Dynamical elastic bodies in Newtonian gravity
aa r X i v : . [ g r- q c ] J un DYNAMICAL ELASTIC BODIES IN NEWTONIAN GRAVITY
LARS ANDERSSON, TODD A. OLIYNYK † , AND BERND G. SCHMIDT Abstract.
Well-posedness for the initial value problem for a self-gravitatingelastic body with free boundary in Newtonian gravity is proved. In the materialframe, the Euler-Lagrange equation becomes, assuming suitable constitutiveproperties for the elastic material, a fully non-linear elliptic-hyperbolic systemwith boundary conditions of Neumann type. For systems of this type, theinitial data must satisfy compatibility conditions in order to achieve regularsolutions. Given a relaxed reference configuration and a sufficiently smallNewton’s constant, a neigborhood of initial data satisfying the compatibilityconditions is constructed. Introduction
In Newtonian physics, the two-body problem is solvable for point particles mov-ing around their common center of gravity on Kepler ellipses. However, if oneconsiders extended bodies, the situations changes drastically. Assuming that a so-lution exists for extended bodies, one can show that the centers of mass of thebodies move as point particles, but existence could not be established for a longtime.The first sucessful attack on the problem was made by Leon Lichtenstein [21]who considered self-gravitating fluid bodies moving on circles about their center ofgravity. In this case the Euler equations for a self-gravitating system become timeindependent in a coordinate system co-moving with the bodies. For the case ofsmall bodies or widely separated large bodies, Lichtensten showed the existence ofsuch solutions.Well-posednedness for the Cauchy problem for fluid bodies with free boundarywas proved only recently. Lindblad [22] proved well-posedness for a non-relativisticcompressible liquid body (i.e. positive boundary density) with free boundary. Inthis paper one can also find references to earlier work. A different proof of theresult of Lindblad that is valid for both the relativistic and non-relativistic caseswas given by Trakhinin [33].The more singular case of fluids with vanishing boundary density is discussed in[12, 17]. Unfortunately, the problem of proving well-posedness for self-gravitatingcompressible fluid bodies with free boundary is still open in general relativity andeven in Newtonian gravity. However, see [23] for the case of self-gravitating in-compressible fluids. See also [29, 10, 18] for results dealing with various restrictedversions of the Cauchy problem for fluid bodies in general relativity.One can argue that the Cauchy problem should be simpler to handle if the bodiesconsist of elastic material, since such a body can, at least in the case of small bodies,be said to ’have a shape of its own’. In this paper we solve the boundary initial valueproblem for self-gravitating deformations of a relaxed body. In particular, consider † Partially supported by the ARC grant DP1094582 and an MRA grant. a relaxed (i.e. with vanishing stress) elastic body in the absence of gravity. Thenwe show existence of a self-gravitating solution for initial data close to those of thestatic relaxed body and for a small gravitational constant. Thus, the motion of thebody will consist of small nonlinear oscillations around the equilibrium solution.The Cauchy problem for a Newtonian elastic body with free boundary is, undersuitable constitutive assumptions on the elastic material, a fully non-linear elliptic-hyperbolic initial-boundary value problem with boundary values of Neumann type.The case of purely hyperbolic problems of this type is covered by a theorem of Koch[19]. The method of proof of this theorem can be adapted to include the non-localelliptic terms in the which appear in the system of differential equations consideredin this paper via the Newtonian gravitational field.In order to achieve a regular evolution for the initial-boundary value problem,the initial data must satisfy certain compatibility conditions. The problem of con-structing an open neighborhood of initial data satisfying the compatibility condi-tions given one such data set is discussed in the work of Koch. The work of Lindbladcontains a related discussion for the fluid case. For the case of a Newtonian elasticbody, we construct, for small values of Newton’s constant, initial data satisfyingsuitable