aa r X i v : . [ m a t h . A P ] A p r Expansion of a compressible gas in vacuum
Denis Serre ∗ August 30, 2018
Dedicated to Tai-Ping Liu, on the occasion of his 70th birthdayEn m´emoire de G´erard Lasseur
Abstract
Tai-Ping Liu [12] introduced the notion of “physical solution” of the isentropic Eulersystem when the gas is surrounded by vacuum. This notion can be interpreted by sayingthat the front is driven by a force resulting from a H¨older singularity of the sound speed.We address the question of when this acceleration appears or when the front just moveat constant velocity.We know from [7, 17] that smooth isentropic flows with a non-accelerated front existglobally in time, for suitable initial data. In even space dimension, these solutions maypersist for all t ∈ R ; we say that they are eternal . We derive a sufficient conditionin terms of the initial data, under which the boundary singularity must appear. As aconsequence, we show that, in contrast to the even-dimensional case, eternal flows with anon-accelerated front don’t exist in odd space dimension.In one space dimension, we give a refined definition of physical solutions. We showthat for a shock-free flow, their asymptotics as both ends t → ±∞ are intimately relatedto each other. We consider a compressible isentropic gas of density ρ , velocity u , pressure p and specificinternal energy e . The flow is governed by the Euler system ∂ t ρ + div( ρu ) = 0 , (1) ∂ t ( ρu ) + Div( ρu ⊗ u ) + ∇ p ( ρ ) = 0 . (2)Classical solutions of (1,2) satisfy in addition the conservation of energy ∂ t ( 12 ρ | u | + ρe ) + div(( 12 ρ | u | + ρe + p ) u ) = 0 . ∗ UMPA, UMR 5669 CNRS-ENS Lyon ; 46, all´ee d’Italie 69364 LYON Cedex 07, FRANCE ∂ t ( 12 ρ | u | + ρe ) + div(( 12 ρ | u | + ρe + p ) u ) ≤ p ( ρ ) = Aρ γ , A > , where γ > adiabatic constant . Then the internal energy is given by(4) ρe = pγ − . The sound speed is c = √ p ′ = √ γA ρ κ where κ := γ − . We recall that a mono-atomic gascorresponds to the choice γ d := 1 + 2 d , which means that the molecules have only d degrees of freedom, all of them associated withtranslations in space. In the physically relevant case d = 3, γ = is the parameter associatedwith mono-atomic gases like Argon. Di-atomic gases, like H , O or the air, obey to a pressurelaw with γ = < γ . If d = 2, then γ = 2 corresponds to the shallow water equations.We are interested in this paper in flows for which the total mass and energy are finite, andthe gas occupies a bounded region Ω( t ) of the ambient space R d at time t . The gas is surroundedby vacuum. We assume a priori that the front Γ( t ) = ∂ Ω( t ) is a smooth or piecewise smoothhypersurface in R d . We denoteΩ = { ( x, t ) | x ∈ Ω( t ) and t ∈ R } , Γ = { ( x, t ) | x ∈ Γ( t ) and t ∈ R } . For weak admissible solutions of the Euler system, the total mass is conserved and the totalenergy is a non-increasing function of time : Z R d ρ ( x, t ) dx ≡ Z R d ρ ( x ) dx =: M < ∞ , ddt Z R d ( 12 ρ | u | + ρe )( x, t ) dx ≤ . The latter inequality is an equality if the solution is classical.Of particular importance is the nature of the boundary condition along Γ( t ). On the onehand, because there should not be any transfer of mass accross the boundary, Γ moves at thevelocity u . More precisely, its normal velocity equals u · ν , where ν is the unit outer normal.The boundary can be viewed as a collection of particles moving at the fluid velocity u ; theirpaths are integrals curves of the ODE(5) dXdt = u ( X ( t ) , t ) .
