Yongcai GENG Yachun LI

1 Introduction

The Euler system of conservation laws for a perfect fluid in special relativity can be written as follows(see,e.g.,[1,18,26–27,33,39,41–47]):

wherenandρare the rest mass density and the mass-energy density,respectively,satisfying

with the specific internal energye,andprepresents the pressure.The constantcis the speed of light,v=denotes the particle speed,andv=|v|satisfies the relativistic constraintAll these variables are the functions of(t,x)∈R+×R3.

For the classical Euler system which is the non-relativistic version of(1.1),Makino,Ukai and Kawashima[31]introduced a new symmetrization to deduce the local-in-time solution even for the case with vacuum states.

If the pressurepdepends only on the mass-energyρ,and the system of energy and momentum conservation laws is closed,(1.1)reduces to the following subsystem(see,e.g.,[1,18,26–27,33,39,41–47]):

Great progress has been made with(1.3),yet mainly for the 1-dimensional or spherically symmetric 3-dimensional cases(see[2–4,6,11–13,15–17,20,22–25,32,34,37–38,40,48–49]and the references therein).

For general multi-dimensional cases of(1.3),Makino-Ukai[29–30]constructed a suitable symmetrizer if a strictly convex entropy exists,and then by applying Friedrichs-Lax-Kato’s theory(see[14,28]),the authors established the local existence of solutions with the data away from the vacuum.For the vacuum case of(1.3),since the coefficient matrix in[29–30]is degenerate near the vacuum,Lefloch-Ukai[19]introduced a different symmetrization based on the generalized Riemann invariants and the normalized velocity,and then established the local existence results of smooth solutions by also using Friedrichs-Lax-Kato’s theory(see[14,28]).Moreover,for(1.3),the singularity formation of smooth solutions is studied in[10,35,37].

In this paper,we consider the system of isentropic relativistic Euler equations,which corresponds to the conservation of the baryon numbers and momentum and reads as(see,e.g.,[1,18,26–27,33,39,41–47]):

We know that,formally,the Newtonian limit of(1.4)is the following classical system of non-relativistic isentropic Euler equations(see[7,26,39]):

which is one of the motivation for our study on(1.4).Another motivation for our study is that some special relativistic effects are revealed for 3-dimensional relativistic equations(see[8]),which do not appear in the corresponding non-relativistic case.

We consider(1.4)with the equation of the state

satisfying

where 0≤≤∞are any non-negative constants subject to the subluminal condition≤.Note that ifp(ρ)=then=∞forγ=1 andρ∗=forγ>1.

The first law of thermodynamics(Gibb’s Equation)reads as

with temperatureθand the specific entropyS.For isentropic fluids(S≡const.),we have

Denote

For simplicity,wein the sequel.From(1.8),we have

withρmbeing any fixed number inandC=

We consider the Cauchy problem(1.4)with initial data

Research results of(1.4)is not so rich as that of(1.3),and all results are about 1-dimensional case(see[21,23,36,39,48]).Naturally,we are interested in the local existence of smooth solutions to the Cauchy problem(1.4)and(1.10)for both vacuum and non-vacuum cases.

The main result of our paper is as follows.

Theorem 1.1Suppose that the initial dataand there exists a positive constant δ0which is sufficiently small,such that

and

whereis the uniformly local Sobolev space defined in[14],are determined by(1.9)withorThen,the Cauchy problem(1.4)and(1.10)admits a unique solution(n(t,x),v(t,x))with n∗≤n(t,x)≤n∗,|v(t,x)|

where T depends only on δ and thenormof the initial data.

We shall prove the main theorem by symmetrizing(1.4)and applying the Friedrichs-Lax-Kato theory(see[14,28])of symmetric hyperbolic systems.Thus for both the non-vacuum and vacuum cases,the construction of a suitable symmetrizer is necessary and important.We borrow some ideas from[19,29–30],which are more complicated due to the structure of the system itself,and different to some extent since we involve variablesnand v instead ofρand v.

More precisely,in Section 2,we firstly solve out the strictly convex entropy function of(1.4)for the non-vacuum case according to[9],then we construct a symmetrizer as in[29–30].In Section 3 for the vacuum case,due to the degeneracy of the symmetrized system near the vacuum,as in[19],we transform(1.4)into a symmetric form in terms of the generalized Riemann invariants and the normalized velocity.

2 Non-vacuum Case

In this section,we will establish the existence of local smooth solutions to the Cauchy problem(1.4)and(1.10)for the non-vacuum case in Theorem 1.1.For clarity,we will divide it into two subsections:Strictly convex entropy function and symmetrization.

