Lagrangian formulation and a priori estimates for relativistic fluid flows with vacuum
aa r X i v : . [ m a t h . A P ] N ov LAGRANGIAN FORMULATION AND A PRIORI ESTIMATESFOR RELATIVISTIC FLUID FLOWS WITH VACUUM
JUHI JANG , PHILIPPE G. L E FLOCH , AND NADER MASMOUDI Abstract.
We study the evolution of a compressible fluid surrounded by vac-uum and introduce a new symmetrization in Lagrangian coordinates that al-lows us to encompass both relativistic and non-relativistic fluid flows. Theproblem under consideration is a free boundary problem of central interest incompressible fluid dynamics and, from the mathematical standpoint, the mainchallenge to be overcome lies in the loss of regularity in the fluid variablesnear the free boundary. Based on our Lagrangian formulation, we establishthe necessary a priori estimates in weighted Sobolev spaces which are adaptedto this loss of regularity. Introduction
We study here the Euler equations describing the evolution of a relativistic com-pressible fluid, that is, the system ∂ t (cid:0)e ρ − ǫ p ( ρ ) (cid:1) + div( e ρ u ) = 0 ,∂ t (cid:0)e ρ u (cid:1) + div (cid:0)e ρ u ⊗ u ) + grad (cid:0) p ( ρ ) (cid:1) = 0 , (1.1)in which the mass density ρ = ρ ( t, x ) and the velocity vector of the fluid u = u ( t, x )(with t ≥ x ∈ R ) are the main unknowns and satisfy the physical constraints ρ ≥ , | u | < /ǫ. (1.2)The parameter 1 /ǫ represents the light speed and, in (1.1), the pressure p = p ( ρ )is a given function of the density, while the “modified density” e ρ is defined by e ρ = e ρ ( ρ, u ) := ρ + ǫ p − ǫ | u | . (1.3)We also set x = ( x i ) ≤ i ≤ and use the standard notation for the divergence div u := P i ∂ x i u i = ∂ i u i (with implicit summation on i ) and for the gradient grad( p ) = (cid:0) ∂ i p (cid:1) ≤ i ≤ .Under the standard physical assumption that p ′ ( ρ ) ≥ ρ = 0), the Euler equations (1.1) away from the vacuum state form astrictly hyperbolic system of four conservation laws, which, however, is non-strictlyhyperbolic at the vacuum ρ = 0. We are interested in the evolution of a compressible Department of Mathematics, University of Southern California, Los Angeles CA 90058,USA. Email: [email protected]. Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Universit´ePierre et Marie Curie, 4 Place Jussieu, 75252 Paris, France. Email: contact@philippelefloch.org. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, NY 10011New York, USA. Email: [email protected]
AMS Subject Classification.
Primary: 35L65. Secondary: 76L05, 76N.
Keywords. Relativis-tic fluid, vacuum state, free boundary, Lagrangian formulation, weighted energy . E FLOCH, AND N. MASMOUDI fluid region surrounded by vacuum, in particular when a fluid is continuously incontact with vacuum. This is a classical problem in fluid dynamics and, from themathematical standpoint, the main technical challenge to be overcome lies in theloss of regularity in the fluid variables near the free boundary between the fluidand the vacuum region. Specifically, we require that the normal acceleration of thefluid near the boundary is non-vanishing and bounded: C ≤ | ∂ ν p ′ ( ρ ) | ≤ C (1.4)for some constants 0 < C ≤ C < + ∞ , where ν ∈ R denotes the normal unitvector to the free fluid-vacuum boundary. This vacuum boundary condition, theso-called “physical vacuum” boundary, can be realized by some self-similar solutionsand stationary solutions for different physical systems such as Euler equations withdamping and Euler-Poisson systems for gaseous stars [4, 6, 12, 13, 15].Let us mention several earlier works on the above problem which attracted a lotof attention in recent years. Coutand and Shkoller [1, 2] successfully establishedan existence result for non-relativistic compressible fluids (that is, the system (1.1)with ǫ = 0) by degenerate parabolic regularizations, while, independently, Jang andMasmoudi developed a hyperbolic-type weighted energy estimates for all spatialderivatives including normal derivatives in order to prove the existence of solutionsin one space dimension [5] as well as in several space dimensions [7]. We alsomention that in a recent work [13], Makino addressed some existence result for theEuler-Poisson system based on the Nash-Moser-Hamilton theory. We refer to [6, 7]for a historical background and a bibliography on the subject.As far as relativistic fluids are concerned, earlier investigations on compactlysupported solutions to the relativistic Euler equations include Makino and Ukai[14] and, for the equations in several space variables, LeFloch and Ukai [11]. Inthese works, a stronger regularity property is implied on the fluid variables nearthe free boundary between the fluid and the vacuum. A general existence theoryfor relativistic compressible fluids encompassing the above vacuum condition (1.4)is therefore still lacking.The goal of this article is to present a new Lagrangian formulation of the rela-tivistic Euler equations and is to derive the necessary a priori bounds satisfied bysolutions subject to (1.4) based on such our formulation.An outline of the paper is as follows. In Section 2, we present the compressiblefluid flow equations, discuss its reduction to a second–order hyperbolic system inLagrangian variables, and also derive the relativistic vorticity equation. In Sec-tion 3, we introduce the free boundary problem of interest and present the a prioriestimates. Section 4 includes discussion on non-relativistic flows as well as theexistence theory for special cases such as radially symmetric flows.2. Lagrangian formulation for relativistic fluid flows
Equations of state.
The Euler equations in Minkowski spacetime read ∂ t (cid:16) ρ + ǫ p − ǫ | u | − ǫ p (cid:17) + ∂ k (cid:16) ρ + ǫ p − ǫ | u | u k (cid:17) = 0 ,∂ t (cid:16) ρ + ǫ p − ǫ | u | u j (cid:17) + ∂ k (cid:16) ρ + ǫ p − ǫ | u | u j u k + p δ jk (cid:17) = 0 , (2.1) PRIORI ESTIMATES FOR RELATIVISTIC FLUID FLOWS WITH VACUUM 3 where ( ρ, u ) : [0 , T ] × R → R + × R is the main unknown, defined on some timeinterval [0 , T ). As pointed out in the introduction, it is convenient to introduce thevariable e ρ , so that the Euler equations read ∂ t (cid:0)e ρ − ǫ p (cid:1) + ∂ k (cid:0)e ρ u k (cid:1) = 0 ,∂ t (cid:0)e ρ u j (cid:1) + ∂ k (cid:0)e ρ u j u k + p δ jk (cid:1) = 0 . (2.2)Observe that, by letting formally ǫ → e ρ → ρ and we recover thenon-relativistic Euler equations.As it is required by the physics of the problem, the sound speed c ( ρ ) := p ′ ( ρ ) / is assumed to be real and smaller than the light speed, that is,0 < c ( ρ ) < ǫ − provided ρ > . (2.3)For concreteness, the pressure p is assumed to be a power-law of the particle number N , that is, p = aγ − N γ , where N is related to the energy density ρ = N + ǫ aγ − N γ and γ ∈ (1 ,
2) is a constant referred to as the adiabatic exponent of gases and a > ρ = ǫ p + κ p /γ , κ γ := γ − a . It is easily checked that the sound speed does not exceed the light speed, since c = p ′ ( ρ ) = 1 ǫ + γ − aγ N − γ ≤ ǫ . We also define the function h = h ( ρ ) by dh := 1 N dp and, from now on, adopt the normalization a := γ −
1, so that the equation of stateof the fluid finally reads p ( ρ ) = N γ , h ( ρ ) = γγ − N γ − , with ρ = N + ǫ N γ , (2.4)where γ ∈ (1 , The energy equation and the number density equation.