2n the other hand, the Rankine–Hugoniot relations reduce on Γ to [ p ( ρ )] = 0, where ρ vanisheson the vacuum side ; therefore ρ must vanish on the interior side, meaning that ρ is continuousacross Γ( t ) (see also [11]). We mention in passing that this property might be violated at isolatedtimes because an isolated (in space and time) discontinuity is not a shock front. An explicitexample of that possibility was given by Greenspan & Butler [8], see Section 3 (“sub/supersonic”example). More generally, we allow discontinuities that occur on a codimension-1 subset of Γ,because they are not shock waves.Once we know that ρ ≡ t ). Here one can think of two approaches,complementary to each other. Both are based on the quasi-linear symmetric form of the system, ∂ t ¯ c + ( u · ∇ )¯ c + κ ¯ c div u = 0 , (6) ∂ t u + ( u · ∇ ) u + κ ¯ c ∇ ¯ c = 0 , (7)where ¯ c := cκ is a renormalized version of the sound speed. We point out that system (6,7)removes the singularity at ρ = 0, inherent to the Euler system, whose hyperbolicity degeneratesat vacuum. These approaches distinguish two regimes, depending on whether the quantity ¯ c ∇ ¯ c vanishes or not. If it does, then the particles at the front move freely ; we say that the frontis not accelerated, or that the flow is smooth up to vacuum . If instead c ∇ c = 0, then the frontexperiences a normal acceleration. This is what T.-P. Liu called a physical singularity at theboundary. Of course it requires that c be -H¨older at the boundary. Beware that a given flowcan be smooth up to vacuum in some region, for instance on some time interval, and displaya physical singularity elsewhere. The transition between both regimes is still not completelyunderstood. Outline of the paper.
Section 2.2 deals with flows for which the sound speed is smoothenough at the vacuum, so that the front is not accelerated. Such flows arise for convenientinitial data, but might exist only for a finite time. When the boundary regularity is lost, itis expected that a H¨older-type singularity of c develops at the boundary. These singularities,called physical by T.-P. Liu, are investigated in Section 3.Our main results are Theorem 2.5 (non-existence in odd space dimension of eternal flowsthat are smooth up to vacuum) and Theorem 3.1 on the relation between the asymptotics as t → ±∞ in one space-dimension. The first approach considers a solution ( c, u ) of (6,7), smooth in the entire space R d , with acompactly supported component c . The following result is a classical application of the theoryof symmetric hyperbolic systems, which can be found in [4].3 heorem 2.1 Let the initial data ( c , u ) belong to the Sobolev space H s ( R d ) for some param-eter s > d . Then there exists an open time interval I containing t = 0 , and a uniquelocal-in-time solution in the class C ( I ; H s ) ∩ C ( I ; H s − ) . Thanks to Sobolev embedding, this solution is classical : ( c, u ) ∈ C ( I × R d ). If c is compactlysupported, then c ( · , t ) is too, and c vanishes at the boundary of its support.The following argument is adapted from Liu & Yang [13], where it was designed in thecontext of damped flows. For a solution given by Theorem 2.1, the equation (7) reduces to(8) ∂ t u + ( u · ∇ ) u = 0 along Γ . Since Γ is transported by the flow, this exactly means that the trajectories defined by (5) andoriginating on the boundary (that is, X (0) ∈ Γ(0)) remain on Γ and have a constant velocity u ( X ( t ) , t ) ≡ cst . We summarize this argument into
Proposition 2.1
Let ( c, u ) be a flow given by Theorem 2.1 with Ω( t ) bounded. Then the front Γ( t ) is obtained from Γ(0) by a (uniform in time) transport: Γ( t ) = ψ t (Γ(0)) where ψ t ( x ) = x + tu ( x ) . We point out that ψ is not the flow map of the gas, although it coincides with it on the vacuumboundary.Because they are classical ones, the solutions provided by Theorem 2.1 yield perfectly ad-missible solutions of the Euler equations (1,2). At this stage, it seems that we don’t need anextra condition at the boundary. Their only flaw is that they are defined only on a finite timeinterval, but this is something we are accustomate with in the theory of hyperbolic conserva-tion laws. We anticipate that if I = ( − T ∗ , T ∗ ) is bounded for a maximal solution, then