Mathematical theory of fluids in motion

Transcription

1 Mathematical theory of fluids in motion Eduard Feireisl March 28, 27 Institute of Mathematics of the Academy of Sciences of the Czech Republic Žitná 25, CZ-5 67 Praha, Czech Republic Abstract The goal of the course is to present the recent development of the mathematical fluid dynamics in the framework of classical fluid mechanics phenomenological models. In particular, we discuss the Navier-Stokes (viscous) and the Euler (inviscid) systems modelling the motion of a compressible fluid. The theory is developed from fundamental physical principles, the necessary mathematical tools introduced at the moment when needed. In particular, we discuss various concepts of solutions and their relevance in applications. Particular interest is devoted to well-posedness of the initial-value problems and their approximations including possibly certain numerical schemes. All pictures used in the text thanks to wikipedia

2 Contents Continuum fluid mechanics 4. Mass density, velocity, mass conservation Eulerian vs. Lagrangean description Mass transport Momentum equation Cauchy stress in fluids, examples of fluid equations Inviscid fluids, compressible Euler system Viscous fluid, compressible Navier-Stokes system First law of thermodynamics - complete fluid systems Second law of thermodynamics Various concepts of solutions to the equations and systems of mathematical fluid dynamics 2. Possible blow-up of smooth solutions Classical - strong solutions Local existence for the compressible Euler and Navier-Stokes system Rewriting the Euler system as symmetric hyperbolic Global existence for small initial data Weak solutions Weak solutions to the compressible Euler system Dissipative (weak) solutions Relative energy Weak vs. strong solutions 9 3. Weak-strong uniqueness Weak-strong uniqueness for the Euler system Weak-strong uniqueness for the Navier-Stokes system Wild dissipative solutions of the Euler system Admissible weak solutions to the compressible Euler system Stability of the solution set A priori bounds Energy estimates Estimates based on dissipation Stability of strong solutions for the Euler system Weak sequential stability Oscillations - Young measures Measure-valued solutions Weak-strong uniqueness in the class of measure-valued solutions

3 5 Weak sequential stability of solutions to the Navier-Stokes system Integrability Pressure estimates Refined pressure estimates, renormalized equation of continuity Weak compactness and continuity of the convective terms Strong convergence of the density Effective viscous flux

4 Continuum fluid mechanics Our goal is to present a phenomenological theory of fluids in motion - fluid dynamics - based on the principles of classical continuum mechanics. Accordingly, a fluid is described my means of observable macroscopic quantities as mass density, bulk velocity, internal energy depending on the time t and the reference spatial coordinate x R N. Although the physically relevant values are N =, 2, 3, we mostly focus on the most complex case N = 3. If not stated otherwise, the Eulerian reference system will be used attached to the physical domain occupied by the fluid in contrast with the Lagrangean description related to the hypothetical fluid particles and their trajectories - streamlines - in the physical space. Leonhard Euler [77 783]. Mass density, velocity, mass conservation The distribution of a fluid at a given time t is given by the mass density ϱ = ϱ(t, x) - a non-negative scalar function - such that the integral ϱ(t, x) dx = M(B) B gives the total mass of the fluid contained in a given set B at the time t. The motion of (hypothetical) fluid particles is determined by the velocity field u(t, x) R N. trajectory of a bulk of fluid occupying at the initial instant t = the set B is given by X(t, B), t, where X(t, x) = u(t, X(t, x)), X(, x) = x, x B. t The individual trajectories t X(t, x) are called streamlines. The velocity field must enjoy certain regularity in the x variable for the streamlines to be well defined on a time interval I, specifically The x u L (I; L loc(r N, R N )). (.) Remark.. As we shall see below, the question whether or solutions of a fluid model enjoy the desired regularity (.) is mostly an open problem, in particular in higher space dimensions, see the review paper by Fefferman [2]. This partially explains why the Eulerian rather than Lagrangean description is used in the mathematical theory... Eulerian vs. Lagrangean description 4

5 Given a velocity field u generating a family of streamlines X = X(t, x) in the physical space, a quantity Q can be expressed in terms of the Eulerian variables as Q = Q E (t, x), t I, x X(t, B), or, in the Lagrangean variables Joseph-Louis Lagrange Q = Q L (t, Y ), t I, Y B, [736 83] where Q L (t, Y ) = Q E (t, X(t, Y )) or Q L (t, X (t, x)) = Q E (t, x). In particular, the time derivative in the Lagrangean setting corresponds to the material derivative in the Euler coordinates: t Q L (t, Y ) = t Q E (t, X(t, x)) + x Q E (t, X(t, x)) u(t, X(t, x)). At first glance it may seem that the Lagrangean description is simpler; whence more suitable for a mathematical treatment. On the other hand, transforming spatial gradients requires invertibility of the mapping X and therefore certain regularity of the velocity field that is often out of reach of the available analytical methods...2 Mass transport Consider a piece B of the physical space R N (Eulerian) velocity u = u(t, x). containing a fluid of the density ϱ moving with the The physical principle of mass conservation asserts: The change of the total mass of the fluid contained in B during a time interval t < t 2, ϱ(t 2, x) dx dt ϱ(t, x) dx dt, B = (equals) the total out/in flux of the mass through the boundary B t2 ϱ(t, x)u(t, x) n(x) ds x dt, t B where n denotes the outer norm vector to B. B The above relation must hold for any time interval [t, t 2 ] and any volume element B. Assuming that all quantities are sufficiently smooth (differentiable), we may use the Gauss-Green theorem and perform the limit t 2 t to deduce a differential form of the principle of mass conservation - equation of continuity: t ϱ + div x (ϱu) =. (.2) 5

6 It is remarkable that the same principle can be derived from an apparently weaker statement [ϱ t ϕ + ϱu x ϕ] dx dt = for any ϕ Cc (I ), (.3) I which is usually termed the weak formulation of (.2). Indeed it is enough to take ϕ ε Cc (I ) a suitable approximation of the characteristic function [t,t 2 ] B and let ε. If B is smooth, one can consider a family of Lipschitz functions { } { } ϕ ε (t, x) = min dist[x, B]; min ε ε min{t t ; t 2 t };..2 Momentum equation In order to determine the velocity field u we need a relation between the changes of the momentum ϱu and the material forces acting on a volume element of the fluid. We consider two kinds of such forces: (i) stress forces acting on any surface element shared by two adjacent parts of the fluid, (ii) bulk or volumic forces. The stress is characterized by the Cauchy stress tensor T producing the stress T n acting on a unit surface element characterized by a normal vector n. Augustin-Louis Cauchy [ ] Mathematical formulation of Newton s second law reads ϱu(t 2, x) dx ϱu(t, x) dx B B t2 = ϱ(t, x)u(t, x) (u(t, x) n(x)) ds x dt + t t2 B T(t, x) n(x) ds x dt + t2 t B t B ϱ(t, x)f(t, x) dx dt, for any B and any t < t 2. Remark.2. Note that the above relation is vectorial relating N components of the momentum ϱu to the stress forces. Similarly to Section..2, we may deduce the differential form of Newton s second law t (ϱu) + div x (ϱu u) = div x T + ϱf, (.4) 6