We recall theenergy pair (
V, H ) associated with the relativistic Euler equations ∂ t V + ∂ j H j = 0 , (2.5)with V := ǫ − (cid:18) (1 + κǫ )( e ρ − ǫ p ) − N ( ρ )(1 − ǫ | u | ) / (cid:19) ,H j := ǫ − (cid:18) (1 + κǫ ) e ρ u j − N ( ρ ) u j (1 − ǫ | u | ) / (cid:19) , where N ( ρ ) := exp (cid:18)Z ρ dss + ǫ p ( s ) (cid:19) , κ := Z p ( s ) s ds. J. JANG, P.G. L E FLOCH, AND N. MASMOUDI
Here the function N is determined so that, as ǫ →
0, the pair (
V, H ) tends to thestandard energy pair of the non-relativistic Euler equations. Indeed, as ǫ → V ∼ ρ | u | + ρ Z ρ p ( s ) s ds,H j ∼ u j (cid:0) ρ V + p ( ρ ) (cid:1) . It can be checked that the above function is strictly convex [14] in the conservativevariable ω = ( ρ, ρu ), and ∇ ω V ≥ C away from the vacuum,where the constant C is uniform on every compact subset of (cid:8) ρ > , | u | < ǫ (cid:9) ,excluding therefore the vacuum.From the mass density equation in (2.1) and the energy equation (2.5), we deducethat any solution ( ρ, u ) : R + × R → R + × R to the Euler equations (2.1) alsosatisfies the following number density equation ∂ t g + ∂ j (cid:0) g u j (cid:1) = 0 ,g := N Γ , Γ = Γ( u ) := (1 − ǫ | u | ) − / . (2.6)In the following we will work with (2.6) together the second equation in (2.2). Notethat the Cauchy problem is posed by prescribing, at the initial time t = 0, theinitial density ρ and the initial velocity u of the fluid ρ (0 , x ) = ρ ( x ) , u (0 , x ) = u ( x ) , x ∈ R (2.7)with, of course, ρ ≥ , | u | < /ǫ. (2.8)We are interested in the situation where the density is positive in some smoothopen set Ω ⊂ R and vanishes identically outside this set, i.e. ρ ( > , x ∈ Ω , = 0 , x ∈ R \ Ω . (2.9)2.3. Lagrangian coordinates and notation.
We are going to now reformulatethe fluid equations above in terms of the
Lagrangian coordinates η j = η j ( t, x )defined by the following ordinary differential equation with prescribed initial data: ∂ t η j := u j ( t, η ) ,η j (0 , x ) := x j . (2.10)We introduce the Jacobian matrix of this transformation, that is, A − := (cid:0) ∇ x η (cid:1) = (cid:0) ∂ i η j (cid:1) and J := det (cid:0) ∇ x η (cid:1) . We use Einstein’s summation convention and thenotation F, k to denote the k -th partial derivative of F : ∂ k F . Both expressions willbe used throughout the paper. Differentiating the inverse of deformation tensor,since A · [ Dη ] = I , one obtains ∂ t A ki = − A kr ∂ t η r , s A si ; ∂ l A ki = − A kr ∂ l η r , s A si . (2.11)Differentiating the Jacobian determinant, one obtains ∂ t J = JA sr ∂ t η r , s ; ∂ l J = JA sr ∂ l η r , s . (2.12)For the cofactor matrix JA , from (2.11) and (2.12), one obtains the following Piolaidentity: ( JA ki ) , k = 0 . (2.13) PRIORI ESTIMATES FOR RELATIVISTIC FLUID FLOWS WITH VACUUM 5
For a given vector field F , we use DF , div F , curl F to denote its full gradient,its divergence, and its curl: [ DF ] ij ≡ F i , j div F ≡ F r , r [curl F ] i ≡ ǫ ijk F k , j where ǫ ijk is the Levi-Civita symbol: it is 1 if ( i, j, k ) is an even permutation of(1 , , i, j, k ) is an odd permutation of (1 , , η :[ D η F ] ir ≡ A sr F i , s div η F ≡ A sr F r , s [curl η F ] i ≡ ǫ ijk A sj F k , s which indeed correspond to Eulerian full gradient, Eulerian divergence, and Euler-ian curl written in Lagrangian coordinates. In addition, it is convenient to introducethe anti-symmetric curl matrix Curl η F :[Curl η F ] ij ≡ A sj F i , s − A si F j , s . Note that Curl η F is a matrix version of a vector curl η F and that | Curl η F | =2 | curl η F | holds. We will use both curl η and Curl η . We end this section by recallingthe following property of the Lagrangian curl: if ω k = A rk f, r , curl η ω = 0 . Lagrangian formulation.
By introducing the modified velocity χ j := (1 + ǫ h ) Γ ∂ t η j , we arrive at the following equivalent formulation of the Euler equations ∂ t g + gA kj ∂ k u j = 0 , g ∂ t χ j + A kj ∂ k p = 0 . (2.14)Furthermore, from ∂ t J − JA kj ∂ k u j = 0, we deduce that g := gJ = Γ N J .Therefore, N can be expressed in terms of the main unknown η by N = g Γ J = g J (1 − ǫ | ∂ t η | ) − / . In turn, the second equation in (2.14) reads g ∂ t (cid:16) (1 + ǫ h ) Γ ∂ t η j (cid:17) + JA kj ∂ k N γ = 0 , (2.15)in which the spatial derivative terms take also the form JA kj ∂ k N γ = ∂ k (cid:16) g γ A kj J − γ Γ − γ (cid:17) and, using the expression of Γ, JA kj ∂ k N γ = ∂ k (cid:16) g γ A kj J − γ (cid:17) Γ − γ − g γ A kj J − γ γ Γ − γ − ǫ Γ ∂ t η i ∂ t ∂ k η i . J. JANG, P.G. L E FLOCH, AND N. MASMOUDI
On the other hand, the time derivative in (2.15) takes the form ∂ t (cid:16) (1 + ǫ h ) Γ ∂ t η j (cid:17) = (1 + ǫ h ) Γ ∂ t η j + ∂ t η j (cid:16) (1 + (2 − γ ) ǫ h ) ǫ Γ ∂ t η i ∂ t η i − ( γ − ǫ h Γ ∂ t log J (cid:17) = (cid:16) (1 + ǫ h ) δ ji + (1 + (2 − γ ) ǫ h ) ǫ Γ ∂ t η i ∂ t η j (cid:17) Γ ∂ t η i − ( γ − ǫ h Γ ∂ t η j ∂ t log J. (2.16)The latter term can be expressed as a second-order term in η , by writing ∂ t J = JA ki ∂ k ∂ t η i so that g ( γ − ǫ h Γ ∂ t η j ∂ t log J = γg γ Γ − γ J − γ A ki ∂ k ∂ t η i ∂ t η j . (2.17)Plugging (2.16)–(2.17) in the equation (2.15) and collecting the terms, we thusfind a second-order equation in ηg B ji ∂ t η i + g γ C kij ∂ k ∂ t η i + ∂ k (cid:0) g γ A kj J − γ (cid:1) = 0 , in which the coefficients are given by B ji := (cid:16) (1 + ǫ h ) δ ji + (1 + (2 − γ ) ǫ h ) ǫ Γ ∂ t η i ∂ t η j (cid:17) Γ γ +1 ,C kij := − γǫ Γ J − γ (cid:16) A ki ∂ t η j + A kj ∂ t η i (cid:1) . Finally, by letting g = w α , g γ = w α where α = ( γ − − we arrive at the following second–order formulation in Lagrangian coordinates w α B ji ∂ t η i + w α C kij ∂ k ∂ t η i + ∂ k (cid:0) w α A kj J − /α (cid:1) = 0 . (2.18)Importantly, we have the symmetry property B ji = B ij and C kij = C kji , while B ji ispositive definite.In Lagrangian coordinates, we prescribe the reference density function ρ ≥ g = w α ) so that it is positive in some smoothopen set Ω ⊂ R and vanishes identically outside this set (cf. (2.9) above) and wecan then pose the Cauchy problem of interest by requiring that η (0 , x ) = x, η t (0 , x ) = η ( x ) , x ∈ Ω (2.19)for some data η (which is precisely the velocity data u in (2.19) expressed inLagrangian coordinates).2.5. Relativistic vorticity.
One additional set of equations will be required inour analysis. Observe that the second equation in (2.14) can be rewritten asΓ ∂ t χ j + A kj ∂ k h = 0 . (2.20)Note that this equation allows us to control the spatial derivatives ∂ k h by the timederivative ∂ t χ . By taking the curl of that equation in Lagrangian coordinates, weobtain Curl η (Γ ∂ t χ ) = 0, implyingΓCurl η ∂ t χ + [Curl η , Γ] ∂ t χ = 0 , with [Curl η , Γ] ∂ t χ = A li Γ , l ∂ t χ j − A lj Γ , l ∂ t χ i . PRIORI ESTIMATES FOR RELATIVISTIC FLUID FLOWS WITH VACUUM 7
Since [ ∂ t , Curl η ] χ = ∂ t A li χ j , l − ∂ t A lj χ i , l , this equation can be written as ∂ t Curl η χ = [ ∂ t , Curl η ] χ − Γ − [Curl η , Γ] ∂ t χ, (2.21)which we refer to as the Lagrangian relativistic vorticity equation.