2.1 Strictly convex entropy function

If an entropy-entropy flux pair of(1.4)exists,then we may construct a symmetrizer accordingly.According to[9],we first find the strictly convex entropy function of(1.4).To do this,we fit(1.4)into the following general form of conservation laws:

where

and

whereis the Kronecker symbol.

A scalar functionη(u)is called the entropy to(2.1)if there exist scalar functions(j=1,2,3)satisfying

Let z=(n,By direct but tedious computations,we have

and

wherestands for the 3×3 identity matrix.

Similarly,we have

where=

Using(2.4)together with(2.5),we get

where

From(2.2),we have

Settingη=η(n,y)and=Q(n,y),wherey=(2.7)becomes

which yields

where=Furthermore,(2.9)leads to

It follows from the first equation of(2.9)that

whereGdepends only onn.

Inserting this equation into the third equation of(2.9),we have

Furthermore,plugginginto(2.12),we obtain

Assumingq(n,y)=(2.13)becomes

Integrating this equation with respect to the variablen,we have

Substituting(2.15)into(2.13)and separating variables,we get

where the left-hand side of(2.16)depends only onn,and the right-hand side depends only ony.So both sides of(2.16)should be equal to the same constant,assumed asD,which implies

Noting(1.8),we solve the first equation of(2.17)to have

whereis a constant.

From the fact that

it holds that

Solving the second ordinary differential equation of(2.17),we have

whereis a constant.

Inserting(2.20)and(2.21)into(2.15),we have

whereis an integration constant.Notingq(n,y)=we obtain

Inserting(2.22)into(2.14)and using(1.8),we get

Integrating this equation yields

Thus substituting(2.23)–(2.24)into(2.11)leads to

whereD1=andD3is a constant.

Noting that

we define

whereK:=Together with(2.19),we have

Φ(ρ)can be expanded with respect toat 0 as

Similarly,we also expandwith respect toat 0 as

From(2.25)and(2.28)–(2.9),we have

To choose the constantswe consider the entropy function of the corresponding non-relativistic fluid,which is

which can be obtained in exactly the same way as(2.31).

Lettingc→∞in(2.31)and comparing with(2.32),we choose=0.Then it holds that

2.2 Symmetrization

In this subsection,we use the obtained strictly convex entropy function to construct a suitable symmetrization of(1.4)and verify the positive definiteness of the coefficient matrix.

Define

The existence of a strictly convex entropy guarantees that classical solutions to the initialvalue problem depend continuously on the initial data,even within the broader class of admissible bounded weak solutions(see[5,Theorem 5.2.1]or[35,Theorem B]).Thus,if the initial datatake values in any compact subsetDof={z:n∗

Let w=It holds that

whereαstands for one of the argumentst,Then the system(2.1)reduces to

If we set

(2.35)can be rewritten as

whose coefficient matrices(w)(α=0,1,2,3)satisfy the following:

A hyperbolic system(2.37)satisfying(2.38)is called a symmetric hyperbolic system(see[15,29]).

Now we figure out the expressions of(w)and(w)(j=1,2,3),and verify the positive definiteness of(w).w can be written as

then we can compute that

It is not easy to show by direct calculation that

and

(w)

It is obvious that(w)(j=1,2,3)are symmetric forms.Now we prove thatis uniformly bounded.Firstly,we verify that(w)has a strict lower bound.In fact,for any given four-dimensional vector r=it holds

where

Settingwith 0<δ

under the condition that

i.e.,

Since the right-hand side of the above inequality has a positive lower bound,denoted byδ∗,we can takeδ

we have

Thanks to the fact that(n,v)∈Ωz,we get the upper bound of

Then the local existence of smooth solutions to the Cauchy problem(1.4)and(1.10)for the non-vacuum case follows from Friedrichs-Lax-Kato theory(see[14,28]).

3 The Vacuum Case

In this section,when the initial data(n0,v0)are allowed to contain vacuum states,the coefficientsA0(w)for(1.4)will blow-up near the vacuum.Thus the symmetric method to the non-vacuum case will not be valid any longer.To overcome this difficulty,we adopt Lefloch-Ukai’s symmetrization(see[19])for(1.3),however our transformation is about variables ofnand v instead of variables ofρand v in[19].The coefficient matrix of the new system under this transformation is no longer degenerate near vacuum.Then we use Friedrichs-Lax-Kato theory(see[14,28])to prove the local existence of smooth solutions to(1.4)and(1.10)with the initial data of the vacuum case.