7 or its weak formulation [(ϱu) t ϕ + ϱu u : x ϕ] dx dt = I for any ϕ C c (I ; R N ). I T : x ϕ dx dt Remark.3. As the density ϱ satisfies the equation of continuity (.2) we may always write t (ϱq) + div x (ϱqu) = ϱ ( t Q + u x Q), I ϱf ϕ dx dt (.5) where the expression on the right hand side is nothing other than the material derivative of a quantity Q multiplied on the mass density..3 Cauchy stress in fluids, examples of fluid equations The system of equations (.2), (.4) is not closed, a description of the Cauchy stress T is needed in terms of ϱ, u or other quantities as the case may be. The following statement is often used as a mathematical definition of fluid. A fluid is characterized by Stokes law T = S pi, where S is viscous stress tensor and p a scalar quantity called pressure..3. Inviscid fluids, compressible Euler system We start with an example of inviscid (or perfect) fluids for which S =. In the simplest case, the pressure is just a function of the density and we obtain Compressible Euler system t ϱ + div x (ϱu) =, (.6) t (ϱu) + div x (ϱu u) + x p(ϱ) =. (.7) If N =, system (.6, (.7) can be rewritten in the Lagrangian (mass) coordinates by introducing new independent variables [t, x] [t, y(t, x) = x ϱ(t, z) dz]. 7 George Gabriel Stokes [89 93]

8 The Lagrangian velocity v = v(t, y) satisfies ( x ) v t, ϱ(t, z) dz = u(t, x); whence and (.6) reads while (.7) gives rise to t v ϱu y v = t u, ϱ y v = x u, t U y v =, (.8) ( ) t v + y p = (.9) U where U = is the specific volume. Problem (.8), (.9) is called p-system. ϱ The Euler system formally conserves energy. Taking the scalar product of (.7) with u we obtain, by means of straightforward manipulation ( ) [( ) ] t 2 ϱ u 2 + div x 2 ϱ u 2 + p(ϱ) u p(ϱ)div x u =. (.) Moreover, multiplying (.6) on b (ϱ) we deduce the renormalized equation of continuity t b(ϱ) + div x (b(ϱ)u) + [b (ϱ)ϱ b(ϱ)] div x u =, (.) Isaac Newton [ ] which, together with (.), gives rise to the total energy balance ( ) [( ) ] ϱ p(z) t 2 ϱ u 2 + P (ϱ) + div x 2 ϱ u 2 + P (ϱ) + p(ϱ) u =, with P (ϱ) = ϱ dz. (.2) z Viscous fluid, compressible Navier-Stokes system We consider the simplest possible example of a viscous fluid, where the viscous stress tensor is a linear function of the velocity gradient. For isotropic fluid, the associated S is given by Newton s rheological law ( S( x u) = µ x u + t xu 2 ) N div xu + ηdiv x ui, (.3) where µ and η are scalar quantities taken to be constant here for the sake of simplicity. Accordingly, we obtain Compressible Navier-Stokes system t ϱ + div x (ϱu) =, (.4) t (ϱu) + div x (ϱu u) + x p(ϱ) = div x S( x u), (.5) ( S( x u) = µ x u + t xu 2 ) N div xu + ηdiv x ui. (.6) 8

9 Similarly to (.2), we write the total energy balance ( ) [( ) ] t 2 ϱ u 2 + P (ϱ) + div x 2 ϱ u 2 + P (ϱ) + p(ϱ) u S( x u) u = S( x u) : x u. (.7) As an physically admissible system should not produce energy, the source term S( x u) : x u must be non-negative for any admissible process, in particular µ, η. We speak about viscous fluids if µ >..4 First law of thermodynamics - complete fluid systems Claude-Louis Navier [ ] As we have seen in the preceding part, the description by means of purely mechanical quantities like ϱ and u is not complete from the physics point of view; the resulting compressible Navier-Stokes system dissipates (mechanical) energy. For the First law of thermodynamics to hold, one has to introduce a new quantity called (specific) internal energy e. Alternatively, we may consider the absolute temperature ϑ and suppose that both p = p(ϱ, ϑ) and e = e(ϱ, ϑ) are given functions of the state variables ϱ, ϑ. The functional dependence of p and e on ϱ and ϑ is called equation of state and characterizes the material properties of a given fluid. We rewrite the energy balance (.7) as ( ) [( ) ] t 2 ϱ u 2 + div x 2 ϱ u 2 + p(ϱ, ϑ) u S( x u) u (.8) = S( x u) : x u + p(ϱ, ϑ)div x u Jean Baptiste Joseph Fourier [768 83] ( t [ϱ and append the system by a similar relation for the internal energy t (ϱe(ϱ, ϑ))+div x [ϱe(ϱ, ϑ)u]+div x q = S( x u) : x u p(ϱ, ϑ)div x u, (.9) where q is a new quantity that represents the diffuse flux of the internal energy. In accordance with the First law of thermodynamics, the total energy of the system must be conserved; whence )] 2 u 2 + e(ϱ, ϑ) + div x [ϱ ( 2 u 2 + e(ϱ, ϑ) ) ] u + p(ϱ, ϑ)u S( x u) u + q =. (.2) The internal energy flux very often coincides with the heat flux, the latter being given by Fourier s law q = κ x ϑ. (.2) 9

10 Thus we have derived Navier-Stokes-Fourier system with t ϱ + div x (ϱu) =, (.22) t (ϱu) + div x (ϱu u) + x p(ϱ, ϑ) = div x S( x u), (.23) ( )] [ ( ) ] t [ϱ 2 u 2 + e(ϱ, ϑ) + div x ϱ 2 u 2 + e(ϱ, ϑ) u + p(ϱ, ϑ)u S( x u) u + q =, (.24) ( S( x u) = µ x u + t xu 2 ) N div xu + ηdiv x ui, (.25) q = κ x ϑ. (.26) The inviscid version is known as Complete Euler system t ϱ + div x (ϱu) =, (.27) t (ϱu) + div x (ϱu u) + x p(ϱ, ϑ) =, (.28) ( )] [ ( ) ] t [ϱ 2 u 2 + e(ϱ, ϑ) + div x ϱ 2 u 2 + e(ϱ, ϑ) u + p(ϱ, ϑ)u =. (.29).5 Second law of thermodynamics The Second law of thermodynamics encodes the time irreversibility of the fluid evolution characteristic for viscous and heat conducting fluid. We recall the internal energy balance where ϱ t e(ϱ, ϑ) + ϱu x e(ϱ, ϑ) + div x q = S : x u pdiv x u, Willard Gibbs [839 93] ϱ t e(ϱ, ϑ) + ϱu x e(ϱ, ϑ) = ϱde(ϱ, ϑ) [ t ϱ, t ϑ] + ϱu De(ϱ, ϑ) [ x ϱ, x ϑ]. Suppose that e and p are interrelated through Gibbs equation ( ) ϑds(ϱ, ϑ) = De(ϱ, ϑ) + D p(ϱ, ϑ), (.3) ϱ

