Mathematics of complete fluid systems

Transcription

1 Mathematics of complete fluid systems Eduard Feireisl Institute of Mathematics of the Academy of Sciences of the Czech Republic Žitná 25, Praha 1, Czech Republic The scientist only imposes two things, namely truth and sincerity, imposes them upon himself and upon other scientists. Erwin Schroedinger [ ] Contents 1 Introduction Balance laws Balance law - differential form Balance law - weak form Systems of balance laws - constitutive relations Field equations of continuum fluid mechanics Fluid description - thermostatics Transport - bulk velocity

2 2.3 Balance of linear momentum Balance of energy Kinetic energy balance Internal energy balance Entropy production Constitutive equations for the viscous stress and the internal energy flux Boundary conditions Weak formulation Weak continuity Equation of continuity - weak formulation Momentum equation - weak formulation Entropy production Total energy balance Weak vs. strong solutions Vacuum problem Weak-strong compatibility Exercises Maxwell s equation Entropy vs. pressure Monoatomic gas state equation A priori bounds Total mass conservation Energy estimates Dissipation and entropy estimates Thermodynamic stability Ballistic free energy Total dissipation balance Korn inequality More hypotheses imposed on constitutive equations General pressure law Effect of thermal radiation Exercises Thermodynamic stability for monaotomic gases Energy estimates revisited Pressure estimates

3 5.7.1 Renormalized equation of continuity Boundedness of entropy Weak sequential stability Preliminary observations Weak continuity of the convective terms Compactness via Div-Curl lemma Compactness of the remaining convective terms Poinwise a.a.) convergence of the temperature field Pointwise a.a.) convergence of the densities Weak continuity of the effective viscous pressure Commutator estimates Density oscillations Exercise - convexity and strong convergence Oscillation defect measure Boundedness of the oscillations defect measure Everything should be made as simple as possible, but not one bit simpler. Albert Einstein [ ] 1 Introduction This lecture series is a brief introduction to the mathematics of complete fluid systems. By complete we mean the systems that are energetically closed; the total energy of the fluid is a constant of motion. We stay on the platform of classical continuum mechanics - a fluid in motion will be described by means of phenomenological quantities - fields - that are continous in space and time and obey the basic principles of classical physics. We introduce a mathematical theory of complete fluid systems in the framework of weak solutions. The main advantage of this approach is the fact that the associated mathematical problems 3

4 are solvable without any artificial smallness and smoothness) restrictions imposed on the data and on arbitrary lapses of time, including infinite time intervals. We refer to the monographs by Ladyzhenskaya [?], Lions [?], [5], Sohr [?], Temam [?], [?], and to [4], [?], for more details concerning the application of the concept of weak solutions in fluid mechanics. 1.1 Balance laws The time evolution of systems in continuum mechanics is determined by the physical principles expressed mathematically as balance laws. A balance law includes three basic quantities: a density d of a balanced quantity; a flux vector field F; a source s; where all quantities depend on the time t and the spatial variable x R 3. Here, the symbol denotes the physical space - a domain in the Euclidean space R 3 occupied by the fluid. A general form of a balance law reads Balance law for d [total amount of d in B at the time t 2 ] [total amount of d in B at the time t 1 ] = 1.1) B dt 2, x) dx B t2 t2 dt 1, x) dx = Ft, x) n ds x dt + st, x)dx dt t 1 B t 1 B = [total flux through the boundary B during the time interval [t 1, t 2 ]] + [source] Such a description is associated to the Eulerian reference system, where the values of all physical quantities are related to a fixed point x in the physical space, in contrast with the Lagrangean description based on individual particle trajectories, see Truesdell and Rajagopal [?]. 4

5 1.1.1 Balance law - differential form The following argument is classical. Assuming that all quantities in 1.1) are smooth differentiable) we may apply the Gauss-Green theorem to deduce that B dt 2, x) dx furthermore, letting t 2 t 1 t, yielding, finally, B B t2 t2 dt 1, x) dx = div x Ft, x) dx dt + st, x)dx dt, t 1 B t 1 B t dt, x) dx = div x Ft, x) dx + st, x)dx, B B Balance law for d - differential form t dt, x) + div x Ft, x) = st, x) 1.2) Balance law - weak form There are two ways how to view the so-called weak formulation of the balance law 1.1). First, we multiply equation 1.2) by a smooth differentiable) test function ϕ = ϕt, x), with a compact support. Integrating by parts, we obtain Balance law for d - weak form T [ dt, x) t ϕt, x) + Ft, x) x ϕt, x) + st, x)ϕt, x) ] dx dt = 1.3) for any ϕ Cc, T ) ). It seems a bit strange that the weak form was obtained from the strong differentiable form. Another possibility how to see 1.3) is to go back to 1.1) that can be viewed as t2 lim dt, x) t ϕ ε t, x) dx dt ε t 1 B 5

6 B where ϕ ε C c dt 2, x) dx T lim ε B t2 t2 dt 1, x) dx = Ft, x) n ds x dt + st, x)dx dt t 1 B t 1 B Ft, x) x ϕ ε t, x) dx dt + lim ε T t 1, t 2 ) B) is a suitable family of smooth functions, ϕ ε 1, ϕ ε 1 as ε. 1.2 Systems of balance laws - constitutive relations st, x)ϕ ε t, x) dx dt, The mathematical formulation of problems in continuum fluid mechanics consists of a system of balance laws for quantities d 1,..., d N, fluxes F 1,..., F N, and sources s 1,..., s N. Clearly, as we would always have more unknowns than equations, the system must be closed by determining the fluxes and sources in terms of the unknowns d 1,..., d N. These relations characterize material properties of a specific fluid and are termed constitutive relations. 2 Field equations of continuum fluid mechanics The basic equations of continuum fluid mechanics correspond to the physical principles of balance of mass, momentum, and energy. 2.1 Fluid description - thermostatics The problem of time in physics and chemistry is closely related to the formulation of the second law of thermodynamics. I. Prigogine [ ], Nobel Lecture, 8 December 1977 Consider, for a moment, a fluid at rest. We suppose that the state of the fluid is completely determined by two fields: the mass density ϱ and the temperature ϑ. The internal) energy density 6

7 e = eϱ, ϑ) and the pressure p = pϱ, ϑ) are uniquely determined by ϱ and ϑ. In accordance with Second law of thermodynamics, there is another thermodynamic variable s = sϱ, ϑ) called specific) entropy, see Callen [?]. The internal energy e, the pressure p, and the entropy s are interrelated through Gibbs equation ϑdsϱ, ϑ) = Deϑ, ϑ) + pϱ, ϑ)d ) ) ϱ Willard Gibbs [ ] Relation 2.1) should be viewed as the stipulation that [ 1 Deϑ, ϑ) + pϱ, ϑ)d ϑ )] 1 ϱ is a perfect differential. Note that the entropy s is determined modulo an additive constant. The latter can be fixed by means of Third law of thermodynamics that requires the entropy to approach zero for ϑ, see Belgiorno [?], [?]. 2.2 Transport - bulk velocity The observable macroscopic) motion of the fluid is described by means of the velocity field u = ut, x). The streamlines are obtained by solving a system of ordinary differential equations: dx dt = ut, Xt, x)), X) = x. The velocity describes mass transport, the quantity ϱu is the mass flux, and the physical principle of mass conservation reads 7

