Navier-Stokesovy rovnice a související problémy

Transkript

1 Navier-Stokesovy rovnice a související problémy Dr. Matteo Caggio disertační práce k získání akademického titulu doktor (Ph.D.) v oboru Aplikovaná matematika Školitel : RNDr. Šárka Nečasová, DSc. Konzultant : doc. RNDr. Zdeněk Skalák, CSc. Katedra matematiky Plzeň, 7

2

3 Navier-Stokes equations and related problems Dr. Matteo Caggio dissertation thesis for taking academic degree Doctor of Philosophy (Ph.D.) in specialization Applied Mathematics Supervisor: Šárka Nečasová Co-supervisor: Zdeněk Skalák Department of Mathematics Pilsen, 7

4

5 Čestné prohlášení Prohlašuji, že práci, kterou předkládám jako disertační práci je originální prací. Skládá se z jednotlivých kapitol. Tyto kapitoly obsahují vědecké články, ve kterých jsem autorem nebo spoluautorem. V práci jsou uvedeny citace prací, ze kterých jsem čerpal, v seznamu literatury. V rámci doktorandské práce byly použity standardní vědecké postupy. Plzeň,. června, Dr. Matteo Caggio 5

6 6

7 Abstrakt Disertační práce je věnována studiu matematických problémů Navierových - Stokesových rovnic v kontextu rigorózního matematického odvození modelů a jejich matematické analýzy. Zejména je práce zaměřena na problematiku singulárních limit v mechanice tekutin pro stlačitelné tekutiny (režim malého Machova čísla, velkého Reynoldsova čísla, redukce dimenze) a problematice regularity pro nestlačitelné tekutiny. Klíčová slova Navierovy-Stokesovy rovnice, stlačitelné tekutiny, Navierovy-Stokesovy-Fourierovy rovnice, singulární limity, slabé řešení, silné řešení, Eulerovy rovnice, teorie regularity, nestlačitelné tekutiny, anisotropní Lebesgueovy prostory. 7

8 8

9 Abstract The present thesis is devoted to the study of mathematical problems related to the Navier-Stokes equations in the context of mathematical rigorous derivation of models and their analysis. In particular we deal with the problem of singular limits in fluid mechanics for compressible fluids (low Mach number limit and high Reynolds number limit, reduction of dimension) and the problem of global regularity for incompressible fluids. Keywords Navier-Stokes equations, compressible fluids, Navier-Stokes-Fourier equations, singular limits, weak solutions, strong solutions, Euler equations, regularity theory, incompressible fluids, anisotropic Lebesgue spaces. 9

10

11 Estratto Il presente lavoro di tesi è dedicato allo studio di problematiche legate alle equazioni di Navier-Stokes nel contesto della derivazione rigorosa di modelli e della loro analisi. In particolare ci occuperemo dei problemi relativi ai limiti singolari nella meccanica dei fluidi comprimibili (limite di bassi numeri di Mach e alti numeri di Reynolds, riduzione di dimensione) e del problema della regolarità globale per fluidi incomprimibili. Parole chiave Equazioni di Navier-Stokes, fluidi comprimibili, equazioni di Navier-Stokes- Fourier, problemi ai limiti singolari, soluzioni deboli, soluzioni forti, equazioni di Eulero, teoria della regolarità, fluidi incomprimibili, spazi di Lebesgue anisotropi.

12

13 Acknowledgements I would like to thank to Pavel Drábek and Šárka Nečasová for giving me this opportunity to do my doctorate study in the Czech Republic. Also I would like to thanks to Petr Kučera, Jiří Neustupa and Zdeněk Skalák not only for teaching me mathematics but for wonderfull time which I could spent with them and discuss about mathematical problems. Also I would like to thank Milan Pokorný for possibility to collaborate with him. A special thanks to Šárka Nečasová and Zdeněk Skalák for helping me during the research work and supported me in these years. Thanks to all the people who love me and who have taken care of me. A particular thanks to my family for everything. Prague, st June 7. Matteo Caggio 3

14 I would also like to acknowledge all the financial sources that helped me achieve my scientific results. GAČR (Czech Science Foundation) project No. 6-33S in the framework of RVO: Institute of Mathematics of the Czech Academy of Sciences. Students grant of the University of West Bohemia in Pilsen, grant SGS

15 Preface Navier-Stokes equations is a challenging problem in mathematical analysis. During the years several authors have faced different problems related to these equations. Some of these problems concern variations of the Navier-Stokes equations depending on the properties of the fluid and the presence of external forces. The present work deals with the so-called problem of singular limit in fluid mechanics for compressible fluids and the problem of global regularity for an incompressible fluid. The following articles are the result of this work: Guo Z., M. Caggio, Z. Skalák, Regularity criteria for the Navier-Stokes equations based on one component of velocity, Nonlinear Analysis: Real World Application, 35, , 7. Caggio M., Š. Nečasová, Inviscid incompressible limit for rotating fluids, to appear in Nonlinear Analysis. Ducomet B., M. Caggio, Š. Nečasová, M. Pokorný, The rotating Navier- Stokes-Fourier system on thin domains, submitted in Acta Appl. Math; available on arxiv:66.54v. 5

16 6

17 Contents Čestné prohlášení 5 Abstrakt 7 Abstract 9 Estratto Introduction. The problem of singular limits for compressible fluids The inviscid incompressible limit for compressible barotropic fluids The dimension reduction limit for compressible heat conducting fluids The problem of global regularity for incompressible fluids Regularity criteria in terms of one velocity component.. 9 Inviscid incompressible limit for rotating fluids 3. Weak and classical solutions Bounded energy weak solutions Classical solutions Acoustic waves Energy and dispersive estimates Convergence analysis Relative energy inequality Main results Convergence Conclusions Dimension reduction for compressible heat conducting fluids 5 3. Thermodynamics Weak and classical solutions Weak solutions Classical solutions Convergence analysis Relative energy inequality Main results Convergence Conclusions

18 4 Global regularity for incompressible fluids Preliminary results State of art and main results State of art Main results Proofs of main results Proof of Theorem Proof of Theorem Proof of Theorem Proof of Theorem Proof of Theorem Conclusions

19 9

20 Chapter Introduction The present work is devoted to the study of mathematical problems related to models describing the dynamics of fluids. A fluid is a continuous medium whose state is characterized by its velocity, pressure and density fields, and possibly other relevant fields (for example temperature). Most of the fluid dynamics results have been obtained starting from the Navier-Stokes equations. These equations have many variations depending on the properties of the fluid itself, for example compressibility, thermoconductivity, viscosity, etc., and on the forces acting on the fluid, for example the centrifugal force, the Coriolis force, the gravity force etc. (see Nazarenko [79]). Two kind of problems will be under consideration: the problem of singular limits for compressible fluids and the problem of global regularity for incompressible fluids.. The problem of singular limits for compressible fluids The problem of singular limits for compressible fluids can be presented in the following way. One starts from a system of equations describing the motion of a kind of fluid. After a scale analysis the system presents several characteristic parameters whose asymptotic behavior determines a change in the fluid phenomenology and consequently, at least at a formal level, a different system of equations compared to the starting one. The singular limit problem requires to show that the solution of the starting system converges to the solution of the limit (or target) system when these parameters tend to zero or infinity in some sense. In the following we would like to briefly describe the problems we will deal with, postponing a deeper analysis to the next chapters... The inviscid incompressible limit for compressible barotropic fluids The motion of a compressible barotropic fluid is described by means of two unknown fields: the density ϱ = ϱ (x, t) and the velocity u = u (x, t) of the

