Weak Solutions to Steady Navier–Stokes Equations for Compressible Barotropic Fluids with Vorticity–Type Boundary Conditions

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Weak Solutions to Steady Navier–Stokes Equations for

Compressible Barotropic Fluids with Vorticity–Type

Boundary Conditions

Olivier Muzereau

To cite this version:

Olivier Muzereau. Weak Solutions to Steady Navier–Stokes Equations for Compressible Barotropic Fluids with Vorticity–Type Boundary Conditions. Mathematics [math]. Université du Sud Toulon Var, 2009. English. �tel-00474984�

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THÈSE

Cotutelle francotchèque présentée à

L'UNIVERSITÉ DU SUD TOULON-VAR par

Olivier MUZEREAU Pour l'obtention des grades de :

DOCTEUR de l'UNIVERSITÉ DU SUD TOULON-VAR,

DOCTEUR de l' UNIVERSITÉ TECHNOLOGIQUE DE PRAGUE

SPÉCIALITÉ :

Mathématiques Appliquées, Mécanique des uides Mathématique

Analyse des solutions du système des équations de NavierStokes

avec des conditions aux limites de type vorticité

pour les uides barotropiques compressibles

Soutenue le 24 Fevrier 2009 devant le jury composé de : Rapporteurs : R. DANCHIN Professeur Université de Paris XII

M. POKORNÝ : Professeur Université Charles de Prague Examinateurs : A. NOVOTNÝ : Professeur Université du Sud-Toulon-Var

S. KRAMAR : Professeur Université Technologique de Prague Directeurs de thèse : J. NEUSTUPA : Professeur Académie des sciences Tchèque

P. PENEL : Professeur Université du Sud-Toulon-Var

Laboratoire SNC Czech Technical

ISITV-Bat.X-Bureau 136 University in Prague,

Avenue POMPIDOU-B.P. 56 Karlovo nám. 13 ; 121 35

83162 LA VALETTE CEDEX-FRANCE Praha 2, Czech Republic.

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Universit´e du Sud Toulon Var

Czech Technical University in Prague

Doctoral Dissertation Thesis

Weak Solutions to Steady Navier–Stokes

Equations for Compressible Barotropic

Fluids with Vorticity–Type

Boundary Conditions

Olivier MUZEREAU

Universit´e du Sud Toulon Var

Laboratoire de Syst`emes Navals Complexes

FEBRUARY 24th 2009

Supervisors:

Jiˇr´ı NEUSTUPA, Czech Academy of Sciences, Prague,

Czech Technical University in Prague

Patrick PENEL, Universit´e de Sud Toulon Var

Reviewers:

Raphaël DANCHIN, Université de Paris XII, Créteil

Milan POKORN Y,

Charles University Prague

Examiner:

Stanislav KRA ˇ

CMAR, Czech Technical University in Prague

Anton´in NOVOTN ´

Y,

Universit´e de Sud Toulon Var

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`

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L’idée spéculative est l’effectivité infinie qui, dans l’absolue vérité d’elle–même, se résout à laisser aller librement hors d’elle même, le moment de sa particularité ou de la première détermination et altérité, l’Idée immédiate, comme son reflet, elle–même, comme nature. Hegel, Encyclopédie des Sciences Philosophiques,§193

C’est dans l’entendement divin que la vérité se trouve d’une manière propre et première; elle est encore dans l’entendement humain d’une manière propre mais secondaire; enfin dans les choses elle n’est que d’une manière à la fois impropre et secondaire car elle ne peut être appelée ainsi que relativement à l’une ou l’autre de ces deux vérités.

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Acknowledgements

It would not have been possible to write this doctoral thesis without the help and support of several persons.

I would like to thank my wife Emilie for her faithful support, her great patience and her illuminating love.

I would like to thank Patrick Penel for being a great director. His ideas and tremendous support had a major influence on this thesis. He has always been present during these four years. His passion for mathematics and his humanity make him more than a master in mathematics.

I would like to thank my director Jiri Neustupa for his constant disposability during my stays in Prague and his deep remarks concerning many parts of the thesis. “Dˇekuji mnohokr´at” to have made me discover the sumptuous city of Prague, its neighborhoods and... the becherovka.

I would like to thank Milan Pokorn´y and Rapha¨el Danchin for reviewing the thesis. I have appreciated their numerous and interesting remarks. They enable me to precise many points in the thesis.

I would like to thank the Czech Technical University and the Necas Center for their financial supports during my stays in Prague, particularly Jiri Fort, Josef Malek and Edouard Feiresl. I thank the Universit´e du Sud Toulon Var that have financially supported some of my trips in Prague. I thank the diocese of Fr´ejus Toulon that have provided me a place to live with my wife during the last four years. I also thank my laboratory SNC and his director Yves Lacroix.

I would like to thank my family and my wife’s family who have supported me with their love. Especially, I thank my father and my wife’s father.

I would like to thank my friends and colleagues J. Briche, J. Deon, P.G. Gourio–Jewell, R. Estival, M. Legonidec and I. Toulgoat for our friendly and intellectual discussions.

Finally, I thank my students from 2d A and 2d G for their hard work, interest and motivation when they were following my courses in the Lyc´ee de Lorgues.

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In his work concerning history of geometry, J.C. Carrega [7] explains that carefully looking at the proofs developed in the thirteen books called elements of Euclid, the reader can notice that geometrical figures were often an important part of the proof. Actually, the proofs were supposed to talk both to the reason and to the eyes of the reader. In order to convince, a proof had to be presented with a clear geometrical figure effected with well known and universally admitted geometrical instruments. This remark of J.C. Carrega is not so surprising if we recall that Euclid was a disciple of Aristotle and that this philosopher takes an important place in the scholasticism where the truth were defined as the adequatio rei et intellectus, i.e.: the adequation between reality and mind. Aristote was also the author of the book called Physics in which he carefully tried to think what the motion is. Even if the way to do physics has really changed nowadays, it has preserved one of the main ways of thinking always used by Aristote which is to prove by constantly refereing to the reality. For example, if we ask any modern physicist in fluid mechanics what does homogeneous Dirichlet’s boundary condition mean, he will certainly answer that this means that the flow does not ”slip” on the boundary of the domain. The answer of a mathematician would be different: homogeneous Dirichlet’s boundary condition is particularly useful because it implies important simplifications for boundary terms during integration by parts, because we know very well the spectrum basis for the Laplace problem with this boundary condition, because this boundary condition allows the L2-norm of the gradient to be equivalent to the W1,2-norm and so on. We could then think that this difference between physicists and mathematicians would mean that nowadays, the method used by Euclid is closer to the one of physicists than to the one of mathematicians because physicists would keep in mind the object on which they are working on (the fluid). However, this would not be perfectly true. It is clear that nowadays, the object on which mathematicians work is not the fluid but a partial differential system which is supposed to describe the fluid. Nevertheless, a hundred years of mathematical work on incompressible Navier Stokes equations provides the mathematicians with a great number of well known tools in order to deal with mathematical fluids. One of the main differences in comparison with the physicist’s approach is that the object on which mathematicians work (Navier Stokes System) does not determine them in the same way as the physical intuition of a fluid determines physicists. What we can call the mathematical intuition of the fluid is determined by the panel of tools used to treat the equations successively selected during the mathematical history of the mathematicians’ work on the fluid. What we can call the physical intuition of the fluid is determined by the way physicists describe the fluid by observing, experimenting and thinking it.

Our work is clearly a mathematical one and we then affiliate to the mathematical way of thinking fluids.

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Part I

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1 Introduction and notation of function spaces.

1.1 General considerations.

The Navier–Stokes equations are most frequently used as a model for fluid flows as liquids or gases and they have an important part in the analysis of many physical applications (en-gineering, meteorology, oceanography, geophysics, astrophysics, nuclear reactors, ...).

Two equations govern the motion of a viscous compressible fluid. One describes the conser-vation of mass, the other describes the conserconser-vation of momentum. The conserconser-vation of mass leads to the said equation of continuity:

∂tρ + div ρu = 0 (1.1)

where ρ = ρ(t, x) is the density of the fluid at point x and time t, u = u(t, x) = (u1(t, x),

u2(t, x), u3(t, x)) is the velocity of the fluid particle passing through the point x = (x1, x2, x3)

at time t. The components x1, x2, x3 of the reference x are called Eulerian coordinates. The

vector field of the corresponding accelerations is expressed as

Γ(t, x) = Γ = ∂tu + (u· ∇)u. (1.2)

The complexity of the Navier–Stokes equations stands in the last equations. Firstly because of the non linear quadratic term in u (1.2) and secondly because of the hyperbolic structure of the equation of continuity (1.1).

The law of conservation of momentum leads to the equation

ρΓ = DivT + f (1.3)

where f = f (t, x) represents the volume force acting at time t on point x and T is the stress tensor.

An important class of fluids that takes a central place in mathematical theory is represented by linearly viscous fluids for which T depends linearly on the symmetric part of the velocity gradient D(u). These fluids are called Newtonian fluids. Their stress tensor has the form

T_ij = T_ji = 2µD(u)_ij+ λdiv u δ_ij − p δ_ij (1.4)

where µ and λ are two constants, respectively the dynamic viscosity and the volume viscosity, that satisfy µ ≥ 0 and 2µ + 3λ ≥ 0, and p is the characteristic pressure of the considered fluid, which is a scalar function of thermodynamical quantities.