somekind of singularity will appear as t → T ∗ . We have in mind the formation of shock waves asusual, but because the front Γ = ∂ Ω is a place where the hyperbolicity of the Euler systemdegenerates, the singularity might appear at Γ( T ∗ ) and then propagate along Γ. We shall seelater on sufficient conditions that lead to such boundary singularities in finite time. It may happen that T ∗ be infinite, in which case the smooth solution of the forward Cauchyproblem is global-in-time. Such solutions where first constructed in [17] when u is close to alinear field x Ax , and the spectrum (the set of eigenvalues) of the matrix A is contained in C \ ( −∞ , heorem 2.2 (M. Grassin [7].) In Theorem 2.1, assume in addition that the spectrum of ∇ u ( x ) is contained in some fixed compact subset of C \ ( −∞ , . There exists an ǫ > ,depending upon u , such that if k c k H s < ǫ, then the solution exists for all positive time : T ∗ = + ∞ . The idea behind the proof of Theorem 2.2 is that the data is close to (0 , u ). For the latterdata, the solution is ( c ≡ , u ), with u governed by the vectorial Burgers equation(9) ∂ t u + ( u · ∇ ) u = 0 . By assumption, the flow ψ t ( x ) = x + tu ( x ) is one-to-one at every positive time and thereforethe solution of (9) is smooth for t >
0. For a non-zero c , there is a competition between thedispersion induced by ψ t (the distance between two particles tends to increase linearly in time),and the nonlinear effect due to the pressure. Because of the dispersion, the density decays atan algebraic rate, and therefore the role of the pressure force gets smaller and smaller. If itwas weak enough at initial time, we may expect that it will never be strong enough to leadto shock formation. An other important point is that the characteristic cones, which move atvelocity u + ξ with | ξ | = c ( ρ ), will never overtake the boundary Γ( t ), because the slope of c atthe boundary is dominated by the diverging gradient of u . Eternal smooth flows
Theorem 2.2 has the beautiful consequence that smooth eternal compactly supported flowsexist in even space dimension :
Corollary 2.1 ( d even.) Suppose the space dimension d is even. Then in Theorem 2.1, onecan choose the initial data so that the solution is eternal : I = R .Proof Just choose u so that the spectrum of ∇ u ( x ) is contained in a fixed compact subset K of C \ R . Then choose c small enough in H s ( R d ). Because K does not meet ( −∞ , T ∗ = + ∞ . Because K does not meet [0 , + ∞ ), we have T ∗ = −∞ .Corollary 2.1 raises the question of whether there exist eternal solutions with compactsupport, smooth up to vacuum, in odd space dimension. The construction in the proof above isnot possible, because odd-size matrices do have at least one real eigenvalue. It is remarkable thatone meets an obstruction, see Theorem 2.5 below. The non-existence in one space dimensionis a consequence of a calculus due to P. Lax [10] : the Euler system, whose wave velocities are λ ± := u ± c , can be diagonalized in Riemann coordinates r ± = u ± ¯ c, ∂ t + λ ± ∂ x ) r ± = 0 . Each of the transport equations above tells us that r ± keeps a constant value along every char-acteristic curve t X ( t ) defined by (5). Lax showed that there exists a positive function N ( ρ )such that the expressions y ± := N ( ρ ) ∂ x r ± satisfy Ricatti equations along the characteristics(10) ( ∂ t + λ ± ∂ x ) y ± + N − ∂λ ± ∂r ± y ± = 0 . Because of genuine nonlinearity ∂λ ± ∂r ± > , the only eternal solution of (10) is y ± ≡
0. Therefore the global existence implies that r ± ,or equivalently ρ and u , are constant. With the assumption of finite mass, we conclude that ρ ≡ y of(10) exists on a time interval ( T, + ∞ ) (respectively on ( −∞ , T )), then y ≥
0, that is ∂ x r ± ≥ ∂ x r ± ≤ γ = γ = 3, the system above decouples as two independent Burgers equations,since then r ± = λ ± :(11) ( ∂ t + λ ± ∂ x ) λ ± = 0 . In this case, one can take N ≡ Proposition 2.2 ( d =1.) In one space dimension, there does not exist a non-trivial eternalsolution ( c, u ) of (6,7), of class C , with c of compact support. vs smooth boundary The following statement tells us that if the initial velocity field does not drive fast enough theboundary, then a singularity must happen in finite time.
