valued function p=p(x, t) (the pressure). The data are f=f(x, t) (a density of external forces) and u0=u0(x) (an initial velocity field). It is assumed that the mass density of the fluid is equal to 1. The first equality is the conservation of momentum law, i.e. Newton’s second law, written along the trajectories. The second one indicates that the fluid is incompressible, i.e. that the volume occupied by a set of particles is independent of time. We have complemented these equations with boundary conditions on Σ that express that the particles adhere to the wall (and therefore do not slip) and initial conditions at time t= 0. Our first aim in this paper is to recall the main known existence, uniqueness and regularity results that hold for (1), as well as the main ideas needed in their proofs; a complete analysis can be found for instance in [7, 13, 19, 20, 22, 31, 33]. In view of the generality of the concepts and arguments presented below, we believe that this can be of help for the theoretical analysis of the Navier-Stokes and many other PDE’s arising from Sciences and Engineering. From the viewpoint of mechanics, to try to solve (1) is full of sense: we assume that the mechanical state of the fluid at time t= 0 and the external forces acting on the fluid during (0, T) are known and we try to determine the mechanical state for t∈(0, T ). However, the equations in (1) are not always appropriate for the description of the flow of an incompressible fluid. Thus, there are many (realistic) modifications of (1) that lead to related mathematical problems. Let us mention some of them: •The term −ν∆uin (1) is the contribution of viscosity to the motion of particles. In some idealized situations, it may be adequate to assume that ν= 0 (this means that the role of viscosity is negligible). Then we find the so called Euler equations: ut+ (u·∇)u+∇p=f, ∇·u= 0.(2) •Sometimes, it is more accurate to assume that the mass density of the fluid is not a constant. Then we must introduce an additional unknown in (1), the positive real-valued function ρ=ρ(x, t) and we must complete (1) with the mass conservation law. The resulting equations are the following:    ρt+∇·(ρu) = 0, ρ(ut+ (u·∇)u)−µ∆u+∇p=ρf, ∇·u= 0. (3) The Navier-Stokes and other similar equations 47 These are the nonhomogeneous incompressible Navier-Stokes equations. •For more complex flows, νis not a constant but depends on the mechanical state of the fluid. It is then usual to assume that νis a positive function of |Du|, where Du =1 2(∇u+∇ut) is the symmetric part of the gradient of u. This leads to the equations for the so called quasi-Newtonian fluids: ut+ (u·∇)u−2∇·(ν(|Du|)Du) + ∇p=f, ∇·u= 0.(4) •More generally, it may happen that viscous effects depend on uglobally, for instance through the solution of a transport equation governed by u. There are many situations where this is the right way to model the fluid. For instance, this is the case for the so called visco-elastic fluids of the Oldroyd kind, which obey to the following system:    ut+ (u·∇)u−ν∆u+∇p=∇·τ+f, ∇·u= 0, τt+ (u·∇)τ+cτ +ga(∇u, τ) = bDu. (5) Here, τ=τ(x, t) is a symmetric tensor known as the extra elastic stress tensor, c and bare positive constants and ga:RN×N×RN×N7→ RN×Nis an appropriate bilinear tensor-valued function. For more details on these and other equations arising in fluid mechanics, see for instance [6, 11, 15, 22, 27, 29]. See also section 5 for a brief review of known results. There are mainly two reasons to consider problem (1). First, as mentioned above, it can be used for the description of an important family of flows. On the other hand, from a mathematical viewpoint, the analysis of (1) is of high interest, since it leads in practice to the main difficulties one usually encounters when dealing with nonlinear PDE’s. To get an idea of the large number of relevant open questions raised by (1), see [14]. Of course, before presenting the main results, we have to give a sense to (1) and specify the kind of solution we are looking for. We will need some function spaces and basic results. First, let us introduce V:= {ϕ∈C∞ 0(Ω)N:∇·ϕ= 0 in Ω }, where C∞ 0(Ω) stands for the space formed by the functions ϕ: Ω 7→ Rof class C∞with compact support in Ω. Then we have the following well known De Rham’s lemma: Lemma 1 Let S∈ D0(Ω)Nbe given, with hS, ϕi= 0 ∀ϕ∈ V. Then there exists q∈ D0(Ω) such that S=∇q. 48 E. Fern´ andez-Cara Another version of De Rham’s lemma will be given below; see lemma 13 in section 3. We will denote by H(resp. V) the adherence of Vin the Hilbert space L2(Ω)N(resp. in H1 0(Ω)N). Of course, H(resp. V) is a new Hilbert space for the norm of L2(Ω)N(resp. the norm of H1 0(Ω)N), which will be denoted by |·| (resp. k·k). Moreover, we have H={v∈L2(Ω)N:∇·v= 0 in Ω, v ·n= 0 on ∂Ω}.(6) Here, n(x) is by definition the outward normal vector to Ω at x(a point of ∂Ω). It is known that, for any v∈L2(Ω)Nsuch that ∇·v∈L2(Ω), we can give a sense to the normal trace v·non ∂Ω in a space that contains L2(∂Ω). This justifies (6). As a consequence of lemma 1, we also find that V={v∈H1 0(Ω)N:∇·v= 0 in Ω }.(7) Another property of the Hilbert spaces Hand Vis the following: V ,→H≡H0,→V0,(8) with dense and compact embeddings; furthermore, V0can be identified (isomorphically and isometrically) to the quotient space H−1(Ω)N/∇L2(Ω). In other words, the “points” of V0can be viewed as the classes of H−1(Ω)N determined by the following equivalence relation f∼gif and only of f−g=∇qfor some q∈L2(Ω). We will also need to speak of the space of distributions D0(D;X), where D⊂Rmis an open set and Xis a Banach space; very often, we will have D= (0, T). By definition, D0(D;X) is the space of linear sequentially continuous mappings S:D(D)7→ X1. For given S∈ D0(D;X) and ϕ∈ D(D), we will denote by hS, ϕiD0(D;X),D(D) or more simply hS, ϕithe point of Xassigned by Sto ϕ. We will say that the sequence {Sn}converges to Sin D0(D;X) if hSn, ϕi → hS, ϕiin Xfor every ϕ∈ D(D). Every f∈L1 loc(D;X) determines uniquely a distribution Sf∈ D0(D;X) through the formula hSf, ϕi=ZD f(ξ)ϕ(ξ)dξ ∀ϕ∈ D(D). Furthermore, the mapping f7→ Sfis linear, sequentially continuous and oneto-one. Accordingly, L1 loc(D;X) can be identified to a subspace of D0(D;X) 1Recall that ϕn→ϕin D(D) if all the supports of the functions ϕnare contained in the same compact set K⊂Ω and any derivative of any order of ϕnconverges uniformly in Kto the corresponding derivative of ϕ. The Navier-Stokes and other similar equations 49 and Sfcan be denoted by f. This will be made in the sequel. In particular, for any p∈[1,+∞], Lp(D;X) is also a subspace of D0(D;X). Notice however that there are (many) distributions in D0(D;X) that do not belong to L1 loc(D;X). Indeed, if ξis a point of Dand we set hδξ, ϕi=ϕ(ξ)∀ϕ∈ D(D), then δξ∈ D0(D;X), but there is no function f∈L1 loc(D;X) such that δξ=Sf. If S∈ D0(D;X) is given, we can give a sense to any derivative of Sof any order. Thus, if α= (α1,...αm) is a standard multi-index and we set |α|=α1+···+αm,∂αSis by definition the X-valued distribution given as follows: h∂αS, ϕi= (−1)|α|hS, ∂αϕi ∀ϕ∈ D(D). In particular, we can speak of any derivative of a function in L1 loc(D;X). Notice that, for each α, the linear operator ∂α:D0(D;X)7→ D0(D;X) is sequentially continuous. The following results are well known. Their proofs can be found for instance in [8] and [31]. Lemma 2 Let Xbe a Banach space. Assume that 1≤p0, p1≤+∞, f∈Lp0(0, T;X)and ft∈Lp1(0, T;X). Then f∈C0([0, T]; X)and we have the estimate kfkC0([0,T ];X)≤C¡kfkLp0(0,T ;X)+kftkLp1(0,T ;X)¢, where Conly depends on p0and p1. Lemma 3 Let Vand Hbe Hilbert spaces satisfying (8) with dense and continuous embeddings. Assume that 1< p < +∞,f∈Lp(0, T;V)and ft∈Lp0(0, T;V0). Then f∈C0([0, T]; H)and we have the estimate kfkC0([0,T ];H)≤C³kfkLp(0,T ;V)+kftkLp0(0,T ;V0)´, where Conly depends on p. Furthermore, the function t7→ kf(t)k2 His absolutely continuous and one has d dtkf(t)k2 H= 2hft(s), f(s)iV0,V a.e. in (0, T ).(9) Consequently, the following identity holds for any t1, t2∈[0, T]: 1 2kf(t2)k2 H−1 2kf(t1)k2 H=Zt2 t1hft(s), f(s)iV0,V ds. (10) Lemma 4 Let Xand Ybe Banach spaces. Assume that Xis reflexive, X ,→Y with a continuous embedding and v∈L∞(0, T;X)∩C0([0, T]; Y). Then v∈C0 w([0, T]; Y), i.e. for every L∈Y0the real-valued function t7→ hL, v(t)iY0,Y is continuous. 