Yue CAO Yachun LI

Abstract In this paper,the authors study the Cauchy problem of n-dimensional isentropic Euler equations and Euler-Boltzmann equations with vacuum in the whole space.They show that if the initial velocity satisfies some condition on the integral J in the“isolated mass group”(see(1.13)),then there will be finite time blow-up of regular solutions to the Euler system with J≤0(n≥1)and to the Euler-Boltzmann system with J<0(n≥1)and J=0(n≥2),no matter how small and smooth the initial data are.It is worth mentioning that these blow-up results imply the following:The radiation is not strong enough to prevent the formation of singularities caused by the appearance of vacuum,with the only possible exception in the case J=0 and n=1 since the radiation behaves differently on this occasion.

Keywords Euler and Euler-Boltzmann equations,Finite time blow-up,Multidimensional,Regular solutions,Vacuum

1 Introduction

It is well-known that the motion of isentropic inviscid fluid can be described by Euler equations

where t≥0 is the time variable,x=(x1,···,xn)∈Rnis the space variable,ρ(t,x)is the mass density,u(t,x)=(u1,···,un)Tis the fluid velocity,pmis the material pressure satisfying the equation of state

where A>0 is the gas constant,γ>1 is the adiabatic exponent.As is known to all,the radiation effects become remarkable in some physical problems as the temperature increases,for example,in the high-temperature plasma physics[24]and various astrophysical contexts[15].Thus,one needs to consider radiation effects when describing the fluid motion.For the isentropic fluid,the coupling of fluid field and radiation field involves momentum source depending on the specific radiation intensity driven by the so called radiation transfer integro-differential equation(see[24]),and the equations of radiation hydrodynamics result from the balances of particles and momentum.More precisely,the mass density ρ(t,x),the fluid velocity u(t,x)and the specific radiation intensity I(v,Ω,t,x)are governed by the following Euler-Boltzmann equations

where v and Ω are radiation variables,v∈R+is the frequency of photon and Ω∈Sn−1is the travel direction of photon,Sn−1stands for the unit sphere in Rn,and c is the light speed.The impact of radiation on the dynamical properties of the fluid is described by the radiation flux Frand the radiation pressure tensor Pr:

The collision term on the right-hand side of the radiative transfer equation is

which involves emission,absorption and scattering of energy,where I=I(v,Ω,t,x),I′=I(v′,Ω′,t,x);σe=σe(v,Ω,t,x)≥0 is the rate of energy emission due to spontaneous process;σa=σa(v,t,x,ρ)≥0 denotes the absorption coefficient that may depend on the mass density ρ;Similar to absorption,a photon can undergo scattering interactions with matter,and the probability of a photon being scattered from v′to v contained in dv,from Ω′to Ω contained in dΩ,and traveling a distance ds is given by the“differential scattering coefficient”σs(v′→v,Ω′·Ω)dvdΩds(see[17,24]).Moreover,the time rates of outscattering and inscattering within a unit volume element are

and σs,behave like

Noticing that,unlike Euler-Poisson or Euler-Maxwell systems,where Euler equations are coupled with an elliptic or a parabolic equation,(1.3)is a system that Euler equations are coupled with a hyperbolic equation.Thus the study of Euler-Boltzmann equations is challenging due to the high complexity and mathematical difficulty of the system itself.

As has been shown in[17,24],from the assumptions of“induced process”and local thermal equilibrium,σeand σacan be written as

Thus,when σs=0,the radiative transfer equation can be written as

and the isentropic Euler-Boltzmann system(1.3)is reduced to

One of the motivations that we study the radiation system(1.5)lies on the fact that the isentropic Euler system(1.1)can be,to some extent,regarded as the non-radiation limit of the isentropic Euler-Boltzman system(1.5).In fact,since the radiation of the black-body is the smallest one,so I≡implies that the radiation effect is ignored,and formally system(1.5)is reduced to system(1.1).For the rigorous justification of this type of limit,we refer to Ducomet-Neasov[7]for the diffusion limit of the Navier-Stokes-Boltzmann system,when the radiative intensity is driven to equilibrium or non-equilibrium,see also Lowrie-Morel-Hittinger[21],Buet-Desprs[2]for more results in this direction.While,as far as we know,the rigorous diffusion limit of system(1.5)is still unknown,which is worth considering in the future work.

In this paper,we consider the singularity formation of regular solutions to the Cauchy problems of(1.1)and(1.5)with initial data

and

respectively.