Theorem 2.3
We assume γ ≤ d − .Let ( c , u ) ∈ H s ( R d ) be an initial data, with s > d . Assume that c is compactlysupported in Ω(0) , a domain with smooth boundary. Assume that (12) J := Z Ω(0) det ∇ u ( x ) dx ≤ . Then the maximal time T ∗ > of existence provided by Theorem 2.1 is finite: T ∗ < + ∞ . J in (12) depends only upon the restriction of u to the boundary ∂ Ω(0). The identity det ∇ u ( x ) = div( u ∇ u × ∇ u )yields the alternate formula Z Ω(0) det ∇ u ( x ) dx = Z ∂ Ω(0) u det( ν, ∇ u , ∇ u ) ds ( x ) , which shows that J depends on u only through its restriction to the boundary. By symmetry,the same is true for every component u j .To prove Theorem 2.3, we begin with a property which must be rather classical. A simplerversion, can be found in Milnor’s paper [16]. Notice that this lemma is valid for every parameter γ > Lemma 2.1
As a function of time, the volume Ω( t ) is a polynomial of degree ≤ d .Proof A function G ( t ) is polynomial if and only if it is locally a polynomial. It is therefore enoughto prove that G is a polynomial of degree ≤ d over some non-trivial interval ( − ǫ, ǫ ).Let us consider the map ψ t ( x ) = x + tu ( x ). Whenever | t |k u k , ∞ is less than 1, ψ t is adiffeomorphism. Therefore ψ t (Ω(0)) is the bounded open domain whose boundary is ψ t ( ∂ Ω(0)).Because ψ t coincides with the flow of the fluid over ∂ Ω(0), we find that ψ t (Ω(0)) is the boundedopen domain whose boundary is ∂ Ω( t ), and we concludeΩ( t ) = ψ t (Ω(0)) . We infer the formula | Ω( t ) | = Z Ω(0) det( ∇ ψ t ( x )) dx = Z Ω(0) det( I d + t ∇ u ) dx, where the integrand is a polynomial in t of degree at most d .Let us point out that the integral J is precisely the coefficient of the monomial t d in thispolynomial. We now prove our theorem. Proof
Let Q ( t ) be the polynomial t
7→ | Ω( t ) | , Q ( t ) = J t d + l . o . t . Our assumption amounts to(13) | Ω( t ) | ≤ C (cid:0) (1 + | t | ) d − (cid:1) , ∀ t > . J <
0, we actually have Q ( t ) < J = 0.Let us recall the following inequality, which can be found in [1, 17]. It is a direct consequenceof the Euler system (1,2) and of the energy inequality (3) :(14) ddt Z R d (cid:18) ρ | tu − x | + t ρe (cid:19) dx ≤ − dκ ) t Z R d ρe dx. Remark that the quantity in the left integral is a linear combination of the density, the mo-mentum and the mechanical energy with polynomial coefficients :12 ρ | tu − x | + t ρe = | x | ρ + x · ( ρu ) + t (cid:18) ρ | u | + ρe (cid:19) . When γ ∈ (1 , γ d ], the factor 1 − dκ is non-negative. With the Gronwall inequality, we deducethe estimate(15) Z R d ρ γ dx = cst · Z R d ρe dx = O (cid:0) (1 + t ) − dκ (cid:1) . Now, we can estimate the total mass with the help of H¨older Inequality: M = Z Ω( t ) ρ dx ≤ (cid:18)Z R d ρ γ dx (cid:19) /γ | Ω( t ) | − /γ = O (cid:16) (1 + t ) − dκγ +( d − − γ ) (cid:17) = O (cid:16) (1 + t ) γ − (cid:17) . Letting t → + ∞ , we deduce that M = 0. In other words, there is no fluid at all.If instead γ > γ d , then 1 − dκ < Z + ∞ t dt Z R d ρe dx ≤ dκ − Z R d | x | ρ ( x ) dx, wich gives Z + ∞ t dt Z R d ρ γ dx < ∞ . Using the same H¨older inequality as above, we infer M γ Z + ∞ t (1 + t ) ( d − − γ ) dt < ∞ , which implies either M = 0 (no fluid at all) or1 + ( d − − γ ) < − , γ > d − . Q.E.D.Remark that Theorem 2.3 doesn’t fully use that ( c, u ) is a classical solution. We only needon the one hand that ( ρ, ρu ) be a weak solution of the Euler system, satisfying the “entropy”inequality (3) (this is admissibility in the sense of Lax), and on the other hand that ( c, u ) is ofclass C in a neighborhood of the front Γ. There is a statement analogous to Theorem 2.3 inthis context : Theorem 2.3’
We assume γ ≤ d − .Let ( ρ, ρu ) be a locally bounded admissible (in the sense that (3)) solution of the Euler system(1,2) over (0 , + ∞ ) × R d . Assume that ρ is supported in Ω , a domain bounded in space at eachtime, whose boundary is smooth. Assume finally that ( c, u ) is of class C in a neighborhood of ∂ Ω . Then necessarily (16) Z Ω(0) det ∇ u ( x ) dx > . This shows that the development of shock waves is not sufficient to resolve the lack ofsmooth solutions. Many flows actually exhibit also some boundary singularity after some time.This is where T.-P. Liu’s notion of physical singularity comes into play.