11 where s is a new quantity called (specific) entropy. Accordingly, ϱ t e(ϱ, ϑ) + ϱu x e(ϱ, ϑ) = ϱde(ϱ, ϑ) [ t ϱ, t ϑ] + ϱu De(ϱ, ϑ) [ x ϱ, x ϑ] = ϑϱds(ϱ, ϑ) [ t ϱ, t ϑ] + ϑϱu Ds(ϱ, ϑ) [ x ϱ, x ϑ] + p(ϱ, ϑ) ϱ ( tϱ + u x ϱ) whence (.9) can be rewritten as = ϑ [ t (ϱs(ϱ, ϑ)) + div x (ϱs(ϱ, ϑ)u)] p(ϱ, ϑ)div x u; t (ϱs(ϱ, ϑ)) + div x [ϱs(ϱ, ϑ)u] + ϑ div xq = ϑ S( xu) : x u, where, furthermore, ( ) ϑ div xq = div x ϑ q + ϑ q xϑ, 2 and, consequently, we end up with the entropy balance equation ( ) t (ϱs(ϱ, ϑ)) + div x [ϱs(ϱ, ϑ)u] + div x ϑ q = [ S( x u) : x u q ] xϑ. (.3) ϑ ϑ If q = κ x ϑ is given by Fourier s law, the entropy production rate represented by the right-hand side of (.3) is non-negative as long as κ. In other words, the entropy is produced in the course of the process in accordance with the Second law of thermodynamics. 2 Various concepts of solutions to the equations and systems of mathematical fluid dynamics A typical problem for an evolutionary equation is the initial-value or Cauchy problem. Knowing the state of the system at an initial time, say t =, solve the evolutionary equation for this initial data. If the physical system (fluid) is confined to a bounded domain, the boundary behavior must be prescribed. To avoid the difficulties created by the influence boundaries on the fluid motion, we focus on problems posed on the whole physical space R N or on the periodic (flat) torus T N = ( [, ] {,} ) N. In the latter case, we simply imposed the periodic boundary conditions, that may be seen as a useful though not very realistic approximation. Evolutionary problems are expected to be well posed in the sense of Hadamard: solutions exist for any physically admissible choice of the initial data; Jacques Hadamard [ ] solutions are uniquely determined by the initial data; solutions depend in a continuous way on the initial data. Our goal is to address these issues for the evolutionary equations arising in fluid dynamics.

12 2. Possible blow-up of smooth solutions Solutions of non-linear problems similar to the compressible Euler system (.6), (.7) may develop singularities in a finite time no matter how smooth and/or small the data are. Note that small here means sufficiently close to the equilibrium state ϱ = const, u =. The classical example of blow up is provided by the -D Burgers equation Indeed it is easy to observe that smooth solutions must satisfy Obviously, if V (x ) > V (x 2 ) for some x < x 2, we get x = x + tv (x ) = x 2 + tv (x 2 ) for t = t V + V x V =, V () = V. (2.) V (t, x + V (x)t) = V (x), t. (2.2) x 2 x V (x ) V (x 2 ) > ; whence V (t, x) = { V (x ) V (x 2 ) Now we show how this construction can be adapted to the compressible Euler system. We restrict ourselves to the case of one space dimension, where (.6), (.7) can be rewritten in the Lagrangean coordinates as the p system (.8), (.9), namely Bernhard Riemann [ ] t s x v = (2.3) t v x P (s) =, P >. (2.4) This system can be written in terms of the so-called Riemann invariants, namely P (s) t s P (s) x v = t v P (s) P (s) x P (s) =. Introducing Z(s) = s P (z) sz we get where we have introduced the Riemann invariants or t Z A(Z) x v = t v A(Z) x Z =, ( ) R + R 2 t R A x R = 2 ( ) R + R 2 t R 2 + A x R 2 =, 2 R = v + z, R 2 = Z v. Setting either R or R 2 constant, we end up with a non-linear scalar equation of type (2.). We conclude that solutions of the compressible Euler system (.6), (.7) may, and indeed do, develop singularities for a generic class of smooth initial data. 2

13 Remark 2.. Let us point out that similar construction fails for the compressible Navier Stokes system (.4.6). As shown by Kazhikhov [7], this problem admits global in time smooth solutions for smooth initial data if N =. Similar results for N = 2, 3 are not known, see however Kazhikhov and Vaigant [29]. 2.2 Classical - strong solutions Ideally, a system of partial differential equations should admit strong or classical solutions possessing all the necessary derivatives and enjoying certain continuity to attain the initial data. Sometimes we speak about strong solutions if all derivatives involved in equations exist in the generalized sense as integrable functions. Once the existence of a classical solution is established, the questions of uniqueness and/or continuous dependence on the data are usually an easy task given the high regularity of solutions. Unfortunately, classical solutions of most problems in mathematical fluid dynamics are known to exist only on short time intervals - local eistence - or globally in time but for the initial data that are sufficiently close to an equilibrium solution - global existence for small data Local existence for the compressible Euler and Navier-Stokes system Consider the Euler/Navier-Stokes system describing the motion of a compressible fluid introduced in Section.3.2: t ϱ + div x (ϱu) =, (2.5) t (ϱu) + div x (ϱu u) + x p(ϱ) = div x S( x u), (2.6) ( S( x u) = µ x u + t xu 2 ) N div xu + ηdiv x ui, (2.7) with µ = η = for the inviscid (Euler) system, and µ > for the viscous (Navier-Stokes) fluid. For the sake of simplicity, we consider the periodic boundary conditions, meaning the physical space = T N is the flat torus, and prescribe the initial state ϱ(, ) = ϱ, u(, ) = u, (2.8) where ϱ and u are as smooth as needed, and ϱ > to the degenerate vacuum regime. Problem ( ) is known to be locally well posed in the Sobolev scale W m,2 () for m large enough, see e.g. Majda [2], Kleinerman and Majda [8]. 3

14 Theorem 2.2. Let p C (, ), p (ϱ) > for ϱ >, µ, η, and [ ] N ϱ >, ϱ W m,2 (), u W m,2 (; R N ) for m > +. 2 Then there exists a positive time T > such that problem ( ) admits a strong solution [ϱ, u] in (, T ), unique in the class ϱ C([, T ]; W m,2 ()), t ϱ C([, T ]; W m,2 ()), u C([, T ]; W m,2 (; R N )), t u C([, T ]; W m k,2 (; R N )), k = if µ = η =, k = 2 if µ > or η >. Remark 2.3. Obviously, the solution [ϱ, u] is as smooth as we wish thanks to the Sobolev embedding relation [ ] N W k,2 () C(), k >. 2 The Sobolev spaces, based on the concept of generalized derivatives and Lebesgue integration, form a natural framework for the so called energy method used in the proof of local existence. The a priori bounds are usually established in the Sobolev framework rather than the framework of spaces of classical continuous and Hölders continuous functions. The desired regularity is then obtained by means of the embedding theorems. We refer to the monographs by Adams [], Maz ya [23], Pick et al. [26] or Ziemer [3] for the basic properties of the Sobolev spaces Rewriting the Euler system as symmetric hyperbolic Let us show very briefly the leading idea behind the proof of Theorem??, namely how to rewrite the Euler problem as a symmetric hyperbolic system. To this end, we make a simplifying but still physically relevant assumption that the pressure p is given by the isentropic constitutive relation p(ϱ) = aϱ γ, γ >. With the new choice of independent variables Sergej Lvovic Sobolev [98 989] we may rewrite the Euler system in the form r = 2aγ γ ϱ γ 2, t r + u x r + γ rdiv x u = (2.9) 2 4