8 Equation of continuity: t ϱ + div x ϱu) =, 2.2) or, in the weak form, T [ ϱ t ϕ + ϱu x ϕ ] dx dt = 2.3) for any test function ϕ C c, T ) ). 2.3 Balance of linear momentum Similarly to the preceding part, we obtain the balance of linear momentum in the differential) form t ϱu) + div x ϱu u) = div x T + ϱf, where T is the so-called Cauchy stress, and f is a given) external force acting on the fluid. Fluids are characterized among other materials by Stokes law: where S is the viscous stress tensor and p the pressure. T = S pi, 2.4) George Gabriel Stokes [ ] Accordingly, the balance of momentum in an isotropic fluid can be written as 8

9 Balance of momentum - Newton s second law: or, in the weak form T t ϱu) + div x ϱu u) + x pϱ, ϑ) = div x S + ϱf, 2.5) [ ϱu t ϕ + ϱu u) : x ϕ + pϱ, ϑ)div x u ] dx dt = for any test function ϕ C c, T ) ; R 3 ). T [ S : x u ϱf ϕ ] dx dt 2.6) Isaac Newton [ ] We have noticed that, unlike in the mass conservation, the momentum flux consists of two components: the convective one representes by the tensor ϱu u that corresponds, similarly to 2.2) to the mass transport, and the diffusive one S pi. As we shall se below, the energy is transported in a similar way. 2.4 Balance of energy Determining a suitable form of the balance of energy is one of the cornerstones of the theory. Our approach leans on a formulation based on Second law of thermodynamics, specifically, on the entropy production equation formulated below Kinetic energy balance Taking the scalar product of the momentum balance equation 2.5) with u we obtain ) ) 1 1 ) t 2 ϱ u 2 + div x 2 ϱ u 2 u + div x pϱ, ϑ)u divx Su ) = S : x u + pϱ, ϑ)div x u + ϱf u. 2.7) 9

10 Note that, in accordance with the equation of continuity 2.2), we have t ϱu) + div x ϱu u) = t ϱ + div x ϱu) ) u + ϱ t u + ϱu x u. The quantity 1 2 ϱ u 2 is called kinetic energy, whereas equation 2.7) represent the kinetic energy balance. Obviously, equation 2.7) contains a source term S : x u + pϱ, ϑ)div x u + ϱf u. In many cases, the external force f is a gradient of a scalar potential, specifically, f = x F, F = F x). Accordingly, making use of the continuity equation 2.2), we get whence 2.7) reads ) 1 t 2 ϱ u 2 ϱf ϱf u = ϱ x F u = div x ϱf u ) F divx ϱu) = t ϱf ) + divx ϱf u ) ; ) ) 1 ) + div x 2 ϱ u 2 ϱf u + div x pϱ, ϑ)u divx Su ) 2.8) = S : x u + pϱ, ϑ)div x u. The presence of the source term on the right-hand side of 2.8) indicates that the kinetic energy is not conserved, in particular, the total energy balance should also take into account the changes of the internal energy. Note, however, that if the fluid is inviscid idealization), we would have S. If, in addition, the fluid is barotropic, meaning the pressure p = pϱ) depends solely on the density, we have, making once more use of the continuity equation, where pϱ)div x u = t ϱp ϱ) ) divx ϱp ϱ) ), P ϱ) = ϱ 1 In such a case, equation 2.8) would reduce to a conservation law ) ) ) 1 1 ) t 2 ϱ u 2 + ϱp ϱ) ϱf + div x 2 ϱ u 2 + ϱp ϱ) ϱf u + div x pϱ, ϑ)u = 2.9) pz) z 2 dz. for the energy of an inviscid barotropic fluid. 1 2 ϱ u 2 + ϱp ϱ) ϱf 1

11 2.4.2 Internal energy balance In accordance with First law of thermodynamics, the total energy of the system is conserved. Thus the internal energy balance must contain the source term appearing in 2.8) as a sink on the right-hand side. Specifically, the internal energy balance equation reads t ϱeϱ, ϑ)) + div x ϱeϱ, ϑ)u) + div x q = S : x u pϱ, ϑ)div x u. 2.1) In addition to the convective transport term ϱeu, the internal energy can be transported without mass transfer by diffusion, represented by the extra internal energy flux q. Summing up 2.8), 2.1) we obtain the total energy balance equation Entropy production ) ) ) 1 1 t 2 ϱ u 2 + ϱeϱ, ϑ) ϱf + div x 2 ϱ u 2 + ϱeϱ, ϑ) ϱf u +div x pϱ, ϑ)u ) divx Su ) + div x q =. 2.11) Using again the continuity equation 2.2), we can rewrite the internal energy equation 2.11) in the form [ ] [ ] t ϱ x ϱ ϱdeϱ, ϑ) + ϱu Deϱ, ϑ) + div t ϑ x ϑ x q = S : x u pϱ, ϑ)div x u. 2.12) Since e, p are related to the entropy s by Gibbs equation 2.1), we get whence 2.12) reads as ϱdeϱ, ϑ) [ t ϱ t ϑ ] + ϱu Deϱ, ϑ) [ x ϱ x ϑ [ ] [ ] t ϱ x ϱ pϱ, ϑ) = ϱϑdsϱ, ϑ) + ϱϑu Dsϱ, ϑ) + t ϑ x ϑ t ϱ + u x ϱ) ϱ [ ] [ ] t ϱ x ϱ = ϱϑdsϱ, ϑ) + ϱϑu Dsϱ, ϑ) pϱ, ϑ)div t ϑ x ϑ x u; ϱϑdsϱ, ϑ) [ t ϱ t ϑ In other words, we have deduced ] + ϱϑu Dsϱ, ϑ) 11 [ x ϱ x ϑ ] ] + div x q = S : x u.

12 Entropy balance equation: or, in the weak form, T ) q t ϱsϱ, ϑ)) + div x ϱsϱ, ϑ)u) + div x = σ 2.13) ϑ [ ϱsϱ, ϑ) t ϕ + ϱsϱ, ϑ)u x ϕ + q ] ϑ xϕ T dx dt = σϕ dx dt 2.14) for any ϕ C c, T ) ), with the entropy production rate σ = 1 ϑ S : x u q xϑ ϑ ). 2.15) Although the internal energy balance 2.1), the total energy conservation 2.11), and the entropy production equation 2.13) are, in fact, equivalent in the framework of regular solutions, it may not be the case for weak solutions. As we shall see, it is the entropy that provides the most relevant piece of information in the weak formulation. 2.5 Constitutive equations for the viscous stress and the internal energy flux In accordance with Second law of thermodynamics, the entropy production rate σ in 2.15) must be a non-negative quantity for any admissible physical process. This fact imposes certain restrictions on the specific form of S and q. Here, we suppose that S is a linear function of the velocity gradient x u while is a linear function of x ϑ. Specifically, we impose Newton s rheological law: S = µ x u + t xu 2 ) 3 div xui + ηdiv x ui, 2.16) together with 12

13 Fourier s law: q = κ x ϑ. 2.17) Joseph Fourier [ ] Note that the entropy production rate σ takes the form σ = 1 ϑ µ 2 x u + t xu 2 3 div 2 xu + η div x u 2 + κϑ) xϑ 2 ) ; 2.18) ϑ whence, in particular, the shear viscosity coefficient µ, the bulk viscosity coefficient η as well as the heat conductivity coefficient κ must be non-negative. In this lecture, we always assume strict positivity of µ and κ. We remark that the specific form of 2.16) is dictated by the principle of material frame indifference, see Chorin and Marsden [?]. Other examples of more complicated so-called non-newtonian viscous stress tensors may be found in Málek and Rajagopal [?]. 2.6 Boundary conditions In this series of lectures, we always assume that the physical space containing the fluid is a bounded domain R 3. Accordingly, some kind of boundary behavior of certain fields must be specified. Our principal assumption is that the total energy of the fluid is conserved. A short inspection of equation 2.11) reveals that a suitable condition could be impermeability of the boundary: together with u n =, 2.19) 13