Now our initial datan0,v0satisfy the condition(1.12).Before proceeding,we first introduce some notations as in[19].

The modified mass density variablewis

wherecsis the local sound speed in the fluid.

The modified velocity scalar is

and we refer to

as the generalized Riemann invariant variables.

We also introduce the normalized velocityand the associated projection operatorP(v)as follows:

Here we present some useful identities in[19],

Proposition 3.1

For the convenience of the reader,we give a simple proof here.

Proof of Proposition 3.1

(1)

(2)

where we used

(3)Similarly to(2),we get

(4)On the one hand,noting that=1,we have

On the other hand,it holds that

From the definition ofP(v)in(3.4),(4)is proved.

3.1 Symmetric form of Euler equations

In this section,we will deduce a symmetric formulation of(1.4)with respect to the general Riemann invariants and the normalized velocity defined by(3.1)–(3.2).We conclude as follows.

Lemma 3.1In terms of the generalized Riemann invariant variables()and the normalized velocitydefined by(3.3)–(3.4),respectively,the relativistic Euler equations reduce to the following symmetric form:

where z±are real valued andis a unit vector satisfying||=1.

ProofIn Section 1,we know from(1.8)that

whereqis defined asq:=+ρ.

Using(3.7),we expand the conservation equation of baryon numbers in(1.4)as follows:

Multiplying(3.8)bywe have

Expanding the momentum conservation equation of(1.4)and using(3.7),we have

where we used

From(3.9)–(3.10),we have

Moreover,multiplying(3.9)by(ρ),(3.11)can be simplified as

From the definition ofuin(3.2),we have

then(3.12)reduces to

By(3.11)can be rewritten as

Multiplying this equation byand using(3.13),(3.15)becomes

To obtain the expression of,we multiply(3.11)by the projectionP(),and due to(3.5)and the definition ofw,we get

To obtain the expression ofw,we combine(3.14)and(3.16)to give

Using the definition ofuin(3.2)again,we have

Plugging this into(3.16),we have

Substituting(3.19)for(3.18)leads to

To derive our desired symmetric form,∇needs to be transformed,and from(4)in(3.5),we have

Plugging this identity into(3.20)–(3.21),and together with(3.17),we have

Using the generalized Riemann invariant variables=u±w,it is easy to obtain the symmetric formulation(3.6).

Moreover,(3.6)can be written as the symmetric form(2.37),in which(w)andw)are,respectively,

where

Remark 3.1By this lemma,although(1.3)–(1.4)are different,they are symmetrized to the same form(3.6).

The following proof is the same as that in[19]and for the reader’s convenience,here we briefly list the main steps.

Observe that the above matrix(w)allows the density to vanish,since the coefficients remain bounded as the density approaches to zero.

Moreover,from(3.23),we observe that

wheredenotes the Eucilidian inner product inand

From(3.24),the matrix(w)is positively definite as long as the velocity v never vanishes.According to the Friedrichs-Lax-Kato theory(see[14,28]),a local in-time solution exists.

However,the matrix(w)may lose its positive definiteness,since the coefficientc0of∂tin the third equation vanishes at v=0.On this occasion,we apply a well-chosen Lorentz transformation,which allows the Lorentz-transformed velocity not to exceed the light speed and remain bounded away from zero.

For the reader’s convenience,we list some technical results about the Lorentz-transformed velocity(see[19]).

3.2 Lorentz transformation

Assume thatKandare two reference frames,in which(t,x)andrepresent the space-time coordinates corresponding toKandrespectively.Kmoves with respect toat the velocity V.The transformation

is called a Lorentz transformation,where=is the Lorentz factor.

From(3.26),there holds the velocity transformation law

where v=Denote

Then we have the following lemma.

Lemma 3.2(Uniform Bounds for the Velocity)(see[19])Given any∈(0,1)and any vectorV∈satisfying<1,there exist positive constants0<<<1dependingonly on r0and,such that the Lorentz-transformed velocity(3.27)is uniform bound away from both the origin and the light speed,i.e.,

holds for anywhere:={y∈

Using the Lorentz invariance of relativistic Euler equations,(2.37)can also be expressed in the transformed coordinatesdefined by(3.26),that is,

and(3.25)becomes

In view of the upper and lower bounds(3.29),we conclude that the transformed matrixis positively definite in the coordinate systemHence the Friedrichs-Lax-Kato theory(see[14,28])applies to the initial-value problem(3.30),provided that the initial data are imposed on the initial hypersurface=0.In the relativistic setting,the initial plane:t=0 is not preserved under the transformation(3.26).However,in the new coordinate systemthe initial plane becomes

In order to prove the local well-posedness of the oblique initial-value problem(3.30)with the data on,it is convenient to introduce a further change of the coordinates

which maps the hyperplaneto the hyperplane

This transformation puts(3.31)into the following form:

where the matrixis still positively definite(see[19]).