Remark 1.1

In the most general case, the coefficients of viscosity µ and λ are not constant but are quantities that may depend on the values of other state variables such as density (ρ) and temperature (ϑ). Experiments show that the viscosity of fluids is quite sensitive to changes in temperature ϑ. From this point of view, there is a major difference between gases and liquids: viscosity of gases increases with temperature, whereas viscosity of liquids decreases

(see e.g. Chapter 10 in [8]).

Remark 1.2

A state law is needed in order to express the pressure: the barotropic nature of the fluid leads to a relation between ρ and P in the form P = P (ρ) = Cργ under the thermodynamical assumptions that the specific heat is constant in adiabatic evolution. Remark 1.3

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The natural situation is not adiabatic and the well known convection motions (hot air goes up letting its place to cold air) enforce to introduce a third equation describing the temperature

(this one is not taken into account in our work).

As a result, we obtain the following system as a model for compressible barotropic fluids: ρ(∂tu + (u· ∇)u) = µ∆u + (µ + λ)∇div u − ∇P (ρ) + f ,

∂tρ + div (ρu) = 0. (1.5)

It remains to describe the limit conditions on the boundary of Ω, the container of the fluid, and the initial conditions.

The above system is then spatiotemporal and non linear. Its resolution requires to find ρ and u since P (ρ) is already given. A simpler case is the linearized system of Stokes equations in the steady case where the whole physical quantities do not depend on time:

0 = µ∆u + (µ + λ)∇div u − ∇P (ρ) + f ,

div (ρu) = 0. (1.6)

This system is particularly useful in the case of low velocities but it is just a formal approxi-mation of the Navier–Stokes system.

If we introduce the vector field of vorticity w = curlu =∇ × u, it is possible to equivalently rewrite the stationary non linear model as follows:

ρw× u = −µcurl2u + (2µ + λ)∇div u − ∇(P (ρ) + 1 2|u|

2_{) + f ,}

div (ρu) = 0. (1.7)

The famous vortex shedding in the flow of an incompressible fluid following a cylinder which has a uniform translation motion in the fluid is sufficient to point out the importance of the quantity w; as it is well known, turbulent flows are generally strongly rotational.

Classical references for the modelling of fluid flows are obviously the works made by H.Navier [39] (1827) and G.G. Stokes [53] (1845). For more details, please refer to the book of L.D. Landau and E.M. Lifschitz [26] (1959).

In the theory of partial differential equations, the Navier Stokes equations have taken a very important place due to the complex method of analysis they require, the rich and various properties they allow to obtain, and the plurality of open problem they involve. For exam-ple, either the global in time existence of smooth solutions to Navier–Stokes equations, or the possibility of a breakdown in Navier–Stokes solutions has been proposed as one of the seven problem to solve for the millennium by the Clay Mathematical Institute (see e.g. C.L. Fefferman [17]).

75 years ago, Jean Leray wrote the first pages of the modern theory for the Navier Stokes equation [27], [28] and [29] (1931–1934). His contribution has initiated a wide interest for the study of the existence, the unicity and the regularity of solutions to the Navier–Stokes equations. Without claiming for completeness, it appears us essential to quote the contri-butions of E. Hopf [24] (1951), J.L. Lions and G. Prodi [31] (1959), H. Fujita and T. Kato [20] (1962), J. Serrin [50] (1963), K. Masuda [36] (1967), O. Ladyzenskaia [25] (1969), R. Temam [54] (1977), V.A. Solonnikov and A.V. Kazhikhov [52] (1981), L. Caffarelli, R. Kohn and L. Nirenberg [5] (1982), P. Constantin and C. Foias [10] (1988), P.L. Lions [33] (1993), G.P. Galdi [21] (1994),..., and to apologize for the many other unquoted mathematicians. An impressing number of contributions has appeared during the last two decades (see e.g. the bibliography in the two books of P.L. Lions [34] (1996) and [35] (1998) and in the book of E.

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Feireisl [15] (2004)): we can say that today, the works of the French and the Czech Schools enable to understand how to analyze weakly the compressible model.

In order to find a weak solution to the boundary value problems involving the aforementioned systems of equation, the method is to look for an equivalent integral formulation of the prob-lem. For example, let us choose the model (1.5) with ρ = 1 and div u = 0 completed with the Dirichlet boundary condition and the initial condition u|t=0= u0. We have:

dt Z Ω u(t, x)· v(x) dx + 3 X i,j=1 h µ Z Ω ∂jui(t, x)∂jvi(x) dx + Z Ω (uj∂jui)(t, x)vi(x) dx i = Z Ω f (t, x)· v(x) dx with Z Ω u(0, x)· v(x) dx = Z Ω u0(x)· v(x) dx (1.8)

for any test function v ∈ W1,20 (Ω)∩ L2σ(Ω) (see subsection 1.3 for the definitions of these

spaces).

There is an energy inequality that is attached to this integral formulation: when v = u, we have: dt 2ku(t)k 2 0,2+ µk∇u(t)k20,2= Z Ω f _{· u dx.} (1.9)

This provides some a priori estimates for every sequence of approximations approaching u. It is the standard scheme for the studies in some subspaces of L2(Ω). It is possible to consider a similar scheme in some subspaces of Lp(Ω). The properties of weak convergence for subsequences are not sufficient in order to justify all the limit passages in (1.8). This requires a hard and technical work. As a result, we obtain a generalized solution (called a weak solution). Actually, the obtained properties of regularity for u in terms of integrability and differentiability are too weak and we can just interpret u as a generalized solution (L2(Ω)– generalized solution or Lp(Ω)–generalized solution) to the system

∂tu− µ∆u + (u · ∇)u − f = ∇p,

div u = 0, (1.10)

in the sense of distributions on (0, T )× Ω completed by u|∂Ω= 0 and u|t=0= u0. A classical

solution would be of class C2 with respect to the space variable x.

Obtaining a sort of energy inequality for the compressible model is even less simple. We have: dt Z Ω 1 2ρ|u| 2 _{dx +}Z Ω ργ dx+ µk∇u(t)k20,2+ (µ + λ)kdiv uk20,2= Z Ω f · u dx. (1.11) It is easy to deduce from the form of the last inequality the difficulties of deducing sufficient a priori estimates enabling the justification of the limit passages in the integral formulation of the model. This is corroborated by the fact that concerning the non linear terms, one has to treat the terms ρ7→ ργ_{, and (ρ, u)}_{7→ div (ρu ⊗ u)... The work is very hard and technical but}

well documented through the contributions of P.L. Lions [35] (1998), A.Novotn´y, I. Straˇskraba [46] (2003) and E. Feireisl [15] (2004). The control of the oscillations for ρ is crucial.

The choice of the function spaces ensures the required integrability properties (treating the encountered integrals as finite quantities). The treatment of the boundary conditions restricts the function spaces to the domains of the considered operators. For example, D(−∆D) =

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W1,2₀ (Ω)∩ W2,2(Ω) for a Laplacian with the Dirichlet boundary conditions, and D(curl∗₂) = D1,2_σ (Ω) for a curl–operator such as the one that appears in our work.

In the case of the Dirichlet boundary condition, when Ω is sufficiently regular (C2_{), the}

Hilbert space of reference L2(Ω) admits the decomposition PσL2(Ω)⊕ {u ∈ L2(Ω)/u =

∇Π, Π ∈ W1,2_(Ω)_{} where P}

σ is the Leray Helmholtz projector (in the present case it is

orthogonal) on the divergence free vector field of the functions satisfying the impermeability boundary condition, i.e.: such that their normal component on the boundary of the domain is 0. Pσ is an operator with remarkable properties. This is the reason why it is often

used in the theory of Navier–Stokes equations with some decompositions of the form u = w +∇Π for the velocity fields. Nevertheless, Pσ∆D 6= ∆D, whereas Pσcurl∗2 = curl∗2 and

Pσ(curl∗2)2 = (curl∗2)2(= curl∗2curl∗2(·)) as it has been observed by many authors (see e.g.:

H. Bellout, J. Neustupa and P. Penel in [2] (2004)). Then, since Pσu· n|∂Ω = 0 satisfies

the impermeability boundary condition, W1,2₀ (Ω) is not invariant under the action of Pσ and

we have PσW1,20 (Ω) = D1,2σ (Ω). The choice of the boundary conditions is precious for the

contribution of the coherence of the mathematical theory. Remark 1.4

It is interesting to remark that in the inviscid case (µ = λ = 0), the boundary conditions that have to be joined to the equations are precisely uinviscid· n|∂Ω = 0. These conditions

are induced by the conservation laws, not only through mathematical considerations. In the viscous case, one has to introduce additional boundary conditions. Another very important quantity is the named effective pressure (or effective viscous flux) Π = P (ρ)− (2µ + λ)div u. Its importance has been pointed out during the nineties by D. Serre [51] (1991), P.L. Lions [33] (1993), A.Novotn´y and M. Padula [44] (1994), D. Hoff [23] (1995), A.Novotn´y, M. Padula and P. Penel [45] (1996): Π is mathematically more regular than P (ρ) and div u separately. This regularity implies a form of “orthogonality” between the spacial oscillations of Π and ρ that is for example crucially used in the works of P.L. Lions [35] (1998) and E. Feireisl [16] (2000).