14 Thermal insulation: q n =. 2.2) In the presence of viscosity, the impermeability hypothesis 2.19) is not sufficient to determine uniquely the behavior of the fluid. Alternatively, we may take the most common No-slip boundary condition: u =, 2.21) or Complete slip boundary condition: u n =, [Sn] n =. 2.22) Claude Louis Marie Henri Navier [ ] The reader may consult Málek and Rajagopal [?], Priezjev and Troian [?] for a thorough discussion concerning the physical relevance of the slip - no-slip boundary conditions. 14

15 3 Weak formulation The system of equations 2.2), 2.5), 2.13), with the constitutive relations for S and q given by 2.16), 2.17), will be called Navier-Stokes-Fourier system. Our goal in the present section is to introduce a suitable weak formulation of the problem that takes into account the satisfaction of the boundary conditions ). In order to have, at least formally, a well-posed problem, we introduce the initial conditions that characterize the original state of the fluid system at the time t = : ϱ, ) = ϱ, ϱu, ) = ϱ u, ϱsϱ, ϑ) = ϱ sϱ, ϑ ). 3.1) In order to comply with the basic physical principles, we suppose that 3.1 Weak continuity ϱ >, ϑ > in. 3.2) Weak solutions are usually quantities that are only locally) integrable with respect to the space and time variables, satisfying some differential equations in the sense of generalized derivatives. Consider, for instance, the weak formulation of a general conservation law 1.3). Here, it is enough that d, F, and s be integrable functions in the space-time cylinder, T ), d L 1, T ) ), F L 1, T ) ; R 3 ), s L 1, T ) ). Take the quantity ϕt, x) = ψt)φx) as a test function in 1.3) to obtain T ) T ) dt, )φ dx t ψ dt = F x φ sφ) dx ψ dt. 3.3) As relation 3.3) holds for any smooth ψ = ψt), we deduce that the function of time: t dt, )φ dx is absolutely continuous for any φ Cc ). In other words, we are able to identify the instantaneous values of the averages dφ dx provided φ is differentiable and compactly supported. In addition, suppose that d enjoys some extra integrability, say ess sup dt, ) L p ) c for a certain p > 1, t,t ) in other words, d L, T ; L p )). 15

16 Since the smooth functions φ Cc ) are dense in the dual space L p ), 1/p + 1/p = 1, we get t dt, )φ dx is absolutely continuous for any φ L p ) [L p )], in other words, the density d, viewed as a function of the time t with values in the Banach space L p ), is continuous with respect to the weak topology in L p ): d C weak [, T ]; L p )). 3.2 Equation of continuity - weak formulation Suppose that ϱ u and that ϱ L, T ; L γ )), ϱu L, T ; L β ; R 3 )) for certain β, γ > 1. We say that ϱ, ϱu is a weak solution of equation 2.2), supplemented with the boundary condition 2.19) and the initial condition 3.1), if the integral identity [ ϱτ, )ϕτ, ) ϱ ϕ, ) ] dx = τ ϱ t ϕ + ϱu x ϕ ) dx dt 3.4) is satisfied for any τ [, T ] and any test function ϕ C c [, T ] ). It is worth-noting that 3.4) holds in the whole physical space, T ) R 3 provided ϱ was extended to be zero outside. 3.3 Momentum equation - weak formulation In order to formulate a weak analogue of the momentum equation, we need the stress tensor to be al least integrable. As the latter is given by Newton s law 2.16), we need the spatial gradient of u to be at least integrable. Accordingly, we suppose that u L q, T ; W 1,q ; R 3 )) for a certain q > 1. As the Sobolev functions in W 1,q possess well-defined traces, the no-slip boundary condition 2.21) may be incorporated in the function space, requiring u L q, T ; W 1,q ; R 3 )) for a certain q > ) 16

17 Similarly to the preceding section, we say that ϱ, u, ϑ is a weak solution of the momentum equation 2.5), 2.16), with the boundary conditions 2.21) and the initial condition 3.1) if the velocity field belongs to the class specified in 3.5), S is given by 2.16), and if the integral identity [ ϱuτ, ) ϕτ, ) ϱ u ϕ, ) ] dx = τ ϱu t ϕ+ϱu u : x ϕ+pϱ, ϑ)div x ϕ ) dx dt 3.6) τ S : x ϕ ϱ x F ϕ ) dx dt holds for any τ [, T ] and any test function ϕ C c [, T ] ; R 3 ). In 3.6), we tacitly assume that all quantities are at least integrable in the set, T ). If the no-slip condition 2.21) is replaced by the complete slip 2.22), we have to replace 3.5) by and to extend the class of admissible test functions to 3.4 Entropy production u L q, T ; W 1,2 ; R 3 )), u n =, 3.7) ϕ C c [, T ] ; R 3 ), ϕ n =. We have reached the crucial moment of the mathematical theory. In the weak formulation of the entropy production equation 2.14), we replace equality 2.15) by the inequality σ 1 S : x u q ) xϑ. 3.8) ϑ ϑ To this end, we must have ϑ > a.a. in, T ). Accordingly, the weak formulation of the entropy balance reads τ [ ϱsϱ, ϑ)τ, )ϕτ, ) ϱ sϱ, ϑ )ϕ, ) ] τ dx dt τ ϱsϱ, ϑ) t ϕ + ϱsϱ, ϑ)u x ϕ + q ) ϑ xϕ for a.a. τ, T ) and any ϕ C c [, T ] ), ϕ. 1 S : x u q ) xϑ ϕ dx dt ϑ ϑ dx dt 3.9) 17

18 Of course, replacing 2.15) by 3.8) means a crude simplification and considerable extension of the class of possible solutions. In order to compensate for this lack of information, the resulting system must be supplemented by an extra condition formulated in the following section. 3.5 Total energy balance Thanks to our choice of the boundary conditions, we may integrate the total energy balance 2.11) to obtain for a.a. τ, T ). [ ] 1 [ ] 1 2 ϱ u 2 + ϱeϱ, ϑ) ϱf τ, ) dx = 2 ϱ u 2 + ϱ eϱ, ϑ ) ϱ F dx 3.1) To conclude, we recall that the weak formulation of the Navier-Stokes-Fourier system consists of Mass conservation. Equation of continuity in the sense of generalized derivatives. Newton s second law. Momentum equation + boundary conditions satisfied in the weak sense. Entropy production. Entropy inequality with a non-negative production rate. Total energy balance. Conservation of the total energy in time. 4 Weak vs. strong solutions However beautiful the strategy, you should occasionally look at the results. Sir Winston Churchill,

19 Our concept of weak solutions to the Navier-Stokes-Fourier system apparently extends that of the strong solutions, in the sense that, evidently, strong solutions satisfying all equations in the classical sense, including the entropy equation 2.13), 2.15) are also weak solutions in the sense specified in Section Vacuum problem One of the major open problems of the theory is the possibility for ϱ to become zero in a finite time. Although weak solutions can be constructed with ϱ, we are not able to show that, in fact, ϱt, ) > as long as ϱ >. Suppose that solutions are smooth. We can rewrite the equation of continuity 2.2) in the form or, equivalently, Thus, defining the characteristic field we obtain whence where t ϱ + u x ϱ = ϱdiv x u, t logϱ) + u x logϱ) = div x u. dx dt = ut, Xt)), d dt logϱt, Xt)) = div xut, Xt)); exp Lt) inf ϱ x) ϱt, x) explt) sup ϱ x) for all t >, 4.1) x L = x sup div x ut, x). t,x),t ) Unfortunately, uniform bounds on div x u are beyond reach of the present theory so 4.1) remains valid only in the class of strong solutions. 19