By integrating (2.21) in time, we deduce thatCurl η χ = Curl η χ (cid:12)(cid:12) t =0 + Z t [ ∂ t , Curl η ] χ ds − Z t Γ − [Curl η , Γ] ∂ t χ ds. (2.22)For the purpose of the energy estimates, we will need to derive the equation for thecurl of η and estimate them. From the definition of χ , we see thatCurl η χ = Γ(1 + ǫ h )Curl η ∂ t η + [Curl η , Γ(1 + ǫ h )] ∂ t η. Hence, (2.22) reads asCurl η ∂ t η + [Γ(1 + ǫ h )] − [Curl η , Γ(1 + ǫ h )] ∂ t η = [Γ(1 + ǫ h )] − (cid:16) Curl η χ (cid:12)(cid:12) t =0 + Z t [ ∂ t , Curl η ] χ ds − Z t Γ − [Curl η , Γ] ∂ t χ ds (cid:17) . (2.23)We observe that the boxed term in (2.23) is not of lower order. By using (2.20),we rewrite it so that it does not contain two spatial derivatives of η := [Γ(1 + ǫ h )] − (Γ , l (1 + ǫ h ) + Γ ǫ h, l )( A li ∂ t η j − A lj ∂ t η i )= ǫ Γ ∂ t η m ∂ t η m , l (cid:0) A li ∂ t η j − A lj ∂ t η i (cid:1) − ǫ Γ(1 + ǫ h ) − ( ∂ t χ i ∂ t η j − ∂ t χ j ∂ t η i )= ǫ Γ ( ∂ t η m ∂ t η m , l (cid:0) A li ∂ t η j − A lj ∂ t η i (cid:1) − ∂ t η i ∂ t η j + ∂ t η j ∂ t η i ) . By rearranging terms, we write the curl equation (2.23) as[D η ∂ t η ] mi ( δ jm + ǫ Γ ∂ t η j ∂ t η m ) − ( δ mi + ǫ Γ ∂ t η i ∂ t η m )[D η ∂ t η ] mj + ǫ Γ ( ∂ t η j ∂ t η i − ∂ t η i ∂ t η j )= [Γ(1 + ǫ h )] − h Curl η χ (cid:12)(cid:12) t =0 + Z t [ ∂ t , Curl η ] χ ds − Z t Γ − [Curl η , Γ] ∂ t χ ds i ji . (2.24)We next define the symmetric matrix S jm := ( δ jm + ǫ Γ ∂ t η j ∂ t η m )and the anti-symmetric matrices R ji := ǫ Γ ( ∂ t η j ∂ t η i − ∂ t η i ∂ t η j ) X ji := [Γ(1 + ǫ h )] − h Curl η χ (cid:12)(cid:12) t =0 + Z t [ ∂ t , Curl η ] χ ds − Z t Γ − [Curl η , Γ] ∂ t χ ds i ji Then, (2.24) can be written as[D η ∂ t η ] mi S jm − S mi [D η ∂ t η ] mj + R ji = X ji (2.25)Notice that S is symmetric and positive definite, hence, letting U := S − , weget the following equivalent expression to the curl equation (2.25): U mi [D η ∂ t η ] jm − [D η ∂ t η ] im U jm + U mi R lm U jl = U mi X lm U jl . J. JANG, P.G. L E FLOCH, AND N. MASMOUDI
Relativistic Euler equations as a second-order hyperbolic system.
Sofar, we have reformulated the relativistic Euler equations as a second-order quasi-linear hyperbolic system in Lagrangian coordinates, where η = ( η j ( t, x )) ∈ R is the main unknown, and have identified the corresponding curl structure. Wesummarize such formulations in the following proposition. Proposition 2.1.
Suppose ( ρ, u ) are smooth solutions to relativistic Euler equa-tions (2.1) written in Eulerian coordinates. Let w α = N Γ , where N = N ( ρ ) isthe initial particle number density determined by (2.4) and Γ = (1 − ǫ | u | ) − / .Then the solution η to the ODE (2.10) satisfies the following second-order quasi-linear hyperbolic system w α B ji ∂ t η i + w α +1 C kij ∂ k ∂ t η i + ∂ k (cid:0) w α +1 A kj J − /α (cid:1) = 0 , (2.26) where B ji = (cid:16) (1 + ǫ h ) δ ji + (1 + (1 − α ) ǫ h ) ǫ Γ ∂ t η i ∂ t η j (cid:17) Γ /α ,C kij = − (1 + α ) ǫ Γ J − /α (cid:16) A ki ∂ t η j + A kj ∂ t η i (cid:1) , (2.27) and furthermore, admits the following structure U mi [ D η ∂ t η ] jm − [ D η ∂ t η ] im U jm + U mi R lm U jl = U mi X lm U jl , (2.28) where U ji = ( S − ) ji where S jm = ( δ jm + ǫ Γ ∂ t η j ∂ t η m ) ,R ji = ǫ Γ ( ∂ t η j ∂ t η i − ∂ t η i ∂ t η j ) ,X ji = [Γ(1 + ǫ h )] − h Curl η χ (cid:12)(cid:12) t =0 + Z t [ ∂ t , Curl η ] χ ds − Z t Γ − [ Curl η , Γ] ∂ t χ ds i ji . (2.29) Here we recall that χ j = (1 + ǫ h ) Γ ∂ t η j , h = (1 + α ) w (Γ J ) − /α , Γ = (1 − ǫ | ∂ t η | ) − / . (2.30) Conversely, if ( η, η t ) (with J being bounded away from zero and above) are smoothsolutions to the above system, ( ρ, u ) is a solution to the Eulerian system. We observe that this proposition can be justified at least away from vacuum,where smooth solutions are available in Eulerian coordinates; for instance, see [11,14].In the next section, based on the above reformulation (2.26)–(2.30) of the rel-ativistic Euler equations in Lagrangian coordinates, we will establish the a prioriestimates for smooth solutions in the presence of a physical vacuum.3.
The free boundary problem for the relativistic Euler system
Main result.
In this section, we consider a vacuum free boundary problemfor relativistic Euler equations in Lagrangian coordinates. We first prescribe a classof w : w is the prescribed function in Ω with smooth boundary ∂ Ω and it vanishesat the boundary like a distance function: w = 0 on ∂ Ω ,C d ( x, ∂ Ω) ≤ w ≤ C d ( x, ∂ Ω) . (3.1)The regularity for w will be specified in the next subsection. (See (3.3).) PRIORI ESTIMATES FOR RELATIVISTIC FLUID FLOWS WITH VACUUM 9
We can pose the Cauchy problem of interest by requiring that η (0 , x ) = η ( x ) , η t (0 , x ) = η ( x ) , x ∈ Ω (3.2)for some given data η , η . Note that due to degeneracy of w we indeed do not needto impose the boundary condition on ∂ Ω. We are interested in the free boundaryvalue problem associated with the noninear hyperbolic systems (2.26), that is, wesearch for solutions that are supported in a domain Ω with smooth boundary ∂ Ω.It is a moving boundary value problem because η t , the velocity of fluids, is notnecessarily zero along the boundary, and the moving domain in Eulerian coordinatesis given by Ω( t ) = η ( t )(Ω).Observe that the condition imposed near the boundary is singular in nature andspecial care will be required to handle derivatives of η , especially in the directionnormal to the boundary.For simplicity of the presentation, we consider the case when the initial domainis taken as Ω = T × (0 , , where T is a two-dimensional period box in x , x . The result can be extended tothe general case in the same way as done in [7]. The initial boundary is given as ∂ Ω = { x = 0 } ∪ { x = 1 } as the reference vacuum boundary.We use Latin letters i, j, k, . . . to denote 1 , , β, κ, σ, τ to denote 1 ,
2, only. We use ∂ mτ to denote ∂ m ∂ m and | m | to denote | m | = m + m .To any sufficiently regular function η defined on [0 , T ] × Ω, we associate the followingenergy functionals (defined for any integer N ≥ E ( I ) N := X | m | + n ≤ N Z Ω w α + n ∂ mτ ∂ n η jt B ji ∂ mτ ∂ n η it dx =: X | m | + n ≤ N E ( I ) m,n ,E ( II ) N := X | m | + n ≤ N Z Ω w α + n +1 J − /α | div η ∂ mτ ∂ n η | dx =: X | m | + n ≤ N E ( II ) m,n ,E ( III ) N := X | m | + n ≤ N Z Ω w α + n +1 [ D η ∂ mτ ∂ n η ] jm U im [ D η ∂ mτ ∂ n η ] ji dx =: X | m | + n ≤ N E ( III ) m,n ,E ( IV ) N := X | m | + n ≤ N Z Ω w α + n +1 | ∂ mτ ∂ n Curl η χ | dx. Note that E ( II ) N is bounded by E ( III ) N . The total energy of interest is the sum E N = E ( I ) N + E ( III ) N + E ( IV ) N . Furthermore, the regularity of the weight function w is determined by introduc-ing the norms: F M [ w ] := X | m | + n ≤ M Z Ω w α + n +1 | ∂ mτ ∂ n w | dx,F ( I ) M [ w ] := X | m | + n ≤ M Z Ω w α + n +1 | D∂ mτ ∂ n w | dx. (3.3)We now state the result on the a priori estimates for solutions of (2.26) in theabove energy spaces. E FLOCH, AND N. MASMOUDI
Theorem 3.1 (A priori estimates) . Let N ≥ α + 9 be fixed for given exponent α > and let w be given satisfying (3.1) and F N [ Dw ] < ∞ . Suppose η and η t solve (2.26) for t ∈ [0 , T ] with E N = E N ( η, η t ) < ∞ and /C ≤ J ≤ C for some C ≥ for the initial data η , η : R → R in (3.2) satisfying E N [ η , η ] < ∞ . We furtherassume that η and η t enjoy the a priori bound: for any s = 1 , , and , [ N/ X | p | + q =0 | w q/ ∂ pτ ∂ q η r , s | + [ N/ − X | p | + q =0 | w q/ ∂ pτ ∂ q η rt , s | < ∞ . (3.4) Then we obtain the following a priori estimates: ddt (cid:20) E ( I ) m,n + (1 + 1 α ) E ( II ) m,n (cid:21) ≤ F (cid:16) E ( I ) N , E ( III ) N (cid:17) for | m | < N,ddt (cid:20) E ( I ) N, + (1 + 1 α ) E ( II ) N, + G (cid:21) ≤ F (cid:16) E ( I ) N , E ( III ) N (cid:17) for | m | = N, (3.5) where for any δ > | G | ≤ δ E ( III ) N, + C δ E ( III ) N − , as well as E ( III ) N ≤ F (cid:16) E N [ η , η ] , E ( I ) N , E ( III ) N , T (cid:17) ,E ( IV ) N ≤ E ( IV ) N [ η , η ] + F (cid:16) E ( I ) N , E ( III ) N , T (cid:17) , (3.6) where F , F and F are smooth functions in their arguments. Moreover, the apriori assumption (3.4) can be justified. The proof of Theorem 3.1 is a direct consequence of the following two lemmas.