21 fluid, functions of the spatial position x R 3 and the time t R, and satisfying the following Navier-Stokes system of equations. The continuity equation reads The momentum equation is t ϱ + div x (ϱu) =. (..) continuity t (ϱu) + div x (ϱu u) + x p(ϱ) = div x S ( x u) + ϱf, (..) momentum_intro with the stress tensor given by the following relation ( S = S( x u) = µ x u + t xu ) 3 div xui + η div x ui, µ >, η. (..3) stress_intro The system above presents two parameters: the shear viscosity coefficient µ and the bulk viscosity coefficient η. The scalar function p is the pressure, given function of the density, and ϱf represents an external forcing. For each physical quantity X present in the Navier-Stokes system (..) - (..3), we introduce its characteristic value X char and replace X with its dimensionless analogue X/X char. As a result, we obtain the scaled version of the compressible Navier-Stokes system [Sr] t ϱ + div x (ϱu) =, (..4) continuity_scaled [Sr] t (ϱu) + div x (ϱu u) + [Ma] xp(ϱ) = [Re] div xs + ϱf. (..5) [Fr] momentum_scaled The above system presents several characteristic numbers. The Strouhal number [Sr] = length char time char velocity char. The Strouhal number plays a role in oscillating, non-steady flows, as the Kármán vortex street. It is often defined as [Sr] = fl U, where f is the frequency of vortex shedding in the wake of von Kármán, L is the characteristic length of the body invested by the flow and U is the characteristic velocity of the flow investing body. The Mach number [Ma] = velocity char pressurechar /density char. The Mach number is the ratio of the characteristic velocity of the flow to the speed of the sound in the fluid. Low Mach number limit characterizes incompressibility. The Reynolds number [Re] = density charvelocity char length char viscosity char.

22 The Reynolds number is the ratio of the inertial to the viscous forces in the fluid. High Reynolds number is attributed to turbulent flows. The Froude number [Fr] = velocity char lengthchar frequency char. The Froude number is the ratio of the flow inertia to the external field. The latter in many applications simply due to gravity. Redefining the Reynolds number and the Mach number in terms of a nonnegative parameter ε, namely Re := ε and Ma := ε, and setting the other characteristic numbers equal to one, the inviscid incompressible limit aims to show the convergence u v and ϱ, for ε, where v is the solution of the incompressible Euler system t v + v x v + x Π =, div x v = (..6) euler_intro and u is the solution of the compressible Navier-Stokes system. Indeed, in the high Reynolds number limit the viscosity of fluid becomes negligible and in the low Mach number limit the fluid becomes incompressible. The inviscid and/or incompressible limit problem was investigated by several authors in similar and different contexts: in bounded, unbounded or expanding domains, in presence of external forces and for barotropic or heat conductive fluids. For more details we refer to the works of Bardos and Nguyen [], Feireisl [39], Feireisl and Novotný [44], Feireisl, Jin and Novotný [46], Feireisl, Nečasová and Sun [47], Lions and Masmoudi [7] (see also [73, 74]), Masmoudi [75], Sueur [4] and references therein. In the context described above, we will deal with the inviscid incompressible limit for a compressible barotropic fluid in a "fast" rotating frame occupying the whole space R 3. More precisely, we would like to show the convergence of the solution of the compressible Navier-Stokes system t ϱ + div x (ϱu) =, (..7) massi t (ϱu) + div x (ϱu u) = ε xp(ϱ) + εdiv x S( x u) (ϱu ω), (..8) momentumi ( S = S( x u) = µ x u + t xu ) 3 div xui + η div x ui, µ >, η. towards the solution of the rotating incompressible Euler system (..9) stressi t v + v v + v ω + x Π =, div x v =, (..) euleri for large values of the angular velocity ω = [,, ], namely "fast" rotating frame. Above, the shear viscosity coefficient µ and the bulk viscosity coefficient η are assumed to be constant. The quantity (ϱu ω) represents the Coriolis force. The effect of the centrifugal force is neglected. This is a standard simplification adopted, for instance, in models of atmosphere or astrophysics (see [54, 55, 56]). The analysis will be based on the work of Caggio and Nečasová [7]. The problem is a particular case of the Masmoudi [75] result where we will use a different technique (see the discussion below).

23 The technique to reach the convergence will be based on the so-called relative energy method in the framework of the relative energy inequality. The relative energy inequality was introduced by Dafermos [] in the context of the Second Law of Thermodynamics. In the fluid context, it was introduced by Germain [5]. Afterwards, the method was developed by Feireisl, Novotný and co-workers in the framework of the problem of singular limits in fluid mechanics (see for example Feireisl and Novotný [4], [43], Feireisl, Jin and Novotný [45] and Feireisl, Novotný and Sun [5] and references therein). In the following we describe briefly the method, leaving the technical details to the next chapters. The basic idea is to introduce a relative energy functional. This functional plays the role of measuring the stability of two solutions. One with more regularity compared to the other one. In our context, the two solutions will be the weak solution of the Navier-Stokes system and the classical solution of the Euler system respectively. Next, along with the relative energy functional, a relative energy inequality has to be derived. This last will give us the possibility to reach the convergence in terms of a Gronwall type inequality. The compressibility of the fluid allows the propagation of acoustic waves described by the acoustic system related to the Navier-Stokes model. The acoustic waves have to decay in the incompressible limit. Therefore, the analysis requires a technique in order to ensure this decay. In the whole space is common to use the so-called dispersive estimates (see Desjardins and Grenier [], Feireisl and Novotný [4], Masmoudi [75], Schochet [95] and Strichartz [3]). We will introduce the acoustic system and the dispersive estimates during our analysis... The dimension reduction limit for compressible heat conducting fluids The motion of an heat conducting compressible fluid is described by means of three unknown fields: the density ϱ = ϱ (x, t), the velocity field u = u (x, t) and the temperature ϑ = ϑ(x, t) of the fluid, functions of the spatial position x R 3 and the time t R, and satisfying the following Navier-Stokes-Fourier system of equations. The continuity equation reads The momentum equation is t ϱ + div x (ϱu) =. (..) cont_eps_intro t (ϱu) + div x (ϱu u) + x p(ϱ, ϑ) = div x S (ϑ, x u) + ϱf. (..) NSFP_intro with the stress tensor given by the following relation ( S (ϑ, x u) = µ (ϑ) x u + t xu ) 3 div xui + η (ϑ) div x ui. (..3) S_intro The entropy equation is t (ϱs (ϱ, ϑ)) + div x (ϱs (ϱ, ϑ) u) + div x ( q (ϑ, x ϑ) ϑ = ϑ ( S (ϑ, x u) : x u q (ϑ, ) xϑ) x ϑ, (..4) s_intro ϑ ) 3