For the classical well-posedness of Euler equations,there are rich literatures on the existence of strong or classical solutions with vacuum.For the Cauchy problem of non-isentropic Euler equations with compactly supported initial density and velocity,Makino-Ukai-Kawashima[22]established the local existence of regular solutions in 3D space and proved that the life span is finite for any non-trivial solution,see also Liu-Yang[20]for similar results with damping for non-isentropic Euler equations in 3D space and isentropic Euler equations(1.1)in 1D space.For some small initial density with compact support and smooth initial velocity satisfying

where Sp(∇u0)stands for the spectrum of the matrix∇u0and δ is a constant,Grassin[8]and Serre[25]obtained the global existence of smooth solutions for both isentropic and nonisentropic Euler equations(see also[9]).For the Euler equations with radiation,when the radiation fluid is inviscid and the initial density is away from vacuum,Jiang-Zhong[14]obtained the local existence of C1solutions to the Cauchy problem of non-isentropic fluid in multidimensional space,see also Jiang-Wang[13]for the initial-boundary value problem in multidimensional space and Blanc-Ducomet[1]for the global existence of weak solutions in 1D space.When the initial vacuum is allowed,Jiang-Wang[12]obtained the global existence of weak entropy solutions to the Cauchy problem of(1.5)in 1D space,see also Jiang[10]for isothermal fluids.Li-Zhu[19]proved the local existence of Makino-Ukai-Kawashima type(see[22])regular solutions to the Cauchy problem of(1.5)in 3D space.For the Navier-Stokes-Boltzmann equations of viscous radiation fluid,we refer to Ducomet-Feireisl-Neasov[3],Ducomet-Neasov[3–6],Li-Zhu[17–18]and references therein for related existence results.

In this paper,we are interested in the finite time blow-up of regular solutions with initial density containing vacuum state.For the case without initial vacuum,Sideris[27]proved two types of finite time blow-up of C1solutions to the Cauchy problem of Euler equations in 3D space.The first one is for the non-isentropic Euler equations,if the initial data are“large”,where the“large”essentially means that the initial flow velocity must be supersonic in some region,then the singularity formation is detected as a disturbance that overtakes the wave front forcing the front to propagate with supersonic speed.This blow-up result has been generalized to the multi-dimensional non-isentropic Euler-Boltzmann equations by Jiang-Zhong[14]and to the 3D isentropic version(1.5)by Jiang-Wang[11],with the initial radiation satisfying

for some positive constant R.The second one shows that singularity will develop for both isentropic and non-isentropic Euler equations if the fluid,on average,is slightly compressed and out-going near the wave front,which has been generalized to the 3D isentropic Euler-Boltzmann equations(1.5)by Jiang-Wang[11]with the initial radiation I0satisfying(1.8)and in addition

For the case with initial vacuum,Serre[26]proved that the regular solution to the Cauchy problem of Euler system(1.1)will blow up in finite time if the initial density is compactly supported in some bounded region V⊂Rn(n≥1)and the initial velocity satisfies some integral condition,i.e.,

For the Euler-Boltzmann system(1.5),inspired by the“isolated mass group”introduced by Xin-Yan[30]for Navier-Stokes equations and the global existence established by Grassin and Serre[8–9,25]for Euler equations,Li-Zhu[19]identified the following two classes of initial data which contain local vacuum states such that the regular solutions of Cauchy problem blow up in finite time:

(i)(Isolated Mass Group)For some bounded open sets A0⊂B0⊂BR0⊂R3,

where R0is a positive constant and BR0is the ball centered at the origin with radius R0,BCR0is the complementary set of BR0in R3.This class removed the key assumption(1.9)for radiation field in[11].

(ii)(Hyperblic Singularity Set)In some smooth open set V⊂R3,

They also proved that this blow-up mechanism also holds for the corresponding non-radiation version,i.e.,the Euler system(1.1).

For the finite time formation of singularities on the Navier-Stokes-Boltzmann equations of viscous fluid,we refer to[17]and the references cited therein for details.

As have been observed in Li-Zhu[19],according to the definition of regular solutions(see Definition 1.1),one has

which implies that in the vacuum domain,the behavior of the velocity is controlled by a positive and symmetric hyperbolic system,i.e.,the so called multi-dimensional Burgers equations.Thus,in general,the velocity may not be zero in vacuum region.In this paper,based on the similar idea,we will present a scenario(see Definition 1.2)for finite time singularity formation of regular solutions to the Cauchy problems(1.1)with(1.6)and(1.5)with(1.7)for both Euler and Euler-Boltzmann equations.

Before stating our main results,we give some related definitions.The first one is the regular solution of system(1.1)and system(1.5)that we consider in this paper.