Now Friedrichs-Lax-Kato theory(see[14,28])guarantees the existence of a solution defined in a small neighborhood of this hyperplane.Making the transformation back to the original variables,we obtain a solution in a small neighborhood of the initial linet=0.This completes the proof of the main theorem for the vacuum case.

[1]Anile,A.M.,Relativistic Fluids and Magneto-Fluids,Cambridge Monographs on Mathematical Physics,Cambridge University Press,New York,1989.

[2]Chen,G.Q.and Li,Y.C.,Stability of Riemann solutions with oscillation for the relativistic Euler equations,J.Dif f.Eq.,202,2004,332–353.

[3]Chen,G.Q.and Li,Y.C.,Relativistic Euler equations for isentropic fluids:stability of Riemann solutions with large oscillation,Z.Angew.Math.Phys.,55,2004,903–926.

[4]Chen,J.,Conservation laws for relativisticp-system,Commu.Part.Dif f.Eq.,20,1995,1605–1646.

[5]Dafermos,C.M.,Hyperbolic Conservation Laws in Continuum Physics,Spring-Verlag,Berlin,2000.

[6]Frid,H.and Perepelista,M.,Spatially periodic solutions in relativistic isentropic gas dynamics,Commun.Math.Phys.,250,2004,335–370.

[7]Geng,Y.C.and Li,Y.C.,Non-relativistic global limits of entropy solutions to the extremely relativistic Euler equations,Z.Angew.Math.Phys.,61,2010,201–220.

[8]Geng,Y.C.and Li,Y.C.,Special relativistic effects revealed in the Riemann problem for threedimensional relativistic Euler equations,Z.Angew.Math.Phys.,62,2011,281–304.

[9]Godunov,S.K.,An interesting class of quasilinear systems,Dokl.Acad.Nauk.SSSR,139,1961,521–523.

[10]Guo,Y.and Tahhvildar-Zadeh,S.,Formation of singularities in relativistic fluid dynamics and in spherically symmetric plasma dynamics,Cntemp.Math.,238,2009,151–161.

[11]Hao,X.W.and Li,Y.C.,Non-relativistic global limits of entropy solutions to the Cauchy problem of the three dimensional relativistic Euler equations with spherical symmetry,Commun.Pure Appl.Anal.,9,2010,365–386.

[12]Hsu,C.H.,Lin,S.and Makino,T.,Spherically symmetric solutions to the compressible Euler equation with an asymptoticγ-law,Japan J.Indust.Appl.Math.,20,2003,1–15.

[13]Hsu,C.H.,Lin,S.and Makino,T.,On spherically symmetric solutions of the relativistic Euler equation,J.Dif f.Eq.,201,2004,1–24.

[14]Kato,T.,The Cauchy problem for quasi-linear symmetric hyperbolic systems,Arch.Rational Mech.Anal.,58,1975,181–205.

[15]Kunik,M.Qamar,S.and Warnecke,G.,Kinetic schemes for the ultra-relativistic Euler equations,J.Comput.Phys.,187,2003,572–596.

[16]Kunik,M.,Qamar,S.and Warnecke,G.,Second-order accurate kinetic schemes for the relativistic Euler equations,J.Comput.Phys.,192,2003,695–726.

[17]Kunik,M.,Qamar,S.and Warnecke,G.,Kinetic schemes for the relativistic gas dynamics,Numer.Math.,97,2004,159–191.

[18]Landau,L.D.and Lifchitz,E.M.,Fluid Mechanics,2nd edition,Pergamon Press,New York,1987,505–512.

[19]Lefloch,P.and Ukai,S.,A symmetrization of the relativistic Euler equations in sevaral spatial variables,Kinet.Relat.Modles.,2,2009,275–292.

[20]Li,Y.C.,Feng,D.and Wang,Z.,Global entropy solutions to the relativistic Euler equations for a class of large initial data,Z.Angew.Math.Phys.,56,2005,239–253.