Lemma 3.2 (Energy estimates) . Under the assumptions of Theorem 3.1, one hasthe energy inequality (3.5) . Lemma 3.3 (Gradient and curl estimates) . Under the assumptions of Theorem 3.1,one obtains the energy bounds (3.6) . The structure exhibited by the second-order system and the curl system (2.26)–(2.30) is fundamental in order to derive the necessary estimates. The first energyinequality is a consequence of the wave-like structure of the second-order system,while the second energy bounds will follow from the vorticity equations. We observethat the energy functionals incorporate suitable powers of the weight function w .Higher powers are required for normal derivatives, while no such loss is encounteredfor tangential derivatives. The same algebraic structure of the change in weightshas been identified for the non-relativistic flows in [7].While there is some similarity to the proof for the non-relativistic Euler flows asdone in [7], our proof here involves some new ingredients and the estimates are notthe same. The paper [7] derived the estimates for ∂ t η and the full gradient of η fromthe energy estimates of the second-order hyperbolic system at the expense of loosingthe positivity of the curl part in the energy and the method therein compensated alost curl energy by an auxiliary estimate from the curl equation. The curl equationfor the non-relativistic Euler flows is rather simple and elegant (i.e. almost an ODE)in Lagrangian coordinates, which is one of key ingredients used in [7], but such asimple structure does not seem to be available for the relativistic Euler equations.To get around this difficulty present for the relativistic Euler flows even at the PRIORI ESTIMATES FOR RELATIVISTIC FLUID FLOWS WITH VACUUM 11 formal level, we obtain the estimates for ∂ t η and the divergence of η via the energyestimates at the expense of loosing the positivity of the full tangential derivativeterms and recover the full gradient estimates from the relativistic vorticity equation.This new scheme is applied to the non-relativistic Euler equations and it gives analternative way of deriving the estimates.We observe that Theorem 3.1 can be extended to a larger class of quasilinear hy-perbolic systems inheriting the same leading-order structure as in (2.26) and (2.28),so long as the coefficient matrices and tensors satisfy suitable algebraic conditionssuch as symmetry, anti-symmetry, and positive definiteness. For instance, takinginto account lower-order forcing term such as a gravitational coupling or dampingterms would not add any further difficulty at this level.3.2. Hardy inequality and embedding of weighted Sobolev spaces.
Beforewe derive the energy estimates, we recall the following useful Hardy inequality andembedding results. First of all, for the Hardy inequality we have the following [9].
Lemma 3.4. (Hardy inequality) Let k be a real number and g a function satisfying R s k ( g + g ′ ) ds < ∞ .If k > , then we have R s k − g ds ≤ C R s k ( g + | g ′ | ) ds .If k < , then g has a trace at x = 0 and R s k − ( g − g (0)) ds ≤ C R s k | g ′ | ds . Note that using Lemma 3.4 with k = α + 1, we get Z Ω w α − | v | dx ≤ C Z Ω [ w α +1 | ∂ v | + w α +1 | v | ] dx. (3.7)We will also use the following variant of Hardy inequality: for any fixed δ > Z Ω w α − | v | dx ≤ δ Z Ω w α +1 | ∂ v | dx + C δ Z Ω w α +1 | v | dx. (3.8)The above energy functionals induce a family of weighted Sobolev spaces. It isconvenient to introduce the function spaces X α,b , Y α,b , Z α,b to discuss the embed-ding results: X α,b ≡ { w α F ∈ L (Ω) : Z Ω w α + n | ∂ mτ ∂ n F | dx < ∞ , ≤ | m | + n ≤ b } ,Y α,b ≡ { w α D η F ∈ L (Ω) : Z Ω w α + n | D η ∂ mτ ∂ n F | dx < ∞ , ≤ | m | + n ≤ b } ,Z α,b ≡ { w α F ∈ L (Ω) : Z Ω w α + n | ∂ mτ ∂ n F | dx < ∞ , ≤ | m | + n ≤ b } . (3.9)Then as an application of the Hardy type embedding of weighted Sobolev spaces[9], we obtain the embedding of X α,b , Y α,b , Z α,b into the standard Sobolev spaces H s for sufficiently smooth w . Lemma 3.5.
For b ≥ ⌈ α ⌉ , k F k H b − α - k F k X α,b . In particular, for b ≥ [ α ] + 4 , k F k ∞ - k F k X α,b . E FLOCH, AND N. MASMOUDI
We have the similar embeddings for Y α,b and Z α,b : for b ≥ ⌈ α ⌉ + 1 , k DF k H b − α − - k F k Y α,b and k F k H b − α − - k F k Z α,b . We observe that the a priori bound in (3.4) in Theorem 3.1 can be justified inour energy function spaces by using Lemma 3.4 and Lemma 3.5, in other words | w q/ ∂ pτ ∂ q η r , s | and | w q/ ∂ pτ ∂ q η rt , s | for 0 ≤ | p | + q ≤ [ N/
2] are bounded by E N .The remaining part of this section is devoted to the proof of Lemma 3.2 andLemma 3.3.3.3. Proof of Lemma 3.2.