1.2 Introduction to the studied model.

We assume that Ω is a given bounded and simply connected domain in R3 with the boundary of the class C2_{. We introduce the model system which consists of the stationary Navier–Stokes}

equations:

∂j(ρuju) + µ curl2u− (2µ + λ)∇div u + ∇P (ρ) = ρf + g in Ω, (1.12)

div (ρu) = 0 in Ω. (1.13)

where ρ is the unknown density, u = (u1, u2, u3) is the unknown velocity and P (ρ) = ργ is the state law for the pressure (γ ≥ 1). µ and λ are two constants satisfying µ > 0 and 2µ + λ > 0, f = (f1, f2, f3) and g = (g1, g2, g3) are given functions. (f is a specific body force, i.e. the body force related to the unit mass, and g is an external force related to the unit volume.)

For any distribution φ∈ D0(Ω), we have defined

curlφ = ∂φ 3 ∂x2 − ∂φ₂ ∂x3 ,∂φ1 ∂x3 − ∂φ₃ ∂x1 ,∂φ2 ∂x1 − ∂φ₁ ∂x2

and curl2v = curl(curlv).

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boundary conditions for velocity, and vorticity: u_{· n} ∂Ω= 0, (1.14) curl u· n ∂Ω= 0. (1.15)

The Navier–Stokes equations for incompressible fluid with these boundary conditions (we call them the generalized impermeability conditions) were studied in the papers [2] by H. Bellout, J. Neustupa and P. Penel and in [40], [41] and [42] by J. Neustupa and P. Penel. The Navier– Stokes equations are usually considered with the no–slip boundary condition (or in other words homogeneous Dirichlet’s boundary condition)

u

∂Ω= 0, (1.16)

which in fact involves three scalar equations. On the other hand, if we only consider the im-permeability condition for velocity (1.14) and we assume that the law of conservation of linear momentum holds “up to the boundary” of Ω, we can naturally obtain, as a “complementary boundary condition” to (1.14), the following

T· n

τ+ k tnuτ = 0 (1.17)

where the subscript τ denotes the tangential component, T is the stress tensor, k is the coefficient of friction between the fluid and the wall and tn= n·T·n is the normal component

of the normal stress acting on ∂Ω. (Many authors consider only the case k = 0 or they simply put tn = 1.) The conditions (1.14) plus (1.17) are called Navier’s boundary conditions.

Observe that the conditions (1.14) plus (1.17) also represent three scalar equations. Now it is obvious that if we prescribe the velocity u to satisfy the two conditions (1.14) and (1.15) on ∂Ω, we find ourselves between Dirichlet’s condition (1.16) and Navier’s condition (1.14). We will show later that the conditions (1.14) and (1.15) can be naturally completed by the third scalar condition

curl2u· n

∂Ω= 0 (1.18)

so that any sufficiently smooth solution of the problem (1.12), (1.13) in fact also satisfies three scalar conditions on the boundary of Ω (i.e. the conditions (1.14), (1.15)) and (1.18). In order to avoid the under-determination of ρ, we need to add the following conditions:

ρ_{≥ 0,} Z

Ω

ρ dx = m (1.19)

where m is a given real positive number (the total mass of the fluid in Ω).

Let us note that if curl f = 0 in Ω then the problem (1.12)–(1.15) has a solution u = 0 and ρ, where ρ satisfies the equation∇ργ_{= ρf . This equation is equivalent to}_∇ργ−1_{= (γ}_{−1)f /γ}

and the solution has the form ρ(x) = γ − 1

γ F (x) + C _γ−11

+

where F is the potential to f and C is an arbitrary constant.

The study of system (1.12)–(1.15) and (1.18), (1.19) will later show that the condition ρ≥ 0 is in fact not a constraint.

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In our work, we first prove that the main theorems obtained by P.L. Lions [35] and A.Novotn´y, I. Straˇskraba [46] remain true with boundary conditions (1.14) and (1.15).

Let us here briefly explain the main ideas of the proof.

We first use the Helmholtz decomposition of the velocity field in section 2. This changes the model with the unknowns (ρ, u) into a new one with the unknowns (ρ, v, ϕ, Π) by dividing it into a Stokes problem for (v, Π), a Laplace–Neumann problem for ϕ and a degenerate hyperbolic equation of continuity for ρ.

In order to relax the degeneracy and to moreover obtain an elliptic equation, we use in section 3 two auxiliary parameters α and ε.

In order to obtain our solution with a pressure P (ρ) = ργ with minimal value for γ, we change our law of pressure into an artificial pressure P (ρ) = ργ+ δρβ by introducing two other parameters δ and β.

The introduction of the last four parameters is based on an idea we have found in A.Novotn´y, I. Straˇskraba [46].

For each quadruplet (ε, α, δ, β), using a fixed point procedure, we are able to obtain a solution (ρε,α,δ,β, vε,α,δ,β, ϕε,α,δ,β, Πε,α,δ,β) to our approximated model in good Sobolev spaces.

It remains to pass to the limit for ε, α and δ successively tending to zero. The first limit process (ε → 0) automatically involves a lack of regularity for the element ρ0,α,δ,β because

this element, contrary to ρε,α,δ,β (for ε6= 0), does not satisfy any elliptic equation. This lack

of regularity has at least two bad consequences: it implies a similar lack of regularity for the velocity (see Theorem 2.1 for details), and it does not provide with sufficient a priori estimates for the sequences of densities {ρε,α,δ,β}ε to converge strongly when ε→ 0.

Fortunately, the effective pressure Π0,α,δ,β enables through very technical consideration due

to P.L. Lions [35] to nevertheless obtain the strong convergence of{ρε,α,δ,β}ε to ρ0,α,δ,β when

ε → 0. We then repeat this procedure when α → 0. We refer to the introductions of our Subsections 5.5 and 5.10 for more details concerning these limit process.

It remains to let δ → 0. Unfortunately, for γ < 5

3, the integrability of ρ0,0,0,0 (the weak limit

of ρ0,0,δ,β when δ → 0) is not sufficient to use the effective pressure as in the two previous

cases. Using an idea from A.Novotn´y, I. Straˇskraba [46], we then introduce a sort of cut–off function of the density. It is clear that the obtained truncated density belongs to L∞-spaces. What is more interesting is to observe, as it is done in P.-L. Lions ([35], p.67), that the measure of the domain on which ρ0,0,0,0 and the truncated density are different is very small.

This has two important consequences.

Firstly, it enables to use the effective pressure Π0,0,0,0 and to obtain the strong convergence

of the truncated density in a similar way as it is done when {ρε,α,δ,β}ε → ρ0,α,δ,β and when

{ρ0,α,δ,β}α → ρ0,0,δ,β. Since the truncated density and ρ0,0,0,0 are very close, some technical

arguments allow to conclude to the strong convergence of {ρ0,0,δ,β}δ → ρ0,0,0,0 when δ → 0

and complete the extension of the results of P.L. Lions–Novotn´y-Straˇskraba to the generalized impermeability boundary conditions. We refer to the introduction of our Subsection 6.5 for more details concerning this last limit process.

Secondly, it appears reasonable to build a new model of approximation for the compressible barotropic Navier–Stokes Equations in such a way that this model involves bounded densities as solutions. This is done by modifying the classical equations through the introduction of a truncated off function in front of the density. Since we have discovered that the measure of the domain on which the density that solves the classical model and the truncated density are different is very small. The solutions of our new model will be very close to the solutions of the classical model.

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the advantage of the originality, as in the meantime, P.B. Mucha and M. Pokorny developed the same analysis with Navier’s boundary conditions.

We first find a solution (ρε, vε, ϕε, Πε) to our new approximated model in a similar way as

we have solved our first problem of approximations (all the details are given in Section 7). Secondly, we find some a priori estimates independent of ε. More interesting, we find some a priori estimates independent of the truncated off function. This is extremely important because it implies that our model does not just provide us with solutions that are close to solutions of the classical model, but it provides us with some solution (ρ, v, ϕ, Π) of the classical model with a bounded density (actually ρ∈ L∞_{(Ω)). This implies that the result}

obtained by P.B. Mucha and M. Pokorn´y in [37] with the Navier’s boundary conditions remains true with the impermeability boundary conditions. Moreover, we prove that the generalized boundary conditions enable to improve the regularity of the incompressible part of the velocity and of the effective pressure (see Theorem 2.2 for details).

1.3 Definition and basic properties of some function spaces

We denote by capital boldface letters spaces of vector–valued functions. The norm in the Sobolev spaces Wk,p_{(Ω) or W}k,p_{(Ω) is denoted by} _{k k}

k,p. Particularly,

k k0,p denotes the norm in the Lebesgue spaces Lp(Ω) or Lp(Ω).

For any set M denote by C0∞(M ) or D(M) the set of all functions that are infinitely

differentiable with compact support in M . We denote by D0(M ) the dual ofD(M). We recall the usual Sobolev embedding theorem which says that for integers k ≥ 0, Ω a

bounded Lipschitz domain and 1≤ p ≤ ∞, we have the continuous embedding: Wk,p(Ω) ,→ Lq_(Ω) _for _{q =} 3p 3− kp, k < 3 p. (1.20) Wk,p(Ω) ,→ Lq(Ω) for 1≤ q < ∞, k = 3 p. (1.21) Wk,p(Ω) ,→ Lq(Ω) for 1≤ q ≤ ∞, k > 3 p. (1.22)

This particularly implies that there exists a constant C_0,qk,p> 0 (depending only on Ω) such that for all u_{∈ W}k,p(Ω), we have

kuk0,q ≤ (C0,qk,p)

1

2 kuk_k,p. (1.23)

(exponent 1₂ is used in order to simplify the notations for the constants all along the thesis). WNk,p(Ω), k ≥ 2, 1 < p < +∞, is the space of functions ϕ ∈ Wk,p(Ω) that satisfy the

Neumann boundary condition (∇ϕ · n)|∂Ω= 0 in the sense of traces .