50 E. Fern´ andez-Cara Following [20], we can now present a first rigorous formulation of (1): Problem I: Given f∈L2(Q)Nand u0∈H, find uand psuch that      u∈L2(0, T;V)∩L∞(0, T;H), p ∈ D0(Q), ut+ (u·∇)u−ν∆u+∇p=fin D0(Q), u|t=0 =u0. (11) It will be seen below that any function usatisfying u∈L2(0, T;V)∩ L∞(0, T;H) and ut+(u·∇)u−ν∆u+∇p=fin D0(Q) for some p∈ D0(Q) also satisfies ut∈Lσ(0, T;V0) for an appropriate σ > 1. Thus, in view of lemma 2, u∈C0([0, T]; V0) and it is meaningful to speak of u|t=0 and to ask for the initial condition in (11) at least as an equality in V0. Notice that, if uand psolve (11), we automatically have u(·, t)∈Vfor t a.e. in (0, T). Consequently, we have in some sense ∇·u= 0 in Qand u= 0 on the lateral boundary Σ. Let us now give a second formulation of (1): Problem II: Given f∈L2(Q)Nand u0∈H, find usuch that      u∈L2(0, T;V)∩L∞(0, T;H), hut, viV+b(u, u, v) + νa(u, v) = h`, vi ∀v∈V, u|t=0 =u0. (12) Here, h·,·iVstands for the duality pairing associated to Vand V0,`=`(t) with h`(t), viV=ZΩ f(x, t)v(x)dx ∀v∈V, t ∈[0, T] a.e. and we have introduced the bilinear and trilinear forms a(·,·) and b(·,·,·), given as follows: a(u, v) = ZΩ∇u·∇v dx ∀u, v ∈V, (13) b(u, v, w) = ZΩ (u·∇)v·w dx ∀u, v, w ∈V. (14) Since f∈L2(Q)N, we have `∈L2(0, T;V0) and h`, viV∈L2(0, T ) for any v∈V. On the other hand, if u∈L2(0, T ;V)∩L∞(0, T;H), it is not difficult to check that, for each v∈V, we have a(u, v)∈L2(0, T) and, at least, b(u, u, v)∈L1(0, T) and hut, vi ∈ D0(0, T). Consequently, the equalities in (12) can be understood in the sense of D0(0, T). Let us introduce the linear operator A:V7→ V0, with hAu, viV=a(u, v)∀u, v ∈V The Navier-Stokes and other similar equations 51 and the bilinear operator B:V×V7→ V0, with hB(u, v), wiV=b(u, v, w)∀u, v, w ∈V. Then an equivalent formulation of (12) is the following:      u∈L2(0, T;V)∩L∞(0, T;H), ut+B(u, u) + νAu =`in D0(0, T;V0), u|t=0 =u0. (15) It will be seen below that any function u∈L2(0, T;V)∩L∞(0, T;H) that satisfies ut+B(u, u) + νAu =`belongs to C0([0, T ]; V0). Thus, the initial conditions in (12) and (15) again make sense. The main results concerning the existence and uniqueness of solution for problem II are the following: Theorem 5 Assume that f∈L2(Q)Nand u0∈Hare given. Then there exists at least one solution of problem II. Theorem 6 Assume that N= 2 and f∈L2(Q)2and u0∈Hare given. Then problem II possesses exactly one solution. We will see in section 3 that any solution of problem II is, together with some p, a solution of problem I. As a consequence, theorems 5 and 6 show that the original system (1) can be solved in an appropriate class. 2 Proof of uniqueness In this section, we will prove that, when N= 2, problem II possesses at most one solution. We will need some previous results: Lemma 7 Assume that N= 2. Then there exists a constant C > 0such that kvkL4≤C|v|1/2kvk1/2∀v∈H1 0(Ω).(16) The proof can be found in [20]. A consequence of this lemma is the following: Lemma 8 Assume that N= 2. Then for every v∈L2(0, T ;V)∩L∞(0, T;H) one has B(v,v)∈L2(0, T;V0). Furthermore, there exists a constant C > 0such that kB(v,v)kL2(0,T ;V0)≤CkvkL∞(0,T ;H)kvkL2(0,T ;V)(17) for all such v. Proof: Assume that v∈L2(0, T;V)∩L∞(0, T;H). Let us try to estimate hB(v(t), v(t)), wiVfor each w∈V. 52 E. Fern´ andez-Cara We have: hB(v(t), v(t)), wiV=ZΩ (v(t)·∇)v(t)·w dx =−ZΩ (v(t)·∇)w·v(t)dx ≤Ckv(t)k2 L4kwk ≤ C|v(t)|kv(t)kkwk. Hence, kB(v(t), v(t))kV0≤C|v(t)|kv(t)k for ta.e. in (0, T) and we obviously have B(v,v)∈L2(0, T;V0) and (17). Remark 1 When N= 3, the estimates (16) and (17) do not hold. In this case, instead of (16) we only have kvkL4≤C|v|1/4kvk3/4∀v∈H1 0(Ω).