compatibility conditions near data for a relaxed elastic body in the ab-sence of gravity. It should be noted that due to the presence of the Newtoniangravitational potential this problem is non-local.The situation considered in this paper has several interesting generalizations. Ifwe consider a large static, self-gravitating body, a relaxed state may not exist. Re-stricting to the spherically symmetric case, solutions to the field equations can befound by ODE techniques [28]. If we perturb the data slightly, there should exist so-lutions with small oscillations around the equilibrium configuration. Our approachallows one to consider also such more complicated problems; the essential difficultyis the construction of the initial data satisfying the compatibility conditions andnot the time evolution. However, the results in this paper do no immediately coverthese more general situations. In particular, a proof of a suitable version of thewell-posedness result for the Cauchy problem, in these situations, requires a moregeneral treatment of potential theory in Riemannian manifolds. This problem willbe considered in a separate paper.Given a solution u to the Cauchy problem for the Newtonian elastic body whichexists for for times t ∈ [0 , T ], one may conclude from the main result of this paper,see theorem 4.3, that by choosing initial data sufficiently close to that of u , the timeof existence of the resulting solution can be arbitrarily close to T . In particular,given a a static solution, there exist nearby data such that the corresponding solu-tion to the Cauchy problem has a time of existence not less than any given time T .The proof of existence of static self-gravitating bodies in Newtonian theory [6] couldbe used together with a generalization of theorem 4.3 that allows for more generalinitial data, as alluded to above, as well as a generalization of the construction ofinitial data to show that shows that nearby data define solutions which exist up tosome time T .The treatment of two fluid balls in steady rotation due to Lichtenstein, whichwas mentioned above, has been generalized to the case of elastic bodies, see [8].Using these solutions in the manner just described, it is possible to obtain classesof solutions of the two body problem in any prescribed finite time interval. Weleave these problems for later investigations. YNAMICAL ELASTIC BODIES IN NEWTONIAN GRAVITY 3
Global existence for small data is an important question. For unbounded fluidsin 3-dimensions, it is now known that shocks form for a large class of initial data[31, 11], and it is widely believed that for most equations of state shocks will alwaysform for arbitrary small perturbation of constant density state. On the other hand,there do exists small data global existence results for unbounded elastic bodiesprovided the material satisfies certain additional conditions, cf. [32, 14]. For elasticbodies of finite extent, it appears to be an open question whether there existssolutions that are global in time or under what conditions shocks form.
Overview of this paper.
The paper is organized as follows. Section 2 describesNewtonian elasticity, sets up the basic equations and gives the conditions we imposeon the material. We derive the equations in the material frame and in spacetimefrom a variational principle. Section 3 deals with the problem of finding solutionsto the compatibility conditions needed for the proof of local existence. Due to thenon-local terms in the equations, these condition imply conditions on the Cauchydata on the whole initial surface. To find initial data satisfying these conditions, wemake use of some results from potential theory. These are developed in section 3.1.Further, the Poisson equation must be studied in the material frame, see section3.2. The results concerning the linearized elasticity operator which are needed canbe found in section 3.3. Section 4 generalizes Koch’s theorem and proves our maintheorem. Appendix A contains some background material for the function spacesused in this paper. 2.
Newtonian elasticity
Kinematics.