The energy inequality (3.5) is due to the symmetricstructure of the reformulation (2.26). While there is some similarity to the prooffor non-relativistic Euler as done in [7], our proof here is not the same. Unlike in[7], we will not keep the precise curl structure at the level of the energy estimates,but aim to control the divergence part only at this point. Then the full energy willbe recovered by exploiting the curl equations. We notice that the obvious differencelies in that B ji is a symmetric positive definite matrix and C kij is a symmetric tensorfor the current case, while B ji = δ ji and C kij = 0 for the non-relativistic Euler case.The proof consists of three steps: the zeroth order estimate, the derivation ofhigh order equations, and the high order estimates. Let us start with the zerothorder estimate. Step 1 - the zeroth order estimate:
Multiply (2.26) by η jt and integrate to get Z Ω w α η jt B ji η itt dx + Z Ω w α +1 η jt C kij ∂ k η it dx + Z Ω η jt ∂ k (cid:0) w α +1 A kj J − /α (cid:1) dx = 0 . The first and second terms can be written as Z Ω w α η jt B ji η itt dx = 12 ddt Z Ω w α η jt B ji η it dx − Z Ω w α η jt ∂ t B ji η it dx Z Ω w α +1 η jt C kij ∂ k η it dx = − Z Ω η jt ∂ k ( w α +1 C kij ) η it dx by using the symmetry relation B ji = B ij and C kij = C kji . The third term can bewritten as Z Ω η jt ∂ k (cid:0) w α +1 A kj J − /α (cid:1) dx = ddt Z Ω αw α +1 J − /α dx by using (2.12). Thus (3.5) is valid for m = 0 and n = 0 in the energy. Step 2 - the derivation of high order equations:
Let m and n for 1 ≤ | m | + n ≤ N be fixed. Taking ∂ mτ ∂ n of w − α · (2.26) and by multiplying it back by w α + n , wefirst obtain w α + n B ji ∂ mτ ∂ n η itt + X | p | + q< | m | + n c p,q w α + n ∂ m − pτ ∂ n − q B ji ∂ pτ ∂ q η itt + w α + n C kij ∂ k ∂ mτ ∂ n η it + X | p | + q< | m | + n c p,q w α + n ∂ m − pτ ∂ n − q (cid:2) wC kij (cid:3) ∂ k ∂ pτ ∂ q η it + w α + n ∂ mτ ∂ n (cid:16) w∂ k ( A kj J − /α ) + (1 + α ) ∂ k wA kj J − /α (cid:17) = 0 (3.10) PRIORI ESTIMATES FOR RELATIVISTIC FLUID FLOWS WITH VACUUM 13
We first claim that the last double-lined term in (3.10) can be written as follows: w α + n ∂ mτ ∂ n (cid:16) w∂ k ( A kj J − /α ) + (1 + α ) ∂ k wA kj J − /α (cid:17) = − (1 + 1 α ) ∂ k (cid:16) w n + α J − α A kj div η ∂ mτ ∂ n η (cid:17) + (1 + α ) w α + n J − α ∂ w ( A j A σr − A r A σj ) ∂ mτ ∂ n η r , σ + w α + n R m,n , (3.11)where R m,n consists of lower order terms: R m,n = R m,n (cid:16) ∂ m − ( p + q ) τ ∂ n − ( i + j )3 w∂ pτ ∂ i D η∂ qτ ∂ j Dη, ∂ m − ( p + q ) τ ∂ n − ( i + j )3 ∂ σ w∂ pτ ∂ i Dη∂ qτ ∂ j Dη,∂ qτ ∂ j w∂ pτ ∂ i Dη∂ m − ( p + q ) τ ∂ n − ( i + j )3 D∂ σ η, ∂ qτ ∂ j Dw∂ pτ ∂ i Dη∂ m − ( p + q ) τ ∂ n − ( i + j )3 Dη ;0 ≤ | p | + i ≤ | m | + n − ≤ | q | + j ≤ | m | + n ; i + j ≤ n ; p + q ≤ m (cid:17) . (3.12)We observe that the structure encoded in (3.11) is different from the one in [7].A new aspect is that instead of looking at the gradient of the full gradient plusdivergence minus curl as suggested by the following identity ∂ l ( A ki J − /α ) = − J − /α A kr [ D η ∂ l η ] ir − α J − /α A ki div η ∂ l η − J − /α A kr [Curl η ∂ l η ] ri we will make use of the structure of the gradient of the divergence. To make itprecise, first note that ∂ l ( A ki J − /α ) = − J − /α A kr A si ∂ l η r , s − α J − /α A ki A sr ∂ l η r , s = − (1 + 1 α ) J − /α A ki div η ∂ l η + J − /α (cid:2) A ki A sr − A kr A si (cid:3) ∂ l η r , s (3.13)and moreover, ∂ l ∂ k ( A ki J − /α ) = − (1 + 1 α ) ∂ k (cid:16) J − /α A ki div η ∂ l η (cid:17) + ∂ k h J − /α A ki A sr − J − /α A kr A si i ∂ l η r , s (3.14)after the cancelation due to the symmetry in k, s : (cid:2) A ki A sr − A kr A si (cid:3) ∂ k ∂ l η r , s = 0. Weobserve that the second term in (3.14) is lower order. Based on (3.13) and (3.14),we will establish the following equivalent expression to (3.11): ∂ mτ ∂ n (cid:16) w∂ k ( A kj J − /α ) + (1 + α ) ∂ k wA kj J − /α (cid:17) = − (1 + 1 α ) h w∂ k (cid:16) J − α A kj div η ∂ mτ ∂ n η (cid:17) + (1 + n + α ) ∂ k wJ − α A kj div η ∂ mτ ∂ n η i + (1 + α ) J − α ∂ w ( A j A σr − A r A σj ) ∂ mτ ∂ n η r , σ + R m,n (3.15)We will present the details for normal derivatives ( m = 0) on how the weightstructure changes and move onto tangential and mixed derivatives. Our first claimis that ∂ n (cid:16) w∂ k ( A kj J − /α ) + (1 + α ) ∂ k wA kj J − /α (cid:17) = − (1 + 1 α ) h w∂ k (cid:16) J − α A kj div η ∂ n η (cid:17) + (1 + n + α ) ∂ k wJ − α A kj div η ∂ n η i + (1 + α ) J − α ∂ w ( A j A σr − A r A σj ) ∂ n η r , σ + R ,n , (3.16) E FLOCH, AND N. MASMOUDI where R ,n consists of lower order terms: for n ≥ R ,n = R ,n ( ∂ n − ( i + j )3 w∂ i D η∂ j Dη, ∂ n − ( i + j )3 ∂ σ w∂ i Dη∂ j Dη,∂ j w∂ i Dη∂ n − ( i + j )3 D∂ σ η, ∂ j Dw∂ i Dη∂ n − ( i + j )3 Dη ;0 ≤ i ≤ n − ≤ j ≤ n ; i + j ≤ n ) (3.17)We will establish (3.16) inductively. ∗ Case of n = 1 in (3.16). Note that ∂ (cid:16) w∂ k ( A kj J − /α ) + (1 + α ) ∂ k wA kj J − /α (cid:17) = w∂ ∂ k ( A kj J − α ) + ∂ w∂ k ( A kj J − α ) + (1 + α ) ∂ k w∂ ( A kj J − α )+ (1 + α ) ∂ ∂ k wA kj J − α (3.18)The second term in the right hand side of (3.18) is not lower order with respect tothe weight. We rewrite it as ∂ w∂ k ( A kj J − /α )= − (1 + 1 α ) J − /α ∂ wA kj A sr ∂ k η r , s = − (1 + 1 α ) J − /α ∂ k wA kj A sr ∂ η r , s +(1 + 1 α ) J − /α ( ∂ σ wA σj A sr ∂ η r , s − ∂ wA σj A sr ∂ σ η r , s )Now the second and third terms in (3.18) together become ∂ w∂ k ( A kj J − /α ) + (1 + α ) ∂ k w∂ ( A kj J − /α )= − (1 + 1 α )(2 + α ) ∂ k wJ − /α A kj div η ∂ η + (1 + α ) J − /α ∂ k w ( A kj A sr − A kr A sj ) ∂ η r , s + (1 + 1 α ) J − /α ( ∂ σ wA σj A sr ∂ η r , s − ∂ wA σj A sr ∂ σ η r , s )but then, the second term in the right hand side when k = 3 reduces to ∂ w ( A j A sr − A r A sj ) ∂ η r , s = ∂ w ( A j A σr − A r A σj ) ∂ η r , σ since when s = 3, A j A r − A r A j = 0.Hence by using (3.14) for the first term in (3.18), we see that (3.18) can be rewrittenas ∂ (cid:16) w∂ k ( A kj J − /α ) + (1 + α ) ∂ k wA kj J − /α (cid:17) = − (1 + 1 α ) h w∂ k (cid:16) J − α A kj div η ∂ η (cid:17) + (2 + α ) ∂ k wJ − /α A kj div η ∂ η i + (1 + α ) J − α ∂ w ( A j A σr − A r A σj ) ∂ η r , σ + R , , (3.19)where R , := w∂ k [ J − /α A ki A sr − J − /α A kr A si ] ∂ η r , s + (1 + α ) J − α ∂ κ w ( A κj A sr − A κr A sj ) ∂ η r , s +(1 + α ) ∂ ∂ k wA kj J − /α + (1 + 1 α ) J − /α (cid:0) ∂ σ wA σj A sr ∂ η r , s − ∂ wA σj A sr ∂ σ η r , s (cid:1) . Note that R , = R , ( wD η∂ Dη, ∂ σ wDη∂ Dη, ∂ DwDη, ∂ wDηD∂ σ η )which consists of lower order terms. This verifies (3.16) for