(18) Accordingly, it can be proved that for every v∈L2(0, T;V)∩L∞(0, T;H) one has B(v,v)∈L4/3(0, T ;V0) and the estimates kB(v(t), v(t))kV0≤C|v(t)|1/2kv(t)k3/2 a.e. in (0, T) and kB(v,v)kL4/3(0,T ;V0)≤Ckvk1/2 L∞(0,T ;H)kvk3/2 L2(0,T ;V),(19) but nothing better than this. Now, assume that N= 2 and uand u0are two solutions of (15), where the data f∈L2(Q)2and u0∈Hare given. Observe that, in this case, utand u0 tbelong to L2(0, T;V0). Indeed, we have ut=`−B(u, u)−νAu. It is immediate that `, Au ∈L2(0, T;V0) and, on the other hand, in view of lemma 8, we also have B(u, u)∈L2(0, T ;V0). Consequently, ut∈L2(0, T ;V0). A similar argument holds for u0 t. Let us set w=u−u0. Then w∈L2(0, T;V)∩L∞(0, T;H), wt∈L2(0, T;V0), wt+νAw =−B(u, u)−B(u0, u0)≡ −B(u, w)−B(w, u0) (20) and w|t=0 = 0. In view of lemma 3, w∈C0([0, T]; H) and we have 1 2 d dt|w(t)|2=hwt(t), w(t)iVa.e. in (0, T). This, together with (20), yields the following: 1 2 d dt|w(t)|2+νkw(t)k2=−b(w(t), u0(t), w(t)) ≤C|w(t)|kw(t)kku0(t)k ≤ν 2kw(t)k2+Cku0(t)k2|w(t)|2. The Navier-Stokes and other similar equations 53 After integration in time, we deduce at once that |w(t)|2≤CZt 0ku0(s)k2|w(s)|2ds ∀t∈[0, T] (21) and, from Gronwall’s lemma, we find that w≡0 and uand u0must coincide. This ends the proof of theorem 6. Remark 2 With an argument a little more complicate, it can also be proved that the solution uof problem II depends continuously of the data fand u0. For more details, see for instance [31]. Remark 3 In general, the uniqueness of solution of (12) with N= 3 and not necessarily small data fand u0is unknown. Actually, this is a major open problem in Navier-Stokes theory2. When N= 3, we have uniqueness of regular solution. For instance, if the solution of (12) satisfies u∈L2(0, T;V)∩L∞(0, T;H)∩Ls(0, T;Lr(Ω)3) with 2/s + 3/r ≤1 and r > 3, then uis the unique solution in this class. This is a result from [20] (see also [28]) that can be proved as follows: •It is sufficient to consider the case in which 2/s + 3/r = 1 and r > 3. Let uand u0be two solutions with the previous regularity and let us set w=u−u0. Then utand u0 tbelong to L2(0, T;V0). Indeed, we have for instance that |hB(u(t), u(t), viV| ≤ |u(t)|2/sku(t)k3/rku(t)kLrkvk for all v∈Vand the function |u|2/skuk3/rkukLr∈L2(0, T). Consequently, we also have wt∈L2(0, T;V0). •Proceeding as in the proof of theorem 6, we have 1 2 d dt|w(t)|2+νkw(t)k2=−b(w(t), u0(t), w(t)) a.e. in (0, T). But now |hB(u(t), u(t), viV| ≤ |u(t)|2/sku(t)k3/rku(t)kLrkvk, whence we have again (21) and w≡0. Remark 4 More recently, under the assumption u∈L2(0, T;V)∩L∞(0, T;H)∩C0(0, T;L3(Ω)3), the uniqueness of solution of (12) with N= 3 has been established; see [23]. 2When N= 3, regularity results for non necessarily small data are also open; see section 4. 60 E. Fern´ andez-Cara This shows that usolves problem II. Consequently, theorem 5 is proved. Remark 5 It is not difficult to prove that the solution we have found satisfies the energy inequalities 1 2|u(·, t0)|2+νZt0 tku(·, s)k2ds ≤1 2|u(·, t)|2+Zt0 t (f(·, s), u(s)) ds (47) for all t, t0∈[0, T] with t < t0. However, in general it is unknown whether similar equalities hold. For more details about energy inequalities, see [22]. To end this section, let us prove that we have solved in fact problem I. More precisely, let us show that problems I and II are equivalent. Thus, let uand psolve problem I. Then, for any ϕ∈ V and any ψ∈ D(0, T), we have: 0 = hut+ (u·∇)u−ν∆u+∇p−f, ϕψiD0(Q)N,D(Q)N =ZZQ (−u·(ψϕ)t+ (u·∇)u·(ψϕ) + ν∇u·∇(ψϕ)) dx dt −ZZQ f·(ψϕ)dx dt =−ZT 0µZΩ u·ϕ dx¶ψtdt +ZT 0 (b(u, u, ϕ) + νa(u, ϕ)−h`, ϕiV)ψ dt. Hence, d dt µZΩ u·ϕ dx¶+b(u, u, ϕ) + νa(u, ϕ) = h`, ϕiVin D0(0, T) for all ϕ∈ V. This proves that uis a solution of problem II. In order to prove the reciprocal, we will use the following Banach-valued version of De Rham’s lemma: Lemma 13 Let Ebe a Banach space and let S∈ D0(Ω; E)Nbe given, with hS, ϕi= 0 ∀ϕ∈ V. Then there exists q∈ D0(Ω; E)such that S=∇q. Furthermore, if r∈(1,+∞), s∈Zand S∈Ws,r(Ω; E)N, we can choose qin Ws+1,r(Ω; E)and depending continuously of S, i.e. such that the mapping S∈Ws,r(Ω; E)N7→ q∈ Ws+1,r(Ω; E)is continuous. The Navier-Stokes and other similar equations 61 Let ube a solution to problem II and let us set S=ut+ (u·∇)u−ν∆u−f. It is then easy to check that S∈W−1,∞(0, T;H) + Lσ(0, T;H−1(Ω)N)⊂ W−1,∞(0, T;H−1(Ω)N). But this last Banach space is isomorphic and isometric to H−1(Ω; W−1,∞(0, T))N. Therefore, we can apply lemma 13 with E= W−1,∞(0, T) (recall that hS, ϕi= 0 for all ϕ∈ V). The conclusion is that there