15 t u + u x u + r x r =. (2.) To obtain suitable a priori estimates, we differentiate the equations ([N/2] + ) times with respect to the x variable and then multiply (2.9) on x α ϱ, and (2.) on x alpha u, α = [N/2] +. We obtain the relation ( M d x α ϱ 2 + x α u 2 dx c + x α ϱ 2 + x α u dx) 2, M > dt from we deduce local in time boundedness of the Sobolev norm This operation requires the embedding relation α x ϱ L 2 () + α x u 2 L 2 (). W α,2 C for α > N Global existence for small initial data The Navier-Stokes system admits global-in-time solutions provided the initial data are close enough to an equilibrium state. Here, we present a possible result in this direction that was essentially shown by Matsumura and Nishida [2], [22] for N = 3. Theorem 2.4. Let p C (, ), p (ϱ) > for ϱ >, µ >, η. Let a positive constant ϱ > be given. Then there exists ε > such that for any initial data ϱ W 3,2 (), u W 3,2 (; R 3 ), (ϱ ϱ) dx =, ϱ ϱ W 3,2 () + u W 3,2 (;R 3 ) < ε, the Navier Stokes problem ( ) admits a unique strong solution [ϱ, u] defined on the time interval (, ), ϱ C([, T ]; W 3,2 ()), t ϱ C([, T ]; W 2,2 ()), u C([, T ]; W 3,2 (; R 3 )), t u C([, T ]; W,2 (; R 3 )) such that ϱ(t, ) ϱ in W 3,2 (), u(t, ) in W 3,2 (; R 3 ) as t. Remark 2.5. Solutions in Theorem 2.4 are more regular provided we increase accordingly the regularity of the initial data. 5

16 A similar result for the inviscid Euler system is not available. As we shall see below, solutions of the compressible Euler system develop singularities in a finite lap of time for a fairly generic class of smooth and even small initial data. 2.3 Weak solutions Weak solutions satisfy the equations in the sense of distributions. A single equation in the weak formulation is replaced by an (infinite) family of integral identities satisfied by sufficiently smooth test functions. The weak formulation of the Euler/Navier Stokes system (.4), (.5) reads T [ϱ t ϕ + ϱu x ϕ] dx dt =, T for any test function ϕ Cc ((, T ) ), T [ϱu t ϕ + ϱu u : x ϕ + p(ϱ)div x ϕ] dx dt = S( x u) : x ϕ dx dt for any test function ϕ C c ((, T ) ; R N ). Note that for the Euler system S and therefore less regularity for the weak solution is required. It follows from the weak formulation that t ϱ(t, )ϕ dx and t (ϱu)(t, ) ϕ dx can be seen as continuous scalar function of time for any test function ϕ. Accordingly, it is more convenient to write the weak formulation in the form [ ] t=τ2 τ2 ϱϕ dx = [ϱ t ϕ + ϱu x ϕ] dx dt, t=τ τ for any τ τ 2 T and for any test function ϕ Cc ([, T ] ), [ ] t=τ2 T ϱu ϕ dx = [ϱu t ϕ + ϱu u : x ϕ + p(ϱ)div x ϕ] dx dt (2.) t=τ T S( x u) : x ϕ dx dt for any τ τ 2 T and for any test function ϕ C c ([, T ] ; R N ). Remark 2.6. Note that the weak formulation (2.) already includes satisfaction of the initial conditions. Observe that it is more natural to formulate the initial state in terms of the momentum ϱu rather that the velocity u as the former is always weakly continuous Weak solutions to the compressible Euler system The class of weak solutions is obviously larger than that of classical solutions as basically no smoothness is required. The price to pay is a dramatic lost of well posedness as the following result shows (see Chiodaroli [5], EF [3]). 6

17 Theorem 2.7. Let p C 3 (, ) and let T > be given. Let the initial data belong to the class ϱ C 3 (), ϱ >, u C 3 (; R N ), = T N, N = 2, 3. Then the compressible Euler system (.6), (.7 admits infinitely many weak solutions emanating from the initial state [ϱ, u and belonging to the class ϱ L ((, T ) ), ϱ >, u L ((, T ) ; R N ). Theorem 2.7 can be shown by the method of convex integration developed recently in the context of fluid mechanics by DeLellis and Székelyhidi [9], [8], []. It shows that on one hand the class of weak solutions, at leats for the inviscid fluid flows, is large enough to provide positive existence results, on the other hand it is too large to ensure uniqueness. Remark 2.8. We emphasize that results similar to Theorem 2.7 for viscous fluids, meaning for the Navier-Stokes system, are not known. 2.4 Dissipative (weak) solutions The solutions, the existence of which is claimed in Theorem 2.7, are produced in a rather non-constructive way by adding oscillatory components to a quantity called sub-solution. As a result, they typically produce energy, meaning violate the energy balance (.2). Thus we may try to eliminate them by adding a weak counterpart of the energy balance as an integral part of the definition of weak solution. We say the [ϱ, u] is a dissipative weak solution to the Euler/Navier Stokes system if, in addition to the weak formulation (2.), the total energy inequality holds for a.a. τ (, T ). [ ] 2 ϱ u 2 + P (ϱ) (τ, ) dx + S( x u) : x u dx dt [ ] 2 ϱ u 2 + P (ϱ ) dx τ (2.2) Remark 2.9. Recall that for the Euler system S( x u) and (2.2) simply says that the total energy of the system is non-increasing in time. Remember that the Navier-Stokes system is already dissipative so adding inequality in (2.2) does not violate any physical principle. In addition, we have extrapolated this argument also to the conservative Euler system. 7

18 2.5 Relative energy The dissipative solutions introduced in the previous section satisfy an extended version of the energy inequality (2.2) known as relative energy inequality. We introduce the relative energy functional ( ) [ ] E ϱ, u r, U = 2 ϱ u U 2 + P (ϱ) P (r)(ϱ r) P (r) dx. (2.3) If ϱ p(ϱ) is strictly increasing in (, ), then the pressure potential P is strictly convex and E represents a kind of (non-symmetric) distance function between [ϱ, u] and [r, U]. Seeing that ( ) E ϱ, u r, U = ( 2 ϱ u 2 + P (ϱ) ) dx ϱu U dx + we easily observe that wa may evaluate the time difference [ ( E ϱ, u r, U ϱ U 2 )] t=τ t= dx P (r)ϱ dx + p(r) dx as soon as [ϱ, u] is a dissipative (weak) solution to the Euler/Navier-Stokes system, and [r, U] is a pair of sufficiently smooth test functions, r >. Indeed we successively compute [ ( ) ] t=τ τ 2 ϱ u 2 + P (ϱ) dx + S( x u) : x u dx dt t= in accordance with the energy inequality (2.2), keeping in mind that S for the Euler system; taking U as a test function in the weak formulation of the momentum equation: [ ] t=τ2 T ϱu U dx = [ϱu t U + ϱu u : x U + p(ϱ)div x U] dx dt t=τ T S( x u) : x U dx dt; taking 2 U 2 and P (r) as test functions in the weak formulation of the equation of continuity: [ ϱ ] t=τ2 τ2 2 U 2 dx = [ϱu t U + ϱu U x U] dx dt, t=τ τ and [ ϱp (r) dx ] t=τ2 τ2 = t=τ τ [ϱp (r) t r + P (r)ϱu x r] dx dt. 8