n = 1. PRIORI ESTIMATES FOR RELATIVISTIC FLUID FLOWS WITH VACUUM 15 ∗ Case of n ≥ ∂ of (3.16), wefirst obtain ∂ n +13 (cid:16) w∂ k ( A kj J − /α ) + (1 + α ) ∂ k wA kj J − /α (cid:17) = − (1 + 1 α ) h w∂ ∂ k (cid:16) J − α A kj div η ∂ n η (cid:17) + ∂ w∂ k (cid:16) J − α A kj div η ∂ n η (cid:17)i − (1 + 1 α )(1 + n + α ) h ∂ k w∂ (cid:16) J − α A kj div η ∂ n η (cid:17) + ∂ ∂ k wJ − α A kj div η ∂ n η i + (1 + α ) ∂ (cid:16) J − α ∂ w ( A j A σr − A r A σj ) ∂ n η r , σ (cid:17) + ∂ R ,n We rewrite the first three terms in the right hand side after rearrangement as − (1 + 1 α ) h w∂ k (cid:16) J − α A kj div η ∂ n +13 η (cid:17) + (2 + n + α ) ∂ k w (cid:16) J − α A kj div η ∂ n +13 η (cid:17)i − (1 + 1 α ) h w∂ k (cid:16) ∂ ( J − α A kj A sr ) ∂ n η r , s (cid:17) + ∂ w∂ (cid:16) J − α A j A sr (cid:17) ∂ n η r , s i − (1 + 1 α ) h ∂ w∂ σ (cid:16) J − α A σj div η ∂ n η (cid:17) − ∂ σ wJ − α A σj div η ∂ n +13 η i − (1 + 1 α )(1 + n + α ) ∂ k w∂ (cid:16) J − α A kj A sr (cid:17) ∂ n η r , s , where the first line is the main structural expression. Thus, we see that ∂ n +13 (cid:16) w∂ k ( A kj J − /α ) + (1 + α ) ∂ k wA kj J − /α (cid:17) = − (1 + 1 α ) h w∂ k (cid:16) J − α A kj div η ∂ n +13 η (cid:17) + (2 + n + α ) ∂ k wJ − α A kj div η ∂ n +13 η i + (1 + α ) J − α ∂ w ( A j A σr − A r A σj ) ∂ n +13 η r , σ + R ,n +1 , where R ,n +1 = − (1 + 1 α ) h w∂ k (cid:16) ∂ (cid:16) J − α A kj A sr (cid:17) ∂ n η r , s (cid:17) + ∂ w∂ (cid:16) J − α A j A sr (cid:17) ∂ n η r , s i − (1 + 1 α ) h ∂ w∂ σ (cid:16) J − α A σj div η ∂ n η (cid:17) − ∂ σ wJ − α A σj div η ∂ n +13 η i − (1 + 1 α )(1 + n + α ) h ∂ k w∂ (cid:16) J − α A kj A sr (cid:17) ∂ n η r , s + ∂ ∂ k wJ − α A kj div η ∂ n η i + (1 + α ) ∂ (cid:16) J − α ∂ w ( A j A σr − A r A σj ) (cid:17) ∂ n η r , σ + ∂ R ,n which recovers (3.16) and (3.17) for n + 1.We will now move onto the tangential and mixed derivatives in (3.15). We firstverify (3.15) for n = 0. ∗ Case of | m | ≥ n = 0 in (3.15). Let us start with | m | = 1 and n = 0. ∂ τ (cid:16) w∂ k ( A kj J − /α ) + (1 + α ) ∂ k wA kj J − /α (cid:17) = w∂ τ ∂ k ( A kj J − /α ) + ∂ τ w∂ k ( A kj J − /α ) + (1 + α ) ∂ k w∂ τ ( A kj J − /α )+ (1 + α ) ∂ τ ∂ k wA kj J − /α (3.20) E FLOCH, AND N. MASMOUDI
Then the second term in in the right hand side of (3.20) is indeed lower order since ∂ τ w behaves like w . Hence we can rewrite it as ∂ τ (cid:16) w∂ k ( A kj J − /α ) + (1 + α ) ∂ k wA kj J − /α (cid:17) = − (1 + 1 α ) h w∂ k (cid:16) J − α A ki div η ∂ τ η (cid:17) + (1 + α ) ∂ k wJ − α A ki div η ∂ τ η i + (1 + α ) J − α ∂ w ( A j A σr − A r A σj ) ∂ τ η r , σ + R , , (3.21)where R , = w∂ k [ J − α A ki A sr − J − α A kr A si ] ∂ τ η r , s +(1 + α ) J − α ∂ σ w ( A σj A sr − A σr A sj ) ∂ τ η r , s − (1 + 1 α ) J − /α ∂ τ wA kj A sr ∂ k η r , s +(1 + α ) ∂ τ ∂ k wA kj J − /α (3.22)We observe R , can be put into the following form R , = R , ( wD η∂ τ Dη, ∂ σ wDηD η, ∂ τ DwDη )which consists of essentially lower order terms with respect to the derivatives andweights. One can take more tangential derivatives of (3.20) to obtain ∂ mτ (cid:16) w∂ k ( A kj J − /α ) + (1 + α ) ∂ k wA kj J − /α (cid:17) = − (1 + 1 α ) h w∂ k (cid:16) J − α A ki div η ∂ mτ η (cid:17) + (1 + α ) ∂ k wJ − α A ki div η ∂ mτ η i + (1 + α ) J − α ∂ w ( A j A σr − A r A σj ) ∂ mτ η r , σ + R m, , (3.23)where R m, having the form in (3.12) consists of lower order terms. ∗ Case of | m | ≥ n ≥ ∂ τ consecutively of (3.16). The point is that ∂ τ w behaves like w , unlike theaction of ∂ , the weight structure will not change under ∂ τ . Since the procedure issimilar to the previous cases we omit the details. Step 3 - High order energy estimates:
We will now perform the energy esti-mates for (3.10) for 1 ≤ | m | + n ≤ N . The energy inequality will be obtained bymultiplying (3.10) by ∂ mτ ∂ n η jt and integrating over the domain. We will derive theestimates line by line. • The first line in (3.10). The first term in (3.10) yields the energy term corre-sponding m, n in E ( I ) N plus a commutator term Z Ω w α + n ∂ mτ ∂ n η jt B ji ∂ mτ ∂ n η itt dx = 12 ddt Z Ω w α + n ∂ mτ ∂ n η jt B ji ∂ mτ ∂ n η it dx + R = 12 E ( I ) m,n + R , where R = − Z Ω w α ∂ mτ ∂ n η jt ∂ t B ji ∂ mτ ∂ n η it dx - Z Ω w α + n ∂ mτ ∂ n η jt B ji ∂ mτ ∂ n η it dx since | ∂ t BB − | is bounded due to the a priori bound (3.4). The second term inthe first line of (3.10) yields essentially lower order nonlinear terms since | p | + q < | m | + n . By using (3.4), Lemma 3.4 and Lemma 3.5, one can deduce that thoselower order terms are bounded by a continuous function of E ( I ) N and E ( III ) N . PRIORI ESTIMATES FOR RELATIVISTIC FLUID FLOWS WITH VACUUM 17 • The second line in (3.10). The first term can be written Z Ω w α + n ∂ mτ ∂ n η jt C kij ∂ k ∂ mτ ∂ n η it dx = − Z Ω ∂ mτ ∂ n η jt ∂ k (cid:0) w α + n C kij (cid:1) ∂ mτ ∂ n η it dx (by integration by parts) - Z Ω w α + n ∂ mτ ∂ n η jt B ji ∂ mτ ∂ n η it dx, where the last step is due to (3.4). The second term in the second line is lower-orderwith respect to number of the derivatives and the weight and hence by standard non-linear estimates using (3.4), Lemma 3.4 and Lemma 3.5, we see that it is boundedby a continuous function of E ( I ) N and E ( III ) N . • The third line in (3.10). We will use the expression (3.11). Multiplying (3.11)by ∂ mτ ∂ n η jt and integrating, we have Z Ω w α + n ∂ mτ ∂ n η jt ∂ mτ ∂ n (cid:16) w∂ k ( A kj J − /α ) + (1 + α ) ∂ k wA kj J − /α (cid:17) dx = − (1 + 1 α ) Z Ω ∂ mτ ∂ n η jt ∂ k (cid:16) w n + α J − α A kj div η ∂ mτ ∂ n η (cid:17) dx + (1 + α ) Z Ω w α + n ∂ mτ ∂ n η jt J − α ∂ w ( A j A σr − A r A σj ) ∂ mτ ∂ n η r , σ dx + Z Ω w α + n ∂ mτ ∂ n η jt R m,n dx =: ( I ) + ( II ) + ( III )For ( I ), by integration by parts, we obtain( I ) = (1 + 1 α ) Z Ω ∂ k ∂ mτ ∂ n η jt (cid:16) w n + α J − α A kj div η ∂ η (cid:17) dx = 12 ddt (1 + 1 α ) Z Ω w n + α J − α | div η ∂ mτ ∂ n η | dx −
12 (1 + 1 α ) Z Ω w n + α ∂ t ( J − α ) | div η ∂ mτ ∂ n η | dx − (1 + 1 α ) Z Ω ∂ k ∂ mτ ∂ n η j w n + α J − α ∂ t A kj div η ∂ mτ ∂ n ηdx The first term is the energy term E ( II ) m,n in E ( II ) N and the last two terms are com-mutators. Since J , ∂ t J , ∂ t A are bounded due to (3.4), those commutators arebounded by R Ω w n + α | div η ∂ mτ ∂ n η | dx and R Ω w n + α | D∂ mτ ∂ n η | dx , which are inturn bounded by E ( III ) N . For ( II ), we divide into cases. If n ≥