exists p∈L2(0, T;W−1,∞(0, T)) such that S=−∇p. In other words, for some p∈W−1,∞(0, T;L2(Ω)), one has ut+ (u·∇)u−ν∆u+∇p=fin D0(Q)N. This proves that uand psolve problem II. Remark 6 There are other ways to prove the existence of solution of problem II. Some of them are nonconstructive and rely on appropriate fixed point theorems; see for instance [19, 33]. There are also other constructive proofs; see [20, 31, 22]. Remark 7 With similar arguments, the existence of solutions of problems I and II can be proved under slightly more general assumptions. Thus, Ω ⊂RN can be an arbitrary connected open set (not necessarily bounded), the right hand side fcan belong to the space L1(0, T;H−1(Ω)N), etc. Remark 8 Assume that the nonlinear term b(u, u, v) is omitted in (12). Then we can argue as in steps 1 and 2 of the proof of theorem 5 and construct Galerkin approximations umsatisfying (32). This suffices to choose a subsequence satisfying (35) and (37). But now this is enough to pass to the limit in the approximated problems in all the terms. Consequently, the need of the estimates (34) for the approximated solutions of (12) comes from the fact that this system contains nonlinear terms. Remark 9 Lemma 12 is interesting by itself. It provides a criterion to ensure the relative compactness of a family F ∈ Lp0(0, T;X). This subject has been investigated by J. Simon in [30]. There, the following assertion is proved: Assume that Xis a Banach space, 1≤p0<+∞and F ⊂ Lp0(0, T;X)is given. Then Fis relatively compact in Lp0(0, T;X) if and only if one has:  The set {Zt2 t1 f(s)ds :f∈ F } is relatively compact in Xfor any t1, t2∈[0, T]with t1≤t2.  kτhf−fkLp0(0,T −h;X)→0uniformly in f∈ F as h→0. Here, τhf(t)≡f(t+h)for any t∈[0, T −h]and any h∈(0, T). 62 E. Fern´ andez-Cara 4 Some regularity results Besides existence and uniqueness results, it is also interesting to investigate regularity properties of the solutions of problem II. Indeed, it is reasonable to expect that, when the open set Ω and the data fand u0are more regular than in theorem 5, so are the associated solutions. When N= 2, this can be established rigorously. For instance, we have the following: Theorem 14 Assume that N= 2,Ω⊂R2is a bounded connected open set of class C1,1,f∈L2(Q)2and u0∈V. Then the unique solution of problem II satisfies u∈L2(0, T;H2(Ω)2)∩C0([0, T]; V), ut∈L2(0, T;H).(48) For the proof, it suffices to get uniform estimates of the Galerkin approximations umin L2(0, T;H2(Ω)2) and L∞(0, T ;V) and uniform estimates of the time derivatives um tin L2(0, T;H). The estimates of umcan be obtained by taking in (24) v=Aum(t) for each t∈(0, T). Indeed, with the notation (25), we have Aum(t) = m X i=1 λiηim(t)wi and this shows that this choice of vis admissible. We easily deduce that 1 2kum(t)k2+νZt 0|Aum(s)|2ds =1 2ku0mk2+Zt 0 (fm(·, s), Aum(s)) ds −Zt 0 b(um(s), um(s), Aum(s)) ds ≤1 2ku0k2+ν 2Zt 0|Aum(s)|2ds +CZt 0|f(·, s)|2ds +CZt 0|um(s)|2kum(s)k4ds. Here, we have used that u0∈V. These inequalities, together with (32), Gronwall’s lemma and the regularity of Ω, yield the desired bounds for um. For the estimates of um t, we take v=um t(t) in (24). Now, we find that |um t(t)|2+ν 2 d dtkum(t)k2= (fm(·, t), um t(t)) −b(um(t), um(t), um t(t)) ≤1 2|um t(t)|2+|f(·, t)|2+Ckum(t)k2 H2kum(t)k2 and integrating with respect to twe find that um tis uniformly bounded in L2(0, T;H). For more details, see for example [7]. The Navier-Stokes and other similar equations 63 When N= 3, the situation is much more complicated. We can obtain results of the kind of theorem 14 only when the data are small (in appropriate norms). For instance, we have the following result: Theorem 15 Assume that N= 3,Ω⊂R3is a bounded connected open set of class C1,1,f∈L2(Q)3and u0∈V. Then there exists ε > 0(depending on Ω) such that, whenever ku0k+kfkL2(Q)3≤ε, (49) the solution of problem II furnished by theorem 5 satisfies u∈L2(0, T;H2(Ω)3)∩C0([0, T]; V), ut∈L2(Q)3(50) and is unique in this class. For the proof, we try to find the same estimates above for the Galerkin approximations umand their time derivatives um t. In this case, we find that 1 2kum(t)k2+νZt 