20 4.2 Weak-strong compatibility A more delicate task is to show that a weak solution that is sufficiently regular, represents a strong solution. In particular, the entropy inequality 3.9) can be replaced by 2.13), 2.15). In order to see this, we first observe that the kinetic energy balance 2.8) holds as soon as the weak solution is smooth, in particular, ) d 1 dt 2 ϱ u 2 ϱf dx = S : x u + pϱ, ϑ)div x u ) dx. 4.2) The next step is to take ϑϕ, ϕ, ϕ Cc, T ) ), as a test function in 3.9) to obtain: τ τ [ S : x uϕ dx dt ϑ t ϱsϱ, ϑ) + div x ϱsϱ, ϑ)u) ] ϕ q x ϕ ) dx dt. 4.3) On the other hand, Gibbs equation 2.1) yields ϑ [ t ϱsϱ, ϑ)) + div x ϱsϱ, ϑ)u) ] = ϱ [ ϑ t sϱ, ϑ) + ϑu x sϱ, ϑ) ] = ϱ [ t eϱ, ϑ) + u x eϱ, ϑ) ] + pϱ, ϑ)div x u, where we have also used the equation of continuity 2.2), together with the fact that the regular) density remains bounded below away from zero provided we have chosen ϱ >, see 4.1). We infer that 4.3) gives rise to d ϱeϱ, ϑ) dx S : x u pϱ, ϑ)div x u ) dx. 4.4) dt Thus the total energy balance imposed through 3.1) is compatible with 4.2), 4.4) only if the sign is replaced by = in 4.4), meaning relation 3.9) holds with equality sign. Weak solutions of the Navier-Stokes-Fourier system introduced in Section 4 coincide with the strong solutions as soon as they are regular, specifically, if ϱ >, ϑ >, and ϱ, ϑ, u are continuously differentiable in t and twice continuously differentiable in x. As a matter of fact, one can establish the principle of weak-strong uniqueness for the class of weak solutions in the sense specified above. A weak solution coincide with a classical one emanating from the same initial data as long as the latter exists. In other words, the strong solutions are unique within the class of weak solutions, see [?]. 2

21 4.3 Exercises Maxwell s equation Suppose that p = pϱ, ϑ), e = eϱ, ϑ), and s = sϱ, ϑ) satisfy Gibbs relation 2.1). Show Maxwell s equation eϱ, ϑ) = 1 ) pϱ, ϑ) pϱ, ϑ) ϑ. 4.5) ϱ ϱ 2 ϑ Entropy vs. pressure Using Gibbs equation 2.1) and Maxwell s equation 4.5) show that sϱ, ϑ) ϱ = 1 pϱ, ϑ). ϱ 2 ϑ Monoatomic gas state equation A general monoatomic gas is characterized by the relation pϱ, ϑ) = 2 ϱeϱ, ϑ). 4.6) 3 Suppose, in addition to 4.6), that p and e satisfy Gibbs equation 2.1). Show that p takes the form ) ϱ pϱ, ϑ) = ϑ 5/2 P for a certain function P. ϑ 3/2 5 A priori bounds A priori bounds are constraints imposed on the solution set by the data and the differential identities satisfied. As is well known in the standard theory of elliptic equations, a priori bounds determine the function spaces framework in which the solutions are looked for, see Lions [?]. Evolutionary equations arising in continuum fluid mechanics are known to possess only very poor a priori bounds, which makes the rigorous analysis rather delicate. To begin, we point out that a priori bounds are purely formal, derived under the hypothesis of smoothness of all quantities in question. In particular, the spatial domain is bounded and belongs to the class C 2+ν, ϱ, ϑ, u possess all the necessary derivatives and ϱ, ϑ are strictly positive. 21

22 5.1 Total mass conservation The simplest bound follows directly by integrating the equation of continuity 2.2) over the spatial domain. Supposing that the velocity field u obeys the impermeability condition we immediately obtain in other words ϱt, ) dx = u n = d ϱt, ) dx =, dt ϱ dx, for any t [, T ]. 5.1) The physical meaning of 5.1) is obvious - the total mass of the fluid contained in the physical domain is a constant of motion. Note that the same conclusion holds even in the class of the weak solutions satisfying 3.4), where it is enough to take ϕ 1. Since the density is non-negative we get sup ϱt, ) L 1 ) = ϱ L 1 ) for any t [, T ]. 5.2) t [,T ] 5.2 Energy estimates The total energy balance 3.1) yields immediately sup ϱut, ) L 2 ;R 3 ) cinitial data), 5.3) t [,T ] and sup ϱeϱ, ϑ)t, ) L 1 ) cinitial data) 5.4) t [,T ] provided, say, F L ). Of course, we have tacitly assumed that e. Similarly to the preceding part, the energy estimates remain valid even in the class of weak solutions in the sense specified in Section 4. 22

23 5.3 Dissipation and entropy estimates Our next step is to exploit the entropy balance equation 2.13). Integrating by parts and making use of the boundary conditions we deduce that τ [ 1 ϱsϱ, ϑ)τ, ) dx = ) ϱ sϱ, ϑ ) dx + S : x u q. ] xϑ ϑ ϑ Thermodynamic stability dx dt. 5.5) In accordance with Second law of thermodynamics, we can control the total entropy production rate provided we have an upper bound on the total entropy of the system. Since we already know that the total internal energy is bounded, we may attempt to obtain upper bound on ϱsϱ, ϑ) in terms of ϱeϱ, ϑ). To this end, we introduce Hypothesis of thermodynamic stability pϱ, ϑ) ϱ >, 5.6) eϱ, ϑ) ϑ >. 5.7) Condition 5.6) asserts that compressibility of the fluid is always positive while ϑ e is the specific heat at constant volume, see Bechtel, Rooney, and Forest [?] Ballistic free energy Let us introduce a thermodynamic potential H Θ ϱ, ϑ) = ϱeϱ, ϑ) Θϱsϱ, ϑ), with Θ > a constant, 5.8) termed ballistic free energy, see Ericksen [?]. It follows from the hypotheses 5.6), 5.7) that ϱ H Θ ϱ, Θ) is strictly convex; 5.9) 23

24 ϑ H Θ ϱ, ϑ) is decreasing for ϑ < Θ and increasing for ϑ > Θ. 5.1) Total dissipation balance Combining the total energy conservation 3.1) with 5.5) we deduce Total dissipation balance: [ H Θ ϱ, ϑ) ϱ ϱ) H ] Θϱ, Θ) H Θ ϱ, Θ) ϱf τ, ) dx 5.11) ϱ τ Θ + S : x u q ) xϑ dx dt ϑ ϑ [ = ) H Θ ϱ, ϑ ) ϱ ϱ) H ] Θϱ, Θ) H Θ ϱ, Θ) ϱ F dx, ϱ where the constant ϱ > has been chosen so that ϱt, ) ϱ) dx =. Consequently, making use of the coercivity properties of the ballistic free energy H Θ established in Section 5.3.2, we deduce from 5.11) the following a priori bounds: cdata) ϱsϱ, ϑ)τ, ) dx cdata) for all τ [, T ], 5.12) and T 1 S : x u q ) xϑ ϑ ϑ dx dt cdata). 5.13) 24