The mono-atomic case If γ = γ d , Theorem 2.3 can be improved in quantitative way. Suppose that J ≥
0. From ddt Z R d (cid:18) ρ | tu − x | + t ρe (cid:19) dx ≤ , we infer A Z Ω( t ) ρ γ dx = Z Ω( t ) ( γ − ρe dx ≤ γ − t I, I := Z Ω(0) ρ | x | dx. Applying again the H¨older Inequality, this yields AM γ γ − t ≤ | Ω( t ) | /d I. With | Ω( t ) | = J t d + l . o . t . , we deduce that for a solution to be smooth at the boundary for all t >
0, we must have AM γ γ − ≤ J /d I. We therefore have 9 heorem 2.4
We assume γ = 1 + d − .Let ( c , u ) ∈ H s ( R d ) be an initial data, with s > d . Assume that the domain Ω(0) = { c > } is bounded, with smooth boundary. Assume that (17) J := Z Ω(0) det ∇ u ( x ) dx < (cid:18) AM γ ( γ − I (cid:19) d/ . Then the maximal time T ∗ > of existence provided by Theorem 2.1 is finite: T ∗ < + ∞ . The odd-dimensional case
As anounced above, we prove that an obstruction occurs in the much more general context ofan odd space dimension, a case which includes the realistic 3-dimensional space. The followingresult answers a question raised in [18] about eternal solutions.
Theorem 2.5 ( d odd.) We assume that the space dimension d is odd, and that γ ≤ d − .Then there does not exist a non-trivial eternal solution ( c, u ) ∈ C ( R d ) of (6,7) (hencesmooth up to vacuum), with c supported in a compact domain with smooth boundary.Proof Suppose ( c, u ) is such a solution, and denote Q ( t ) = | Ω( t ) | . From Lemma 2.1, Q is apolynomial of degree ≤ d . Because Q takes positive values for every t ∈ R , we find that deg Q must be an even number. Because d is odd, we deduce that actually deg Q ≤ d −
1, whence(18) | Ω( t ) | = O (cid:0) (1 + | t | ) d − (cid:1) . The rest of the proof is exactly the same as in the proof of Theorem 2.3. Q.E.D.Again, Theorem 2.5 doesn’t fully use that ( c, u ) is a classical solution, but only that itis smooth up to the vacuum. It tells us more about the onset of physical singularity at theboundary, which we consider in the next Section.10
Physical singularity
It was first recognized by T.-P. Liu [12] that many flows that are classical solutions in theinterior of Ω( t ) must experience a singularity at the boundary. His first motivation was the factthat for the Euler system with damping, the front travels so slowly that Ω( t ) remains uniformlybounded. For gaseous stars, the gravity has a similar effect [15]. This lack of dispersion comesin conflict with the decay of the integral of ρ γ and the conservation of the total mass. We haveshown in Theorem 2.3’ that such an obstruction is present even without damping. Since the gasmust flow anyway, the way to resolve the obstruction is too admit that c is only H¨older-regular,of exponent , at the boundary. In other words, c is Lipschitz and the normal derivative doesnot necessarily vanish(19) g := − γ − ∂c ∂ν ≥ . That g is non-negative follows from the fact that c is positive in Ω( t ) and vanishes on theboundary. Assuming that u is smooth accross the boundary, we obtain the identity ddt u ( X ( t ) , t ) = gν along a boundary path. Therefore g can be viewed as an acceleration of the front expandingin vaccum. The possibility that g be non-zero resolves of course the obstruction raised byTheorem 2.3’. It allows the volume of the domain to grow fast enough, at least as t d whateverthe initial velocity. Notice that we don’t need that g be positive for every time. For instance,an appropriate initial data in even dimension yields a global smooth solution (Theorem 2.1),for which we just have g ≡ g is strictly positive ; it seems to be an open problem to have an existence result covering bothregimes g = 0 and g >
0, and in particular the transition from one regime to the other. Thelocal existence in several space dimensions is proved in [9, 3].Global existence of a weak entropy solution in one space dimension, in presence of vacuum,has been proved by means of Compensated Compactness, see [5, 6]. However, the method ofproof says nothing about the behaviour of the solution at the front with vacuum. At least,this approach has the merit to deal with both regimes, the accelarated and the non-acceleratedones. One drawback is that the flow is obtained as the limit of approximated solutions, whosesupport grows faster than that expected for the genuine flow.