0|Aum(s)|2ds ≤1 2ku0k2+CZt 0|f(·, s)|2ds +ν 2Zt 0|Aum(s)|2ds +CZt 0kum(s)k6ds and, in order to conclude, the assumption (49) is needed (with εsufficiently small). We refer to [13] and [7] for more details. Remark 10 Using appropriately lemma 13, we can deduce from (48) and (50) further regularity properties for p. In particular, under the assumptions of theorems 14 or 15, we find that p∈L2(0, T;H1(Ω)). Accordingly, the PDE’s in (11) are satisfied a.e. in Q. Remark 11 It is completely unknown whether “large” regular data fand u0 lead to regular solutions when N= 3. In fact, one of the one-million dollars open problems proposed by the Clay Institute in 2000 is the following: Assume that Ω = R3,f≡0and u0∈ V. Prove that problem II possesses a solution uof class C∞. See http://www.claymath.org/millennium/Navier-Stokes_Equations/ for more details. 5 Some other results and open questions In this section, we will take T= +∞,Q= Ω ×(0,+∞) and Σ = ∂Ω×(0,+∞). Here, our aim is to recall very briefly some of the main known results concerning variants of the Navier-Stokes equations. We will only consider fluids modelled by the equations (2), (3), (4) and (5). We believe this is enough to get an idea of the variety and complexity of the subject. 64 E. Fern´ andez-Cara 5.1 The incompressible Euler equations When viscous effects are negligible, it is admissible to take ν= 0 in the motion equation. For example, this is the case when we are considering the flow of the air around an obstacle at high velocity and we observe the behavior of the fluid only at points located far from the obstacle. This leads to the incompressible Euler equations (2). Now, the fluid is modelled by a system of nonlinear first-order equations. Accordingly, it is reasonable to look for solutions satisfying not so many complementary conditions as in (1). On the other hand, it is also realistic to expect that, in principle, the solution be less regular than in the Navier-Stokes case. To fix ideas, we will consider the system    ut+ (u·∇)u+∇p= 0,∇·u= 0 in Q, u·n= 0 on Σ, u(x, 0) = u0(x) in Ω, (51) where u0=u0(x) is given. It will be said that uis (together with some p) a weak solution of (51) if u∈L2(0,+∞;V)∩L∞(0,+∞;H), hut, viV+b(u, u, v) = 0 ∀v∈V(52) and u|t=0 =u0. It should be noticed that this is equivalent to ZZQ u·(ϕt+ (u·∇)Pϕ)dx dt +ZΩ u0(x)·ϕ(x, 0) dx = 0 (53) for any ϕ∈ D(Q)N(recall that P:L2(Ω)27→ His the usual orthogonal projector) and slightly stronger than ZZQ u·(ϕt+ (u·∇)ϕ)dx dt +ZΩ u0(x)·ϕ(x, 0) dx = 0 (54) for any ϕ∈ D(Q)Nsatisfying ∇·ϕ= 0 in Q. Then the following is known: Theorem 16 Let us assume that N= 2,u0∈Hand ∇×u0∈L∞(Ω). Then (51) possesses exactly one weak solution ufurthermore satisfying u∈C0([0,+∞); W1,q(Ω)2)∀q∈(1,+∞).(55) For the proof, see [32]. A similar existence result can also be proved when u0∈Hand ∇×u0∈Lr(Ω) for some r∈(1,+∞), but the uniqueness of weak solution is unknown in this case; for more details, see [22]. When N= 3, the situation is much more complicated (and less understood). In general, only local in time existence results can be ensured for large initial data (even if they are smooth). A lot of appropriate numerical results and the The Navier-Stokes and other similar equations 65 analysis of some similar systems seem to indicate that regular solutions can blow-up at finite time; see for instance [12, 18] and the references therein. But the problem remains open at present. For completeness, let us recall the following result, whose proof can be found in [1]: Theorem 17 Let us assume that N= 3 and u0∈Hs(Ω)3∩Hfor some s > 5/2. Then there exists T∗>0such that (51) possesses exactly one weak solution in Ω×(0, T∗). This solution satisfies u∈C0([0, T]; Hs(Ω)3)∀T∈(0, T∗). Furthermore, if T∗<+∞, then ZT∗ 0k(∇×u)(·, t)kL∞dt = +∞.