25 5.3.4 Korn inequality Our next goal is to use 5.13) in order to derive bounds on x u and x ϑ, respectively. We start with x ϑ seeing that, in accordance with Fourier s law 2.17), Consequently, assuming T κϑ) x ϑ 2 for certain positive constants κ, κ we deduce that T ϑ 2 dx dt cdata). κ1 + ϑ 2 ) κϑ) κ1 + ϑ 2 ) 5.14) x ϑ 2 dx dt + T x logϑ) 2 dx dt cdata) 5.15) Similar estimates on the velocity gradient are more delicate. By virtue of Newton s law 2.16), we get S : x u = µ x u + t xu 2 ) 3 div xui : x u whence, assuming, say, we deduce from 5.13) that T = µϑ) 2 xu + t xu 2 3 div 2 xui ; µ1 + ϑ) µϑ) µ1 + ϑ), 5.16) xu + t xu 2 3 div xui 2 dx dt c 5.17) Suppose, for a moment, that u vanishes on, meaning u satisfies the no-slip boundary condition 2.21). By simple computation, we get xu + t xu 2 3 div 2 xui = x u 2 + t xu x u t xu 4 3 div xu 2, where, by means of a simple by parts integration, x u t xu dx = 25 div x u 2 dx.

26 Thus the bound 5.13) yields T x u 2 dx cdata). 5.18) Finally, since we have assumed that u =, we may combine 5.18) with Poincaré s inequality to conclude that T u 2 + x u 2) dx cdata). 5.19) Henri Poincaré [ ] We note that u 2 + x u 2) dx u 2 W 1,2 ;R 3 ), where W 1,2 stands for the standard Sobolev space of functions having first spatial) partial derivatives square integrable. 5.4 More hypotheses imposed on constitutive equations The bounds derived in the preceding part are the best known) a priori bounds available under the general conditions imposed on the initial data and the existence time T >. Still, they are not strong enough, in general, to render all terms appearing in the weak formulation at least equi-integrable in order to ensure their stability. Besides the general hypothesis of thermodynamic stability 5.6), 5.7), more restrictions will be imposed on the constitutive relations in order to obtain better estimates to make the problem mathematically tractable General pressure law A general pressure law of a monoatomic gas reads pϱ, ϑ) = 2 ϱeϱ, ϑ), 5.2) 3 26

27 see Eliezer, Ghatak, Hora [3]. It is straightforward to check that 5.2) is compatible with the general Gibbs relation 2.1) if the pressure takes the form ) ϱ pϱ, ϑ) = ϑ 5/2 P 5.21) ϑ 3/2 for a certain function P : [, ) R. Now, the hypothesis 5.6) can be interpreted as P = C 1 [, ), P ) =, P Z) > for any Z. 5.22) Going back to 5.2), the specific internal energy reads eϱ, ϑ) = 3 ) ϑ 3/2 2 ϱ P ϱ Now, the second condition of thermodynamic stability 5.7) translates to in particular, the function 5 P Z) P Z)Z 3 Z ) ϱ. 5.23) ϑ 3/2 > for any Z >, 5.24) Z P Z) is non-increasing, Z5/3 5.25) and we set P Z) lim Z Z = p 5/3 >. 5.26) Under the present circumstances, the entropy is determined, up to an additive constant, by the Gibbs equation 2.1), specifically, ) ϱ sϱ, ϑ) = S, ϑ 3/2 5.27) where S Z) = 3 5 Z)Z 3 <. 2 Z ) In accordance with Third law of thermodynamic, we may fix S by postulating lim SZ) =. 5.29) Z 27

28 5.4.2 Effect of thermal radiation The pressure p can be augmented by a component p R = p R ϑ) resulting from thermal radiation, namely, p R = a 3 ϑ4, a >, with a the Stefan-Boltzmann constant. Consequently, we end up with the following specific form of the constitutive equations: Pressure: ) ϱ pϱ, ϑ) = ϑ 5/2 P + a ϑ 3/2 3 ϑ4, 5.3) Internal energy: eϱ, ϑ) = 3 ) ϑ 3/2 2 ϑ P ϱ sϱ, ϑ) = S Entropy: ) ϱ + a ϑ 3/2 ϱ ϑ4, 5.31) ) ϱ + 4a ϑ 3/2 3ϱ ϑ ) 5.5 Exercises Thermodynamic stability for monaotomic gases Show the the hypothesis of thermodynamic stability 5.7) implies relations 5.24), 5.25). 5.6 Energy estimates revisited From now on, we shall assume that the pressure p, the internal energy e, and the entropy s are given through ). Thus we get, as a consequence of 5.26), the following bounds 28

29 sup ϱt, ) L 5/3 ) cdata), 5.33) t [,T ] and, because of the radiation component of the internal energy, sup ϑt, ) L 4 ) cdata). 5.34) t [,T ] 5.7 Pressure estimates The a priori bounds should be at least so strong for all the expressions appearing in the weak formulation to make sense to be at least integrable in the space-time cylinder, T ). As a matter of fact, slightly more is needed, namely the equi-integrability property in order to perform the limits with respect to the weak topology of the Lebesgue space L 1. Note that the estimates 5.33), 5.34) guarantee only that pϱ, ϑ), ϱeϱ, ϑ) L, T ; L 1 )). Better estimates can be obtained by computing the pressure by means of the continuity equation 2.5): pϱ, ϑ) = div x div x S div x div x ϱu u) div x t ϱu), where, for the sake of simplicity, we have take f. Then, very formally indeed, pϱ, ϑ) = div x 1 div x S div x 1 div x ϱu u) 1 div x t ϱu). Let us remark that both ϱu u and S are already known to be bounded in the Lebesgue space L q, T ) for a certain q > 1. Indeed, by virtue of the Sobolev imbedding relation W 1,2 ) L 6 ), 5.35) combined with the uniform bounds established in 5.3), 5.19), and 5.32), we have ϱu u L q ;R 3 3 ) = ϱ ϱu u L q ;R 3 3 ) ϱ L 5/3 ) ϱu L 2 ;R 3 ) u W 1,2 ;R 3 ), with whence 1 γ = , meaning γ = 6 29 > 1; ϱu u L 2, T ; L 3/29 ; R 3 3 )). 5.36) 29

30 Similarly, by virtue of 5.19), 5.34), and hypothesis 5.16) whence S L q,r 3 3 ) c ϑ L 4 ) x u L 2 ;R 3 3 ), q = 4 3 ; S L 2, T ; L 4/3 ; R 3 3 )). 5.37) Seeing that the operator div x 1 div x is of zero-th order, the estimates 5.35), 5.36) would guarantee boundedness of the pressure provided we could handle the time derivative 1 div x t ϱu). In order to make these arguments rigorous, we proceed in a slightly different way. Specifically, we multiply the momentum equation by the expression ϕ x 1 [1 bϱ)], where 1 denotes the characterictic function of thw domain and ϕ Cc ). The obvious advantage of such a choice is the fact that the function vanishes on and one can perform by parts integration by means of the Gauss-Green theorem. The operator x 1 may be viewed as an inverse and can be rigorously defined as a Fourier multiplier div 1 x [ ] xj 1 [v] = Fξ x 1 iξj ξ F xξ[v], j = 1, 2, 3, 2 where F denotes the Fourier transform in the spatial variable x. The following properties of x div 1 x are standard see, for instance, [4, Chapter 5]): x 1 [v] L +L 2 R 3 ;R 3 ) c v L 2 L 1 R 3 ), 5.38) x x 1 [v] L p R 3 ;R 3 3 ) c v L p R 3 ) for any 1 < p <. 5.39) Taking the scalar product of the momentum equation with ϕ x 1 [bϱ)], where b) = and ϱ was extended to be zero outside ; or equivalently, using this quantity as a test function in 3.6), and performing obvious by-parts integration, we obtain τ ϕpϱ, ϑ)bϱ) dx dt 5.4) τ = pϱ, ϑ) x ϕ x 1 [bϱ)] dx dt 3 τ ϕϱu t x 1 [bϱ)] dx dt