Definition 1.1(Regular Solutions)For 1<γ≤3,let T>0 be a finite constant.If

(1)(ρ,u)(t,x)∈C1([0,T)×Rn),∈C1([0,T)×Rn),ρ≥0;

(2)ut+u·∇u=0 holds when ρ(t,x)=0,then(ρ,u)(t,x)is called a regular solution to the Cauchy problem(1.1)and(1.6)in[0,T)×Rn;

If,in addition;

(3)I(v,Ω,t,x)∈L2(R+×Sn−1,C1([0,T)×Rn)),then(I,ρ,u)(t,x)is called a regular solution to the Cauchy problem(1.5)and(1.7)in[0,T)×Rn.

Remark 1.1It is worth pointing out that the local existence of regular solutions to(1.1)and(1.5)in 3D space have been established by Makino-Ukai-Kawashima[22]and Li-Zhu[19],respectively.Actually,these frameworks in[22]and[19]are applicable to arbitrary space dimension with some minor modifications.

Now we give the definition of“isolated mass group”in this paper.

Definition 1.2(Isolated Mass Group)Let A0,B0be two smooth,bounded and connected open sets in Rn,and⊂B0⊆BR0,whereis the closure of A0under standard Euclidean norm,R0is a positive constant and BR0is the ball centered at the origin with radius R0.If

we say that(ρ0,u0)(x)has an isolated mass group(A0,B0).

Remark 1.2This definition is inspired by Li-Zhu[19]and Serre[26],we replace the condition on u0in[19](i.e.,u0=0 when ρ0=0 in(1.11))by the one introduced in[26](i.e.,the condition on the integral J in(1.13)),where the integral J depends only on the value of u0on the boundary∂A0.In fact,by direct calculation,one has

for any j=1,···,n,where Uij(i=1,···,n)are cofactors of the matrix

in which,1=(1,···,1)T∈Rn.Thus

where ν is the unit outward normal vector of∂A0,this implies that J only depends on the value ofon the boundary∂A0.For example,when n=3,one has

when n=1,one has

Now we state main theorems in this paper,the first one is the finite time blow-up for Euler equations.

Theorem 1.1Let 1<γ≤3.Assume that the initial data(ρ0,u0)(x)has an isolated mass group(A0,B0),and(ρ,u)(t,x)is the corresponding regular solution to the Cauchy problem(1.1)and(1.6)in[0,Tm)×Rnwith maximal existence time Tm.Then for all n≥1,under either J<0 or

we both have Tm<+∞.

The second one is the finite time blow-up for Euler-Boltzmann equations.

Theorem 1.2Let 1<γ≤3.Assume that the initial data(ρ0,u0)(x)has an isolated mass group(A0,B0),|u0(x)|L∞(Rn)

we both have Tm<+∞.

Remark 1.3Compared with the blow-up result obtained in[26]for system(1.1),we remove the assumption that the initial density is compactly supported,i.e.,the vacuum can appear in local domain.Compared with the two blow-up results obtained in[19]for system(1.5),we remove the assumption in the first result(see(1.11))that the initial velocity vanishes where initial density vanishes,and we only need the information of initial velocity on the local vacuum boundary∂A(t)instead of the vacuum domain(see(1.12))assumed in the second result of[19].

Remark 1.4For the case without radiation,Theorem 1.1 shows that the appearance of vacuum will cause finite time blow-up of regular solutions to(1.1)in arbitrary dimensional space.Moreover,even with the radiation effect,Theorem 1.2 shows that the vacuum still leads to finite time blow-up of regular solutions to(1.5)in arbitrary dimensional space when J<0 or in multi-dimensional space(n≥2)when J=0,which implies that the radiation is not strong enough to prevent the formation of singularities caused by vacuum in multi-dimensional space.

In the rest section,we first provide some preliminary lemmas required for the proofs of Theorems 1.1–1.2,then we prove Theorem 1.1 and Theorem 1.2,respectively.

2 Finite Time Blow-up

Hereinafter,it is always assumed that initialdata(ρ0,u0)(x)and(I0,ρ0,u0)(x)satisfy assumptions in Theorems 1.1–1.2,(ρ,u)(t,x)and(I,ρ,u)(t,x)are regular solutions to the corresponding Cauchy problems of(1.1)and(1.5)in[0,Tm)×Rn,respectively.

2.1 Preliminaries

Now,we introduce some necessary quantities and preliminary lemmas.First,we define the particle path generated by the velocity u(t,x).