W_N2,p(Ω, R3), 1 < p < +∞, is the space of functions of the form Eϕ, where ϕ ∈ W_N2,p(Ω) and E is the extension operator which is a bounded linear operator from W2,p(Ω) to W2,p(R3), described e.g. in L. C. Evans [13, p. 257].

We denote by W1,p_{(Ω)/R the quotient space of class of functions ˙}_ϕ_{∈ W}1,p_{(Ω) defined up}

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Lpdiv(Ω) :={g ∈ L

p_{(Ω); div g}

∈ Lp_{(Ω) in the sense of distributions}_}.

The norm in Lp_div(Ω) isk kLp_div =k k0,p+kdiv ( )k0,p. It is proved in R. Temam [54] that

if Ω is a bounded Lipschitz domain, 1 < p <∞ and g ∈ Lpdiv(Ω), there exists a continuous

operator of traces from the space Lp_div(Ω) which assigns to each function g from Lp_div(Ω) the normal component of g on ∂Ω: g· n ∈ (W1−p01,p0_(∂Ω))∗_{= W}−

1

p,p_(∂Ω).

Lp0,div(Ω) is the closure of C∞0 (Ω) in L p

div(Ω). We have the following characterization:

g∈ Lp0,div(Ω) iff g ∈ L p

div(Ω) and g· n = 0 on ∂Ω in the sense of traces.

C∞0,σ(Ω) is the space of divergence–free vector function from C∞0 (Ω).

Lp

σ(Ω), 1 < p < +∞, is the closure of C∞0,σ(Ω) in Lp(Ω). It is the Banach space of

divergence–free (in the sense of distributions) vector functions u in Lp(Ω) such that u·n = 0 on ∂Ω in the previous sense. We also denote by Pσ the orthogonal projection from Lp(Ω)

onto Lp_σ(Ω). D1,p

σ (Ω), 1 < p < +∞, is the space of functions u ∈ W1,p(Ω)∩ Lpσ(Ω), that satisfy the

additional boundary condition curl u· n = 0 on ∂Ω in the previous sense of traces (note that since div curlu = 0 in the sense of distributions, curlu ∈ Lpσ and it has a normal

trace). D1,p

σ (Ω, R3), 1 < p < +∞, is the space of functions in W1,p(R3) that have the form Eu

(for u ∈ D1,p

σ (Ω)), where E is again the extension operator, see e.g. L. C. Evans [13,

p. 254].

D−1,pσ (Ω) is a dual to D1,pσ (Ω).

C∞curl(Ω) is the space of the functions from ψ∈ C∞(Ω) that moreover satisfy ψ· n|∂Ω=

curlψ· n|∂Ω= 0. These functions admit the decomposition ψ = ψ1+∇ψ2 for (ψ1, ψ2)∈

D1,2_σ (Ω)_{∩ C}∞(Ω)_{× W}_N2,2(Ω)_{∩ C}∞(Ω) and they clearly form a dense subset of the space D1,2_σ (Ω) + W_N2,2(Ω).

We denote by curl∗pthe restriction of operator curl to the space D1,pσ (Ω). It was proved by

Z. Yosida, Y. Giga [55], R. Picard [47] and H. Bellout, J. Neustupa and P. Penel [2] that operator curl∗₂ is selfadjoint in D1,2_σ (Ω). Moreover, the norm _kcurl∗₂._k0,2 is equivalent

to the norm k . k1,2 in D1,2σ (Ω). In particular, there exists CBN P > 0 such that for all

u∈ D1,2 σ (Ω), we have kuk1,2 ≤ C 1 2 BN P √ µ kcurl∗2uk0,2 (1.24)

(√µ is used in order to simplify the notations for the constants all along the thesis). We shall not further use the notation curl∗_p consistently and we shall often write only curl, even if the argument of curl belongs to D1,p_σ (Ω).

For 1 ≤ q ≤ +∞, we denote by q0_{the conjugate exponent and by q the Sobolev embedding}

exponent, i.e. q0:=      q q−1 if 1 < q <∞ 1 if q =∞ ∞ if q = 1 q :=      3q 3−q if q < 3 arbitrary finite≥ 1 if q = 3 ∞ if q > 3. (1.25)

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2 The studied model and the main results of the thesis.

2.1 From a classical formulation to the weak formulation.

We now assume that ρ, u is a classical solution of the problem (1.12)–(1.15), (1.18). (The classical solution is a solution which is continuous in Ω together with all derivatives which appear in the equations (1.12), (1.13) and it satisfies these equations at all points of Ω. Moreover, the classical solution also admits the continuous extension of the expressions on the left–hand sides of (1.14) and (1.15) to the boundary of Ω so that (1.14), (1.15), (1.18) hold.)

Formally, the Helmholtz decomposition of u is

u = v +∇ϕ (2.1)

where ϕ is smooth and solves the Neumann problem

∆ϕ = div u in Ω, (2.2) ∂ϕ ∂n ∂Ω= 0 (2.3)

and v := u− ∇ϕ is also smooth. We note that the requirement for v and ϕ to be smooth is just formal (see Definition 2.1 below for the spaces in which we look for v and ϕ).

This decomposition guarantees that div v = 0 in Ω and v also satisfies the boundary condi-tions (1.14), (1.15) and (1.18). If we use decomposition (2.1) of u in equacondi-tions (1.12) and (1.13), we obtain

∂jρ (vj+ ∂jϕ) (v +∇ϕ) + µ curl2v +∇−(2µ + λ)∆ϕ + P (ρ)

= ρf + g in Ω, (2.4)

divρ (v + ∇ϕ)) = 0 in Ω. (2.5)

This system is now completed by the boundary conditions v· n

∂Ω= 0, (2.6)

curl v· n

∂Ω= 0 (2.7)

for function v, and by condition (2.3) for function ϕ.

We moreover complete this system with the third scalar boundary condition: curl2v_{· n}

∂Ω= 0 (2.8)

This condition is actually satisfied for every “sufficiently smooth” solution of system (2.3)-(2.7) (see Subsection 2.2.2 below).

Let us first consider a test function φ from space D1,2_σ (Ω) and let us multiply equation (2.4) by function φ and integrate on Ω. Since v is in the domain of the operator (curl∗₂)2 and φ is in the domain of curl∗₂ (i.e. curl∗₂φ = curlφ by definition), we can use the selfadjointness of operator curl∗₂ and we can transform the integral of curl2v· φ (which equals (curl∗2)2v· φ)

to the integral of curl∗₂v· curl∗2φ. Using also the integration by parts, we obtain the integral

identity Z Ω −ρ (vj_{+ ∂} jϕ) (v +∇ϕ) · ∂jφ + µ curl∗2v· curl∗2φ dx = Z Ω (ρf + g)· φ dx. (2.9)

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(We have used the fact the any function in L2(Ω) that is a gradient of another function is orthogonal to D1,2_σ (Ω) in the scalar product of L2(Ω).)

Secondly, our next step is to consider a “smooth” test function which has the form of a gradient with the normal component on the boundary of Ω being equal to zero. I.e. it equals∇ψ with ∂ψ/∂n

∂Ω= 0. (The set of all these test functions is dense in the orthogonal

complement to L2_σ(Ω) in L2(Ω).) Multiplying equation (2.4) with ∇ψ, integrating on Ω and applying the integration by parts, we obtain

Z Ω −ρ (vj_{+ ∂} jϕ) (v +∇ϕ) · ∂j∇ψ + (2µ + λ) ∆ϕ ∆ψ − P (ρ) ∆ψ dx = Z Ω ρf · ∇ψ + g · ∇ψ dx. (2.10)

(The orthogonality of∇ψ to L2σ(Ω) implies that the integral of the product curl2v· ∇ψ on

Ω equals zero because curl2v = (curl∗₂)2v ∈ L2

σ(Ω), see e.g. the development in Technical

Lemma A.1 (A.1) in Appendix for details.) The main advantage of integral identity (2.10) is to take into account the terms ∆ϕ and P (ρ).

Thirdly, considering a “smooth” scalar test function ζ in Ω (not necessarily having a compact support in Ω), multiplying the equation of continuity (2.5) by ζ and integrating on Ω, we arrive at

Z

Ω

ρ (v +∇ϕ) · ∇ζ dx = 0. (2.11)

(We have used the boundary conditions (2.3) and (2.6) in the integration by parts and the surface integral on the boundary of Ω therefore equals zero.)

Now we can write down the weak formulation of the classical problem (2.3)–(2.8).

Definition 2.1 (the weak formulation) Given f ∈ L∞(Ω), g∈ L6/5(Ω), γ ≥ 1, m ∈ R (m > 0) and function P (τ ) = τγ (for τ ≥ 0).

The triplet (v, ϕ, ρ) is said to be a weak solution to the problem (2.3)–(2.8) if • v ∈ D1,2σ (Ω),

• ϕ ∈ W2,2_{(Ω), ϕ satisfies the boundary condition (2.3),}

• ρ ∈ Lγ_(Ω)_{∩ L}3/2_{(Ω), ρ}_{≥ 0 a.e. in Ω,} R

Ωρ(x) dx = m

and the integral identities (2.9)–(2.11) hold for all test functions φ ∈ D1,2

σ (Ω)∩ C∞(Ω),

ψ_{∈ C}∞(Ω) such that ∂ψ/∂n

∂Ω= 0 and ζ∈ C∞(Ω).