The body B is an open, connected, and bounded set with a C ∞ boundary in Euclidean space R B . We refer to R B as the extended body . We consider configurations , i.e. maps f : R S → B , and deformations φ : B → R S with f ◦ φ = id B . (2.1)Thus, the physical body is the domain in space R S given by f − ( B ) = φ ( B ).We shall make use of the extension ˜ φ of φ to a map R B → R S and let ( X A ) A =1 , , and ( x i ) i =1 , , denote global Cartesian coordinates on the body R B and the config-uration R S spaces, respectively. Since we shall consider the Newtonian dynamicsof a body, we let f, φ depend on time, denoted by t . Equation (2.1) gives f A ( t, φ ( t, X )) = X A in B and φ i ( t, f ( t, x )) = x i in f − ( B ) . (2.2)Writing x µ = ( t, x i ), we introduce f Aµ = ∂ µ f A and φ kA = ∂ A φ k . In particular, f A = ∂ t f A . We have φ kA f Aℓ = δ kℓ and f Bk φ kA = δ BA (2.3)where these expressions are defined. This implies ∂φ iA ∂f Bk = − φ iB φ kA , and ∂f Bk ∂φ iA = − f Bi f Ak . (2.4)Let χ f − ( B ) = (cid:26) f − ( B )0 in R S \ f − ( B ) (2.5) L. ANDERSSON, T.A. OLIYNYK, AND B. SCHMIDT be the indicator function of the support of the physical body. Using the aboveidentities, one may calculate the variation of χ f − ( B ) with respect to f A , ∂χ f − ( B ) ∂ ( f A ) = φ iA ∂ i χ f − ( B ) (2.6)Let H AB = f A,i f B,j δ ij and define H AB by H AB H BC = δ AC . Then H AB = φ iA φ jB δ ij . We have ∂H AB ∂ ( f Ck ) = 2 δ kn δ ( AC f B ) n . (2.7)Differentiating (2.3) gives the usual formulas for the derivative of the inverse, ∂ µ φ iA = − φ iB ∂ µ f Bk φ kA , (2.8) ∂ B f Ai = − f Aj ∂ B φ jC f Ci . (2.9)We let v µ ∂ µ = ∂ t + v i ∂ i be defined by v µ f Aµ = 0. This determines the vectorfield v i ( x ) on R S uniquely in terms of f . The velocity field v µ ∂ µ describes thetrajectories of material particles. From the relation v µ ∂ µ f A = 0, we get v i = − φ iA f A . On the other hand, time differentiating (2.2) gives v i ( t, x ) = ( ∂ t φ i )( t, f ( t, x )) . Thus, we have ∂ t φ k ( t, X ) = ∂ t ( v k ( t, φ ( t, X )))= ( ∂ t v k )( t, φ ( t, X )) + ∂ ℓ v k ( t, φ ( t, X )) ∂ t φ ℓ ( t, X )= ( v µ ∂ µ v k )( t, φ ( t, X )) . Further, v µ v ν ∂ µ ∂ ν f A = − v µ ∂ µ v k f Ak , (2.10)which shows that H AB v µ v ν ∂ µ ∂ ν f B = − v µ ∂ µ v m δ mn φ nA . (2.11)The body B carries a reference volume element defined by a 3-form V ABC on B .The number density n is defined, cf. [3, eq. (3.2)], by f Ai f Bj f Ck V ABC ( f ( x )) = n ( x ) ǫ ijk ( x )where ǫ ijk is the volume element of the Euclidean metric δ ij on R S . Since we areconsidering the Newtonian case with Euclidean geometry, we have simply n = det Df.
The mass density is ρ = nm (2.12)where m is the specific mass of the particles, i.e. mV is the mass density for thematerial in its natural state, see [3]. See also [7] where the specific mass is denoted ρ . We have the relation ∂n∂f Ai = nφ iA . (2.13) Our techniques allow for m to be a function on the body, m = m ( X ), but for simplicity, wewill assume that m is constant. YNAMICAL ELASTIC BODIES IN NEWTONIAN GRAVITY 5
The following equations for n follow from the above definitions, using (2.13), ∂ µ ( nv µ ) = 0 , (2.14)and ∂ k ( nφ kA ) = 0 . (2.15)The elastic material is described by the stored energy function ǫ = ǫ ( f A , H AB ) . The various forms of the stress tensor are defined from the stored energy functionvia τ AB = +2 ∂ǫ∂H AB , τ ij = nf Ai f Bj τ AB ,τ iA = f Bi τ BC H CA , τ Ai = nτ AB f Bj δ ji . We have ∂ǫ∂ ( f Ai ) = n − τ Ai ,∂ǫ∂ ( φ iA ) = − τ iA . The elasticity tensor L iAkB is defined by L iAkB = ∂τ iA ∂ ( φ kB ) . (2.16)It follows from this definition and the assumptions above that the elasticity tensorhas the symmetries L iAjB = L jBiA = L iABj = L AijB (2.17)where L iAjB = δ il δ jk L Al kB . Clearly, we also have ∂ A τ iA = L iAkB ∂ A ∂ B φ k . Variational formulation.
We derive the field equations for a self-gravitatingelastic body from the elastic action, supplemented by a term giving Newton’s forcelaw and Newton’s law of gravitation. The Lagrange density is of the form Λ ǫ where ǫ is the 4-volume element on spacetime R × R S in Cartesian coordinates,and Λ = Λ grav + Λ pot + Λ kin + Λ elast where Λ grav = |∇ U | πG , Λ pot = ρU χ f − ( B ) , Λ kin = 12 ρv χ f − ( B ) , Λ elast = − nǫχ f − ( B ) , The sign of the stress tensor here is the opposite of that of σ AB defined in [3, eq. (3.5)], butagrees with the usage in [7, equation (4.2)]. L. ANDERSSON, T.A. OLIYNYK, AND B. SCHMIDT with χ f − ( B ) given by (2.5), |∇ U | = ∂ i U ∂ j U δ ij , and v = v i v j δ ij . Eulerian picture.
