Hyperbolic conservation laws and spacetimes with limited regularity
aa r X i v : . [ m a t h . A P ] N ov Hyperbolic conservation lawsand spacetimes with limited regularity
Philippe G. LeFloch
Laboratoire Jacques-Louis LionsCentre National de la Recherche ScientifiqueUniversit´e de Paris 6, 4, Place Jussieu, 75252 Paris, France.Email:
October 29, 2006
Hyperbolic conservation laws posed on manifolds arise in many applica-tions to geophysical flows and general relativity. Recent work by the authorand his collaborators attempts to set the foundations for a study of weaksolutions defined on Riemannian or Lorentzian manifolds and includes aninvestigation of the existence and qualitative behavior of solutions. Themetric on the manifold may either be fixed (shallow water equations onthe sphere, for instance) or be one of the unknowns of the theory (Einstein-Euler equations of general relativity). This work is especially concernedwith solutions and manifolds with limited regularity. We review here re-sults on three themes: (1) Shock wave theory for hyperbolic conservationlaws on manifolds, developed jointly with M. Ben-Artzi (Jerusalem); (2)Existence of matter Gowdy-type spacetimes with bounded variation, de-veloped jointly with J. Stewart (Cambridge). (3) Injectivity radius estimatesfor Lorentzian manifolds under curvature bounds, developed jointly withB.-L. Chen (Guang-Zhou). 1
Conservation laws on a Riemannian manifold
In the present section, ( M n , g ) is a compact, oriented, n -dimensional Rie-mannian manifold. As usual, the tangent space at a point x ∈ M n isdenoted by T x M n and the tangent bundle by T M n : = S x ∈ M n T x M n , whilethe cotangent bundle is denoted by T ⋆ M n = T ⋆ x M n . The metric structure isdetermined by a positive-definite, 2-covariant tensor field g .A fluxon the manifold M n is a vector field f = f x ( ¯ u ) depending smoothlyupon the parameter ¯ u . The conservation law associated with f reads ∂ t u + div g ( f ( u )) = , (2.1)where the unknown u = u ( t , x ) is defined for t ≥ x ∈ M n and thedivergence operator is applied to the vector field x f x ( u ( t , x )) ∈ T x M n .We say that the flux is geometry-compatible ifdiv g f x ( ¯ u ) = , ¯ u ∈ R , x ∈ M n . (2.2)We propose to single out this class of conservation laws as an importantcase of interest, which leads to robust L p estimates that do not depend onthe geometry of the manifold. The equations arising in continuum physicsdo satisfy this condition.Equation (2.1) is a geometric partial di ff erential equation which de-pends on the geometry of the manifold only. All estimates must take acoordinate-independent form; in the proofs however, it is often convenientto introduce particular coordinate charts. We are interested in solutions u ∈ L ∞ ( R + × M n ) assuming a prescribed initial condition u ∈ L ∞ ( M n ): u (0 , x ) = u ( x ) , x ∈ M n . (2.3)We extend Kruzkov theory of the Euclidian space R n (see [8]) to the caseof a Riemannian manifold, as follows.Let f = f x ( ¯ u ) be a geometry-compatible flux on the Riemannian man-ifold ( M , g ). A convex entropy / entropy-flux pair is a pair ( U , F ) where U : R → R is convex and F = F x ( ¯ u ) is the vector field defined by F x ( ¯ u ) : = Z ¯ u ∂ u ′ U ( u ′ ) ∂ u ′ f x ( u ′ ) du ′ , ¯ u ∈ R , x ∈ M n . u ∈ L ∞ ( M n ) a function u ∈ L ∞ (cid:16) R + , L ∞ ( M n ) (cid:17) is called an entropysolution to the initial value problem (2.1), (2.3) if the following entropyinequalities hold Z Z R + × M n (cid:16) U ( u ( t , x )) ∂ t θ ( t , x ) + g x (cid:16) F x ( u ( t , x )) , grad g θ ( t , x ) (cid:17)(cid:17) dV g ( x ) dt + Z M n U ( u ( x )) θ (0 , x ) dV g ( x ) ≥ , for every convex entropy / entropy flux pair ( U , F ) and all smooth function θ = θ ( t , x ) ≥ , ∞ ) × M n . Theorem 2.1 (Well-posedness theory on a Riemannian manifold. I) . Letf = f x ( ¯ u ) be a geometry-compatible flux on a Riemannian manifold ( M n , g ) .Given u ∈ L ∞ ( M n ) there exists a unique entropy solution u ∈ L ∞ ( R + × M n ) tothe problem (2.1)–(2.3). Moreover, for each ≤ p ≤ ∞ , k u ( t ) k L p ( M n ; dV g ) ≤ k u k L p ( M n ; dV g ) , t ∈ R + , and, given two entropy solutions u , v associated with initial data u , v , k v ( t ) − u ( t ) k L ( M n ; dV g ) ≤ k v − u k L ( M n ; dV g ) , t ∈ R + . The framework proposed here allows us to construct entropy solutionson a Riemannian manifold via the vanishing di ff usion method or the finitevolume method [1, 4]. Following DiPerna [6] we can introduce the (larger)class of entropy measure-valued solutions ( t , x ) ∈ R + × M n ν t , x . Theorem 2.2 (Well-posedness theory on a Riemannian manifold. II) . Letf = f x ( ¯ u ) be a geometry-compatible flux on a Riemannian manifold ( M n , g ) . Let ν be an entropy measure-valued solution to (2.1)–(2.3) for some u ∈ L ∞ ( M n ) .Then, for almost every ( t , x ) , ν t , x = δ u ( t , x ) , where u ∈ L ∞ ( R + × M n ) is the uniqueentropy solution to the problem. Finally we can relax the geometry compatibility condition and considera general conservation law associated with an arbitrary flux f . Moregeneral conservation laws solely enjoy the L contraction property andleads to a unique contractive semi-group of entropy solutions.3 heorem 2.3 (Well-posedness theory on a Riemannian manifold. III) . Letf = f x ( ¯ u ) be an arbitrary (not necessarily divergence-free) flux on ( M n , g ) , satis-fying the linear growth condition | f x ( ¯ u ) | g . + | ¯ u | , ¯ u ∈ R , x ∈ M n . Then there exists a unique contractive, semi-group of entropy solutions u ∈ L ( M n ) u ( t ) : = S t u ∈ L ( M n ) to the initial value problem (2.1), (2.3). For the proofs we refer to [4]. See [1] for the convergence of the finitevolume schemes on a manifold. Earlier material can be found in Panov [12]( n -dimensional manifold) and in LeFloch and Nedelec [10] (Lax formulafor general metrics including the case of spherical symmetry). Motivated by the application to general relativity, we can extend the the-ory to a Lorentzian manifold. Let ( M n + , g ) be a time-oriented, ( n + g being a metric tensor with signature( − , + , . . . , + ). Tangent vectors X can be separated into time-like vectors( g ( X , X ) < g ( X , X ) = g ( X , X ) > ∇ be the Levi-Cevita connection associated with theLorentzian metric g .A flux on the manifold M n + is a vector field x f x ( ¯ u ) ∈ T x M n + ,depending on a parameter ¯ u ∈ R . The conservation law on ( M n + , g )associated with f is div g (cid:16) f ( u ) (cid:17) = , u : M n + → R . (3.1)It is said to be geometry compatible ifdiv g f x ( ¯ u ) = , ¯ u ∈ R , x ∈ M n + . (3.2)Furthermore, f is said to be a time-like flux if g x (cid:16) ∂ u f x ( ¯ u ) , ∂ u f x ( ¯ u ) (cid:17) < x ∈ M n + , ¯ u ∈ R .Note that our terminology here di ff ers from the one in the Riemanniancase, where the conservative variable was singled out. We are interested4n the initial-value problem associated with (3.1). We fix a space-like hy-persurface H ⊂ M n + and a measurable and bounded function u definedon H . Then, we search for u = u ( x ) ∈ L ∞ ( M n + ) satisfying (3.1) in thedistributional sense and such that the (weak) trace of u on H coincideswith u : u | H = u . (3.3)It is natural to require that the vectors ∂ u f x ( ¯ u ) are time-like and future-oriented.We assume that the manifold M n + is globally hyperbolic, in the sensethat there exists a foliation of M n + by space-like, compact, oriented hy-persurfaces H t ( t ∈ R ): M n + = S t ∈ R H t . Any hypersurface H t is referredto as a Cauchy surface in M n + , while the family H t ( t ∈ R ) is called anadmissible foliation associated with H t . The future of the given hyper-surface will be denoted by M n + + : = S t ≥ H t . Finally we denote by n t thefuture-oriented, normal vector field to each H t , and by g t the induced met-ric. Finally, along H t , we denote by X t the normal component of a vectorfield X , thus X t : = g ( X , n t ).A flux F = F x ( ¯ u ) is called a convex entropy flux associated with theconservation law (3.1) if there exists a convex U : R → R such that F x ( ¯ u ) = Z ¯ u ∂ u U ( u ′ ) ∂ u f x ( u ′ ) du ′ , x ∈ M n + , ¯ u ∈ R . A measurable and bounded function u = u ( x ) is called an entropysolutionof the geometry-compatible conservation law (3.1)-(3.2) if Z M n + + g ( F ( u ) , grad g g θ ) dV g + Z H g ( F ( u ) , n ) θ H dV g ≥ . for all convex entropy flux F = F x ( ¯ u ) and all smooth θ ≥ M n + + . Theorem 3.1 (Well-posedness theory on a Lorentzian manifold) . Considera geometry-compatible conservation law (3.1)-(3.2) posed on a globally hyperbolicLorentzian manifold M n + . Let H be a Cauchy surface in M n + , and u : H → R be measurable and bounded. Then, the initial-value problem (3.1)-(3.3) admits aunique entropy solution u = u ( x ) ∈ L ∞ ( M n + ) . For every admissible foliation H t ,the trace u H t exists and belong to L ( H t ) , and k F t ( u H t k L ( H t ) is non-increasing in ime, for any convex entropy flux F. Moreover, given any two entropy solutionsu , v, k f t ( u H t ) − f t ( v | H t ) k L ( H t ) is non-increasing in time. We emphasize that, in the Lorentzian case, no time-translation propertyis available in general, contrary to the Riemannian case. Hence, no time-regularity is implied by the L contraction property. Vacuum Gowdy spacetimes are inhomogeneous spacetimes admitting twocommuting spatial Killing vector fields. The existence of vacuum space-times with Gowdy symmetry is well-known and the long-time asymptoticsof solutions have been found to be particularly complex. In comparison,much less emphasis has been put on matter spacetimes. Recently, LeFloch,Stewart and collaborators [3, 11] initiated a rigorous mathematical treat-ment of the coupled Einstein-Euler system on Gowdy spacetimes. Theunknowns of the theory are the density and velocity of the fluid togetherwith the components of the metric tensor. The existence for the Cauchyproblem in the class of solutions with (arbitrary large) bounded total varia-tion is proven by a generalization of the Glimm scheme. Our theory allowsfor the formation of shock waves in the fluid and singularities in the ge-ometry. The first results on shock waves and the Glimm scheme in specialand general relativity are due to Smoller and Temple [14] (flat Minkowskispacetime) and Groah and Temple [7] (spherically symmetric spacetimes).The novelty in [3, 11] is the generalization to a model allowing for bothgravitational waves and shock waves.The metric is given in the polarized Gowdy symmetric form ds = e a ( − dt + dx ) + e b ( e c dy + e − c dz ) , (4.1)where the variables a , b , c depend on the time variable t and the spacevariable x , only. We consider Einstein field equations G αβ = κ T αβ forperfect fluids with energy density µ > p = µ c s . Here, the6ound speed c s is a constant with 0 < c s < G αβ denotes the Einsteintensor, while κ is a normalization constant.The 4-velocity vector u α of the fluid is time-like and is normalized tobe of unit length and we define the scalar velocity v and relativistic factor ξ = ξ ( v ) by ( u α ) = e − a ξ (1 , v , ,
0) and ξ = (1 − v ) − / . The matter is describedby the energy-momentum tensor T αβ = ( µ + p ) u α u β + p g αβ , from which weextract the fields τ , S and Σ : T = e − a (cid:16) ( µ + p ) ξ − p (cid:17) = : e − a τ, T = e − a ( µ + p ) ξ v = : e − a S , T = e − a (cid:16) ( µ + p ) ξ v + p (cid:17) = : e − a Σ . After very tedious calculations we arrive at the constraint equations a t b t + a x b x + b t − b xx − b x − c t − c x = κ e a τ, − a t b x − a x b t + b tx + b t b x + c t c x = κ e a S , and the evolution equations a tt − a xx = b t − b x − c t + c x + κ e a ( − τ + Σ − p ) , b tt − b xx = − b t + b x + κ e a ( τ − Σ ) , c tt − c xx = − b t c t + b x c x . The evolution equations for the fluid, ∇ β T αβ =
0, are the
Euler equations τ t + S x = T , S t + Σ x = T , in which the source terms T , T are nonlinear in first-order derivatives ofthe metric and fluid variables.We propose to reformulate the Einstein-Euler equations in the form ofa nonlinear hyperbolic system of balance laws with integral source-term,in the variables ( µ, v ) and w : = (cid:16) a t , a x , β t , β x , c t , c x (cid:17) , where β = e b . It isconvenient to also set α = e a . The functions α, b (and a , β ) are determinedby α ( t , x ) = e a ( t , x ) , a ( t , x ) = R x −∞ w ( t , y ) dy , b ( t , x ) = ln β ( t , x ) , β ( t , x ) = + R x −∞ w ( t , y ) dy . β remains positive.The equations under consideration consist of three sets of two equa-tions associated with the propagation speeds ±