19 Summing up the previous observations we obtain Relative energy inequality [ ( )] t=τ τ E ϱ, u r, U + t= + τ τ τ2 τ S( x u) : x u dx ϱ [ t U + u x U] (U u) dx dt [S( x u) : x U p(ϱ)div x U] dx dt [(ϱ r)p (r) t r + P (r)ϱu x r] dx dt, (2.4) where we have used the identity p (r) = rp (r). 3 Weak vs. strong solutions Our goal in this section is to show the weak-strong uniqueness property for the dissipative solutions of the Euler/Navier-Stokes system. In other words, the weak and strong solutions emanating from the same initial data coincided as long as the latter exists. 3. Weak-strong uniqueness Suppose that the Euler/Navier-Stokes system admits a strong solution [ϱ = r, u = U] on a time interval (, T ), r >. Let [ϱ, u] be a dissipative weak solution of the same problem with the same initial data. Taking [r, U] as test functions in the relative energy inequality (2.4), and using the fact that t U + U x U = r xp(r) + r div xs( x U), Thomas Hakon Grönwall [98 989] 9

20 we obtain ( ) E ϱ, u r, U (τ) + + τ τ τ τ τ Furthermore, as r satisfies (S( x u) S( x U)) : ( x u x U) dx ϱ x U u U 2 dx ϱp (r) x r (U u) dx dt [(ϱ r)p (r) t r + P (r)ϱu x r + p(ϱ)div x U] dx dt [( ϱ ) ] div x S( x U) (u U) r dx dt. t r + U x r = rdiv x U, we get, performing several by-parts integrations, τ ϱp (r) x r (U u) dx dt τ [(ϱ r)p (r) t r + P (r)ϱu x r + p(ϱ)div x U] dx dt τ = [(ϱ r)p (r) t r + P (r)ϱu x r + p(ϱ)div x U] dx dt τ ( ) = div x U p(ϱ) p (r)(ϱ r) p(r) dx dt. Thus we conclude that ( ) τ E ϱ, u r, U (τ) + + τ τ τ (S( x u) S( x U)) : ( x u x U) dx ϱ x U u U 2 dx ( ) div x U p(ϱ) p (r)(ϱ r) p(r) [( ϱ ) ] div x S( x U) (u U) r dx dt dx dt. (3.) 3.. Weak-strong uniqueness for the Euler system Relation (3.) implies ϱ = r, u = U by means of the standard Gronwall lemma. More specifically, in the case of the Euler system, the strong solution must be at least globally Lipschitz and we need p(ϱ) p (r)(ϱ r) p(r) to be dominated by P (ϱ) P (r)(ϱ r) P (r). 2

21 Note that this is true provided, for instance, p C[, ) C 2 (, ), p (ϱ) > for ϱ >, lim inf ϱ We have shown the following result. p (ϱ) >, lim inf ϱ P (ϱ) p(ϱ) >. (3.2) Theorem 3.. Let the pressure p satisfy (3.2). Let the compressible Euler system (.6), (.7) admits a solution [ ϱ, ũ], ϱ > that is Lipschitz in [, T ], = T N, N =, 2, 3. Let [ϱ, u], ϱ be a dissipative weak solution of the same problem, ϱ(, ) = ϱ(, ), ϱu(, ) = ϱũ(, ). Then ϱ = ϱ, u = ũ a.a. in (, T ) Weak-strong uniqueness for the Navier-Stokes system If viscosity is present, we have to handle the last integral in (3.), namely τ [( ϱ ) ] div x S( x U) (u U) dx dt. r Clearly, this integral can be dominated by the relative entropy functional on the set, where < ϱ ϱ ϱ, where ϱ, ϱ are positive constants. Now, we distinguish two cases: (i) ϱ is large ϱ ϱ, (ii) ϱ is small ϱ ϱ. If ϱ is large, specifically ϱ > 2 max{r}, we get ( ϱ ) (u U) c ϱ ( ϱ(u U) c ) 2 ϱ + ϱ u U 2, r where the right-hand side is dominated by the relative energy functional. If ϱ is small, specifically ϱ ϱ < min{r}, then 2 [( ϱ ) ] div x S( x U) (u U) dx dt ϱ ϱ r τ ( δ u U 2 dx + c(δ) div x S( x U) 2 ϱ ) 2 dx dt ϱ ϱ r τ δ ϱ u U 2 + (S( x u) S( x U)) : ( x u x U) dx ( + c(δ) div x S( x U) 2 ϱ ) 2 dx dt r ϱ ϱ 2

22 for any δ >, where the last integral can be absorbed by the left-hand side of (3.) provided δ > is small enough. We have shown an analogue of Theorem 3. for the Navier-Stokes system. Theorem 3.2. Let the pressure p satisfy (3.2). Let the compressible Navier-Stokes system (.4.6) admits a solution [ ϱ, ũ], ϱ > that is Lipschitz in [, T ], = T N, u L (, T ; W 2, (; R N )), N =, 2, 3. Let [ϱ, u], ϱ be a dissipative weak solution of the same problem, ϱ(, ) = ϱ(, ), ϱu(, ) = ϱũ(, ). Then ϱ = ϱ, u = ũ a.a. in (, T ). 3.2 Wild dissipative solutions of the Euler system In the light of the weak-strong uniqueness result established in Theorem 3.2, it may seem that imposing the energy inequality eliminates the wild solutions, the existence of which is claimed in Theorem 2.7. However, this is true only to certain extent as shown in the following result, see [3]. Theorem 3.3. Let = T N, N = 2, 3, T >, and let p C 3 (, ). Let be a given density distribution. Then there exists u, ϱ C 3 (), ϱ > in, u L (; R N ) such that the corresponding initial-value problem for the Euler system (.6), (.7) admits infinitely many dissipative (weak) solutions in (, T ). Remark 3.4. Note carefully that there is no contradiction with the weak-strong uniqueness result as the initial velocity u is not regular. We also repeat that a similar result in the context of viscous fluids, specifically for the compressible Navier-Stokes, is not available. 22

23 3.2. Admissible weak solutions to the compressible Euler system In the context of hyperbolic conservation law, it is customary to strengthen the global energy inequality to a local one introduced in (.2). We may say that [ϱ, u] is an admissible weak solution to the Euler system if [ [ ] t=τ2 τ2 ϱϕ dx = [ϱ t ϕ + ϱu x ϕ] dx dt, t=τ τ for any τ τ 2 T and for any test function ϕ Cc ([, T ] ), [ ] t=τ2 T ϱu ϕ dx [ϱu t ϕ + ϱu u : x ϕ + p(ϱ)div x ϕ] dx dt t=τ = for any τ τ 2 T and for any test function ϕ Cc ([, T ] ; R N ) [ ] ] t=τ τ [( ) ] 2 ϱ u 2 + P (ϱ) ϕ dx t= 2 ϱ u 2 + P (ϱ) t ϕ dx dt τ [( ) ] + 2 ϱ u 2 + P (ϱ) + p(ϱ) u x ϕ dx dt for a.a. τ T and for any test function ϕ C c ([, T ] ), ϕ. Note that the notion of admissible weak solution is stronger than the dissipative weak solution as a local version of the energy inequality is required. Still the following result holds, see Chiodaroli [5]. (3.3) Theorem 3.5. Let = T N, N = 2, 3, and let p C 3 (, ). Let be a given density distribution. Then there exist T > and u, ϱ C 3 (), ϱ > in, u L (; R N ), such that the corresponding initial-value problem for the Euler system (.6), (.7) admits infinitely many admissible weak solutions in (, T ). The previous result is local in time. However, we also have the following, see [3]. 23