The action in the Eulerian picture then takes the form L = Z Λ ǫ dx dx dx dx with Λ = Λ( U, ∂ i U, f A , f A , f Ai ). The Euler-Lagrange equations are of the form E A = 0, E U = 0 with −E A = ∂ µ ∂ Λ ∂ ( ∂f Aµ ) − ∂ Λ ∂ ( f A ) , −E U = ∂ i ∂ Λ ∂ ( ∂ i U ) − ∂ Λ ∂U . We have −E U = ∆ U πG − ρχ f − ( B ) . Next, we consider the Euler-Lagrange terms generated by variations with respectto the configuration f A . A calculation using (2.15) shows that the factor ρχ f − ( B ) gives no contribution to the Euler-Lagrange equations, and hence the kinetic termin the action gives −E kinA = ∂ µ ∂ Λ kin ∂ ( f Aµ ) − ∂ Λ kin ∂ ( f A )= [ ∂ k ( 12 ρφ kA v ) − ∂ t ( ρδ mn v m φ nA ) − ∂ k ( ρδ mn v m v k φ nB )] χ f − ( B ) , which after some calculations, using (2.14) and (2.15), gives −E kinA = − ρv µ ∂ µ v m δ mn φ nA χ f − ( B ) use (2.11) = ρH AB v µ v ν f A χ f − ( B ) . The elastic term gives, using (2.15) and (2.6), −E elastA = ∂ i Λ elast ∂ ( f Ai ) − ∂ Λ elast ∂ ( f A )= − ∂ i ( τ Ai χ f − ( B ) ) . In view of [3, lemma 2.2], we have that the divergence ∂ i ( τ Ai χ f − ( B ) ) is integrableonly if the zero traction boundary condition τ Aj n j (cid:12)(cid:12) ∂f − ( B ) = 0holds, in which case the identity ∂ i ( τ Ai χ f − ( B ) ) = ∂ i τ Ai χ f − ( B ) is valid. YNAMICAL ELASTIC BODIES IN NEWTONIAN GRAVITY 7
Finally, the potential term gives, using (2.15) and (2.6), −E potA = ∂ i Λ pot ∂ ( f Ai ) − ∂ Λ pot ∂f A = ∂ i ( ρφ iA U χ f − ( B ) ) − φ iA ∂ i χ f − ( B ) = ρφ iA ∂ i U χ f − ( B ) . Adding the terms, we have − E A = ρH AB v µ v ν ∂ µ ∂ ν f A − ∂ i τ Ai + ρφ iA ∂ i U in f − ( B ) , (2.18)subject to the boundary condition τ Aj n j (cid:12)(cid:12) f − ( B ) = 0 . Hence, we find that the Euler-Lagrange equations E A = 0 , E U = 0 are equivalent tothe system − ρH AB v µ v ν ∂ µ ∂ ν f B + ∂ i τ Ai = ρφ iA ∂ i U, in f − ( B ) (2.19a)∆ U = 4 πGρχ f − ( B ) , (2.19b) τ Ai n i (cid:12)(cid:12) ∂f − ( B ) = 0 (2.19c)where n i is the unit outward pointing normal to ∂f − ( B ). These are the field equa-tions for a dynamical elastic body in Newtonian gravity displayed in the Eulerianframe. We note that, by using (2.11) and multiplying by f Ai , equations (2.19a)and (2.19c) take the form ρv µ ∂ µ v i + ∂ j τ ij = ρ∂ i U in f − ( B ) and τ ij n j (cid:12)(cid:12) ∂f − ( B ) = 0 . (2.20)2.2.2. Material frame.