24 with q = κ (ϑ) x ϑ. (..5) flux_intro In the system above the shear viscosity coefficient µ (ϑ), the bulk viscosity coefficient η (ϑ) and the heat conductivity coefficient κ (ϑ) are functions of the temperature. The scalar functions p(ϱ, ϑ) and s(ϱ, ϑ) are the pressure and the entropy respectively, functions of the density and the temperature, and ϱf represents an external forcing. In analogy with the arguments presented before, we can obtain the scaled version of the compressible Navier-Stokes-Fourier system [Sr] t ϱ + div x (ϱu) =, (..6) cont_nsf_scal [ [Sr] t (ϱu) + div x (ϱu u) + [ ] = div x S (ϑ, x u) + Re [ t (ϱs (ϱ, ϑ)) + div x (ϱs (ϱ, ϑ) u) + Pe = ϑ ([ Ma Re ] S (ϑ, x u) : x u Ma [ Fr [ Pe where the Péclet number [Pe] is defined as follows ] x p(ϱ, ϑ) ] ϱf, (..7) mom_nsf_scal ] ( q (ϑ, x ϑ) div x ϑ ] q x ϑ ϑ ) ). (..8) s_nsf_scal (Poisson) (eps) [Pe] = pressure charvelocity char length char heat conductivity char temperature char. Similarly to Reynolds number, high Péclet number corresponds to low heat conductivity of the fluid that may be attributed to turbulent flows. Redefining the Froude number in terms of a non-negative parameter ɛ, namely Fr = ɛ β, with β arbitrary non-negative number, and setting the other characteristic numbers equal to one, the dimension reduction limit aims to show the convergence [ϱ, u, ϑ] [r, w, Θ], for ɛ, where the couple [ϱ, u, ϑ] is the solution of the three-dimensional Navier-Stokes-Fourier system and the couple [r, w, Θ] is the solution of the corresponding two-dimensional system. Indeed, in the low Froude number limit the gravitational effects become predominant forcing the fluid to a two-dimensional dynamics. The analysis will be based on the work of Ducomet, Caggio, Nečasová and Pokorný [5] and it aims the extension of the result of Feireisl, Novotný and co-workers []. Remark. For the sake of clarity, in the presence of gravity force, the system describing an heat conducting fluid is given by the Navier-Stokes-Fourier-Poisson system of equations. Remark. It is possible to read ɛ as follows ɛ = l L. Here, l is the horizontal length and L the vertical length. Consequently, the limit can be also seen, more easily, in terms of a pure geometric reduction. 4

25 In the context describe above, we will deal with the dimension reduction limit for a compressible heat conducting fluid in a rotating frame occupying a bounded domain in R 3 where the external forcing is given by the gravity force. More precisely, we consider a fluid confined in a straight layer ɛ = ω (, ɛ) where ω is a two-dimensional domain. We rescale to a fix domain as follows (x h, ɛx 3 ) ɛ (x h, x 3 ), where x h = (x, x ) ω and x 3 (, ). Above, we denoted ɛ = ( h, ɛ ) x3, h = ( x, x ), (..9) scal_i div ɛ u = div h u h + ɛ x 3 u 3, u h = (u, u ), div h u h = x u + x u, (..) scal_i The continuity equation reads now as follow ɛ = x x + x x + ɛ x 3x 3. (..) scal_3i the momentum equation is t ϱ + div ɛ (ϱu) =, (..) cont_epsi t (ϱu) + div ɛ (ϱu u) + ϱu χ + ɛ p(ϱ, ϑ) = div ɛ S (ϑ, ɛ u) + ɛ β ϱ ɛ φ + ϱ ɛ x χ, (..3) NSFPI the entropy equation is ( ) q (ϑ, ɛ ϑ) t (ϱs (ϱ, ϑ)) + div ɛ (ϱs (ϱ, ϑ) u) + div ɛ ϑ = ( S (ϑ, ɛ u) : ɛ u q (ϑ, ) ɛϑ) ɛ ϑ, ϑ ϑ (..4) si with and ( S (ϑ, ɛ u) = µ (ϑ) ɛ u + t ɛu ) 3 div ɛui + η (ϑ) div ɛ ui (..5) S q = κ (ϑ) ɛ ϑ. (..6) fluxi The quantities ϱu χ and ϱ ɛ x χ represent the Coriolis force and the centrifugal force respectively with χ = [,, ] angular velocity and ɛ x χ = ( ) h x χ, = (x, x, ). x + x The gravitational force is expressed by ϱ ɛ φ where the potential φ satisfies the Poisson s equation 5

26 ɛ φ = 4πG(αϱ + ( α)g) in (, T ). (..7) PoissonI Here, G is the Newton constant and α a positive parameter. The first contribution on the right-hand side of the relation (3..6) corresponds to self-gravitation while in the second one g is a given function modeling the external gravitational effects. Here and hereafter, we assume that the function ϱ is extended by zero outside of. Supposing further that g is such that the integral below converges, we have φ (t, x) = G K (x y) (αϱ (t, y) + ( α) g (y)) dy, R 3 where K (x y) = x y and the parameter α may take the values or. For α = the gravitation only acts as an external field, for α = only the selfgravitation is present. Since we also work with ɛ φ (t, x), we have to further assume that ɛ K (x y) (αϱ (t, y) + ( α) g (y)) dy <. R 3 In particular, the gravitational force is given by the following relation (see [5] and [6]) ɛ φ (t, x) = ɛ αϱ(t, ξ) (x ξ, x ξ, ɛ (x 3 ξ 3 )) ɛ ( x h ξ h + ɛ x 3 ξ 3 ) dξ 3/ + ( α) g(y) (x y, x y, ɛ (x 3 y 3 )) R ( x 3 h y h + ɛ x 3 y 3 ) dy 3/ = ɛαφ + ( α) Φ. (..8) phi_gravi In our analysis we will distinguish two cases with respect to the behavior of the Froude number, namely Fr = ɛ for β = / and Fr = for β =. According to the choice of the Froude number, we have to consider the correct form of the gravitational potential. In the former the self-gravitation, namely α =, and in the latter the external gravitation force, namely α =. In the latter, we could also include the self-gravitation, it would, however disappeared after the limit passage. Taking Fr = ɛ for β = / the momentum equation reads as follow t (ϱu) + div ɛ (ϱu u) + ϱu χ + ɛ p(ϱ, ϑ) = div ɛ S (ϑ, ɛ u) + ϱφ + ϱ ɛ x χ. (..9) NSFPI_phi_ While, taking Fr = for β =, we have t (ϱu) + div ɛ (ϱu u) + ϱu χ + ɛ p(ϱ, ϑ) = div ɛ S (ϑ, ɛ u) + ϱφ + ϱ ɛ x χ. (..3) NSFPI_phi_ For Fr = ɛ and β = /, the corresponding two-dimensional momentum equation reads as follows 6