31 τ [ ) + ϱu u + S : x ϕ x 1 [bϱ)] ) ϕϱ x F x 1 [bϱ)] ] dx dt [ ϕϱu x 1 [bϱ)]τ, ) ϕϱ u x 1 [bϱ ) ] dx. Our goal is to take bϱ) = min{ϱ, ϱ α } with α >, where α > is chosen so small that all terms on the right-hand side of 5.4) may be bounded in terms of the previously established energy estimates. To this end, however, we have to compute t x 1 [bϱ)] Renormalized equation of continuity Multiplying equation 2.2) on b ϱ) we obtain Renormalized equation of continuity: t bϱ) + div x bϱ)divx u ) + b ϱ)ϱ bϱ) ) div x u =. 5.41) Ronald DiPerna [ ] The renormalized equation was introduced by DiPerna and Lions [2]. It can be derived from the equation of continuity as long as all quantities are sufficiently smooth. In the framework of weak solutions, equation 5.41) can be taken as a kind of supplementary condition to be satisfied by the weak solutions. We point out that, since u vanishes on, equation 5.41) holds a.a.) in the whole physical space provided ϱ, u were extended to be zero outside. In particular, we may compute t x 1 [bϱ)] = x 1 [div x bϱ)u)] + x 1 [ bϱ) b ϱ)ϱ ) div x u ]. 5.42) Going back to 5.4) we observe that all terms on the right-hand side will be bounded in terms of the energy estimates 5.19), 5.32), 5.34) as soon as we take bϱ) = min{ϱ, ϱ α }, with α > 31

32 sufficiently small. Indeed, given q > 1, we can find α > such that, in view of 5.38), 5.39), x 1 [bϱ)] L, T ; W 1,q ; R 3 )), with W 1,q ) C) for q > ) Similarly, for α = αq) > small enough, we have where we have used the embedding relation Finally, by the same token x 1 [div x bϱ)u)] L 2, T ; L q ; R 3 )) for any q < 6, 5.44) W 1,2 ) L 6 ). x 1 [ b ϱ)ϱ bϱ) ) div x u ] L 2, T ; L q ; R 3 )) for any q < ) Thus we conclude that all integrals on the right-hand side of 5.4) can be bounded in terms of the available estimates; whence we conclude that T K pϱ, ϑ)ϱ α dx dt ck) for any compact set K and for a certain α >. 5.46) In accordance with the hypotheses 5.22), 5.25), we have pϱ, ϑ) cϑ 5/3, c > ; whence T ϱ α+ 5 3 dx dt ck) for any compact K. 5.47) K Finally, by the same token pϱ, ϑ) c1 + ϑ 4 + ϱ 5/3 ), where ϑ 4 = ϑ 3 ϑ, with ϑ 3 bounded in L, T ; L 4/3 )), ϑ bounded in L 2, T ; L 6 )). Therefore we conclude that T ϑ 4+β dx dt cdata) for a certain β >. 5.48) As a matter of fact, the pressure estimates can be extended up to the boundary by means of the methods developed in [?] and by Kukučka [?]. 32

33 5.8 Boundedness of entropy We stengthen 5.24) to 5 3 < P Z) P Z)Z < c for all Z, 5.49) Z meaning that the specific heat at constant volume is positive and uniformly bounded. Assuming the entropy complies with Third law of thermodynamics 5.29) we obtain and ϱsϱ, ϑ) ϱsϱ, 1) ϱ logϱ) for < ϑ 1, ϱsϱ, ϑ) ϱlogϱ) + [logϑ)] + ) for any ϱ, ϑ >. 5.5) Consequently, the uniform bounds established in the previous section are strong enough in order to ensure integrability of all terms appearing in the entropy balance 3.9). Indeed the entropy diffusive flux reads κϑ) ϑ xϑ; where κϑ) satisfies 5.14) and the desired bounds follow from 5.15). 6 Weak sequential stability The problem of weak sequential stability may be stated as follows: Weak sequential stability: Let {ϱ ε, u ε, ϑ ε } ε> be a family of strong or weak) solutions of the Navier-Stokes-Fourier system, emanating from the initial data ϱ ε, ) = ϱ,ε, ϱu), ) = ϱu),ε, ϑ ε, ) = ϑ,ε, and bounded by the a priori bounds derived in the previous section. The goal is to show that ϱ ε ϱ, u ε u, ϑ ε ϑ as ε in a certain sense, where ϱ, u ϑ is a weak solution of the same system. 33

34 Although showing weak sequential stability does not provide an explicit proof of existence of the weak solutions, its verification represents one of the prominent steps towards a rigorous existence theory for a given system of equations, see Lions [?]. 6.1 Preliminary observations In view of the uniform bounds established in the previous section, specifically 5.15), 5.19), 5.32), and 5.34), we may assume that ϱ ε ϱ weakly-*) in L, T ; L 5/3 )), 6.1) u ε u weakly in L 2, T ; W 1,2 ; R 3 )) 6.2) ϑ ε ϑ weakly-*) in L, T ; L 4 )) and weakly in L 2, T ; W 1,2 )), 6.3) passing to suitable subsequences as the case may be. Moreover, since ϱ ε satisfies the equation of continuity 2.3), convergence in 6.1) may be strengthened to ϱ ε ϱ in C weak [, T ]; L 5/3 )). 6.4) Let us recall that 6.4) simply means { t ϱ ε t, )ϕ dx } { t for any ϕ L 5/2 ). Finally, combining 5.3), 5.33) we deduce that ϱt, )ϕ dx } in C[, T ] ϱ ε u ε ϱu weakly-*) in L, T ; L 5/4 ; R 3 )). 6.5) Here and hereafter, we use the symbol bv) to denote a limit of a composition of a nonlinear) function b with a weakly in L 1 ) converging sequence v ε v. As is well know, in general, bv) bv) unless an extra piece of information on oscillations of v ε is available. 6.2 Weak continuity of the convective terms Our next goal is to establish convergence of the convective terms, specifically, to show that ϱu = ϱu, ϱu u = ϱu u, and ϱsϱ, ϑ)u = ϱsϱ, ϑ)u. 34

35 This can be done by means of several rather similar arguments. Suppose we want to show that ϱu = ϱu. This can be observed in several ways. Seeing that W 1,2 ) L q ) compactly for 1 q < 6, we deduce that L p ) W 1,2 ) compactly whenever p > ) In particular, relation 6.4) yields ϱ ε u ε ϱu in C[, T ]; W 1,2 ))), which, combined with 6.2), gives rise to the desired conclusion ϱu = ϱu. Relation can be shown in a similar way. ϱu u = ϱu u Compactness via Div-Curl lemma Div-Curl lemma, developed by Murat and Tartar [6], [8], represents an efficient tool for handling compactness in non-linear problems, where the classical Rellich-Kondraschev argument is not applicable. Luc Tartar 35