Definition 2.1(Particle Path)Let X(t;0,x0)be the particle path starting from x0at t=0,i.e.,

and denote A(t),B(t),B(t)A(t)as the images of A0,B0,B0A0under(2.1),respectively,then,

This definition implies the evolution of A0.Moreover,for any t∈[0,Tm),according to the mass equation(1.1)1or(1.5)2,one has

which,along with the condition in(1.13):ρ0(x)=0 in B0A0,we have ρ(t,x)=0 in B(t)A(t).Then,together with the definition of regular solutions,one has

Thus for each x∈B0A0,it yields that

which implies that

This,with the aid of(2.1),gives

and it immediately has

Here,(2.2)shows the evolution of the set A0.

To show the finite time formation of singularities,in the region A(t),we define the following physical quantities:

which is used to define the following functional(see also[16,29]):

We also denote by

the total mass in A(t),then it is clear that m(0)>0 according to(1.13).

Now,we give some useful lemmas.The first one is the Reynolds transport lemma(see[28]).

Lemma 2.1Considering any part of the fluid in τ(t)and with velocity u,for any G(t,x)∈C1(R+×Rn),one has

where ν is the unit outward normal vector and dS is the surface elements of∂τ(t).

With the help of Lemma 2.1,the second lemma implies the conservation of mass in A(t).

Lemma 2.2The mass is conserved in A(t)for both Euler and Euler-Boltzmann systems,i.e.,

ProofAccording to(1.1)2or(1.5)2and Lemma 2.1,one has

which implies that m(t)=m(0)for any t∈[0,Tm).

The third lemma implies that the volume of the region A(t)is a polynomial function of t with degree no more than n(see[23 Theorem 1,26,Lemma 2.1]).

Lemma 2.3(Polynomial Growth of Volume)In[0,Tm),the volume of A(t)is a polynomial with degree no more than n,and the coefficient of the highest order term tnin the polynomial|A(t)|is exactly J.

Moreover,the volume of A(t)has strictly positive lower bound.

Lemma 2.4For both Euler and Euler-Boltzmann systems,the volume of A(t)satisfies

for some nonnegative integer k≤n and positive constant C0that depends on u0.More precisely,if J=0,then k≤n−1.

ProofFirst,if|A(t)|=0 for some t0∈[0,Tm),then m(t0)=0,which contradicts to the conservation of mass in A(t),i.e.,m(t0)=m(0)>0(see Lemma 2.2).Thus

Moreover,there exists a constant C0>0 such that

otherwise,for any T*

thus there exists a limit t1∈[0,Tm)of the bounded sequence{tj},such that|A(t1)|=0,which contradicts to Lemma 2.2.

Second,according to Lemma 2.3,|A(t)|is a polynomial with degree no more than n,thus with the aid of(2.4),there exists a integer 0≤k≤n,such that

for some positive constant C0that depends on u0.

Especially,when J=0,|A(t)|is a polynomial with degree no more than n−1,thus(2.5)holds for some nonnegative integer k≤n−1.

In the following lemma,we give some basic estimates on H(t).

Lemma 2.5For 1<γ≤3 and t∈[0,Tm),it holds for both Euler and Euler-Boltzmann systems that

for J=0 and some nonnegative integer k≤n−1,where C is a positive constant that depends on C0,m(0),A and γ.

ProofFirst,according to the definition of H(t),one has

thus by the non-negativity of the first term on the right-hand side of(2.7),it is easy to show that

where we have used Lemma 2.4 and the following estimate

and C is a positive constant that depends on C0,m(0),A and γ.

2.2 Blow-up of Euler equations

Now,we are ready to prove Theorem 1.1.

Proof of Theorem 1.1We divide the proof into two steps.

Step 1The case of J=0.We will control H(t)by using Gronwall’s inequality for t∈[0,Tm).On one hand,from the definition of H(t),one has

in which,from the definition of M(t)and(1.1)1,one has

where we have used the fact that ρ=0 on∂A(t).Similarly,

where we have used(1.1)2,integrating by parts and ρ=0 on∂A(t).For

noticing that

with the help of integrating by parts and(1.1)2,it arrives at

where we have used the equalities

Thus,submitting(2.9)–(2.11)into(2.8),it arrives at

With the help of(2.13)and the Gronwall’s inequality,we immediately have

This,together with(2.6),gives

for some constant C>1 that depends on C0,m(0),A and γ.Since for k≤n−1 and n≥1,it holds that

thus(2.14)implies Tm<+∞,otherwise,there is a contradiction to(2.14).