27 with the formula and r t w + rw h w + h p(r, Θ) + r (w χ) = div h S(Θ, h w) + r h φ h + r h x χ, (..3) mom_ti φ h (t, x h ) = ω r(t, y h ) x h y h dy h (..3) grav_t_i S h (Θ, h w) = µ ( h w + t hw div h w ) ( + η + µ ) div h wi h (..33) ShI 3 where I h is the unit tensor in R in the domain (, T ) ω. While, for Fr = and β =, we have g(y) φ h (t, x h ) = G dy. (..34) grav_t_i R 3 x h y h + y3 As in the previous discussion, the technique to reach the convergence will be based on the relative energy method in order to show the convergence of the weak solution of the three-dimensional Navier-Stokes-Fourier system to the classical solution of the corresponding two-dimensional system. In particular, we will follow the framework developed in [43]. The main point of the analysis will be the treatment of the gravitational force. From a phenomenological point of view, this limit concerns the rigorous derivation of the equations describing astrophysical objects called accretion disk which are thin structures observed in various places in the universe. These disks are indeed three-dimensional but their thickness is usually much smaller than their extension, therefore they are often modeled as two-dimensional structures. Indeed, if a massive object attracts matter distributed around it through Newtonian gravitation in presence of an angular momentum, this matter is not accreted isotropically around the central object but forms a thin disk around it. For further details we refer to the work of Choudhuri [7], Montesinos Armijo [78], Ogilvie [87], Pierens [89], Pringle [9] and Shore [96].. The problem of global regularity for incompressible fluids The motion of an incompressible fluid is described by means of its velocity field u = u (x, t), functions of the spatial position x R 3 and the time t R, and satisfying the following Navier-Stokes system of equations t u + u x u µ x u + x p = f, div x u =. (..) NS_intro In the system above µ is the shear viscosity coefficient. The scalar function p is the pressure, functions of the spatial position x R 3 and the time t R, and f represents a given external forcing. An open problem in applied analysis concerns the global regularity of the solution of the Navier-Stokes equations in the whole space R 3. Over the years, several authors have faced the problem (see, for example, [8], [9], [4], [63], 7

28 [64], [66], [7], [7], [9], [], [7], [8], [9]). It is known that for the initial data u L σ (solenoidal functions in L ) the problem (..) possesses at least one global weak solution u satisfying the energy inequality T u(t ) / + x u(t) dt u / (..) eii for every T (see [53], [67] and [93]). Such solutions are called Leray-Hopf solutions. More precisely (see [93]), given u L σ, a weak solution of (..) on [, T ) is a function u L (, T ; L σ) L (, T ; W,) such that T (u, ϕ t ) ( u, ϕ) ((u ) u, ϕ) = (u, ϕ) (..3) weak_ns for every ϕ D ( [, T ), R 3) (, the set of all functions in C ) [, T ), R 3 that are also divergence free, and the following existence Theorem holds (see [93]). existence_ns Theorem 3. For any u L σ there exists at least one weak solution of (..). This solution is weakly continuous into L, namely for any v L, lim (u (t), v) = (u (t ), v) t t for all t [, T ), and in addition it satisfies the energy inequality (..) for every t [, T ). Moreover, u(t) u in L as t. Remark 4. Above we used ( ) to denote the inner product in L. Nevertheless, the uniqueness, regularity, and continuous dependence on initial data for weak solutions are still open problems ([]). If u Wσ, (solenoidal functions from the standard Sobolev space W, ), then strong solutions exist for a short interval of time whose length depends on the physical data of the initial-boundary value problem. Moreover, this strong solution is unique in the larger class of weak solutions ([9], [63], [], [7]). In fact, a strong solution is a weak solution with the additional regularity ([93]) u L (, T ; W,) L (, T ; W,). From the pioneer works of Prodi [9] and of Serrin [98], many results were presented in providing sufficient conditions for the global regularity (see for example Chae and Lee [3], Constantin [8], Doering and Gibbon [4], Ladyzhenskaya [63, 64], Lemarié-Rieussett [66], Lions [7, 7], Sohr [] and Temam [7, 8, 9] and references therein). Some of these conditions provide regularity criteria for the velocity field (see for example Escauriaza, Seregin and Šverák [3], Fabes, Jones and Riviere [3] and Serrin [98]): if a Leray-Hopf weak solution u satisfies u L r (, T ; L s (R 3 )) for some r + 3 s, 3 s then u is regular. Others involve analogous criteria for the pressure (see for example Berselli [5], Berselli and Galdi [6], Cao and Titi [9], Kukavica [59], Seregin and Šverák [97], Zhou [5]): if the pressure p satisfies 8

29 or p L r (, T ; L s (R 3 )) for some r + 3 s, s > 3 p L r (, T ; L s (R 3 )) for some r + 3 s 3, s then u is regular. An analogical situation occurs for u. It was proved in [3] that u is regular if where s (3/, ) and u L r (, T ; L s (R 3 )) r + 3 s =. Still others state sufficient conditions for regularity in terms of the vorticity (see for example Beirao da Veiga [4]): if the vorticity ω = u of a Leray-Hopf weak solution u belongs to the space L r (, T ; L s (R 3 )) for some r + 3 s, s > then u is regular. The result above concerns the regularity of the solution u when conditions are imposed on all the components of the vorticity vector. Chae and Choe [] obtained regularity by imposing the conditions ω j L r (, T ; L s ), j =,, for some r + 3, s (3/, ) s namely, on only two components of the vorticity vector, while the problem with one vorticity component is an outstanding open problem... Regularity criteria in terms of one velocity component The above mentioned criteria are based on the entire velocity vector or on the entire gradient. In the last two decades many authors have studied the regularity criteria where additional conditions were imposed only on some velocity components or on some items of the velocity and pressure gradients. The first contribution in this direction was done by Neustupa and Penel [8]. After, over the years, several authors have obtained important results in that direction (see for example Kukavica and Ziane [6], Zhou and Pokorný [6], [7] and reference therein). In this context described, we are interested in criteria based on only one velocity component. More specifically, we will study criteria based on u 3, u 3 and u 3, and prove, for example, that the condition where p (, ] and u 3 L β (, T ; L p ), /β + 3/p = 7/4 + /(p), 9

30 yields the regularity of u on (, T ]. The analysis will be based on the work of Guo, Caggio and Skalák [5] in the framework of anisotropic Lebesgue spaces. The anisotropic Lebesgue spaces framework seems to be convenient for our purposes, since it differentiates between different directions. It can be useful in the situations where regularity conditions are imposed only on one velocity component. Indeed, in Theorems 38-4 we will see that the use of the anisotropic Lebesgue spaces framework can improve some results from the literature. 3

31 Chapter Inviscid incompressible limit for rotating fluids We consider the scaled compressible Navier-Stokes system for a barotropic rotating fluid in the whole space R 3 already mentioned in Introduction. The continuity equation reads the momentum equation is t ϱ + div x (ϱu) =, (..) mass t (ϱu) + div x (ϱu u) = ε xp(ϱ) + εdiv x S( x u) (ϱu ω), (..) momentum with the stress tensor given by the following relation ( S = S( x u) = µ x u + t xu ) 3 div xui + η div x ui, µ >, η. The system is supplemented by the initial conditions (..3) stress ϱ (x, ) = ϱ (x), u (x, ) = u (x) (..4) ic_ and by the following far field conditions for the density and the velocity field lim ϱ(x, t) =, lim x u(x, t) =. (..5) bound x The first relation in (..5) means the mass of the fluid is infinite. As mentioned in the previous chapter, we want to show that the weak solution of the Navier-Stokes system converges to the classical solution of the corresponding rotating incompressible Euler system t v + v v + v ω + x Π =, div x v =, (..6) euler for large values of ω, namely "fast" rotating frame. 3