We wish to give an accurate definition of what is an admissible flow, from the point of view ofphysical singularity, at least in the one-dimensional case. Let Ω( t ) = ( a ( t ) , b ( t )) be the domain11ccupied by the fluid (where ρ ( · , t ) is positive) . We have b ′ ( t ) = u ( b ( t ) , t ) and therefore b ′′ = g ≥ b is convex. Likewise a is a concave function. Notice that the discontinuitypoints of a ′ or b ′ form sets that are either finite or denumerable.Let us begin with the mono-atomic case, where γ = 3. We have already seen that thecorresponding Euler system splits into two coupled Burgers equation. Therefore, as long asthe flow is smooth, the characteristics are straight lines. Even more, the tangent lines to theboundary are characteristics. The tangent at a point P actually splits into two halves, one beinga 1-characteristic and the other a 2-characteristic. This is clear when considering a point Q close to the boundary, say to the right component Γ r : because b is convex, there are exactly twotangents to Γ r passing through Q , which are the 1- and the 2-characteristics respectively. Weobserve that a 2-characteristic cannot emanate from Γ r , and a 1-characteristic cannot terminatein Γ r .When γ > Definition 3.1 ( d = 1 .) Let γ be > . An admissible flow surrounded by vacuum is a measur-able bounded field ( ρ ≥ , u ) , which satisfies the following requirements:1. The domain Ω is bounded at left and right by two curves t a ( t ) , b ( t ) , with − a and b convex,2. The flow is a distributional solution of (1,2) in Ω ,3. It satisfies the energy inequality (3),4. For almost every boundary point P ∈ ∂ Ω , lim ( x,t ) → P ρ ( x, t ) = 0 ,
5. If a -characteristic β reaches Γ r at some point P , then P is the terminal point of β ; if β reaches Γ ℓ at some point Q , then Q is its initial point. Symmetrically, if a -characteristic β reaches Γ r at some point P , then P is the initial point of β ; if β reaches Γ ℓ at some point Q , then Q is its terminal point. The reader will have noticed that the definition above is somewhat sloppy : without someregularity, characteristic curves may not be well-defined. We shall therefore use it only insituations where ( ρ, u ) are smooth enough. This will be the case in Theorem 3.1. Noticethat our definition is consistent with the rarefaction waves displayed in the “sub/supersonic”example below ; this is a case where infinitely many characteristics emanate from the sameboundary point. We do not consider the case where the number of connected components varies with time. xamples We give below two explicit examples of eternal flows with physical singularity at vacuum. Theyare built in one-space dimension for a mono-atomic gas ( γ = γ = 3), for which the Euler systemdecouples as a pair of Burgers equations away from shock waves. Without loss of generality,we set A = , so that c = ρ . The wave velocities λ ± = u ± ρ satisfy(20) ∂ t λ + λ∂ x λ = 0 . In a domain where λ is Lipschitz, the characteristic curves, on which λ is constant, are lines ofslope λ .We point out that in both examples, u vanishes identically. Because our solutions aresmooth in Ω, this implies the reversibility : ρ ( x, − t ) = ρ ( x, t ) , u ( x, − t ) = − u ( x, t ) . An accelerated case.
In the following example, the front between the gas and vacuum is thehyperbola defined by the equation x = 1 + t . In the gas, the density and velocity of the flow are given by ρ ( x, t ) = √ t − x t , u ( x, t ) = tx t . We leave the reader verifying that each of the functions λ ± ( x, t ) = tx ± √ t − x t satisfies the Burgers equation. Obviously, c = 1 + t − x (1 + t ) vanishes at the boundary, where it is Lipschitz, and the acceleration g ( √ t , t ) = 1(1 + t ) / is positive for all time. Mind however that this expression is integrable in time. Thedomain Ω( t ) behaves asymptotically as the interval ( −| t | , | t | ), and the flow is asymptoticto ρ R ( x, t ) = 1 | t | r − x t , u R ( x, t ) = xt . | t | → ∞ ; the acceleration g decays like | t | − .Let us point out that this example does not contradict Proposition 2.2, because evenafter extending c by zero outside of Ω, the regularity of the solution ( c, u ) is too low toimplement Lax’ calculation. As a matter of fact, the Cauchy–Lipchitz theorem does notapply to the characteristic flow because the wave velocities are not Lipschitz : Γ is theenvelop of the characteristic lines. A discontinuous example.