24 Theorem 3.6. Let = T N, N = 2, 3, p C 3 (, ), ϱ >, and T > be given. Then there exists ε > such that for any there is u, ϱ C 2 (), ϱ ϱ < ε in u L (; R N ), such that the corresponding initial-value problem for the Euler system (.6), (.7) admits infinitely many admissible weak solutions in (, T ). In the light of the above results it may still seem that one could save the game by taking smooth initial data and working in the class of admissible solutions for the Euler system. Unfortunately, even in this case the presence of wild solutions cannot be avoided, see Chiodaroli, DeLellis, Kreml [6]. Theorem 3.7. Let = R 2, N = 2, 3, p C 3 (, ). Then there exist such initial data ϱ W, (), u W, (; R 2 ), ϱ > uniformly in R 2 such that the corresponding initial-value problem for the Euler system (.6), (.7) admits infinitely many admissible weak solutions in (, ). Remark 3.8. Note that the initial data belong to the class, where the weak-strong uniqueness holds. Thus the corresponding solution is smooth (Lipschitz) on some interval [, T crit ), looses regularity at T crit and bifurcates into branches of admissible wild solutions after the critical time. 4 Stability of the solution set Our goal is to identify stability properties of the solution set, in particular, the available a priori bounds and weak sequential stability of families of solutions. 4. A priori bounds A priori bounds reflect the piece of information transferred on the solutions from the data. Optimally, the solutions may stay at least as regular as the data for any positive time but this is may not be true in general. 24

25 As a matter of fact, only little regularity is known to survive for the compressible Euler/Navier-Stokes system characterized by the energy estimates. 4.. Energy estimates Dissipative solutions of the Euler/Navier-Stokes system satisfy the global energy inequality (2.2), [ ] τ 2 ϱ u 2 + P (ϱ) (τ, ) dx + S( x u) : x u dx dt [ ] 2 ϱ u 2 + P (ϱ ) dx, (4.) which, on condition that the initial energy [ ] 2 ϱ u 2 + P (ϱ ) dx is bounded, give rise to a priori bounds sup ϱu(t, ) L 2 (;R N ) c(data) t (,T ) sup P (ϱ)(t, ) L (;R) c(data) t (,T ) (4.2) that are the same for the Euler and Navier-Stokes system. For the sake of simplicity, suppose that the pressure p is given by the isentropic state equation in which case the pressure potential P (ϱ) can be taken in the form and the second estimate in (4.2) reads p(ϱ) = aϱ γ, a >, γ >, (4.3) P (ϱ) = a γ ϱγ, sup ϱ(t, ) L γ (;R) c(data). (4.4) t (,T ) We remark that (4.2), (4.4) are basically all available a priori estimates for the compressible Euler system Estimates based on dissipation For the Navier-Stokes system, we get an important extra bound xu + t xu 2 N div xui c(data). (4.5) L 2 ((,T ) ;R N N ) 25

26 As u satisfies the periodic boundary conditions, it is easy to deduce from (4.5) that x u L 2 ((,T ) ;R N N ) c(data). (4.6) Now, we show that (4.6), together with (4.4) and the mass conservation principle ϱ(t, ) dx = ϱ dx = M > (4.7) that can be easily deduced from (2.), imply that T u(t, ) 2 W,2 (;R N ) dt c(data). (4.8) The desired estimate will follow from a version of Poincaré s inequality that may be of independent interest. Lemma 4.. Let r k be a bounded measurable function such that r dx M >. Then for any v W,2 (). [ ] v L 2 () c(k, M) r v 2 dx + x v 2 dx Proof: Assuming a contrary we construct a sequence {v n } n= W,2 () such that [ ] v n L 2 () = and v n L 2 () n r n v n 2 dx + x v n 2 dx, where r n satisfies the hypotheses of the lemma. Using compactness of the Sobolev embedding W,2 () L q (), q < 6 if N 3, we may suppose v n v in L 2 (), x v n in L 2 (; R N ); whence v = v a constant. (4.9) On the other hand, by the same token v n 2 v 2 = v 2 in L q (), q < 3, r n r weakly-(*) in L (), and r n v n 2 dx = r v 2 dx > M v 2 in contrast with (4.9) r dx > M >, 26

27 Thus, in order to apply Lemma 4., it is enough to check that (4.4), (4.7) implies the existence of k such that r k = min{ϱ, k} satisfies r k dx M. To see this, we write M = ϱ dx = r k dx + ϱ k ϱ dx = r k dx + r k 4.2 Stability of strong solutions for the Euler system ϱ k ϱ γ ϱ γ dx + k γ c(data). The a priori bounds (4.2) based on finiteness of the total energy available for the Euler system are rather poor but almost optimal. Indeed we do not expect to get any bounds on the derivatives that would prevent formation of shocks characteristic for the inviscid fluids, see e.g. Smoller [27]. Thus we concentrate on families of (weak) solutions to the Euler system satisfying solely (4.2), For technical reasons, it will be more convenient to work with the density ϱ and the momentum m = ϱu as independent variables. The Euler system then reads t ϱ + div x m =, (4.) ( ) m m t m + div x + x p(ϱ) =, (4.) ϱ with the convention that m = whenever ϱ = for the convective term in (4.) to be well defined. We consider the dissipative solutions of (4.), (4.), namely [ [ ϱϕ dx [ m ϕ dx ] t=τ2 = t=τ ] t=τ2 = t=τ ] m 2 t=τ + P (ϱ) dx 2 ϱ t= 4.2. Weak sequential stability + τ2 τ τ2 τ τ2 τ dx [ϱ t ϕ + m x ϕ] dx dt, (4.2) [ m t ϕ + m m ϱ p(ϱ)div x ϕ dx dt, ] : x ϕ dx dt (4.3). (4.4) Suppose we have a family {ϱ ε, m ε } ε> of solutions to ( ) defined in a space-time cylinder (, T ). The energy estimates just yield all nonlinearities uniformly bounded in the (non-reflexive) space L ((, T ) ), or, better, in L (, T ; L ()). There are two disturbing phenomena that may occur to bounded sequences in L. 27

28 Oscillations. A sequence {h ε } ε> behaves like ( t h ε (t, x) h ε, x ) for some h periodic in t and x. ε Concentrations. A sequence {h ε } ε> behaves like h ε (t, x) ( t ε h N+ ε, x ) for some h Cc ({ (t, x) < }). ε Both phenomena may, of course, occur simultaneously. In the subsequent analysis, oscillations in the family {ϱ ε, m ε } ε> will be controlled by the equations (4.2), (4.3), while concentrations will be handled by means of the energy inequality (4.4). It is important to observe that all non-linearities in (4.2), (4.3) are dominated by the energy for large values of arguments, specifically ( ) m m ϱ c m 2, p(ϱ) cp (ϱ) for all ϱ >>. (4.5) 2 ϱ As the space L ((, T ) ) is embedded in the space of signed measures M([, T ] ) that happens to be the dual to C([, T ] ), we may assume that ϱ ε {ϱ} weakly-(*) in M([, T ] ), m ε {m} weakly-(*) in M([, T ] ; R N ), p(ϱ ε ) {p} weakly-(*) in M([, T ] ), P (ϱ ε ) {P } weakly-(*) in M([, T ] ), m ε m ε {M} weakly-(*) in M([, T ] ; R N N ) ϱ ε passing to suitable subsequences as the case may be. Consequently, we may let ε in (4.2), (4.3) obtaining T ϱ ϕ dx = [{ϱ} t ϕ + {m} x ϕ] dx dt, for all ϕ Cc ([, T ) ), T m ϕ dx = [{m} t ϕ + {M} : x ϕ + {p}div x ϕ] dx dt, for all ϕ Cc ([, T ) ; R N ), where ϱ, m stand for the limit of the initial data. As a matter of fact, the limit for ϱ ε, m ε can be improved. It follows from (4.4) that (4.6) (4.7) ϱ ε ϱ weakly-(*) in L (, T ; L γ ()), 28