32 . Weak and classical solutions In the following, we introduce the definition of weak solutions for the compressible Navier-Stokes system ( ) and we discuss the global-in-time existence. In particular, we define the so-called bounded energy weak solution (see [38], [48] and [86]) and we discuss the global-in-time existence. Then, we discuss the global existence of the classical solution of the incompressible Euler system (..6). For the discussion on weak solutions we will consider an arbitrary open set R 3. The introduction of the bounded energy weak solution is motivated by the following discussion. In [] it was shown the existence of weak solutions to the compressible Navier-Stokes equations on unbounded domain satisfying the differential form of the energy inequality (and consequently the integral form) for a barotropic fluid with finite mass. While the existence of weak solutions for a fluid with infinite mass remains an open question. Weak solutions satisfying the differential form of the energy inequality are usually termed finite energy weak solutions (see [], [45], [49], [6] and [86]), while weak solutions satisfying the integral form of the energy inequality are usually termed bounded energy weak solutions (see [38], [48] and [86]). Because our analysis will be performed in the whole space R 3 under the condition that the mass of the fluid is infinite (see relation..5), we have to use the integral form of the energy inequality and consequently to deal with bounded energy weak solutions... Bounded energy weak solutions Multiplying (formally) the equation (..) by u and integrating by parts, we deduce the energy inequality in its integral form T E(T ) + ε S ( x u) : x u dxdt E (..) ei where the total energy E is given by the formula E = E [ϱ, u] (t) = with E the initial energy, and H(ϱ) = ϱ the Helmholtz free energy (see [4] and [86]). ϱ u + H(ϱ) ε dx, (..) e p (z) dz (..3) z Remark 5. Here and hereafter the Helmholtz free energy will have the following form (see Novotný and Straškraba [86]): H(ϱ) = γ (ϱγ γϱ + γ ). The parameter γ is the adiabatic index or heat capacity ratio. Now, we define the so-called bounded energy weak solution of the compressible Navier-Stokes system ( ) (see Feireisl, Novotný and Petzeltová [48] and Novotný and Straškraba [86]). 3

33 be Definition 6. (Bounded energy weak solution) Let R 3 be an arbitrary open set. We say that [ϱ, u] is a bounded energy weak solution of the compressible Navier-Stokes system ( ) in the time-space cylinder (, T ) if ϱ L ((, T ), L γ loc ()), ϱ a.e. in (, T ), H(ϱ) L ((, T ), L ()), ( ) ) 3 u L ((, T ), D, (), ϱ u L ( (, T ), L () ). The continuity equation (..) holds in D ((, T ) ). The momentum equation (..) holds in (D ((, T ) )) 3. The energy inequality (..) holds for a.a. t (, T ) with E defined by and E defined by E = E = ϱu ϱ {x;ϱ>} + H(ϱ) ε dx (..4) e_r ϱ u {x;ϱ>} + H(ϱ ) ϱ ε dx. (..5) e_r Remark 7. Here, the space D, () is a completion of D(), the space of smooth functions compactly supported in, with respect to the norm u D, () = u dx. Now, the following theorem concerns with the global-in-time existence of the bounded energy weak solution (see [38] and [48]). thm: Theorem 8. (Global-in-time existence of bounded energy weak solution) Let R 3 be an arbitrary open set. Let the pressure p be given by a general constitutive law satisfying p C [, ), p() =, with a ϱγ b p (ϱ) aϱ γ + b, for all ϱ > (..6) pressure Let the initial data ϱ, u satisfy a >, b, γ > 3. ϱ L (), H(ϱ ) L (), ϱ a.e. in, ϱ u ( L () ) 3 such that ϱ u {x;ϱ>} L () ϱ 33

34 and such that ϱ u = whenever x {ϱ = }. (..7) id Then the problem ( ) admits at least one bounded energy weak solution [ϱ, u] on (, T ) in the sense of Definition 6. Moreover [ϱ, u] satisfy the energy inequality (..). Remark 9. The first existence result for problem ( ) was obtained by Lions [7] on condition that R 3 is a domain with smooth and compact boundary and that p(ϱ) ϱ γ with γ 9 5. This result was relaxed to γ > 3 by Feireisl, Novotný and Petzeltová [49] on condition that is a bounded smooth domain. Existence for certain classes of unbounded domains was shown in Novotný and Straškraba [86] (see also Lions [7]). Remark. The existence result in Feireisl [38] and Feireisl, Novotný and Petzeltová [48] holds in the presence of the Coriolis force (see for example Feireisl and Novotný [44] and Feireisl, Jin and Novotný [46] and reference therein)... Classical solutions For the solvability of the system (..6) with the initial data v() = v, we report the following result (see Takada [5]): thm: Theorem. Let s R satisfy s > 3 +. Then, for < T < and v W ( s, R 3) satisfying div x v =, there exists a positive parameter = (s, T, v W s,) such that if ω then the system (..6) possesses a unique classical solution v satisfying v C ( [, T ] ; W s, (R 3 ; R 3 ) ), t v C ( [, T ] ; W s, (R 3 ; R 3 ) ), Π C ( [, T ] ; W s, (R 3 ; R 3 ) ). (..8) reg Remark. The global existence stated above was proved by Kho, Lee and Takada [57] for the initial data in W s, ( R 3) with s > 7/. Remark 3. Theorem deals with inviscid flows in a rotating frame under the condition of fast rotation. In terms of scale analysis (see Nazarenko [79]), if we define by U and L the characteristic velocity and length scale of the fluid, we can estimate the order of magnitude of the non-linear term and the rotational term in the equation (..6) as follows ( ) U v v O, (..9) vel L where v ω O (U), (..) om ω O () O ( ) U, (..) omega L with characteristic angular velocity. Comparing (..9) and (..), we have 34

35 Fast rotation implies U L. (..) comp U L (..3) fast and we can neglect the non-linear term in (..6), obtaining t v + v ω + x Π =, div x v =. (..4) euler_lin These are linear equations. In other words, fast rotation leads to averaging mechanism that weakens the nonlinear effects. This of course prevents singularity allowing the life span of the solution to extend (see Chemin, Desjardines, Gallagher and Grenier [6] and references therein).. Acoustic waves In the following, we introduce the acoustic system related to the equations (..) and (..). Then, we briefly discuss the acoustic energy introducing appropriate energy estimates. Finally, we discuss the decay of acoustic waves in the limit of Mach number tends to zero introducing the dispersive estimate mentioned before. We assume the perturbation of the density of the first order and small compared to the given ambient fluid density. Therefore, we can write the acoustic system related to the equations (..) and (..) by the following linear relations (see Feireisl and Novotný [4], Feireisl, Nečasová and Sun [47] and Lighthill [68, 69]): with the initial data ε t s + Ψ =, ε t Ψ + a x s =, a = p () >, (..) ac_ s() = ϱ (), xψ() = x Ψ = u v (..) ac_ where v = H[u ] and H denotes the Helmholtz projection into the space of solenoidal functions and Ψ is a potential. Here, s is defined as the change in density for a given ambient fluid density. In other words, the density perturbation. The sound velocity squared is represented by a. For more detail physical discussion concerning acoustics, we refer to the book of Falkovich [33] and Landau-Lifshitz [65]... Energy and dispersive estimates The total change in energy of the fluid caused by the acoustic wave is given by the integral ( a s + ) xψ dx, (..3) den_ac R 3 35