1, since( II )1 + α = Z Ω w α + n ∂ mτ ∂ n η jt J − α ∂ w ( A j A σr − A r A σj ) w α + n ∂ σ ∂ mτ ∂ n − η r , dx we deduce that it’s bounded by R Ω w n + α | ∂ mτ ∂ n η t | dx and R Ω w n + α | ∂ σ ∂ mτ ∂ n − Dη | dx ,which are in turn bounded by E ( I ) N and E ( III ) N . If n = 0, however, it is not immedi-ate to see that it can be controlled by our energy because it involves full tangential E FLOCH, AND N. MASMOUDI derivatives with only w α weight. For 1 ≤ | m | ≤ N − (cid:12)(cid:12)(cid:12)(cid:12) ( II )1 + α (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z Ω w α ∂ mτ η jt J − α ∂ w ( A j A σr − A r A σj ) w α ∂ mτ η r , σ dx (cid:12)(cid:12)(cid:12)(cid:12) - Z Ω w α | ∂ mτ η t | dx + Z Ω w α | ∂ m +1 τ η | dx - Z Ω w α | ∂ mτ η t | dx + Z Ω w α +2 | ∂ m +1 τ Dη | dx + Z Ω w α +1 | ∂ m +1 τ η | dx - E ( I ) N + E ( III ) N (since | m | ≤ N − , where we have used Hardy inequality (3.7). Now let n = 0 and | m | = N , namely fulltangential derivatives. The previous trick via Hardy inequality would not directlywork for this case. We will aim to show the second inequality in (3.5) with the newterm G . We will write it as two terms first( II )1 + α = Z Ω ∂ mτ η jt w α J − α ∂ wA j A σr ∂ mτ η r , σ dx − Z Ω ∂ mτ η jt w α J − α ∂ wA r A σj ∂ mτ η r , σ dx =: ( II ) − ( II ) (3.24)and rewrite the first term ( II ) by performing integration by parts in time andthen space:( II ) = ddt Z Ω ∂ mτ η j w α ∂ wJ − α A j A σr ∂ mτ η r , σ dx − Z Ω ∂ mτ η j w α ∂ wJ − α A j A σr ∂ mτ η rt , σ dx − Z Ω ∂ mτ η j w α ∂ w∂ t ( J − α A j A σr ) ∂ mτ η r , σ dx = ddt Z Ω ∂ mτ η j w α ∂ wJ − α A j A σr ∂ mτ η r , σ dx + Z Ω ∂ mτ η j , σ w α ∂ wJ − α A j A σr ∂ mτ η rt dx + Z Ω ∂ mτ η j ∂ σ ( w α ∂ wJ − α A j A σr ) ∂ mτ η rt dx − Z Ω ∂ mτ η j w α ∂ w∂ t ( J − α A j A σr ) ∂ mτ η r , σ dx Note that the boxed term is the same as the other term ( II ) in (3.24), so theycancel out. Hence( II )1 + α = ddt Z Ω ∂ mτ η j w α ∂ wJ − α A j A σr ∂ mτ η r , σ dx + Z Ω ∂ mτ η j ∂ σ ( w α ∂ wJ − α A j A σr ) ∂ mτ η rt dx − Z Ω ∂ mτ η j w α ∂ w∂ t ( J − α A j A σr ) ∂ mτ η r , σ dx =: ddt ( i ) + ( ii ) − ( iii ) PRIORI ESTIMATES FOR RELATIVISTIC FLUID FLOWS WITH VACUUM 19
We can now employ the Hardy inequality (3.7) and (3.8) for ( i ), ( ii ) and ( iii ). Wewill present the detail for ( i ). | ( i ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z Ω w α − ∂ mτ η j ∂ wJ − α A j A σr w α +12 ∂ mτ η r , σ dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C θ Z Ω w α − | ∂ mτ η | dx + θ Z Ω w α +1 | ∂ mτ Dη | dx by Cauchy-Swartz ≤ C θ δ Z Ω w α +1 | ∂ mτ Dη | dx + C δ C θ Z Ω w α +1 | ∂ mτ η | dx + θ Z Ω w α +1 | ∂ mτ Dη | dx (by (3.8)).We can choose θ and δ small if necessary. This justifies the existence and estimateof G in the second inequality in (3.5). Estimation of ( ii ) and ( iii ) follows similarlyby Hardy inequality: | ( ii ) | + | ( iii ) | - E ( I ) N + E ( III ) N . For (
III ) containing lower order terms, by using (3.4), Lemma 3.4 and Lemma 3.5,one can deduce that it bounded by a continuous function of E ( I ) N and E ( III ) N . Thisconcludes the proof of Lemma 3.2.3.4. Proof of Lemma 3.3.
Let G = ∂ mτ ∂ n η be given for fixed m and n . By takinga number of derivatives of (2.28), we obtain U ri [D η ∂ t G ] jr − [D η ∂ t G ] ir U jr + ǫ Γ U ri ( ∂ t G l ∂ t η r − ∂ t G r ∂ t η l ) U jl = T m,n , (3.25)where T m,n := ∂ mτ ∂ n h U ri X lr U jl i − X | p | + q ≥ ∂ pτ ∂ q [ U ri A sr ] ∂ m − pτ ∂ n − q ∂ t η j , s − X | p | + q ≥ ∂ pτ ∂ q (cid:2) U jr A sr (cid:3) ∂ m − pτ ∂ n − q ∂ t η i , s − X | p | + q ≥ ∂ pτ ∂ q (cid:2) ǫ Γ U ri U jr ∂ t η r (cid:3) ∂ m − pτ ∂ n − q ∂ t η l + X | p | + q ≥ ∂ pτ ∂ q (cid:2) ǫ Γ U ri U jr ∂ t η l (cid:3) ∂ m − pτ ∂ n − q ∂ t η r (3.26)In turn, we integrate in time (3.25) to get U ri [D η G ] jr − [D η G ] ir U jr + ǫ Γ U ri ( ∂ t G l ∂ t η r − ∂ t G r ∂ t η l ) U jl = S m,n , (3.27)where S m,n consists of lower order terms: S m,n := (cid:16) U ri [D η G ] jr − [D η G ] ir U jr + ǫ Γ U ri ( ∂ t G l ∂ t η r − ∂ t G r ∂ t η l ) U jl (cid:17) (cid:12)(cid:12)(cid:12) t =0 + Z t T m,n dt + Z t ∂ t ( U ri A sr ) G j , s dt − Z t ∂ t ( U jr A sr ) G i , s dt + Z t ∂ t ( ǫ Γ U ri U jr ∂ t η r ) ∂ t G l dt − Z t ∂ t ( ǫ Γ U ri U jr ∂ t η l ) ∂ t G r dt (3.28)We will derive the estimates for G by taking the matrix scalar product of (3.27)with w α + n [D η G ]. The first term gives a control of [D η G ]: Z Ω w α + n [D η G ] jr U ri [D η G ] ji dx (3.29) E FLOCH, AND N. MASMOUDI
The second term can be integrated by parts: Z Ω w α + n [D η G ] ir U jr [D η G ] ji dx = Z Ω w α + n A sr G i , s U jr A ki G j , k dx = − Z Ω w α + n A sr A ki G i , sk U jr G j dx − Z Ω ( w α + n ) , k A ki A sr G i , s U jr G j dx − Z Ω w α + n ( A sr U jr A ki ) , k G i , s G j dx =: ( a ) + ( b ) + ( c ) . (3.30)For ( a ), we integrate by parts again to get( a ) = Z Ω w α + n div η G U jr [ D η G ] jr dx + Z Ω ( w α + n ) , s A sr A ki G i , k U jr G j dx + Z Ω w α + n ( A sr A ki U jr ) , s G i , k G j dx and hence, Z Ω w α + n [D η G ] ir U jr [D η G ] ji dx = ( a ) + ( b ) + ( c )= Z Ω w α + n div η G U jr [ D η G ] jr dx + Z Ω ( w α + n ) , s A sr A ki G i , k U jr G j dx − Z Ω ( w α + n ) , k A ki A sr G i , s U jr G j dx + Z Ω w α + n ( A sr A ki U jr ) , s G i , k G j dx − Z Ω w α + n ( A sr U jr A ki ) , k G i , s G j dx It is clear that the the last two terms are lower-order. The first term in the right-hand-side is bounded by (cid:12)(cid:12)(cid:12)(cid:12)Z Ω w α + n div η G U jr [ D η G ] jr dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Ω w α + n U ri [D η G ] jr [D η G ] ji dx + C Z Ω w α + n | div η G | dx. The middle two terms need a special attention because they may not have the rightweights, for instance the second term when s = 3 and the third term when k = 3would have stronger weight w α + n than the desired weight w α + n . This can beovercome through the Hardy inequality. Here is the estimate of the third termwhen k = 3.