Remark 2.1

From this Subsection, adding (2.9) and (2.10) we particularly conclude that a strong formu-lation of the form

∂jρ (vj+ ∂jϕ) (v +∇ϕ) + µ curl2v +∇−(2µ + λ)∆ϕ + P (ρ)

= ρf + g in Ω, implies a weak one of the form

Z Ω−ρ (v j_{+ ∂} jϕ) (v +∇ϕ) · ∂jψ + µ curl∗2v· curl∗2ψ + Π ∆ψ dx = Z Ω

(ρf + g)· ψ dx. for all ψ ∈ C∞curl(Ω)

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2.2 From the weak formulation to the classical formulation.

Now we assume that (v, ϕ, ρ) is a weak solution of the problem (2.3)–(2.8) according to Definition 2.1. We show in this Subsection that if this solution is “sufficiently smooth” then the pair u≡ v + ∇ϕ, ρ is a classical solution of the problem (1.12)–(1.15), (1.18),(1.19). 2.2.1 Equations (1.12) and (1.13).

Let us at first consider the integral identities (2.9), (2.10) with the test functions φ and ψ that have a compact support in Ω. Integrating by parts in both the identities and summing them, we obtain Z Ω ∂jρ (vj+ ∂jϕ) (v +∇ϕ) + µ curl2v +∇−(2µ + λ)∆ϕ + P (ρ) · (φ + ∇ψ) dx = Z Ωρf + g · (φ + ∇ψ) dx (2.12) The set of all test functions φ, that have a compact support in Ω, is dense in the space L2_σ(Ω). The set of all functions of the type ∇ψ, where ψ is a test function with a compact support in Ω, is dense in the orthogonal complement to L2_σ(Ω) in L2(Ω). Thus, the sums φ +∇ψ are dense in L2(Ω). Consequently, (2.12) implies that the functions v, ϕ and ρ satisfy equation (2.4) a.e. in Ω. According to (2.1), this is exactly the equation (1.12) for u. Considering test functions ζ with a compact support in Ω in the integral identity (2.11), we can show in the same way that the functions v, ϕ and ρ also satisfy the equation of continuity (2.5) everywhere in Ω, which is exactly equation (1.13) for u.

2.2.2 Boundary conditions (1.14), (1.15) and (1.18).

The fact that function ϕ satisfies the Neumann boundary condition (2.3) is already involved in the definition of the weak solution. The validity of the two conditions (2.6) and (2.7) directly follows from the condition v _{∈ D}1,2_σ (Ω). This implies that u satisfies the boundary conditions (1.14) and (1.15).

Thus, we still need to show that “sufficiently smooth” weak solutions v + ∇ϕ (actually, conditions v _{∈ W}2,2(Ω) and_{∇((2µ+λ)∆ϕ−P (ρ)) ∈ L}2(Ω) are needed) satisfy the boundary condition (2.8). We therefore return to the integral identities (2.9), (2.10) with general test functions φ, ψ. (General in the sense that they satisfy all the requirements from Definition 2.1, but they do not need to have a compact support in Ω.) Integrating again by parts, summing the identities, using (2.3), (2.6), (2.7) and finally, using also the already derived information that v, ϕ, ρ satisfy equation (2.4), we arrive at

Z

∂Ω

µ curl v· (n × φ) dS = 0. This equality can be rewritten in the form

Z

∂Ω

(φ× curl v) · n dS = 0. (2.13)

Since φ∈ D1,2σ (Ω), it coincides on ∂Ω with a gradient of some function η∈ W2,2(Ω) : φ =∇η

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as in [2]: 0 = Z ∂Ω (∇η × curl v) · n dS = Z Ω div (∇η × curl v) dx = Z Ω∇η · curl 2_v − curl ∇η · curl v dx = Z Ω ∇η · curl2v dx = Z Ω div (curl2v η) dx = Z ∂Ω n· curl2_{v η dS.}

Since φ is arbitrary, η is arbitrary, it follows that the normal component of curl2v is zero on the boundary of Ω, which means that v satisfies (2.8) and u satisfies (1.18).

The described procedure ensures that the weak formulation of the problem (2.1)–(2.7) contains the whole relevant information on the functions v, ϕ and ρ. In other words: no important information on the functions v, ϕ, ρ was lost during the formal derivation of the weak formulation from the classical one. This concerns especially the third boundary condition (2.8) imposed on function v: although this condition does not explicitly appear in the weak formulation, this formulation involves it implicitly.

Remark 2.2

Looking more at v, the incompressible part of the velocity field u, we observe that the chosen variational form in (2.9) with v and φ in D1,2_σ (Ω) leads to the third boundary condition curl2v· n|∂Ω= 0, and consequently, curl2u· n|∂Ω= 0. We now observe the same integral

identity (2.9) with v and φ in L2_σ(Ω)∩ W1,20 (Ω): in this situation, the Dirichlet boundary

conditions for v are equivalent (following [43]) to the three scalar conditions v· n|∂Ω= 0, curlv· n|∂Ω= 0, ∂nv· n|∂Ω= 0.

so, it is clear that the condition curl2u· n|∂Ω= 0 cannot be obtained.

2.3 The main theorems.

The main results of this text are formulated in the following theorems on the existence of weak solutions of the problem (2.1)–(2.7). (The weak solution is introduced in Definition 2.1.)

Theorem 2.1 For Ω a C2 _{and simply connected bounded domain in R}3_{, γ >} 3

2, µ > 0,

2µ + λ > 0, given f ∈ Lr(Ω)∩ Lz(Ω) and g ∈ Lr(Ω) with 3 < r ≤ ∞ and z > 3γ6−5 if 5

3 < γ < 7

3, z = r on the contrary, such that curlf = 0 if 3

2 < γ < 5

3, there exists a weak

solution (ρ, v, ϕ) ∈ Lγ_(Ω)_{∩ L}3/2_(Ω)_{× D}1,2

σ (Ω)× W 2,2

N (Ω) to problem (2.1)–(2.7) such that

moreover ρ_{∈ L}s(γ)_{(Ω) where} s(γ) := ( 3(γ− 1) if 3 2 < γ < 3 2γ if γ ≥ 3

Theorem 2.2 For γ > 3, µ > 0, 2µ + λ > 0, given f ∈ Lr_{(Ω) and g}_{∈ L}r_{(Ω) (3 < r} _{≤ ∞),}

there exists a weak solution (ρ, v, ϕ)∈ Lγ_(Ω)_∩L3/2_(Ω)_×D1,2

σ (Ω)×W 2,2 N (Ω) to problem (2.1)– (2.7) such that (ρ, v, ϕ)∈ L∞(Ω)× D1,pσ (Ω)× W 2,p N (Ω), 1≤ p < ∞. Moreover, v ∈ D 2,r σ (Ω) and Π := (2µ + λ)∆ϕ− ργ_{∈ W}1,r_{(Ω) if r <}_{∞ and v ∈ D}2,p σ (Ω) and Π := (2µ + λ)∆ϕ− ργ∈ W1,p(Ω), for all 1≤ p < ∞ if r = ∞.

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Remark 2.3

General thermodynamical considerations while deriving the momentum equation (1.12) im-pose the conditions µ > 0 and 2µ + 3λ≥ 0. Conditions imposed on µ and λ from the point of view of existence theory in the case of Dirichlet boundary conditions (1.16) are µ > 0, 4µ + 3λ > 0 (see e.g. A. Novotný and I. Straˇskraba [46] Lemma 4.32); those in the case of Navier’s boundary conditions (1.14), (1.17) they are µ > 0 and 2µ + 3λ > 0 (see e.g. P. B. Mucha and M. Pokorný [37] Theorem 1). In our case, we impose µ > 0 and 2µ + λ > 0. It is interesting to see that this restriction is weaker than the three others. The proof of Theorem 2.1 follows the analysis published in the books by P.-L. Lions [35] and A. Novotný and I. Straˇskraba [46], see Sections 3–6.

The proof of Theorem 2.2 is given in Sections 7 and 8 and follows the recent works by P. B. Mucha and M. Pokorn´y ([37]).

The appropriate approximations constructed in Section 3 and in Section 7 are strong solu-tions of mentioned boundary value problems, while the resulting limits, respectively obtained as the auxiliary parameters tend to zero, only represent weak solutions.

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3 Mathematical preliminaries.

3.1 Useful tools from functional analysis and theory of partial differential equations

Lemma 3.1 Lebesgue dominated convergence theorem in Lp-spaces

Consider a sequence of functions fn ∈ Lp(Ω) that converges a.e. to a function f. Suppose

moreover that there exists a function g∈ Lp_{(Ω) such that sup}

n∈N |fn(x)| ≤ g(x) for a.a. x ∈

Ω. Then fn−→ f in Lp(Ω).

This Lemma is a general formulation of the Lebesgue dominated convergence theorem in Lp-spaces (see J. P. Aubin [1, Theorem A], p. 123).