36 where the integrand may be regarded as the density of sound energy (see Landau-Lifshitz [65]). It is easy to verify (see Landau-Lifshitz [65]) that the density of sound energy is conserved in time, namely [ R 3 ( a s + ) ] t=t xψ (t, ) dx =. (..4) ac_en t= In addition, we have the following energy estimates (see Feireisl and Novotný [4]) x Ψ (t, ) W k, (R 3 ;R 3 ) + s (t, ) W k, (R 3 ) ( ϱ c x Ψ W k, (R 3 ;R 3 ) + () W k, (R 3 ) ), k =,,..., (..5) en_est for any t >. Instead, concerning the decay of the acoustic waves in the incompressible limit, the following dispersive estimates hold x Ψ (t, ) W k,p (R 3 ;R 3 ) + s (t, ) W k,p (R 3 ) c( + t ( ε ) ( q p) ϱ x Ψ W k,q (R 3 ;R 3 ) + () p, p + =, k =,,... q W k,q (R 3 ) ), (..6) disp_est For the purpose of our analysis and the use of the estimates (..5) and (..6), it is convenient to regularize the initial data (..) in the following way ( ϱ () = ϱ (),η = χ η ψ η ϱ () ), x Ψ = x Ψ,η = χ η (ψ η x Ψ ), η >, (..7) smooth where {χ η } is a family of regularizing kernels and ψ η C (R 3 ) are standards cut-off functions. Consequently, the acoustic system possesses a (unique) smooth solution [s, Ψ] and the quantities x Ψ and s are compactly supported in R 3 (see Feireisl and Novotný [4])..3 Convergence analysis For the purpose of the convergence analysis, we introduce the relative energy functional and the relative energy inequality associated to the system ( ) already mentioned in the Introduction..3. Relative energy inequality The relative energy functional associated to the system ( ) is given by the following relation [ E(ϱ, u r, U) = ϱ (u U) R 3 36

37 ] + ε (H (ϱ) H (r) (ϱ r) H (r)) dx (.3.) entr_funct along with the relative energy inequality [E(ϱ, u r, U)] t=t t= T T +ε S ( x u x U) : ( x u x U) dxdt R(ϱ, u, r, U)dt, R 3 (.3.) entr_ineq where the remainder R is expressed as follows R(ϱ, u, r, U) = ϱ ( t U + u x U) (U u)dx R 3 +ε S( x U) : ( x U x u)dx R 3 + ε R 3 ((r ϱ) t H (r) + x H (r) (ru ϱu)) dx ε R 3 (p(ϱ) p(r)) div x Udx. + (ϱu ω) (U u) dx := I I 5 R 3 (.3.3) rem Here, r and U are sufficiently smooth functions such that r >, r C c ( [, T ] R 3 ), U C c ( [, T ] R 3 ; R 3). (.3.4) test It can be shown (see Feireisl, Jin and Novotný [45] for different type of domains and boundary conditions) that any weak solution [ϱ, u] to the compressible Navier-Stokes system ( ) satisfies the relative energy inequality for any pair of sufficiently smooth test functions r, U as in (.3.4). The particular choice of [r, U] will be clarified later..3. Main results The following theorem is the main result of this chapter. thm: 3 Theorem 4. Let M > be a constant. Let the pressure p satisfy p C [, ) C 3 (, ), p() =, for all ϱ >, with a ϱγ b p (ϱ) aϱ γ + b, (.3.5) pressure a >, b, γ > 3. 37

38 Let the initial data [ϱ, u ] for the Navier-Stokes system ( ) be of the following form ϱ() = ϱ,ε = + εϱ (),ε, u() = u,ε, (.3.6) well data ϱ (),ε + u,ε L L (R 3 L (R ) 3 ;R 3 ) M. (.3.7) data bound Let all the requirements of Theorem be satisfied with the initial datum for the Euler system v = H[u ]. Let [s, Ψ] be the solution of the acoustic system (..) with the initial data (..7). Then, + ϱ (t, ) s(t, ) ε ϱ (u v Ψ) (t, ) L (R 3 ;R 3 ) L (R 3 ) ( c u,ε u ϱ L (R 3 ;R 3 ) + () + ϱ s(t, ) (t, ) ε/γ,ε ϱ() L (R 3 ) ε (/γ) γ L γ (R 3 ) ), t [, T ], (.3.8) th for any weak solutions [ϱ, u] of the compressible Navier-Stokes system ( ). pert Remark 5. The first relation in (.3.6) refers to the first-order perturbation of the density, namely εϱ (),ε, respect to the ambient fluid density settled equal one. A consequence of the above Theorem is the following Corollary. cor: 4 Corollary 6. Let all the requirements of Theorem 4 be satisfied. Assume that Then ϱ (),ε ϱ() in L (R 3 ), u,ε u in L (R 3 ; R 3 ) when ε. ess sup ϱ (u v) (t, ) L (R 3 ;R 3 ) when ε, t [,T ] ess ess sup ϱ L (R 3 ) when ε, t [,T ] sup ϱ γ L γ (R 3 ) when ε, t [,T ] for any weak solutions [ϱ, u] of the compressible Navier-Stokes system ( ) and [r, U] sufficiently smooth test functions. 38

39 .3.3 Convergence The following discussion is devoted to the proof of Theorem 4. Here and hereafter, the symbol c will denote a positive generic constant, independent by ε, usually found in inequalities, that will not have the same value when used in different parts in the analysis. We start with the a priori bounds. In accordance with the energy inequality (..), we have ess ess From (.3.9) and (.3.), we obtain sup ϱ(t, ) L γ L (R 3 ) c(m), (.3.9) unif_bound t [,T ] sup ϱu(t, ) L (R 3 ;R 3 ) c(m). (.3.) unif_bound t [,T ] ϱu(t, ) L q (R 3 ;R 3 ) = ϱ ϱu(t, ) L q (R 3 ;R 3 ) with ϱ(t, ) L γ (R 3 ) ϱu(t, ) L (R 3 ;R 3 ), (.3.) interp We conclude that q = γ γ +. (.3.) q ess sup ϱu(t, ) L q (R 3 ;R 3 ) c(m), q = γ. (.3.3) unif_bound t [,T ] γ + Moreover, introducing (see Germain [5]) I(ϱ, r) = H (ϱ) H (r) (ϱ r) H (r), (.3.4) I we observe that the map ϱ I(ϱ, r) is, for any fixed r >, a strictly convex function on (, ) with global minimum equal to at ϱ = r, which grows at infinity with the rate ϱ γ. Consequently, the integral R 3 I (ϱ, r) (t, x) dx in (.3.) provides a control of (ϱ r) (t, ) in L over the sets {x : ϱ r (t, x) < } and in L γ over the sets {x : ϱ r (t, x) }. So, for any r in a compact set (, ), there holds I(ϱ, r) ϱ r { ϱ r <} + ϱ r γ { ϱ r }, ϱ, (.3.5) I_ in the sense that I(ϱ, r) gives an upper and lower bound in term of the right-hand side quantity (see Bardos and Nguyen [], Feireisl, Novotný and Sun [5] and Sueur [4]). Indeed, is possible to show (see Bardos and Nguyen [], Lemma.) that for the quantity I(ϱ, r) the following approximation holds I(ϱ, r) ϱ (H (ϱ) H (r)) r (ϱ r) H (r), where the right-hand-side is of order ϱ r when ϱ r, and of order ϱ r γ when ϱ r. Therefore, we have the following uniform bounds 39