1, the speed of light (afternormalization). The principal part of the fluid equations are the standardrelativistic fluid equations in a Minkowski background, with wave speeds λ ± = ( v ± c s ) / (1 ± v c s ). To formulate the initial-value problem it is natural toprescribe the values of µ, v , w on the initial hypersurface at t =
0, denotedby ( µ , v , w ).Our main result is: Theorem 4.1 (Existence of Gowdy spacetimes with compressible matter) . Consider the ( µ, v , w ) -formulation of the Einstein-Euler equations on a polarizedGowdy spacetime with plane-symmetry. Let the initial data ( µ , v , w ) be ofbounded total variation, TV ( µ , v , w ) < ∞ , satisfying the constraints, andsuppose that the corresponding functions α , b are measurable and bounded, sup | ( α , b ) | < ∞ . Then the Cauchy problem admits a weak solution µ, v , w suchthat for some increasing C ( t ) TV ( µ, v , w )( t ) + sup | ( α, b )( t , · ) | C ( t ) , t ≥ , and are defined up to a maximal time T ∞ . If T < ∞ then either the geometryvariables α, b blow up: lim t → T (cid:16) sup R | α ( t , · ) | + | b ( t , · ) | (cid:17) = ∞ , or the energy densityblows up : lim t → T sup R | µ ( t , · ) | = ∞ . Hence, the solution exists until either a singularity occurs in the geom-etry (e.g. the area β of the 2-dimensional space-like orbits of the symmetrygroup vanishes) or the matter collapses to a point. To our knowledge thisis the first global existence result for the Euler-Einstein equations.If a shock wave forms in the fluid, then µ, v will be discontinuousand, as a consequence, w x and w x might also be discontinuous. In fact,Theorem 5 allows not only such discontinuities in second-order derivativesof the geometry components (i.e. at the level of the curvature), but alsodiscontinuities in the first-order derivatives which propagate at the speedof light. The latter correspond to Dirac distributions in the curvature ofthe metric. 8 Lower bounds on the injectivity radius of Loren-tzian manifolds
Motivated by the application to spacetimes of general relativity and byearlier results established by Anderson [2] and Klainerman and Rodni-anski [13], we investigate in [5] the geometry and regularity of ( n + M , g ). Under curvature and volumebounds we establish new injectivity radius estimates which are valid ei-ther in arbitrary directions or in null cones. Our estimates are purely localand are formulated via the “reference” Riemannian metric b g T associatedwith an arbitrary future-oriented time-like vector field T .Our proofs are based on suitable generalizations of arguments fromRiemannian geometry and rely on the observation that geodesics in theEuclidian and Minkowski spaces coincide, so that estimates for the ref-erence Riemannian metric can be carried over to the Lorentzian metric.Our estimates should be useful to investigate the qualitative behavior ofspacetimes satisfying Einstein field equations.We state here one typical result from [5] encompassing a large class ofLorentzian manifolds. Fix a point p ∈ M , and let us assume that a domain Ω ⊂ M containing p is foliated by spacelike hypersurfaces Σ t with normal T , say Ω = S t ∈ [ − , Σ t . Assume also that the geodesic ball B Σ ( p , ⊂ Σ is compactly contained in Σ . Consider the following assumptions where K , K , K and v are positive constants: K ≤ − (cid:12)(cid:12)(cid:12) ∂∂ t (cid:12)(cid:12)(cid:12)(cid:12) g ≤ / K in Ω , (5.1) | L T g | b g T ≤ K in Ω , (5.2) | Rm g | b g T ≤ K in Ω , (5.3)vol g ( B Σ ( p , ≥ v , (5.4)We prove in [5] : Theorem 5.1 (Injectivity radius estimate for Lorentzian manifolds) . Let ( M , g ) be a Lorentzian manifold satisfying (5.1)–(5.4) at a point p ∈ M. Then, thereexists a positive constant i depending on the foliation bounds K , K , curvaturebound K , volume bound v , and dimension n so that the injectivity radius at p isbounded below by i , that is Inj ( M , g , p ) ≥ i . cknowledgments The author was partially supported by the A.N.R. grant 06-2-134423 enti-tled “Mathematical Methods in General Relativity” (MATH-GR).
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