29 where, obviously, ϱ can be identified with {ϱ}. Similarly, m ε = ϱ ε m ε ϱε, where, by virtue of (4.2), { } mε ϱε ε> is bounded in L (, T ; L 2 ()); whence, applying Hölder s inequality and (4.4), we may infer that Finally, we may consider m ε m( {m}) weakly-(*) in L 2γ γ (; R N )). ϕ δ = ψ δ (t)ϕ, ϕ C ([, T ] ) { for t τ, ψ δ C (R), ψ δ, ψ δ (t) = fort τ + δ as a test function in (4.6), (4.7) and perform the limit δ to conclude that τ ϱϕ(τ, ) dx ϱ ϕ dx = [ϱ t ϕ + m x ϕ] dx dt, m ϕ(τ, ) dx m ϕ dx = for a.a. τ (, T ), and for all ϕ C ([, T ] ), τ [m t ϕ + {M} : x ϕ + {p}div x ϕ] dx dt, for a.a. τ (, T ), and for all ϕ C ([, T ] ; R N ), where τ correspond to the Lebesgue points of the mappings (4.8) (4.9) t ϱ(t, ) L γ (), t m(t, ) in L 2γ/(γ ) (; R N ). Finally, we may apply the same arguments to the energy inequality (4.4) deducing [ ] N trace{m} + {P } (τ, ) dx E dx for a.a. τ (, T ), (4.2) 2 where E denotes the limit of the initial energy, and [ ] N τ+δ [ ] N trace{m} + {P } (τ, ) dx = lim trace{m} + {P } 2 δ 2δ τ δ 2 dx dt (4.2) for a.a. τ (, T ), where the limit is understood in the space of measures M(). 29

30 4.2.2 Oscillations - Young measures There is an efficient tool to describe oscillations in weakly convergent sequences - the associated Young measure. We report the following fundamental result, see Pedregal [25, Chapter 6, Theorem 6.2] (also Ball [3]). Theorem 4.. Let {v n } n=, v n : Q R N R M be a sequence of functions bounded in L (Q; R M ), where Q is a domain in R N. Then there exist a subsequence (not relabeled) and a parameterized family {ν y } y Q of probability measures on R M depending measurably on y Q with the following property: For any Caratheodory function Φ = Φ(y, z), y Q, z R M such that Φ(, v n ) Φ weakly in L (Q), we have Φ(y) = ψ(y, z) dν y (z) for a.a. y Q. R M Our goal is to rewrite the limit system in terms of the Young measure associated to the family [ϱ ε, m ε ] R +N. Accordingly, the Young measure is a parameterized family of probability measures supported on the set [, ) R N (the densities are supposed non-negative) with the barycenter ν t,x : (t, x) (, T ) P([, ) R N ), ν L weak ( )((, T ) ; P(R +N )), ν t,x ; ϱ = ϱ(t, x), ν t,x ; m = m(t, x) for a.a. (t, x) (, T ). For a function H = H(ϱ, m) we define provided the latter exists. H(ϱ, m)(t, x) ν t,x ; H(ϱ, m) for a.a. (t, x) (, T ). (4.22) Remark 4.2. Here, we should distinguish between ν t,x ; F (ϱ, m meaning the application of the Young measure to the Borel function F : R R N R, and the composition F (ϱ, m) : (, T ) R. 3

31 From Theorem 4. we know that H(ϱ, m) is well defined L function (meaning for a.a. (t, x)) provided H(ϱ ε, m ε ) H(ϱ, m) weakly in L ((, T ) ). (4.23) We show that H(ϱ, m) can be defined through formula (4.22), meaning that the function H(ϱ, m), [ϱ, m] R +N is ν t,x integrable for a.a. (t, x), and the function whenever T (t, x) ν t,x ; H(ϱ, m) is integrable in (, T ), H(ϱ ε, m ε ) dx dt c(data) uniformly for ε. (4.24) Note that (4.24) in general does not imply (4.23). Since the Lebesgue integral is absolutely convergent, it is enough to consider H, or, equivalently, to assume that H. We take a family of cut-off functions T k (Z) = min{z, k}. As T k (H) are bounded, the functions T k (H) are ν t,x integrable for a.a. (t, x), uniformly for k. However, (t, x) ν t,x ; T k (H(ϱ, m)) L ((, T ) ), (t, x) ν t,x ; T k (H(ϱ, m)) L ((,T ) ) c(data) T k (H(ϱ, m)) H(ϱ, m) for any [ϱ, m] R N+ as k ; whence, by monotone convergence theorem, H is ν t,x integrable (the integral may equal ). On the other hand, T k (H(ϱ, m)) H(ϱ, m) in L ((, T ) ); whence ν t,x ; T k (H(s, v)) H(ϱ, m)(t, x) = ν t,x ; H(ϱ, m) for a.a. (t, x). Thus ν t,x ; H(ϱ, m) is finite for a.a. (t, x) (, T ). Remark 4.3. The function H(ϱ, m) - the limit of T k (H(ϱ, m)) for k is known as the biting limit of {H(ϱ ε, u ε )} ε>, cf. Ball and Murat [4]. We infer that for any H such that weak-(*) limit in M of H(ϱ ε, m ε ) {H} = H(ϱ, m) + D H (4.25) T H(ϱ ε, m ε ) dx dt c(data), 3

32 where D H M([, T ] ) is called concentration defect measure. From the previous discussion, we deduce a useful estimate for the concentration defect measure D H, namely, D H M(Q) = sup ϕ C(Q) D H ϕ dx dt ( lim k lim ε H(ϱ ε,m ε) k, (t,x) Q Q ) (4.26) ( H(ϱ ε, m ε ) k) dx dt Consequently, equations (4.8), (4.9) can be written in the form τ ϱϕ(τ, ) dx ϱ ϕ dx = [ϱ t ϕ + m x ϕ] dx dt, for a.a. τ (, T ), and for all ϕ C ([, T ] ), m ϕ(τ, ) dx m ϕ dx τ [ ( ) ] m m = m t ϕ + : x ϕ + p(ϱ)div x ϕ + D m : x ϕ dx dt, ϱ for a.a. τ (, T ), and for all ϕ C ([, T ] ; R N ), together with the energy inequality [ ( ) ] m 2 + P (ϱ) + D (τ, ) dx 2 ϱ (4.27) (4.28) E dx for a.a. τ (, T ), (4.29) where D L (, T ) is the energy concentration defect satisfying, in view of (4.5), (4.26), τ τ D m dx dt c D(t) dt for a.a. τ (, T ). (4.3) Measure-valued solutions At this stage, we may forget the way how the Young and dissipative measures were obtained and define measure-valued solution of the Euler system (4.), (4.) as the pair ( ) the parameterized measure ν t,x L weak ( ) (, T ) ; P([, ) R N ) ; the dissipation defect D L (, T ) satisfying ν τ,x ; ϱ ϕ dx ν,x ; ϱ ϕ dx = τ [ ν t,x ; ϱ t ϕ + ν t,x ; m x ϕ] dx dt, for a.a. τ (, T ), and for all ϕ C ([, T ] ); (4.3) 32