40 ess ess sup [(ϱ ) (t, )] L { ϱ <} c(m)ε, (R 3 ) (.3.6) unif_bound3 t [,T ] ( [(ϱ 3)) sup ) (t, )] { ϱ } L γ(r c(m)ε /γ, (.3.7) unif_bound4 t [,T ] where we have set r = and U = in the relative energy inequality (.3.). Now, the basic idea is to apply (.3.) to [r, U] = [ + εs, v + x Ψ]. The particular choice of the test functions is motivate by the regularity of the solutions of the Euler (..6) and acoustic (..) system. In the following, η will be fixed. For the initial data we have [E(ϱ, u r, U)]() = R ϱ,ε u,ε u dx 3 [ ( ) ( ) ( ) ( )] + R ε H + εϱ (),ε εh + εϱ () ϱ (),ε ϱ() H + εϱ () dx, 3 (.3.8) initial data conv where u = H[u ] + Ψ. Given (.3.6) and (.3.7), for the first term on the right hand side of the equality (.3.8), we have R ϱ,ε u,ε u dx 3 R 3 R 3 + εϱ () R 3 u,ε u dx + u,ε u dx + ε,ε u,ε u dx R 3 εϱ (),ε ϱ (),ε L (R 3 ) u,ε u dx R 3 u,ε u dx c(m) ( + ε) u,ε u L (R 3 ;R 3 ). (.3.9) initial data conv For the second term on the right hand side of the equality (.3.8), setting a = + εϱ (),ε and b = + εϱ(), and observing that H(a) = H(b) + H (b)(a b) + H (ξ)(a b), ξ (a, b), we have H(a) H (b)(a b) H(b) c a b, R 3 ε [ ( H + εϱ (),ε ) ( ) ( ) ( εh + εϱ () ϱ (),ε ϱ() H ( ε ( ) ) c(m) R ε ϱ (),ε ϱ() dx 3 + εϱ () )] dx 4

41 c(m) ϱ (),ε ϱ(). (.3.) initial data conv L (R 3 ) Finally, we can conclude [E(ϱ, u r, U)]() c(m)[( + ε) u,ε u L (R 3 ;R 3 ) + ϱ () Now, we decompose I into T I dt = T T,ε ϱ() R 3 ϱ [( t U + U x U) (U u)] dxdt L (R )]. 3 R 3 ϱ x U (U u) (U u)dxdt. (.3.) conv For the second term on the right hand side of (.3.), thanks to the Sobolev imbedding theorem, the Minkowski inequality, (..8) and the dispersive estimate (..6), we have T T ϱ x U (U u) (U u)dxdt R 3 ϱ x U (U u) dxdt R 3 T T E x v + xψ L (R 3 ;R 3 ) dt T E x v L (R 3 ;R 3 ) dt + E x Ψ dt L (R 3 ;R 3 ) T c Edt (.3.) The first term on the right hand side of (.3.) can be rewritten as follows T T = ϱ [( t U + U x U) (U u)] dxdt R 3 ϱ(u u) ( t v + v x v) dxdt R 3 T + ϱ(u u) t x Ψdxdt R 3 ϱ(u u) x Ψ : x vdxdt R 3 T + ϱ(u u) v : xψdxdt R 3 T + 4

42 T + ϱ(u u) x x Ψ dxdt. (.3.3) conv3 R 3 In view of uniform bounds (.3.3), (..8) and dispersive estimate (..6), the last three integrals can be estimated as follows T c(m) T ϱ(u u) x Ψ : x vdxdt = ϱ(v+ x Ψ u) x Ψ : x vdxdt R 3 R 3 T = (ϱv) x Ψ : x vdxdt R 3 T + (ϱ x Ψ) x Ψ : x vdxdt R 3 T (ϱu) x Ψ : x vdxdt R 3 T c ϱ L v L x Ψ L x v L dt T +c ϱ L x Ψ L x Ψ L x v L dt [ +c T ε (log (ε + T ) log (ε)) + Similarly to (.3.4), T c(m) and T c(m) R 3 ϱ(u u) v : xψdxdt = [ ε (log (ε + T ) log (ε)) + R 3 ϱ(u u) x x Ψ dxdt = ϱu γ x Ψ γ x v L γ+ L γ L dt ) (ε ε + ε + T T ( ( γ (ε + T ) ε+t ) /γ ε γ R 3 ϱ(v + x Ψ u) v : xψdxdt ) (ε ε + ε + T T ( ( γ (ε + T ) ε+t ) /γ ε γ )] γε. γ (.3.4) th )] γε γ (.3.5) th R 3 ϱ(v + x Ψ u) x x Ψ dxdt [ ( ) ( ) ( ε ε ε + ε + T ε 3 ( ) /γ ε + T (ε + T ) + γ (ε + T ) εγ)]. ε (.3.6) 3th 4

43 Using (..6), for the first term of (.3.3), we have = T T R 3 ϱ(u u) ( t v + v x v) dxdt T T = ϱ(u u) x Πdxdt (U u) (ω ϱv) dxdt R 3 R 3 (.3.7) conv4 Regarding the first integral on the right hand side of (.3.7), as a consequence R 3 ϱu x Πdxdt T R 3 ϱu x Πdxdt T R 3 (U u) (ω ϱv) dxdt. of the estimate (.3.3), we have ( ) ϱu w weakly-(*) in L, T ; L γ/γ+ (R 3 ; R 3 ), (.3.8) press_conv where w denotes the weak limit of the composition. Now, taking the limit in the weak formulation of the continuity equation T ε R 3 ( ϱ ε ) T t ϕdxdt + ϱu x ϕdxdt = (.3.9) weak_cont R 3 for sufficiently smooth ϕ, thanks to the estimate (.3.6) and (.3.7), we deduce that T w x ϕdxdt = R 3 (.3.3) weak_cont_ when ε. We may infer that T R 3 ϱu x Πdxdt T R 3 w x Πdxdt =. (.3.3) conv_ For the second integral on the right hand side of (.3.7), we have T T ϱu x Πdxdt R (ϱ ) U x Πdxdt 3 R 3 T + U x Πdxdt. (.3.3) split R 3 For the first integral on the right-hand side of (.3.3), thanks to (..8), the estimate (..6) and the uniform bounds (.3.6) and (.3.7), we have T cε T R 3 (ϱ ) U x Πdxdt [ϱ ] { ϱ <} v + x Ψ ε L (R L(R3) 3 ;R 3 ) xπ L (R 3 ;R 3 ) dt 43