Lemma 3.2 (on the Bogovskii operator) (A. Novotn´y and I. Straˇskraba [46, p. 174, Corrolary 3.20]) Let Ω be a bounded Lipschitz domain in R3 and 1 < p < +∞. Then there exists a linear operatorBΩ = (BΩ1,B2Ω,B3Ω) with the following properties:

BΩ : Lp(Ω) := n f ∈ Lp(Ω); Z Ω f dx = 0o −→ W1,p0 (Ω), (3.1) divBΩ(f ) = f a.e. in Ω f or f ∈ Lp(Ω), (3.2) k∇BΩ(f )k0,p ≤ C(p, Ω) kfk0,p. (3.3)

If, moreover, f = div g, where g∈ Lp0,div(Ω), then

kBΩ(f )k0,p ≤ C(p, Ω) kgk0,p. (3.4)

This Lemma enables us to test the momentum equation (2.2) with elements of the form B(ρs₋R

Ωρ

s _{dx) and to obtain estimates on ρ, v and ϕ independently of some parameters}

(see e.g. Subsections 6.1 and 8.2).

The proof of the following Lemma can be found e.g. in A. Novotn´y and I. Straˇskraba [46, p. 211, Lemma 4.27])

Lemma 3.3 (the Neumann problem for the Laplacian) Let Ω be a C2 bounded do-main, 1 < p <∞ and b ∈ Lp_0,div(Ω). Then there exists ρ∈ W1,p_{(Ω) satisfying}

Z Ω ∇ρ · ∇ζ dx = − Z Ω b· ∇ζ dx ∀ ζ ∈ C∞(R3). (3.5)

Moreover, ∇ρ ∈ Lp0,div ∩ W1,p(Ω) and

||∇ρ||0,p ≤ C(p, Ω) ||b||0,p, (3.6) ||∇ρ||1,p ≤ C(p, Ω) ||b||Lpdiv. (3.7)

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The lemma gives us the information on the weak solvability of the Neumann problem

∆ρ = −div b in Ω,

∂ρ

∂n = 0 on ∂Ω (3.8)

where b is a given function in Lp_0,div(Ω). We will use it later in order to obtain a solution of equation (4.3). We shall therefore have to choose function b so that div b =−αρ + αh − div [ρ(v +∇ϕ)]. (See the proof of Lemma 4.1.)

Lemma 3.4 (on positive and negative part of ρ) Let 1 < p < +∞ and ρ ∈ W1,p_(Ω).

Then

(i) ρ+ ∈ W1,p_{(Ω) and ρ}− _{∈ W}1,p_(Ω),

(ii) if F ∈ C1_{(R) is such that F}0 _{is bounded, then F (ρ)}_{∈ W}1,p_(Ω),

(iii) ∂jρ+= ∂jρ a.e. in Ωρ>0 and 0 a.e. in Ωρ≤0,

(iv) ∂jρ−=−∂jρ a.e. in Ωρ<0 and 0 a.e. in Ωρ≥0,

(v) ∂jF (ρ) = F0(ρ)∂jρ a.e. in Ω.

Here, we denote by ρ+ the positive part of ρ and by ρ− the negative part (i.e. the maximum of 0, −ρ). Thus, ρ = ρ+_{− ρ}− _{in Ω. We further denote by Ω}

ρ>0 the set of all points x ∈ Ω

such that ρ(x) > 0. The sets Ωρ≥0, etc, are defined by analogy.

Lemma 3.5 (Lemma of convolution.) (H. Brezis [3, IX 1 and IV.15]) Let 1 ≤ p ≤ ∞. Then, for h∈ L1_(R3_{) and φ}_{∈ L}p_(R3_{), we have that φ}_{∗ h ∈ L}p_(R3_{) and}

kφ ∗ hk0,p,R3 ≤ kφk_0,p,R3 khk_0,1,R3

where ∗ denotes the convolution. Moreover, if h_{∈ W}1,1_{(Ω), then}

φ∗ h ∈ W1,1(R3)∩ Lp(R3) and ∇(φ ∗ h) = φ ∗ (∇h)

Remark 3.1

We use this Lemma in the proof of Lemma 4.2.

The proof of the following Lemma can be found e.g. in H. Brezis [3, IX 19]. Lemma 3.6 the Poincar´e inequality for functions in W1,p₀ (Ω), 1_{≤ p < ∞.}

Let 1≤ p ≤ ∞ and Ω be a bounded Lipschitz domain. Then, for each u ∈ W1,p0 (Ω), we have

the following inequality: kuk0,p ≤ CPk∇uk0,p

for CP = C(p, Ω) (P reads for Poincar´e).

Lemma 3.7 (the Poincar´e inequality) (H. Brezis [3], p. 194, 3 A) Let 1≤ p ≤ ∞ and Ω be a C1-domain. Then, for each u∈ W1,p(Ω), we have the following inequality:

Z Ω |u|pdx ≤ C(p, Ω) Z Ω |∇u|pdx + Z Ω u dx p .

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Lemma 3.8 (equivalence of norms) (V. Girault and P.–A. Raviart [22, Corollary 3.7]) Let Ω be a bounded domain with a C2 boundary. There exists a constant Cnr such that for

any u∈ L2_{(Ω) with div u}_{∈ L}2_{(Ω), curl u}_{∈ L}2_{(Ω) and u}_{· n ∈ W}1₂,2_{(∂Ω), we have:}

kuk1,2 ≤ Cnr kuk0,2+kdiv uk0,2+kcurl uk0,2+ku · nk1 2,2; ∂Ω.

Here we denote byk . k1

2,2; ∂Ω the norm in W 1

2,2(∂Ω).

Lemma 3.9 (H¨older’s inequality.) For Ω a bounded domain, f ∈ Lq1_{(Ω) and g}_{∈ L}q2_(Ω)

with _q1

1 +

1 q2 =

1

r, the following inequality holds:

kf · gk0,r≤ kf k0,q1kgk0,q2.

Lemma 3.10 (Young’s inequality.) For Ω a bounded domain, f _{∈ L}q(Ω), g _{∈ L}q0(Ω) and > 0 (recall 1_q+ _q10 = 1), the following inequality holds:

kf · gk0,1≤ 1 q q_{kf k}q 0,q+ 1 q0 −q0 kgkq_0,q0 0.

Two last inequalities are classical one (see e.g. Brezis [3]).

3.2 The Leray–Schauder fixed point theorem

Theorem 3.1 (the Leray–Schauder fixed point theorem) Let (X,k kX) be a Banach

space and D _{⊂ X a bounded open set. Let H : D × [0, 1] −→ X be a mapping with these} properties:

(i) H(, t) : D −→ X is a compact operator for any t ∈ [0, 1],

(ii) for any κ1> 0, and for any ball B ⊂ D, there exists κ2 > 0 such that:

kH(x, t) − H(x, s)k ≤ κ1, ∀x ∈ B, ∀s, t ∈ [0, 1] such that |s − t| < κ2,

(iii) 0 /∈ (I − H(, t))(∂D), for all t ∈ [0, 1],

(iv) there exists at least one u0 such that H(u0, 0) = u0.

Then, for any t ∈ [0, 1], the problem to find ut ∈ D such that H(ut, t) = ut admits at least

one solution.

This theorem is proved e.g. in the book of I. Fonseca and W. Gangbo [19]. Since this theorem plays an important role for us in order to obtain existence of a triple (ρε,α,δ,β, vε,α,δ,β, ϕε,α,δ,β)

solving an approximate system of problem (2.3)-(2.7), and since we did not find any straight-forward proof of it, we hereafter repeat the main steps of its proof. This allow us to understand deeply the assumptions and the conclusions of this Theorem.

Proof.

(We adopt secondary marking tools ?, and ∗ to distinguish the steps of the proof. We will do likewise in all sections of this thesis.)

F In this whole ?, we always denote by φ a function in C1(D) and by D a domain in RN (N ∈ N). Actually, φ : D ⊂ RN _{→ R}N_.

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A point x_{∈ D is called a critical point of φ if Jφ(x) = 0 (for Jφ(x) := det∇φ(x)).}

We denote by Zφ := {x ∈ Ω, Jφ(x) = 0}. If y /∈ φ(Zφ), y is said to be a regular value of φ. ♦ For any regular value y of φ, φ−1(y) has a finite number of elements.

Otherwise, we could choose any sequence xk → x in the finite dimension compact D with

xk⊂ φ−1(y).

Continuity of φ would give x⊂ φ−1(y). φ∈ C1(D) would imply

0 = φ(xk)− φ(x) = ∇φ(x)(xk− x) + (xk− x)o(kxk− xk2)

and then, for any number ι > 0, there would exist k∈ N, ∇φ(x) xk− x kxk− xk2 2 ≤ ι 2 what contradicts y /_{∈ φ(Zφ).}

♦ First enable us to define the degree of φ at y /_{∈ φ(Zφ) ∪ φ(∂D) with respect to D:} d(φ, D, y) := X

x∈φ−1(y)

sign(Jφ(x)).

♦ Given y a regular point for φ, there exists a C1-neighborhood Bφ of φ such that for all

ψ ∈ Bφ, d(φ, D, y) = d(ψ, D, y).

If φ−1(y) = ∅, then ∃ δ > 0, infx∈D|φ(x) − y|1 ≥ δ. Consequently, for all ψ ∈ C1(D)

satisfying_{kφ − ψk}_C1 ≤ ₂δ, we have: inf_x∈D|ψ(x) − y|₁ ≥ δ₂, i.e.: d(φ, D, y) = 0 = d(ψ, D, y).