36 Div-Curl lemma: Lemma 6.1 Let B R M be an open set. Suppose that v n v weakly in L p B; R 3 ), as n, where Let, moreover, for a certain s > 1. Then w n w weakly in L q B; R 3 ) 1 p + 1 q = 1 r < 1. {div[v]} n=1 be precompact in W 1,s B), {curl[w]} n=1 be precompact in W 1,s B, R M M ) v n w n v w weakly in L r B). We apply Div-Curl lemma in order to show compactness of all convective terms. Let us start with ϱ ε u ε. In order to comply with the hypotheses, we introduce cut-off functions: T k z) C R), T k z) = T k z), T k z) concave in, ), T k z) = z whenever z k, T k strictly increasing in R, lim z T k z) = k + 1. We use Div-Curl lemma in the four-dimensional) space-time cylinder, T ) setting Consequently, On the other hand, we have v ε = [ϱ ε, ϱ ε u ε ], w ε = [T k [u j ε],,, ], j = 1, 2, 3. ϱ ε T k [u j ε] ϱt k [u j ] weakly in L 5/4, T ) ). u j ε T k [u j ε] q L q ) 2 36 { u j ε k} u j ε q dx 6.7)

37 Consequently, = 2 u j ε 6 q u j ε 6 dx 2 { u j ε k} k q 6 uj ε q L 6 ) whenever 1 q < 6. and we may write ) 1 1/q u j ε T k [u j ε] q L 2,T ;L q )) u j k ε 6 q L 2,T ;L 6 )), 1 q < 6, ϱ ε u j ε = ϱ ε T k [u j ε] + ϱ ε u j ε T k [u j ε] ) ϱt k [u j ] + χ k = ϱu weakly in L r, T ) ) for a certain r > 1, where and, by the same token Thus we may infer that χ k L r,t ) ) as k, ϱt k [u j ] ϱu j L r,t ) ) as k. ϱu = ϱu Compactness of the remaining convective terms The Div-Curl lemma argument can be successively applied to and to conclude that v ε = [ϱ ε u j ε, ϱu ε u j ε S,j + pδ,j ], j = 1, 2, 3, w ε = [T k [u i ε],,, ], v ε = [ϱ ε sϱ ε, ϑ ε ), ϱ ε sϱ ε, ϑ ε )u ε κϑ ε) x ϑ ε ϑ ε ], w ε = [T k [ϑ ε ],,, ], ϱ ε u ε u ε ϱu u weakly in L r, T ) ; R 3 3 ), 6.8) ϱ ε sϱ ε, ϑ ε )u ε ϱsϱ, ϑ)u weakly in L r, T ) ; R 3 ) 6.9) for a certain r > 1. The convergence stated in 6.9) deserves some comments. Here, we have [ DIV t,x ϱ ε sϱ ε, ϑ ε ), ϱ ε sϱ ε, ϑ ε )u ε κϑ ] ε) x ϑ ε = σ ε, ϑ ε 37

38 with, in accordance with the total dissipation balance 5.11), {σ ε } ε> bounded in L 1, T ) ). However, since we have W 1,p, T ) ) compactly C[, T ] ) for p > 4, L 1, T ) ) W 1,s, T ) ), s < Poinwise a.a.) convergence of the temperature field Our goal is to show that ϑ ε ϑ a.a. in, T ). The pointwise convergence of the temperature is necessary in order to pass to the the limit in the non-linear terms. As a matter of fact, we show a stronger statement, specifically, ϑ ε ϑ in L 4, T ) ). 6.1) In order to show 6.1), we use monotonicity of the entropy sϱ, ϑ) with respect to ϑ, together with compactness of {ϑ ε } ε> in the space variable. To begin, we claim that it is enough to show that T ϱε sϱ ε, ϑ ε ) ϱ ε sϱ ε, ϑ) ) ϑ ε ϑ ) dx dt 6.11) as ε. Indeed, since the entropy is given by 5.32), we have = T ϱ ε S ϱε ϑ 3/2 ε ) T ϱ ε S We proceed by several steps. Step 1: ϱε sϱ ε, ϑ ε ) ϱ ε sϱ ε, ϑ) ) ϑ ε ϑ ) dx dt ϱε ) ) ) 4a ϑε ϑ ε dx dt + 3 4a T ϑ ε ϑ) 4 dx dt. 3 ϑ 3/2 T ϑ 3 ε ϑ 3) ϑ ε ϑ ) dx dt 38

39 The argument based on Div-Curl Lemma can be applied in a similar way as in the preceding section to obtain: ϱsϱ, ϑ)ϑ = ϱsϱ, ϑ)ϑ, in other words, T Thus it remains to show that Step 2: T ϱ ε sϱ ε, ϑ ε ) ϑ ε ϑ ) dx dt as ε. 6.12) ϱ ε sϱ ε, ϑ) ϑ ε ϑ ) dx dt as ε. 6.13) Since bϱ ε ) satisfy the renormalized equation 5.41), we can apply once more the Div-Curl lemma argument in order to show that bϱ)ϑ = bϱ)ϑ for any bounded and smooth Lipschitz) b, in particular, 6.13) would follow should s depend only on ϱ. As a matter of fact, the same argument yield a slightly better result, namely, Step 3: bϱ)gϑ) = bϱ) gϑ) for any b, g bounded and Lipschitz. 6.14) In view of 6.14), the desired conclusion 6.13 follows from the following result - fundamental theorem of the theory of Young measures, see Ball [1], Pedregal [7]. 39

40 Fundamental Theorem on Young measures Theorem 6.1 Let v ε : Q R N R M be a sequence of vector fields bounded in L 1 Q; R M ). Then there exists a subsequence not relabeled) and a family of probability measures {ν y } y Q on R M such that: For any Carathéodory function Φ = Φy, Z), y Q, Z R M such that Φ, v ε ) Φ weakly in L 1 Q) we have Φy) = Φy, Z) dν y Z) for a.a. y Q. R M Laurence Chisholm Young [195-2] Taking ϱ ϱsϱ, ϑt, x)) as a Caratheodory function of the argument ϱ we deduce from 6.14) the desired conclusion ϱsϱ, ϑt, x))ϑ = ϱsϱ, ϑt, x))ϑ yielding 6.13). Thus we have shown 6.11). 6.4 Pointwise a.a.) convergence of the densities Our goal is to show that ϱ ε ϱ a.a. in, T ). 6.15) 4

41 We start by introducing a linear zero-th) order operator or, in terms of Fourier mutlipliers, R i,j = xi 1 xj, [ ] ξi ξ j R i,j [v] = F ξ x ξ ξ[v] Weak continuity of the effective viscous pressure We start with a celebrated result of Lions [5] on the weak continuity of the quantity pϱ, ϑ) 4 3 µϑ) + ηϑ) ) div x u termed effective viscous pressure, see Lions [5]. We introduce ϕ ε = φ x 1 [T k ϱ ε )], ϱ ε outside, φ C c where T k are the cut-off functions defined through 6.7), to be used as test functions in the momentum equation 3.6). Similarly to 5.4), we obtain τ2 τ2 φpϱ ε, ϑ ε )T k ϱ ε ) dx dt φs ε : R[T k ϱ ε )] dx dt 6.16) τ2 = τ 1 τ2 + τ 1 where we have set τ 1 pϱ ε, ϑ ε ) x φ x 1 [T k ϱ ε )] dx dt + τ2 + τ 1 τ 1 τ2 φϱ ε u ε u ε ) : R[T k ϱ ε )] dx dt τ 1 ), φϱ ε u ε t x 1 [T k ϱ ε )] dx dt [ ϱε u ε u ε + S ε ) x φ x 1 [T k ϱ ε )] ) φϱ ε x F x 1 [T k ϱ ε )] ] dx dt [ φϱε u ε x 1 [T k ϱ ε )]τ 2, ) φϱ ε u ε x 1 [T k ϱ ε )]τ 1, ) ] dx, S ε = µϑ ε ) x u ε + t xu ε 2 ) 3 div xu ε I + ηϑ ε )div x u ε I. We remark that, in accordance with 6.1), S ε S µϑ) x u + t xu 2 ) 3 div xui + ηϑ)div x ui weakly in L r, T ) ; R 3 ) 41