33 where ν τ,x ; m ϕ dx ν,x ; m ϕ dx τ [ = ν t,x ; m t ϕ + ν t,x ; m m ϱ for a.a. τ (, T ), and for all ϕ C ([, T ] ; R N ), ν τ,x ; ( ) m 2 + P (ϱ) dx + D(τ) 2 ϱ ν,x ; ( ) m 2 2 ϱ τ ] : x ϕ + ν t,x ; p(ϱ) div x ϕ + D m : x ϕ + P (ϱ) dx dt, dx for a.a. τ (, T ), (4.32) (4.33) τ D m dx dt c D(t) dt for a.a. τ (, T ). (4.34) Remark 4.4. Note that the measure ν plays the role of initial conditions. If we assume that the data were controlled in the limit ε, the initial measure coincides with Remark 4.5. The function is defined as u,x = δ ϱ (x);m (x) for a.a. x. H : [ϱ, m] m 2 ϱ H[ϱ, ] = for all ϱ R, H[ϱ, m] = for ϱ, m. Then H is convex, lower-semi continuous and can be approximated from below by a sequence of continuous functions Weak-strong uniqueness in the class of measure-valued solutions The proof of the weak-strong uniqueness property in the context of measure-valued solutions follows the same lines as in Section 3. We introduce an analogue of the relative energy: ( ) E ν = ν t,x (ϱ, m) r, U = ν t,x ; ( ) (4.35) m ϱu 2 + P (ϱ) P (r)(ϱ r) P (r) dx 2ϱ for C smooth functions r and U. Next, we successively compute ν τ,x ; m U dx τ [ = ν t,x ; m t U + τ + D m : x U dx dt; ν ; m U dx ν t,x ; m m ϱ 33 ] : x U + ν t,x ; p(ϱ) div x U dx dt

34 and 2 ν τ,x; ϱ U 2 dx ν,x ; ϱ U 2 dx 2 τ = [ ν t,x ; ϱ U t U + ν t,x ; m U x U] dx dt; ν τ,x ; ϱ P (r) dx = τ ν,x ; ϱ P (r) dx [ ν t,x ; ϱ t P (r) + ν t,x ; m x P (r)] dx dt. Summing up the previous relations, we obtain a measure-valued analogue of the relative energy inequality (2.4): [ + + ν t,x ; 2ϱ τ τ τ τ ( ) m ϱu 2 + P (ϱ) P (r)(ϱ r) P (r) [ ν t,x ; m t U + ν t,x ; m m ϱ [ ν t,x ; ϱ U t U + ν t,x ; m U x U] dx dt ] t=τ dx t= + D(τ) ] : x U + ν t,x ; p(ϱ) div x U dx dt [ ν t,x ; ϱ t P (r) + ν t,x ; m x P (r) r t P (r)] dx dt D m : x U dx dt, where, furthermore, the expression on the right-hand side can be rewritten as τ [p(r) ν t,x ; p(ϱ) ] div x U dx dt τ + ν t,x ; ϱu m [ t U + U x U] dx dt τ [ ν t,x ; ϱ t P (r) + ν t,x ; m x P (r) r t P (r) + p(r)div x U] dx dt τ τ (m ϱu) (ϱu m) + D m : x U dx dt + ν t,x ; : x U dx dt. ϱ Now we assume in addition that r, U is a smooth solution of the Euler system, t r + U x r = rdiv x U, t U + U x U = p (r) r xr. (4.36) 34

35 Consequently, [ + + ν t,x ; 2ϱ τ τ τ = + τ τ ( ) m ϱu 2 + P (ϱ) P (r)(ϱ r) P (r) [p(r) + ν t,x ; ϱ p (r) ν t,x ; p(ϱ) ] div x U dx dt [r t P (r) p(r)div x U] dx dt τ D m : x U dx dt + ν t,x ; ν t,x ; p(ϱ) p (r)(ϱ r) p(r) div x U dx dt τ D m : x U dx dt + ν t,x ; ] t=τ dx + D(τ) t= (m ϱu) (ϱu m) : x U dx dt ϱ (m ϱu) (ϱu m) ϱ : x U dx dt. (4.37) Thus, in view of (4.34), the integral on the right-hand side of (4.37) can be absorbed by the left-hand side by means of a Gronwall argument as soon as ν ; ( ) m ϱu 2 + P (ϱ) P (r)(ϱ r) P (r) dx, 2ϱ meaning ν,x = δ r(x),r(x)u(x) for a.a. x. We have obtained the following results, cf. Gwiazda, Swierczewska-Gwiazda, Wiedemann [6]. Theorem 4.6. Let ϱ, U be a C solution of the compressible Euler system (.6), (.7) in [, T ]. Let {ν t,x } (t,x) (,T ), be a measure-valued solution of the same system (in terms of ϱ and the momentum m), with a dissipation defect D and such that ν,x = δ r(,x),ru(,x) for a.a. x. Then D = and ν t,x = δ r(t,x),ru(t,x) for a.a. (t, x) (, T ). Remark 4.7. Regularity of the strong solution can be relaxed to globally Lipschitz in (, T ) by a simple density argument. Remark 4.8. A similar result holds for the compressible Navier-Stokes system, see [4]. We point out that not all measure-valued solutions to the compressible Euler system can be identified as asymptotic limits of families of weak solutions, see [7] for a counterexample. 35

36 5 Weak sequential stability of solutions to the Navier-Stokes system Our goal will be to show that, in contrast with the Euler system, a family of (weak, dissipative) solutions {ϱ ε, u ε } ε> to the compressible Navier-Stokes system is weakly compact provided the initial data are. In particular, any accumulation point of such a family is again a weak (dissipative) solution of the same problem. 5. Integrability We recall the a priori bounds that follow directly from the energy inequality, see Section 4.: We get immediately sup sup ϱ ε (t, ) L γ () c(data), t (,T ) sup ϱ ε u ε (t, ) L 2 (;R N ) c(data) t (,T ) t (,T ) T and, by virtue of Hölder s inequality, ϱ ε u ε (t, ) L 2γ/(γ+) (;R N ) c(data) u ε (t, ) 2 W,2 (;R N ) dt c(data). (5.) sup p(ϱ ε )(t, ) L () c(data), (5.2) t (,T ) ϱ ε u ε u ε L p (;R N N ) ϱ ε L 2γ () ϱ ε u ε L 2 (;R N ) u ε L q (;R N ), where p = 2γ { arbitrary finite if N = 2, q, q = 6 if N = 3, where q corresponds to the Sobolev embedding relation, Consequently, we may take p > and conclude W,2 () L q (). ϱ ε u ε u ε L 2 (,T ;L p (;R N )) c(data) if γ = { > if N = 2, γ > 3 2 if N = 3, (5.3) in which case p < 2γ γ + if N = 2, p 6γ 4γ