If φ−1(y) = {x1, . . . , xm}, let r > 0 and c > 0 be such that the xi are centers of disjoint

open balls B_|·|₁(xi, r) on which |Jφ(x)| > c. r is also chosen for the balls not to intercept

∂D and not to contain any critical point. Let δ > 0 be such that elements ψ of the C1_-ball

BC1(φ, δ) satisfy: sup_x∈D|J_φ(x)− J_ψ(x)| ≤ 1₂c, and then, particularly |J_ψ(x)| ≥ 1₂c. We

define m mappings Wi(·) : B| |1(0, r)→ B| |1(0, r) as Wi(·) := ∇ψ(xi)

−1

y− ψ(xi+· ) +

∇ψ(xi)(·). It can be shown that these mappings are contractive for r and δ sufficiently

small. Consequently, each of them has a unique fixed point, which means that for each ψ ∈ BC1(φ, δ), there exists a unique x0_i ∈ B_|·|

1(xi, r), (1 ≤ i ≤ m) such that ψ(x0i) = y.

Since |φ(·) − y|1 > 0 in D\∪mi=1B|·|1(xi, r) , we have |ψ(x) − y|1 > 0 on the last set for δ

sufficiently small.

Summarizing: ]ψ−1(y) = ]φ−1(y). Finally, |Jφ(x)− Jψ(x)| ≤ 1₂c and |Jφ(x)| > c for x in

the balls B_|·|₁(xi, r) imply sign(Jφ(xi))=sign(Jψ(x0i)), 1≤ i ≤ m i.e.: d(φ, D, y)=d(ψ, D, y).

♦ For any y /∈ φ(∂D)∪Z(φ), there exists ι > 0 that can be chosen as small as necessary such that for scalar functions fι∈ Cc(Rn) satisfying

R

Rnfι(x)dx = 1 and supp(fι)⊂ B|·|1(0, ι), we

have: d(φ, D, y) = R

Dfι(φ(x)− y)Jφ(x)dx. It is an equivalent definition for the degree of

φ at y with respect to D.

If φ−1(y) =∅, we choose 0 < ι < |y − φ(D)|1.

If φ−1(y) = {x1, . . . , xm}. Jφ 6= 0 on each ball B|·|1(ai, r) from previous . From inverse

function theorem, we choose ι0 > 0 such that B_|·|₁(y, ι0) ⊂ φ(B|·|1(xi, r)), 1 ≤ i ≤ m. From

previous we choose 0 < ι < ι0 such that ∀x ∈ B|·|1(y, ι), φ

−1_{(x) is a singleton. We then} have: Z D fι(φ(x)− p)Jφ(x)dx = Z |φ(x)−y|1<ι fι(φ(x)− p)|Jφ(x)|sgn(Jφ(x))dx

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= m X i=1 sgn(Jφ(xi)) Z B_|·|1(0,ι) fι(z)dz = m X i=1 sgn(Jφ(xi)) = d(φ, D, y). ♦ Given f ∈ C1

c(Rn) with support K and ℘ ∈ C1([0, 1]) such that K℘ := {k + ℘(s) : k ∈

K, s∈ [0, 1]} ⊂ D, ∃v ∈ C1

c(D) satisfying div v(x) = f (x− ℘(0)) − f(x − ℘(1)).

For fixed z ∈ R3_{, it is easy to verify that v}

℘(x) := z

R1

0 f (x− sz)ds satisfies the above

assumptions for ℘(s) := sz, s∈ [0, 1].

For general ℘, we define the equivalence relation R on [0, 1]: t1R t2 ⇔ ∃v ∈ C1(D), div v(x) =

f (x− ℘(t1))− f(x − ℘(t2)). Let ˙t be a class for R, t1 ∈ ˙t, yt:= ℘(t)− ℘(t1), t∈ [0, 1] and

Kt1 := suppf (x−℘(t1)). Fix ι :=

1

2dist(Kt1, D

c_{) > 0. Since ℘ is continuous,}_{∃η > 0, ∀t} 2,|t2−

t1| < η, |yt2|1 < ι. Define K℘t1 := {k + syt2 : k∈ Kt1, s ∈ [0, 1]}. We have K℘_t1 ⊂ D and

we have already built v such that div v(x) = f (x− ℘(t1)− 0yt2)− f(x − ℘(t1)− 1yt2), i.e.:

div v(x) = f (x− ℘(t1))− f(x − ℘(t2)). This means that ∀t ∈ [0, 1], ˙t is open. Connexity of

[0, 1] then implies [0, 1] = ˙t and 0R 1.

♦ Given D an open connected component of Rn_{\φ(∂D), and y}

1, y2 ∈ D\φ(Zφ), we have

d(φ, D, y₁) = d(φ, D, y₂).

For φ∈ C2(D). SinceD is path connected, let ℘ : [0, 1] → D with ℘(0) = y1 and ℘(1) = y2.

From fourth we choose ι < 1

2dist|·|1(℘,D c_{) and f} ι such that: d(φ, D, y₁) = Z D fι(φ(x)− y1)Jφ(x)dx and d(φ, D, y2) = Z D fι(φ(x)− y2)Jφ(x)dx From previous, ∃v ∈ C1

c(Ω) such that div v(x) = fι(x− y1)− fι(x− y2). For u∈ C1c(D)

defined by ui(x) :=Pnj=1vj(φ(x))Aj,i(x) where (Ai,j(x))i,jis the adjugate matrix of∇φ, one

can algebraically verify that div u(x) = Jφ(x)div v(φ(x)) = Jφ(x)(fι(x− y1)− fι(x− y2)).

We then have d(φ, D, y₁)− d(φ, D, y2) = Z D fι(φ(x)− y1)− fι(φ(x)− y2)Jφ(x)dx = Z D Jφ(x)div v(φ(x))dx = Z D div u(x)dx = Z ∂D u(x)· n(x)dS = 0.

For φ∈ C1(D), third and a density argument enable us to take ψ ∈ C2(D) in a sufficiently small (in order not to intercept φ(∂D)∪ ψ(∂D)) C1_{-neighborhood B}

φ of φ and we obtain

d(φ, D, y₁) = d(ψ, D, y₁) = d(ψ, D, y₂) = d(φ, D, y₂). ♦ φ(Zφ) is a set of measure zero (Sard’s Lemma).

This is a particular case of the Morse–Sard Theorem (see e.g. Evans [14], Section 3.4.2). ♦ Last two enable us to extend definition of degree for y ∈ φ(Zφ)∩ φ(∂D)c:

d(φ, D, y):=d(φ, D, q) for any q /∈ φ(Zφ)∪ φ(∂D) such that |y − q|1 <dist|·|1(p, φ(∂D)).

Due to this extension, it is clear that d(φ, D,·) is constant on each connected component of Rn\∂D

F In this ?, we denote by φ a function in C(D) and by D a domain in RN (N ∈ N). Actu-ally, φ : D⊂ RN _{→ R}N_.

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♦ By a density argument, for y ∈ Rn_{\∂D we define d(φ, D, y):=d(ψ, D, y) for any}

ψ ∈ C1_{(D) such that}_{|ψ(x) − φ(x)| <dist}

|·|1(y, φ(∂D)) for every x∈ D. We note that the

function ψ can moreover be chosen such that y /∈ ψ(Zψ).

♦ H : D ×[0, 1] → Rn_{is a continuous homotopy between φ and ψ} _{∈ C(D) if H is continuous}

on D× [0, 1], H(x, 0) = φ(x) and H(x, 1) = ψ(x) for every x ∈ D. ♦ Let y /∈ φ(∂D). We have the two following properties:

• for every ψ ∈ C(D) such that |ψ − φ|C0 < dist_|·|

1(y, φ(∂D)), we have:

d(φ, D, y)=d(ψ, D, y).

• if H(x, t) is a continuous homotopy and y /∈ H(∂D, t) for every t ∈ [0, 1] then, d(H(·, t), D, y) does not depend on t ∈ [0, 1].

First • , we choose χ ∈ C1(D) such that |ψ − χ|C0 < dist_|·|

1(y, φ(∂D))− |ψ − φ|C0 and we

use properties from first ?.

For second•, from continuity of H ∃δ > 0, |H(·, t) − H(·, s)|C0 < δ implies d(H(·, t), D, y) =

d(H(·, s), D, y). This means that u : [0, 1] → Z with u(t) := d(H(·, t), D, y) is continuous. Since [0, 1] is a connected, u is constant on [0, 1], i. e. : d(φ, D, y) = d(ψ, D, y).

♦ Let y /∈ φ(∂D) be such that d(φ, D, y)6= 0. Then, there exists x ∈ D such that φ(x) = y. Conversely, assume y /∈ φ(D). Since y /∈ φ(∂D), this is y /∈ φ(D). And then, since φ(D) is compact, this yields dist_|·|₁(y, φ(D)) > 0.

We then choose ψ ∈ C1_{(D) such that y /}_{∈ (ψ(∂D) ∪ ψ(Z}

ψ)) and |ψ(x) − φ(x)|1 < 1

2dist|·|1(y, φ(D)) > 0 for every x∈ D. It is clear that y /∈ ψ(D), and then,

d(φ, D, y):=d(ψ, D, y)= 0, which is a contradiction.

F In this ?, we denote by φ : D ⊂ RN → RN−k_{, k, N} _{∈ N, k ≤ N a function in C(D),}

actually, φ = (φ1, . . . , φn−k, 0n−k+1, . . . , 0n).