(1 + α + n ) (cid:12)(cid:12)(cid:12)(cid:12)Z Ω w α + n ∂ wA i [D η G ] ir U jr G j dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Ω w α + n U ri [D η G ] jr [D η G ] ji dx + C Z Ω w α + n − | G | dx (by the Cauchy-Schwarz inequality) ≤ Z Ω w α + n U ri [D η G ] jr [D η G ] ji dx + δ Z Ω w α + n | DG | dx + C δ Z Ω w α + n | G | dx by (3.8) ≤ Z Ω w α + n U ri [D η G ] jr [D η G ] ji dx + C δ Z Ω w α + n | G | dx, PRIORI ESTIMATES FOR RELATIVISTIC FLUID FLOWS WITH VACUUM 21 where the last step can be achieved by choosing δ > (cid:12)(cid:12)(cid:12)(cid:12)Z Ω w α + n ǫ Γ U ri ( ∂ t G l ∂ t η r − ∂ t G r ∂ t η l ) U jl [D η G ] ji dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Ω w α + n U ri [D η G ] jr [D η G ] ji dx + C Z Ω w α + n | ∂ t G | dx It now remains to estimate the right-hand-side of (3.27): Z Ω w α + n [D η G ] S m,n dx ≤ Z Ω w α + n [D η G ] jr U ri [D η G ] ji dx + C Z Ω w α + n | S m,n | dx Note that S m,n consists of initial data and the time integral of lower order terms.Standard nonlinear estimates by using (3.4), Lemma 3.4 and Lemma 3.5 and theintegration by parts in time when necessary (for instance, see [7]) yield Z Ω w α + n | S m,n | dx ≤ F ( E N [ η , η ] , E ( I ) N , E ( III ) N , t ) , where F is a smooth function. This establishes the first inequality of (3.6).We further examine the dependence of initial data on S m,n . It contains someterms depending on the initial data: [ U ri [D η G ] jr − [D η G ] ir U jr + ǫ Γ U ri ( ∂ t G l ∂ t η r − ∂ t G r ∂ t η l ) U jl ] (cid:12)(cid:12) t =0 plus some functions of ∂ pτ ∂ q Curl η χ | t =0 · t for 0 ≤ | p | ≤ m and0 ≤ q ≤ n , which come from X in T m,n – see (3.28), (3.26), and (2.29). Thus weneeded the initial boundedness of not only E ( I ) N and E ( III ) N but also E ( IV ) N . Weobserve that E ( IV ) N contains one more time derivative than E ( III ) N and we cannotrecover it by the estimates that have been presented so far. In order to estimate E ( IV ) N , we will directly use (2.22). Then since ∂ mτ ∂ n Curl η χ = ∂ mτ ∂ n Curl η χ (cid:12)(cid:12) t =0 + Z t ∂ mτ ∂ n [ ∂ t , Curl η ] χ ds − Z t ∂ mτ ∂ n (cid:0) Γ − [Curl η , Γ] ∂ t χ (cid:1) ds by performing integration by parts in time for the second and third terms whennecessary [7], one can deduce that Z Ω w α + n | ∂ mτ ∂ n Curl η χ | dx ≤ Z Ω w α + n (cid:12)(cid:12) ∂ mτ ∂ n Curl η χ (cid:12)(cid:12) t =0 (cid:12)(cid:12) dx + F (cid:16) E ( I ) N , E ( III ) N , t (cid:17) , which completes the proof the lemma.4. Concluding observations
The Euler equations of non-relativistic fluids.
The new a priori esti-mates in Theorem 3.1 are trivially valid for solutions to the Euler equations ofnon-relativistic fluids: ∂ t ρ + ∂ k ( ρ u k ) = 0 ,∂ t ( ρ u j ) + ∂ k (cid:0) ρ u j u k + p δ jk (cid:1) = 0 . (4.1) E FLOCH, AND N. MASMOUDI
Note that the second–order formulation above is simplified drastically when ǫ = 0:we find C kij | ǫ =0 ≡ w α B ji | ǫ =0 ∂ t η i + ∂ k (cid:0) w α A kj J − /a (cid:1) = 0 , with B ji | ǫ =0 := δ ji Γ γ +1 = δ ji , which leads us to the second–order formulation in Lagrangian coordinates for non-relativistic fluids w α ∂ t η j + ∂ k (cid:0) w α A kj J − /α (cid:1) = 0 . (4.2)Similarly, the curl equation (2.22) reduces to the non-relativistic curl equation when ǫ = 0 Curl η ∂ t η = Curl u + Z t [ ∂ t , Curl η ] ∂ t ηds. (4.3)The non-relativistic fluids enjoy much elegant structure as it can be seen from(4.2) and (4.3). We observe that based on the new estimates obtained in Lemma 3.2and 3.3 (of course the proof for the non-relativistic case is much simpler), one canestablish the existence of the solutions to (4.2) justifying Theorem 3.1 correspondingto ǫ = 0 by a duality argument similar to [7].4.2. The non-relativistic limit ǫ → . Theorem 3.1 is valid for any fixed number ǫ ≥ ǫ → ǫ to the solutions ofthe non-relativistic Euler equations when ǫ → ǫ bound for all sufficiently small ǫ , andthey allure the validity of the non-relativistic limit ǫ → Final remark.
As presented in the previous sections, the relativistic Eulerequations exhibit an intriguing structure and it is highly non-trivial to establishthe existence of the solutions satisfying the a priori estimates given in Theorem3.1. In the case where the curl becomes trivial, there is no need to keep track ofthe evolution of the curl and the control of the divergence energy would sufficeboth for getting the estimates and for the existence theory. In that situation, theexistence result follows from our a priori estimates by a similar argument as done in[5, 7]. Those cases cover, for instance, 1+1 dimensional flows and 1+3 sphericallysymmetric flows. However, the existence question for the general relativistic fluidsin vacuum still remains open and we will leave it for future study.
Acknowledgements.
The first author (JJ) was supported in part by NSFgrants DMS-1212142 and DMS-1351898. This work was done when the secondauthor (PLF) enjoyed the hospitality of the Courant Institute of MathematicalSciences, New York University. PLF also acknowledge financial support from theANR grant SIMI-1-003-01 andthe European grant ITN-642768. The third author(NM) was partially supported by the NSF grant DMS-1211806.
PRIORI ESTIMATES FOR RELATIVISTIC FLUID FLOWS WITH VACUUM 23
References [1]
D. Coutand and S. Shkoller,
Well-posedness in smooth function spaces for the moving-boundary 1-D compressible Euler equations in physical vacuum, Comm. Pure Appl. Math.64 (2011), no. 3, 328-366.[2]
D. Coutand and S. Shkoller,
Well-posedness in smooth function spaces for the moving-boundary 3-D compressible Euler equations in physical vacuum, Arch. Ration. Mech. Anal.206 (2012), 515–616.[3]
C.M. Dafermos,
Hyperbolic conservation laws in continuum physics,
Grundlehren der Math-ematischen Wissenschaften, Vol. 325, Springer Verlag, Berlin, 2010.[4]
J. Jang,
Nonlinear instability theory of Lane-Emden stars, Comm. Pure Appl. Math. 67(2014), 1418–1465.[5]
J. Jang and N. Masmoudi N.,
Well-posedness for compressible Euler equations with physicalvacuum singularity, Comm. Pure Appl. Math. 62 (2009), 1327–1385.[6]
J. Jang and N. Masmoudi,
Vacuum in gas and fluid dynamics, in “Nonlinear conservationlaws and applications”, IMA Vol. Math. Appl., Vol. 153, Springer Verlag, New York, 2011,pp. 315–329.[7]
J. Jang and N. Masmoudi,
Well-posedness of compressible Euler equations with a physicalvacuum, Comm. Pure Appl. Math. 68 (2015), 61–111.[8]
T. Kato,
The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. RationalMech. Anal. 58 (1975), 181–205.[9]
A. Kufner, L. Malgranda, and L.-E. Persson,
The Hardy inequality,
Vydavatelsk´y Servis,Plzen Press, 2007.[10]
L. D. Landau and E. M. Lifshitz,
Fluid mechanics,
Course in theoretical physics, Vol. 6,Pergamon Press, 1959.[11]
P.G. LeFloch and S. Ukai,
A symmetrization of the relativistic Euler equations in severalspatial variables, Kinetic & Related Models 2 (2009), 275–292.[12]
T.-P. Liu and T. Yang,
Compressible flow with vacuum and physical singularity, MethodsAppl. Anal. 7 (2000), 495–509.[13]
T. Makino , On spherically symmetric motions of a gaseous star governed by the Euler-Poisson equations, Osaka J. Math. 52 (2015), 545–580.[14]
T. Makino and S. Ukai,
Local smooth solutions of the relativistic Euler equation, J. Math.Kyoto Univ. 35 (1995), 105–114.[15]