[21]Li,Y.C.and Geng,Y.,Non-relativistic global limits of Entropy solutions to the isentropic relativistic Euler equations,Z.Angew.Math.Phys.,57,2006,960–983.

[22]Li,Y.C.and Ren,X.,Non-relativistic global limits of entropy solutions to the relativistic euler equations withγ-law,Commun.Pure Appl.Anal.,5,2006,963–979.

[23]Li,Y.C.and Shi,Q.,Global existence of the entropy solutions to the isentropic relativistic Euler equations,Commun.Pure Appl.Anal.,4,2005,763–778.

[24]Li,Y.C.and Wang,A.,global entropy solutions of the cauchy problem for the nonhomogeneous relativistic Euler equations,Chin.Ann.Math.,27B(5),2006,473–494

[25]Li,Y.C.and Wang,L.,Global stability of solutions with discontinuous initial containing vaccum states for the relativistic Euler equations,Chin.Ann.Math.,26B(4),2005,491–510.

[26]Li,T.T.and Qin,T.,Physics and Parital Differential Equations(in Chinese),2nd edition,Higher Eudcation Press,Beijing,2005.

[27]Liang,E.P.T.,Relativistic simple waves:Shock damping and entropy production,Astrophys.J.,211,1977,361–376.

[28]Majda,A.,Compressible fluid flow and systems of conversation laws in several space variable.Comm.Pure Appl.Math.,28,1975,607–676.

[29]Makino,T.and Ukai,S.,Local smooth solutions of the relativistic Euler equation,J.Math.Kyoto Univ.,35,1995,105–114.

[30]Makino,T.and Ukai,S.,Local smooth solutions of the relativistic Euler equation II,Kodai Math.J.,18,1995,365–375.

[31]Makino,T.,Ukai,S.and Kawashima,S.,Sur la solutionssupport compact de lquation d’Euler compressible.,Japen J.Appl.Math.,3,1986,249–257.

[32]Min,L.and Ukai,S.,Non-relativistic global limits of weak solutions of the relativistic Euler equation,J.Math.Kyoto.Univ.,38,1995,525–537.

[33]Misner,C.W.,Thorne,K.S.and Wheeler,J.A.,Gravitation,Freeman,San Fransisco,1973.

[34]Mizohata,K.,Global solutions to the relativistic Euler equation with spherical symmetry,Japan J.Indust.Appl.Math.,14,1997,125–157.

[35]Pan,R.and Smoller,J.,Blowup of smooth solutions for relativistic Euler equations,Commun.Math.Phys.,262,2006,729–55.

[36]Pant,V.,Global entropy solutions for isentropic relativistic fluid dynamics,Commu.Part.Dif f.Eq.,21,1996,1609–1641.

[37]Rendall,A.,Local and global existence theorems for the Einstein equations,Living Rev.Rel.,3,2000,1–35.

[38]Ruan,L.and Zhu,C.,Existence of global smooth solution to the relativistic Euler equations,Nonlinear Analysis,60,2005,993–1001.

[39]Shi,C.C.,Relativistic Fluid Dynamics,Science Press,Beijing,1992,161–232.

[40]Smoller,J.and Temple,B.,Global solutions of the relativistic Euller equation,Commun.Math.Phys.,156,1993,67–99.

[41]Taub,A.H.,Relativistic Rankine-Hugoniot equations,Phys.Rev.,74,1948,328–334.

[42]Taub,A.H.,Approximate solutions of the Einstein equations for isentropic motions of plane symmetric distributions of perfect fluids,Phys.Rev.,107,1957,884–900.

[43]Taub,A.H.,Relativistic hydrodynamics,relativistic theory and astrophysics 1,Relativity and Cosmology,Ehlers,J.(ed.),A.M.S.,Providence,RI,1967,170–193.

[44]Thompson,K.,The special relativistic shock tube,J.Fluid Mech.,171,1986,365–375.

[45]Thorne,K.S.,Relativistic shocks:The Taub adiabatic,Astrophys.J.,179,1973,897–907.

[46]Weinberg,S.,Gravitation and Cosmology:Applications of the General Theory of Relativity,Wiley,New York,1972.

[47]Whitham,G.B.,Linear and Non-linear Waves,Wiley,New York,1974.

[48]Xu,Y.and Dou,Y.,Global existence of shock front solutions in 1-dimensional piston problem in the relativistic equations,Z.Angew.Math.Phys.,59,2008,244–263.

[49]Yin,G.and Sheng,W.,Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations,Chin.Ann.Math.,29B(6),2008,611–622.