♦ Define ψ(x) := x + φ(x) = (x1+ φ1(x), . . . , xn−k+ φn−k(x), xn−k+1, . . . , xn)∈ C(D) and

χ(x) := ψ(x)|x∈D∩Rn−k = (x1+ φ1(x), . . . , xn−k+ φn−k(x)) : D∩ Rn−k → Rn−k.

For y∈ Rn−k\ψ(∂D), we have: d(ψ, D, y) = d(χ, D ∩ Rn−k, y).

Due to the above assumptions, one can verify that ψ−1(p) = χ−1(p). We then distinguish three cases.

• If D ∩ Rn−k =∅, then d(χ, D ∩ Rn−k, y) = 0 and

∅ = (D ∩ Rn−k)_{∩ χ}−1(y) = D_{∩ ψ}−1(y) = D_{∩ ψ}−1(y). Therefore, d(ψ, D, y) = 0.

• If φ ∈ C1(D), D∩ Rn−k 6= ∅, and if p /∈ χ(Zχ), we are able to explicit the gradient of ψ:

∇ψ(x) = ∇χ(x) B 0_k,n−k Ik where B = ∂φ1 ∂xn−k+1 . . . ∂φ1 ∂xn ∂φn−k ∂xn−k+1 . . . ∂φn−k ∂xn ! .

Therefore, Jψ(x) = Jφ(x) for every x∈ D ∩ Rn−k. Since y /∈ χ(Zχ) and ψ−1(y) = χ−1(y),

this equality particularly implies that there can not exists x ∈ D such that ψ(x) = y and Jψ(x) = 0 are true together. Then, y /∈ ψ(Zψ) and d(ψ, D, y) =

P

x∈ψ−1_(y)sgn(Jψ(x)) =

P

x∈χ−1(y)sgn(Jχ(x)) = d(χ, D∩ Rn−k, y).

• If φ ∈ C(D), second ? enables us to choose ˆφ_{∈ C}1(D) such that

∀x ∈ D, |ˆφ(x)− φ(x)|1 <

1

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We moreover have |ˆχ(x)− χ(x)|D∩Rn−k_,1< 1 2dist|·|1(y, χ(∂D∩ R n−k₎₎ and d( ˆχ, D∩ Rn−k, y) = d(χ, D∩ Rn−k, y). Our second• enables us to conclude.

F In this ?, we denote by X a normed linear space, by T : D ⊂ X → X a compact mapping, by Tι a finite range mapping such that |T (x) − Tι(x)|X < ι for every x∈ D (see e.g. Brezis

[3], Corollary 6.2). We define φ := I− T and φι = I− Tι and use notations Sι:=spanTι(D)

and Dι := D∩ Sι = D∩ span Tι(D).

♦ For y /∈ φ(∂D), and for 0 < ι1 ≤ ι2 < 1₂dist_|·|X(y, φ(∂D)), with notations Sν := span Sι1∪

Sι2 and Dν := D∩ Sν, we have d(φι1, Dι1, y) = d(φι2, Dι2, y) = d(φι2, Dν, y).

From previous ?, we have d(φ_ι₁, Dι1, y)=d(φι1, Dν, y). We define H(x, t) := tφι1(x) + (1−

t)φ_ι₂(x), t ∈ [0, 1], x ∈ Dν. This is a continuous homotopy between φι1 and φι2. We can

verify that y /∈ H(∂Dν, t) for every t ∈ [0, 1]. Due to the third in the second ?, we then

have d(φ_ι₁, Dν, y)=d(φι2, Dν, y), and then d(φι1, Dι1, y)=d(φι2, Dι2, y).

♦ This enable us to extend definition of the degree to compact perturbations of identity: for y /∈ φ(∂D). The Leray-Schauder degree of φ at y with respect to D is defined by d(ˆφ, DV, y),

where ˆφ := I− ˆT , ˆT : D→ X is a compact mapping such that | ˆT− T |C0 < dist_X(y, φ(∂D)),

ˆ

T (D) is of finite dimension and DV := D∩ V where V is any linear space of finite dimension

containing y and ˆT (D).

♦ Assume that H : D × [0, 1] → X satisfies assumptions (i), (ii), (iii) of the Lemma. Set φ_t:= I− H(·, t) for t ∈ [0, 1]. Then d(φt, D, y) is independent of t.

There exists r > 0, dist_|·|_X(y, φ_t(∂D)) > r independently of t.

On the contrary we could choose (tn, xn)∈ [0, 1] × ∂D such that limn→∞|y − φtn(xn)|X = 0.

We then choose a subsequence (tn)→ τ in the compact [0, 1]. Then, since H(·, τ ) is a compact

operator, we again choose a subsequence of (H(xn, τ )) which converges to some z ∈ X. We

have:

y = lim

n→∞φtn(xn) = lim_n→∞(xn− H(xn, τ )) + (H(xn, τ )− H(xn, tn)) = lim_n→∞xn− z.

Hence, y + z = lim_n→∞xn∈ ∂D, and by continuity of H(·, τ ) we deduce that y = (y + z) −

H(y + z, τ ) = φ_τ(y + z), which yields a contradiction.

Let R be the equivalence relation defined on [0, 1] by tRs iff d(φ_t, D, p) = d(φ_s, D, p). We chose ˙t a class for R and t∈ ˙t. We then choose 0 < κ2< r₄. We choose a compact mapping

hκ2 : D→ X such that hκ2(D) is of finite dimension and|hκ2(x)− H(x, t)|X < κ2 for every

x ∈ D. From definition of homotopy of compact transformations, there exists κ1 > 0 such

that|t − s| < κ1 implies|H(x, t) − H(x, s)|X < κ2 for every x∈ D.

Let Vκ2 be a space of finite dimension containing y and hκ2(D), and set Dκ2 := D∩ Vκ2.

For|t − s| < κ1, we both have:

|hκ2(x)− H(x, t)|X < κ2 < dist|·|X(p, φ(∂D)) and

|hκ2(x)− H(x, s)|X < 2κ2 < dist|·|X(p, φ(∂D)).

Due to previous, this gives d(I − H(·, t), D, p) := d(I − hκ2, Dκ2, p) =: d(I− H(·, s), D, p).

Therefore, tRs. We conclude that ˙t is an open set for every t ∈ [0, 1], and by connexity argument, we have 0R1.

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♦ End of the proof of the Leray Schauder fixed point Theorem.

From point (iv) of assumptions of the Leray Schauder fixed point Theorem, we obtain at least one solution to the problem φ₀(x) = 0. Due to successive definitions for degrees (first ?-second , second ?-first , and fourth ?-second ) , this means that d(φ0, D, 0)6= 0, and

from previous , ∀ t ∈ [0, 1], d(φt, D, 0)6= 0.

∀ n such that dist|·|X(0, ∂D) >

1

n,∃ ˆHn,t: D→ X which satisfies |H(x, t) − ˆHn,t(x)|X < 1 n

for every x∈ D with ˆHn,t(D) of finite dimension.

Let Sn:= span{0, ˆHn,t(D)} and Dn:= D∩ Sn.

We know that 06= d(φt, D, 0) = d(ˆφn,t, Dn, 0). Last from second ? implies that for each n,

there exists xn∈ Dn such that ˆφn,t(xn) = 0. The sequence (xn) remains in the bounded set

D. Since H(·, t) is a compact operator, there is a subsequence such that H(xn, t) converges

to some x∈ X. Hence, since xn= ˆHn,t(xn), and using the continuity of φtat x, we obtain:

φ_t(x) = lim

n→∞φt(xn) = limn→∞xn− H(xn, t) = 0.

Since 0 /∈ ∂D, we just have proved that for each t ∈ [0, 1], equation φt(x) = 0 admits the

Weak Solutions to Steady Navier–Stokes Equations for Compressible Barotropic Fluids with Vorticity–Type Boundary Conditions

HAL Id: tel-00474984

https://tel.archives-ouvertes.fr/tel-00474984

Weak Solutions to Steady Navier–Stokes Equations for

Compressible Barotropic Fluids with Vorticity–Type

Boundary Conditions

Olivier Muzereau

To cite this version:

THÈSE

Analyse des solutions du système des équations de NavierStokes

avec des conditions aux limites de type vorticité

pour les uides barotropiques compressibles

Universit´e du Sud Toulon Var

Czech Technical University in Prague

Doctoral Dissertation Thesis

Weak Solutions to Steady Navier–Stokes

Equations for Compressible Barotropic

Fluids with Vorticity–Type

Boundary Conditions

Olivier MUZEREAU

Universit´e du Sud Toulon Var

Laboratoire de Syst`emes Navals Complexes

FEBRUARY 24th 2009

Supervisors:

Jiˇr´ı NEUSTUPA, Czech Academy of Sciences, Prague,

Czech Technical University in Prague

Patrick PENEL, Universit´e de Sud Toulon Var

Reviewers:

Raphaël DANCHIN, Université de Paris XII, Créteil

Milan POKORN Y,

Charles University Prague

Examiner:

Stanislav KRA ˇ

CMAR, Czech Technical University in Prague

Anton´in NOVOTN ´

Y,

Universit´e de Sud Toulon Var

Contents

Part I

1

Introduction and notation of function spaces.

2

The studied model and the main results of the thesis.

3

Mathematical preliminaries.

Part II

Approximate model I based on the

Lions–Novotn´

y–Straˇ

skraba

Analyse des solutions du système des équations de NavierStokes

pour les uides barotropiques compressibles