Step 2The case of J<0.According to Lemma 2.3,the polynomial|A(t)|can be expressed as

where l.o.t denotes the lower order terms in the polynomial.When J<0,for any n≥1 and t>0 large enough,we have|A(t)|≤0,then m(t)=0,which contradicts to the conservation of mass in Lemma 2.2,thus Tm<+∞.

The proof of Theorem 1.1 is finished.

2.3 Blow-up of Euler-Boltzmann equations

Proof of Theorem 1.2We divide the proof into two steps.

Step 1When J<0,the proof is the same as the case of the Euler system,see Step 2 in the proof of Theorem 1.1.This implies that the radiation has no effect when J<0.

Step 2When J=0,we need to consider the behavior of radiation effect.First,one knows that

in which,by using(1.5)2and the fact that ρ=0 on∂A(t),one has

Similarly,with the help of(1.5)3and integrating by parts,one has

and

for all t∈[0,Tm).

Now,we need to consider the radiation effect.

We first claim the following:In the multi-dimensional space,the assumption on I0in(1.14)results in the phenomena that the impact of radiation on dynamical properties of the fluid in A(t)vanishes after some time Tb,i.e.,for n≥2,one has

Noticing that,if Tm≤Tb,then Tmis finite and Theorem 1.2 follows immediately,thus we only consider the case that Tb

Second,combining the claim(2.20)with(2.18)–(2.19),it arrives at

Submitting(2.17)and(2.21)into(2.16),one has

With the help of(2.23)and the Gronwall’s inequality,one has

where H(Tb)is independent of Tm.This,together with(2.6)in Lemma 2.5,gives

for some constant C>1 that depends on C0,m(0),A,γ and Tb.Then for 1<γ≤1+and any nonnegative integer k≤n−1,(2.24)implies Tm<+∞.

Now it remains to prove the claim(2.20)to finish the proof of Theorem 1.2,which will be proved in the following lemma.

Lemma 2.6(Behavior of Radiation Effect)Let n≥2.For the regular solution to the Cauchy problem(1.5)and(1.7),there exists a time Tb<+∞,such that

ProofSinceis independent of t and x,for n≥2,according to(1.5)1,one has

If we denote by y(t;y0)the photon path starting from y0at t=0,i.e.,

then it is easy to show that y0=y−cΩt.According to(2.25),along the photon path,we have

Considering the photon positioned in A(t),since the light speed c>|u0|L∞(Rn),thus taking

and combining with(2.2),A0⊂BR0and|Ω|=1 for n≥2,one has

which implies that after the time Tb,the photon positioned in A(t)comes from BCR0.Thus,together with(2.26)and the assumption on I0in(1.14),it arrives at

The proof of this lemma is completed.

Remark 2.1We emphasize that for 1D case,the Euler-Boltzmann system(see[3,6,17])is deduced from the multi-dimensional case by considering only one single space variable.More precisely,one can consider the three-dimensional case with specific radiation intensity I=I(v,Ω,t,x)that depends only on the single spatial coordinate x3and the single angular coordinate φ,the angle between Ω and x3axis.Introducing ω=cos φ,since I=I(v,ω,t,x3),we have

So,the one-dimensional radiation hydrodynamics equations read as(see[17,24])

where ω∈S0=[−1,1]stands for the angular variable(we emphasize here that|ω|is not just equal to 1,which is different from the multi-dimensional case,where|Ω|=1 for n≥2).

Moreover,according to(2.29)1,under the assumptions in this paper,the radiative effect in A(t)will never disappear,in fact,due to|ω|≤1,we are not able to find a uniform time Tbsuch that(2.28)holds.Thus the radiation related terms on right-hand side of(2.18)–(2.19)will never disappear,and we are not able to deduce finite time blow-up for 1D case under the framework in this paper.

Remark 2.2We also emphasize that,for 1D space,if we denote A0=(a1,a2),B0=(b1,b2)for some constants b1

Compared with the two blow-up results in[19],the first one needs u0(x)=0 on(b1,a1]∪[a2,b2),which plays an important role in proving the finite time blow-up of regular solutions to(2.29),and is much stronger than our condition(2.30);The second one needs Sp(∇u0)∩R−∅in the vacuum domain,which is equivalent to(u0)x<0 on(b1,a1]∪[a2,b2),here,we only need the condition on the boundary points a1,a2.

Remark 2.3Based on Theorems 1.1–1.2,a natural question that we are working on is to consider whether there exists a global regular solution to the Euler system or the Euler-Boltzmann system for all dimensions n≥1 when J>0,as well as to the Euler-Boltzmann system for dimension n=1 when J=0.

AcknowledgementThe authors sincerely appreciate Dr.Shengguo Zhu for his very helpful suggestions and discussions.