44 T +cε and T cε [ϱ ε [ϱ ] { ϱ <} v ε L (R L(R3) 3 ;R 3 ) xπ L (R 3 ;R 3 ) dt ] { ϱ <} x Ψ L (R L(R3) 3 ;R ) xπ 3 L (R 3 ;R 3 ) dt c(m)ε (.3.33) press_conv- T R 3 (ϱ ) U x Πdxdt T c [ϱ ]{ ϱ } L (v + γ (R 3 ) xψ) x Π L T c [ϱ ] L { ϱ } v γ (R 3 ) xπ L T +c [ϱ ]{ ϱ } Lγ (R 3 ) xψ x Π L Thanks to the following interpolation inequalities γ dt γ (R 3 ;R 3 ) γ dt γ (R 3 ;R 3 ) γ γ (R 3 ;R 3 ) dt. (.3.34) press_conv- x Ψ x Π γ L γ xψ x Π (R 3 ;R 3 ) L (R 3 ;R 3 ) xψ x Π γ γ γ γ L (R 3 ;R 3 ) γ γ x Ψ γ γ L (R 3 ;R 3 ) xπ L (R 3 ;R 3 ) xψ x Π /γ L (R 3 ;R 3 ) c(m) x Ψ x Π /γ L (R 3 ;R 3 ) c(m) xψ /γ L (R 3 ;R 3 ) xπ /γ L (R 3 ;R 3 ) c(m) x Ψ /γ L (R 3 ;R 3 ), (.3.35) int_ γ γ v x Π γ v L γ xπ (R 3 ;R 3 ) L (R 3 ;R 3 ) v xπ γ γ v γ γ L (R 3 ;R 3 ) xπ L (R 3 ;R 3 ) v xπ /γ L (R 3 ;R 3 ) γ γ L (R 3 ;R 3 ) c v x Π /γ L (R 3 ;R 3 ) c v /γ L (R 3 ;R 3 ) xπ /γ L (R 3 ;R 3 ) c, (.3.36) int_ and the estimate (..6), for the integral in (.3.34) we have, T [ϱ ] Lγ { ϱ } v (R 3 γ ) xπ dt L γ (R 3 ;R 3 ) 44

45 T + [ϱ ] L { ϱ } γ (R 3 ) xψ x Π L c(m)ε /γ + c(m)ε /γ T γ dt γ (R 3 ;R 3 ) x Ψ /γ L (R 3 ;R 3 ) dt ( ( γ (ε + T ) ε+t ) /γ ) c(m)ε /γ + c(m)ε /γ ε γε. (.3.37) press_conv-3 γ γ For the second integral on the right-hand side of (.3.3), we have T R 3 U x Πdxdt = T R 3 v x Πdxdt+ T R 3 x Ψ x Πdxdt. (.3.38) Performing integration by parts in the first term on the right-hand side of (.3.38), we have T div x v Πdxdt = R 3 thanks to incompressibility condition div x v =. For the second term on the right-hand side of (.3.38) using integration by parts and the acoustic equation (..), we have T T x Ψ x Πdxdt = Ψ Πdxdt R 3 R 3 T = ε t s Πdxdt R 3 [ ] t=t T = ε s Πdx ε s t Πdxdt, (.3.39) phi_p R 3 R 3 t= that it goes to zero for ε. For the second term of (.3.3), we have where T press_conv-4 ϱ(u u) t x Ψdxdt R 3 T T = ϱu t x Ψdxdt + ϱv t x Ψdxdt R 3 R 3 + T ϱ t x Ψ dxdt, (.3.4) u_phi_ R 3 T R 3 ϱv t x Ψdxdt 45

46 T T = (ϱ ) v t x Ψdxdt + v t x Ψdxdt. (.3.4) u_phi_ R 3 R 3 We use the acoustic equation (..) to rewrite the first term above as follows T (ϱ ) v t x Ψdxdt R 3 T ϱ = a v x sdxdt, R ε 3 (.3.4) s_phi where, thanks to (..8), (..6), (.3.6) and (.3.7), we have T T ϱ R ε 3 v x sdxdt [ϱ ] { ϱ <} v ε L L(R3) (R 3 ;R 3 ) xs L (R 3 ;R 3 ) dt and c(m)ε (log (ε + T ) log (ε)) (.3.43) s_rho_v T ϱ R ε 3 v x sdxdt T [ϱ ] { ϱ } v γ ε L γ (R Lγ(R3) xs 3 ;R 3 ) L (R 3 ;R 3 ) dt c(m)ε γ (log(ε + T ) log(ε)), (.3.44) vs where we used the following interpolation inequality for v γ v γ v γ L γ (R 3 ;R 3 ) γ γ γ v L (R 3 ;R 3 ) v γ L (R 3 ;R 3 ) L (R 3 ;R 3 ) v /γ γ γ v L (R 3 ;R 3 ) L (R 3 ;R 3 ) c. For the second term in (.3.4), performing integration by parts, we have T div x v t Ψdxdt = (.3.45) div_phi R 3 thanks to incompressibility condition, div x v =. Regarding I, we have T I dt ε T R 3 (S( x U) S( x u)) : ( x U x u) dxdt T +cε S( x U) dxdt, (.3.46) diss R 3 46

47 where we used the Young inequality and the following Korn inequality x U x u dx c (S( x U) S( x u)) : ( x U x u) dx. R 3 R 3 The first term on the right-hand side of (.3.46) can be absorbed by the second term on the left-hand side in the relation (.3.). For the second term on the right-hand side of (.3.46), in view of (..8) and (..5), we have T cε S( x U) dxdt c(m)ε. R 3 (.3.47) diss3 Regarding the terms I 3 and I 4 we deal with the following analysis. First, we have x H (r) rudx = R 3 p(r)div x Udx R 3 (.3.48) grad_h that it will cancel with its counterpart in I 4. Next, = T T ε x H (r) (ϱu) dxdt = T H (r) x s (ϱu) dxdt R ε 3 R 3 Observing that H ( + εs) H () R ε 3 x s (ϱu) dxdt + ε T H ( + εs) H () = H (ξ)s, ξ (, + εs), ε H ( + εs) H () ε cs, R 3 p () x s (ϱu) dxdt. the first term on the right-hand side of (.3.49) can be estimated in the following way T (.3.49) grad_h_p H ( + εs) H () x s (ϱu) dxdt R ε 3 T c s L x s γ ϱu L γ (R 3 ;R 3 ) L γ dt γ+ (R 3 ;R 3 ) ( ( ) /γ ε + T c(m) γ (ε + T ) εγ). (.3.5) H3 ε For the second integral on the right-hand side, using the acoustic equation (..), we get T p () x s (ϱu) dxdt ε R 3 47