42 for a certain r > 1. Now, we can let ε in 3.6) and take ϕ = φ x 1 [T k ϱ)] as a test function in the resulting integral identity to obtain τ2 τ2 φpϱ, ϑ) T k ϱ) dx dt φs : R[T k ϱ)] dx dt 6.17) τ2 = τ 1 τ2 + τ 1 τ 1 pϱ, ϑ) x φ x 1 [T k ϱ)] dx dt + τ 1 τ2 τ 1 φϱu t x 1 [T k ϱ)] dx dt τ2 + φϱu u) : R[T k ϱ)] dx dt τ 1 [ ) ϱu u + S x φ x 1 [T k ϱ)] ) φϱ x F x 1 [T k ϱ)] ] dx dt [ φϱu x 1 [T k ϱ)]τ 2, ) φϱu x 1 [T k ϱ)]τ 1, ) ] dx. At this stage, we use the renormalized equation 5.41) to compute and, similarly, t x 1 [T k ϱ ε )] = x 1 [div x T k ϱ ε )u ε )] + x 1 [T k ϱ ε ) T kϱ ε )ϱ ε ) div x u ε ] t x 1 [T k ϱ)] = x 1 [div x T k ϱ)u ε )] + x 1 [T k ϱ) T k ϱ)ϱ) div xu]. Seeing that, by virtue of 5.38), 5.39), x 1 [T k ϱ ε )] x 1 [T k ϱ)] in C[, T ] ), we can let ε in 6.16) and compare the resulting expression with 6.17) to conclude τ2 + lim ε τ 1 τ2 τ 1 τ2 lim ε τ 1 = τ2 τ 1 τ2 φpϱ ε, ϑ ε )T k ϱ ε ) dx dt lim φs ε : R[T k ϱ ε )] dx dt 6.18) ε τ 1 φpϱ, ϑ) T k ϱ) dx dt φϱ ε u ε u ε ) : R[T k ϱ ε )] dx dt lim φϱu u) : R[T k ϱ)] dx dt + 42 τ2 τ 1 τ2 ε τ 1 τ2 τ 1 φs : R[T k ϱ)] dx dt φϱ ε u ε x 1 [div x T k ϱ ε )u ε )] dx φϱu x 1 [div x T k ϱ)u)] dx.

43 Rewriting φϱε u ε u ε ) : R[T k ϱ ε )] φϱ ε u ε x 1 [div x T k ϱ ε )u ε )] ) dx = = i,j=1 i,j=1 3 φϱε u i εu j ε : xi 1 xj [T k ϱ ε )] φϱ ε u i ε xi 1 xj T k ϱ ε )u j ε) ) dx 3 φϱε u i εu j ε : xi 1 xj [T k ϱ ε )] xj 1 xi [φϱ ε u i ε]t k ϱ ε )u j ε), ) dx we examine the bilinear form 3 [v, w] = v i R i,j [w j ] w i R i,j [v j ] ), i,j=1 where we may write 3 v i R i,j [w j ] w i R i,j [v j ] ) where = 3 i,j=1 i,j=1 v i R i,j [v j ])R i,j [w j ] w i R i,j [w j ])R i,j [v j ] ) = U V W Z, U i = v i R i,j [v j ]), W i = j=1 j=1w i R i,j [w j ]), div x U = div x W =, and 3 3 V i = xi 1 x jw j, Z i = xi 1 x jv j, i = 1, 2, 3. j=1 j=1 Thus a direct application of Div-Curl lemma Lemma 6.1) yields [v ε, w ε ] [v, w] weakly in L s R 3 ) and whenever v ε v weakly in L p R 3 ; r 3 ), w ε w weakly in L q R 3 ; R 3 ), 1 p + 1 q = 1 s < 1. 43

44 Seeing that T k ϱ ε ) T k ϱ) in C weak [, T ]; L q )) for any 1 q <, ϱ ε u ε u in C weak [, T ]; L 5/4 )) we conclude that T k ϱ ε t, )) x 1 [div x ϱ ε u ε )t, )] ϱ ε u ε )t, ) x 1 x [T k ϱ ε t, ))] 6.19) T k ϱ)t, ) x 1 [div x ϱu)t, )] ϱu)t, ) x 1 x [T k ϱ)t, )] weakly in L s ; R 3 ) for all t [, T ], 1 s < 5/4. Thus we conclude that the convergence in 6.19) takes place in the space L q, T ; W 1,2 ; R 3 )) for any 1 q < ; whence, going back to 6.18) we may infer that Effective viscous flux weak continuity: pϱ, ϑ)t k ϱ) S : R[T k ϱ)] = pϱ, ϑ) T k ϱ) S : R[T k ϱ)] for any k ) Pierre-Louis Lions [*1956] Commutator estimates Our aim is to show the commutator estimates for the vector field 3 3 Z i = xi 1 xj [νy j ] ν xi 1 xj Y j, i = 1, 2, 3. j=1 j=1 where Y = [Y 1, Y 2, Y 3 ] is a vector field in Cc R 3 ; R 3 ) and ν Cc R 3 ). Computing div x Z = x ν [Y x 1 [div x Y] ], 44

45 we may use the bound curl Z = { xi ν xj 1 [div x Y] } 3 i,j=1, to deduce that x Z L q R 3 ;R 3 3 ) c div x Z L q R 3 ) + curl Z L q R 3 ;R 3 3 )), 1 < q < x Z L q R 3 ;R 3 3 ) c x ν L p R 3 ;R 3 ) Y L r 1 R 3 ;R 3 ), Moreover, by means of the standard interpolation argument, Z L q R 3 ;R 3 ) c x ν L p R 3 ;R 3 ) Y L r 2 R 3 ;R 3 ), 3 p 3p + 1 r 2 = 1 q Finally, interpolating 6.21), 6.22) we obtain the following result: Commutator lemma: 1 p + 1 r 1 = 1 q, 1). 6.21), 1), 1 p < ) Lemma 6.2 Let ν C c R 3 ), Y C c R 3 ; R 3 ) be given fields, and let 1 < p < 3, 1 q 1 p < 1 r < 1 q 3 p, q, 1), 3p and Z = x 1 div x [νy] ν x 1 div x [Y]. Then Z D α,q R 3 ) c ν D 1,p R 3 ) U L r R 3 ;R 3 ) for a certain α >. Now, if {Z ε } ε> is a sequence of functions bounded in L 1, T ; D α,q R 3 )) L r, T ) ) for certain α >, q, r > 1, we may use compactness of the embedding and the convergence D α,q ) L q ), T k ϱ ε ) T k ϱ) in C weak [, T ; L p )) for any 1 p < 45