(56) 5.2 The variable density Navier-Stokes equations In practice, it is frequent to find fluids for which mass-density is not a constant but a function of space and time. This can be the case of a river or a portion of an ocean. The resulting equations are (3), where µ > 0. Observe that, in (3), the new variable ρsatisfies a (first-order hyperbolic) transport equation governed by uwhich is called the continuity equation: ρt+u·∇ρ= 0. Accordingly, it may happen that ρ(·, t) be a piecewise regular discontinuous function and the discontinuities of ρ(·, t) be transported by u. This is readily understood by rewriting the continuity equation in the form d dt ρ(X(x, t), t) = 0, where X(x, ·) is the trajectory determined by uand x, i.e. ½Xt(x, t) = u(X(x, t), t), t ∈(0, T), X(x, 0) = x(57) (the components of Xare also known as the Lagrangian coordinates; in fact, X(x, t) is the position at time tof the particle located at xat time t= 0). For simplicity, let us consider the following system for (3):        ρt+∇·(ρu) = 0 in Q, ρ(ut+ (u·∇)u)−µ∆u+∇p= 0,∇·u= 0 in Q, u= 0 on Σ, (ρu)(x, 0) = m0(x), ρ(x, 0) = ρ0(x) in Ω, (58) 66 E. Fern´ andez-Cara where µ > 0 is a constant and m0and ρ0are given. We will say that {ρ, u}is (together with some p) a weak solution of (3) if ½ρ∈L∞(Q)∩C0([0,+∞); Lq(Ω)) ∀q∈[1,+∞), u∈L2(0,+∞;V), ρ|u|2∈L∞(0,+∞;L1(Ω)) (59) the continuity equation ρt+∇·(ρu) = 0 holds in Qin the distributional sense, ρ|t=0 =ρ0in the Lqsense for all q∈[1,+∞) and        ZZQ (ρu ·ϕt+ρ uiuj∂iϕj−µ ∂iuj∂iϕj)dx dt +ZΩ m0(x)·ϕ(x, 0) dx = 0 (60) for any ϕ∈ D(Ω ×[0,+∞))Nsatisfying ∇·ϕ= 0 in Q. Then the following result holds: Theorem 18 Let us assume that N= 2 or N= 3,ρ0∈L∞(Ω),ρ0≥0a.e. in Ω,m0∈L2(Ω)N,m0= 0 a.e. when ρ0= 0 and |m0|2/ρ0∈L1(Ω). Then (58) possesses at least one weak solution {ρ, u}. For the proof, see for instance [30] and [22]. In this last reference, the result is proved in a more general case and it is found that the solution satisfies appropriate energy inequalities. It is also proved there that the distribution function of ρ(·, t) is independent of t; in other words, for any α, β ∈Rand any t > 0, one has meas {x∈Ω : α≤ρ(x, t)≤β}= meas {x∈Ω : α≤ρ0(x)≤β} (in fact, this property can be viewed as a reformulation of the mass conservation law). In general, the uniqueness of weak solution of (58) is unknown even when N= 2. The same can be said for regularity results. However, if N= 2 and the initial data also satisfy ρ0≥a > 0 a.e. in Ω,1 ρ0m0∈V, the regularity of the solutions (and therefore uniqueness) can be obtained. More precisely, under these assumptions one has u∈L2(0,+∞;H2(Ω)2)∩C0([0,+∞); V), ut∈L2(0,+∞;H). 5.3 Some quasi-Newtonian fluids For a general incompressible fluid of constant density ρ= 1, the motion equation states that ut+ (u·∇)u=∇·σ+f, (61) The Navier-Stokes and other similar equations 67 where σ=σ(x, t) is the stress tensor. This means that, for any regular open set W⊂Ω, the resultant of the forces exerted on the particles in Wby the other fluid particles at time tis given by I(W, t) = Z∂W σ(x, t)·n(x)dΓ(x). In the case of the Navier-Stokes equations, it is assumed that σis furnished by the so called Stokes law. This means that the stress tensor σis proportional to the strain or deformation tensor Du =1 2(∇u+∇ut). More precisely, we have σ= 2νDu =ν(∇u+∇ut) (62) for some constant ν > 0. Combining (62) and (61), we find at once the first equation in (1). The fluids satisfying this property are called Newtonian. Sometimes, it is more accurate to assume a more general constitutive law for σ. Thus, instead of (62), we can assume that σ=ν(|Du|)(∇u+∇ut),(63) where ν:R7→ R+is a given function. This leads to the system (4). The fluids governed by (4) are called quasi-Newtonian. In practical problems, many possible functions νare found. Here, we will only consider the choice ν(s) = αsr−2,(64) where r≥1 and αis a positive constant. When 1 ≤r < 2, r= 2 or r > 2, we are respectively considering a visco-plastic,Newtonian or dilatant fluid; in particular, in the limit r= 1, (63) must be understood as follows: σ=2α |Du|Du if Du 6= 0; |σ| ≤ 2αif Du = 0. In this case, we are considering a visco-plastic Bingham fluid (see [4, 5, 9] and the references therein). There are many real phenomena in chemistry, glaciology, biology, etc. where the previous constitutive laws are appropriate; see [25] and the references therein. For simplicity, let us consider the following initial-boundary value problem:    ut+ (u·∇)u−2∇·(ν(|Du|)Du) + ∇p= 0,∇·u= 0 in Q, u= 0 on Σ, u(x, 0) = u0(x) in Ω, (65) where u0is prescribed. Depending on the value of r, several different existence and/or uniqueness results can be established. In principle, as rincreases, better results are found. Thus, for very small r, it can only be proved that a local regular solution exists for regular initial data; for moderate r, global in time 68 E. Fern´ andez-Cara weak solutions exist. for larger r, the uniqueness of strong solution holds, etc. For a complete summary and a list of open questions, see [26]. In order to illustrate the situation, we will recall now one of these results. We will assume that 3N N+ 2 <r<2.(66) Let Vrbe the adherence of Vin the Sobolev space W1,r 0(Ω)N. Obviously, Vris a separable reflexive Banach space for the norm kvkr=µZΩ|∇v|rdx¶1/r ∀v∈Vr. It will be said that uis (together with some p) a weak solution of (65) if u∈Lr(0,+∞;Vr)∩L∞(0,+∞;H) and              ZZQ (ρu ·ϕt+ρ uiuj∂iϕj)dx dt −1 2ZZQ ν(|Du|)(∂iuj+∂jui)(∂iϕj+∂jϕi)dx dt +ZΩ u0(x)·ϕ(x, 0) dx = 0 (67) for any ϕ∈ D(Ω ×[0,+∞))Nsatisfying ∇·ϕ= 0 in Q. Theorem 19 Assume that νis given by (64) with rsatisfying (66) and u0∈H. Then (65) possesses at least one weak solution. It may be also meaningful to consider quasi-Newtonian viscous incompressible fluids with variable density. Roughly speaking, they lead to problems that need an analysis inspired by the arguments used in the proofs of theorems 18 and 19. See for instance [2, 10] for further details. 5.4 Viscoelastic Oldroyd models Sometimes, the molecular structure of the fluid under consideration is so complicated that the constitutive law (63) does not suffice to provide a good description of the flow. In particular, this is the case if elastic efforts among particles are relevant. Then, it has to be assumed that the stress tensor σis of the form σ=σ0+τ, where σ0(the viscous-stress tensor) is given by Stokes law and τ(the elasticstress tensor) satisfies an additional equation that is coupled to the conservation law (61) and serves to close the system. The equation for τcan be a differential or an integral equation. Accordingly, it leads to a differential or to an integro-differential model. In differential The Navier-Stokes and other similar equations 69 models, τis determined by ∇uthrough a system of PDE’s. It is assumed that this system must satisfy the material objectivity or frame indifference principle (in other words, the law must be invariant under time-dependent proper rotations Q=Q(t)). As a consequence, the resulting system must involve objective time derivatives. By this, we mean first order operators of the form ∂t+pi(x, t)∂i such that we always have Q·(∂t+pi∂i)τ·tQ= (∂t+pi∂i)¡Q·τ·tQ¢. The usual material derivative ∂t+ui∂idoes not satisfy the principle of material objectivity. On the contrary, the so called Oldroyd derivatives Daτ Dt=τt+ (u·∇)τ+ga(∇u, τ) (68) are objective derivatives. Here, ga(∇u, τ) = τW(u)−W(u)τ−a(D(u)τ+τD(u)) (69) (a∈[−1,1] is a constant). As before, D(u) = 1 2(∇u+∇ut) and W(u) is the vorticity tensor, i.e. W(u) = 1 2(∇u−∇ut). When a= 0, the corresponding derivative is known as the Jaumann’s or corotational derivative. It is the following: D0τ Dt=τt+ (u·∇)τ+τW(u)−W(u)τ. We will be concerned in this paragraph with the differential Oldroyd model. In dimensionless variables, this is (5), where ga(∇u, τ) is given by (69). This provides a good description of the behaviour of some materials that have in part properties found for elastic solids and, also in part, properties similar to those of viscous fluids (this is why they are called viscoelastic). For a complete presentation and analysis, see for instance [15, 29]. Let us consider the problem        ut+ (u·∇)u−ν∆u+∇p=∇·τ, ∇·u= 0 in Q, τt+ (u·∇)τ+cτ +ga(∇u, τ) = bDu in Q, u= 0 on Σ, u(x, 0) = u0(x), τ(x, 0) = τ0(x) in Ω, (70) where u0and τ0are prescribed. For convenience, let us denote by L2 sthe space of the symmetric tensors τ∈L2(Ω)N×N. It will be said that {u, τ}is (together with some p) a weak solution of (70) if u∈L2(0,+∞;V)∩L∞(0,+∞;H), τ∈L∞(0,+∞;L2 s), hut, viV+b(u, u, v) + νa(u, v) = h∇·τ, vi ∀v∈V, (71)