This one is mimics a situation described in [8]. Let us chooseinstead the initial data ρ ≡ , u ≡ − , . The solution ( c, u ) of (6,7) is Lipschitz, except at the points ( x, t ) = ( ± ,
0) where adiscontinuity happens (obvious in the data above). At time t >
0, the gas occupies thedomain Ω( t ) = ( − t − , t + 1) and the solution is given by u + ρ = (cid:26) x +1 t if − t − < x < t − , t − ≤ x < t + 1 . u − ρ = (cid:26) − − t − < x ≤ − t + 1 , x − t if − t + 1 < x < t + 1 . The flow is a rarefaction wave in the domain defined by | t − | < x < t + 1, as well as inthat defined by − t − < x < −| t − | . The front is accelerated only at time t = 0, butthis acceleration is an impulse : the velocity of the front flips from − Let us consider an eternal flow surrounded by vacuum in the sense of definition 3.1. We supposein addition that ρ, u are smooth in Ω. We are interested in the growth of | Ω( t ) | as t → + ∞ .Because of the decay of the integral of ρ γ , we already know that | Ω( t ) | is bounded below by C t d for some constant C > t ) = ( a ( t ) , b ( t )) is an interval where t
7→ − a, b are convex functions. The derivative b ′ has limits q − ≤ q + as t → ±∞ . Likewise a ′ has limits p + ≤ p − . Because of the lower bound of the volume, we know that p + < q + , q − < p − . We actually have
Theorem 3.1 ( d = 1 .) Let a flow be smooth (shock-free) in its domain Ω( t ) , where Ω(0) =( a , b ) is a bounded interval. We assume that the flow is admissible in the sense of Definition3.1.Then p + = q − and p − = q + . roof We consider 2-characteristics, which are integral curves of the ODE dXdt = λ ( X, t ) , λ := u + c. Because the flow is smooth in Ω, we know that r := u + cκ is constant along each 2-characteristic.We choose the origin of the time arrow in such a way that a ′ and b ′ be continuous at t = 0.Denote Ω + = Ω ∩ { t > } . By Cauchy–Lipschitz, Ω + is foliated by the 2-characteristics. Onsuch a curve β , the time goes from T in ≥ T fin . At t = T in , β reaches the boundary ∂ Ω + at some point m . By admissibility, we actually have m ∈ Γ + := Γ + ℓ ∪ ( a , b ) , Γ + ℓ := Γ ℓ ∩ { t ≥ } . We order the set of 2-characteristics in Ω + from left to right, and denote it ( −∞ , β ), where β = { ( b , } and −∞ is the “point at infinity” along Γ + ℓ .Consider two such characteristics β ≺ β ′ . If T fin ( β ) is finite, the terminal point β ( T fin ) ison Γ r . Then β ′ is contained in the bounded set delimited by β at left and Γ r at right, and weinfer T fin ( β ′ ) ≤ T fin ( β ) < ∞ . We deduce that β T fin is non-increasing, and there exists β ∗ ∈ [ −∞ , β ] such that ( β ≺ β ∗ ) = ⇒ ( T fin ( β ) = + ∞ ) , ( β ∗ ≺ β ) = ⇒ ( T fin ( β ) < + ∞ ) . In the limit case where T fin is finite for every β (unlikely), then β ∗ = −∞ . If instead T fin = + ∞ for every β (unlikely), then β ∗ = b .When β ≺ β ∗ , the Ricatti equation along β , plus the fact that ∂λ∂r = γ + 14 > , imply that ∂ x r ≥
0. This shows that β r | β is non-decreasing up to β ∗ . On the other hand,if β ∗ ≺ β , we have seen that β T fin ( β ) is non-increasing. Since r equals u = b ′ over theboundary, and b ′ is non-decreasing, we deduce that β r | β is non-increasing beyond β ∗ . If β ∗ ≺ b , we have q + = sup Γ + r u ≤ sup Γ + r r ≤ sup Ω + r = r | β ∗ = sup β ∗ ≺ β r | β ≤ sup Γ + r u = q + . If instead β ∗ = b , then q + = sup Γ + r u ≤ sup Γ + r r ≤ sup Ω + r = lim β → β r | β = lim x → b r ( x,
0) = r ( b ,
0) = b ′ (0) ≤ q + , where the last equality stands because b ′ is continuous at t = 0. In all cases, we infersup Ω + r = r | β ∗ = q + . − = Ω ∩ { t < } by the 2-characteristics, wefind a β ∗ ∈ [ α , + ∞ ] such that ( β ≺ β ∗ ) = ⇒ ( T in ( β ) > −∞ ) , ( β ∗ ≺ β ) = ⇒ ( T in ( β ) = −∞ ) . We have that β r | β is non-decreasing over ( a , β ∗ ), non-increasing over ( β ∗ , + ∞ ), and satisfiessup Ω − r = r | β ∗ = p − . We point out that those 2-characteristics passing through a point m ∈ ( α , b ) belong toboth sets, associated with either Ω + or Ω − .Suppose now that p − = q + . Without loss of generality, we have p − < q + . Then r | β ∗ > r | β forevery 2-characteristic β in Ω − . Thus β ∗ does not emanate from a point m ∈ ( a , b ) ; we musthave β ∗ ≺ α . Thus β ∗ emanates from some point ( a ( t ) , t ) of Γ + ℓ and we have r | β ∗ = r ( a ( t ) , t ).This gives q + = a ′ ( t ) ≤ p − , a contradiction.This ends the proof of the Theorem. Open problems
One important issue is of course to give a multi-dimensional version of Definition 3.1. The diffi-culty here is that we cannot distinguish between two characteristic families : the characteristiccurves passing through a point P are tangent to a cone, which is a connected set. This coneshrinks to a line at boundary points.Because of the dispersion estimate and the conservation of mass, we know that | t | d = O ( | Ω( t ) | ) as | t | → + ∞ . We suspect that the converse inequality | Ω( t ) | = O ( t d ) is also true.Theorem 3.1 provides a positive answer in the special case where d = 1 and the flow is shock-free. It is also true for flows that are smooth up to the boundary, see Lemma 2.1.Actually, in both cases, the domain Ω( t ) behaves asymptotically as t O for the same O as t → ±∞ . This latter property is doubtful for flows that display shocks, but is it true forshock-free flows with physical singularity when d ≥ Acknowledgments.
I thank warmly Alexis Vasseur who welcame me at the Departmentof Mathematics of the University of Texas at Austin ; I came back to the topic of eternalsolutions when I learnt from him about Levermore’s construction of global Maxwellians. I amgrateful to ´Etienne Ghys and Bruno S´evennec, who told me about the similarity of one of myarguments above with the elegant proof by J. Milnor of the hairy ball theorem. I am indebted toConstantin Dafermos, who refreshed my memory about Liu’s work on the physical singularity.16 eferences [1] J.-Y. Chemin. Dynamique des gaz `a masse totale finie.
Asymptotic Analysis , (1990), pp215–220.[2] D. Coutand, S. Shkoller. Well-posedness in smooth function spaces for moving-boundary 1-D compressible Euler equations in physical vacuum. Comm. Pure Appl. Math. , (2011),pp 328–366.[3] D. Coutand, S. Shkoller. Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum. Arch. Rat.Mech. Anal. , (2012), pp 515–616.[4] C. Dafermos. Hyperbolic conservation laws in continuum physics . Grundlehren der math-ematischen Wissenschaften , 3rd edition. Springer–Verlag (2010).[5] R. J. DiPerna. Convergence of the viscosity method for isentropic gas dynamics.
Commun.Math. Phys. , (1983), pp 1–30.[6] Xiaxi Ding, Gui-Qiang Chen, Luo Peizhu. Convergence of the fractional step Lax–Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics. Com-mun. Math. Phys. , (1989), pp 63–84.[7] M. Grassin. Global smooth solutions to Euler equations for a perfect gas. Indiana Univ.Math. J. , (1998), pp 1397–1432.[8] H. P. Greenspan, D. S. Butler. On the expansion of a gas into vacuum. J. Fluid Mech. , (1962), pp 101–119.[9] Juhi Jang, N. Masmoudi. Well-posedness for compressible Euler in a physical vacuum. Preprint arXiv:1005.4441 (2010).[10] P. D. Lax. Development of singularities of solutions of nonlinear hyperbolic partial difer-ential equations.
J. Math. Physics , (1964), pp 611–613.[11] Tai-Ping Liu, J. Smoller. On the vacuum state for isentropic gas dynamics equations. Advances in Math. , (1980), pp 345–359.[12] Tai-Ping Liu. Compressible flow with damping and vacuum. Japan J. Indust. Appl. Math. , (1996), pp 25–32.[13] Tai-Ping Liu, Tong Yang. Compressible Euler equations with vacuum. J. Differential Eq. , (1997), pp 223–237.[14] Tai-Ping Liu, Tong Yang. Compressible flow with vacuum and physical singularity. Methodsand Appl. of Anal. , (2000), pp 495–509.1715] Tetu Makino. Blowing up solutions of the Euler–Poisson equation for the evolution ofgaseous stars. Transport Theory Statist. Phys. , (1992), pp 615–624.[16] J. Milnor. Analytic proofs of the “hairy ball theorem” and the Brouwer fixed point theorem. The American Mathematical Monthly , (1978), pp 521–524.[17] D. Serre. Solutions classiques globales des ´equations d’Euler pour un fluide parfait com-pressible. Annales de l’Institut Fourier (1997), pp 139–153.[18] D. Serre. Five open problems in compressible mathematical fluid dynamics. Methods andApplications in Analysis ,20