Similarly, the action in the material frame is given by L mat = Z φ ∗ (Λ ǫ ) dX dX dX dX = Z J Λ mat dX dV ABC dX A dX B dX C where J = ( φ ∗ n ) − = det( ∂ A φ i )is the Jacobian of φ , and Λ mat = Λ mat ( U, φ i , φ i , φ iA ) is given by the relation J Λ mat = J |∇ ¯ U | H πG + ( m ¯ U + 12 mv − ¯ ǫ ) χ B (2.21)where χ B is the indicator function of the support of the body and ¯ U , ¯ ǫ are definedalong the lines of [3]. In particular, ¯ U = U ◦ φ , and |∇ ¯ U | H = H AB ∂ A ¯ U ∂ B ¯ U is the pullback to R B of |∇ U | where we recall that H AB = f Ai f Bj δ ij and f iA = ( ∂ A φ i ) − . Similarly, ¯ ǫ = ǫ ◦ φ so that ¯ ǫ = ¯ ǫ ( f Ai , H AB ). We note also thatdet( H AB ) = det( f Ai ) = J . L. ANDERSSON, T.A. OLIYNYK, AND B. SCHMIDT
The Euler-Lagrange equations in material frame are E i = 0 , E ¯ U = 0 where −E i = ∂ A ∂ L mat ∂ ( φ iA ) − ∂ L mat ∂ ( φ i ) , −E ¯ U = ∂ A ∂ L mat ∂ ( ∂ A ¯ U ) − ∂ L mat ∂ ( ¯ U ) . This gives the system of equations − m∂ t φ i + ∂ A (¯ τ iA ) = mδ ij f Aj ∂ A ¯ U in B , (2.22a)∆ H ¯ U = 4 πGJ − mχ B in R B , (2.22b) ν A ¯ τ iA | ∂ B = 0 (2.22c)where ν A is the unit outward pointing normal to ∂ B ,¯ τ iA = δ ik J ( f Aj τ kj ) ◦ φ is the Piola transform of τ ij , and H = H AB dX A dX B = φ ∗ ( δ ij dx i dx j )is the pull back of the Euclidean metric δ ij dx i dx j under the map φ . Observe thatin (2.22b), we could also have used the notation ¯ ρ = J − m since J − = n ◦ φ .Rescaling time and the Newtonian potential, we can write the evolution equa-tions (2.22a)-(2.22c) in the form − ∂ t φ i + ∂ A ¯ τ Ai − λ δ ij f Aj ∂ A ¯ U = 0 in B , (2.23a)∆ H ¯ U = J − mχ B in R B , (2.23b) ν A ¯ τ Ai | ∂ B = 0 (2.23c)where λ = 4 πmG. We note that the Poisson equation (2.23b) can be written out more explicitly as ∂ A (cid:0) JH AB ∂ B ¯ U (cid:1) = mχ B . (2.24)It follows from the symmetry properties (2.17) of the elasticity tensor that¯ L iAjB = ∂ ¯ τ iA ∂∂ B φ j (2.25)satisfies ¯ L iAjB = ¯ L jDiB = ¯ L iABj = ¯ L AijB (2.26)where ¯ L iAjB = δ jk ¯ L iAjB . Since we will be working in the material representation for the remainder of thearticle, we will drop the B from the body space R B and denote it simply by R .For latter use, we define the following nonlinear functionals: E i ( φ ) = ∂ A (cid:0) ¯ τ iA ( ∂φ ) (cid:1) , and E i∂ ( φ ) = tr ∂ B ν A ¯ τ iA ( ∂φ ) (2.27)where ∂φ = ( ∂ A φ i ). YNAMICAL ELASTIC BODIES IN NEWTONIAN GRAVITY 9
Constitutive conditions.
We make the following assumptions on the elasticmaterial:(1) ¯ τ iA is a smooth function of its arguments ( ∂φ ) in the neighborhood of theidentity map ψ i : R −→ R : ( X i ) ( ψ i ( X ) = X i ) , (2) the identity map is an equilibrium solution to (2.23a)-(2.23c) for λ = 0,i.e. (cid:0) E i ( ψ ) , E i∂ ( ψ ) (cid:1) = (0 , , and(3) a iAjB = ¯ L iAjB (cid:12)(cid:12)(cid:12) ∂φ = ∂ψ satisfies the following properties:(a) there exists an ω > a iAjB ( X ) ξ i ξ A η j η B ≥ ω | ξ | | η | for all ξ, η ∈ R , and(b) there exists a γ > a iAjB σ iA σ jB ≥ γ | σ | for all σ = ( σ iB ) ∈ M × with σ iB = σ Bi .We remark that condition (2) above is satisfied for a stored energy function whichhas a minimum at some reference configuration. See [3, equations (3.20)-(3.22)].3. Construction of initial data
In order to prove the existence of dynamical solutions to the evolution equations(2.23a)-(2.23c), we first need to construct initial data that satisfy the compatibilityconditions to a sufficiently high order. The compatability conditions are defined asfollows.
Definition 3.1.
Fixing s > / φ i | t =0 , ∂ t | t =0 φ i ) = ( φ i , φ i ) ∈ W s +1 , ( B , R ) × W s, ( B , R )satisfies the compatibility conditions to order r (0 ≤ r ≤ s ) if there exists maps φ iℓ ∈ W s +1 − ℓ, ( B , R ) ℓ = 2 , , . . . , r that satisfy ∂ ℓ − t (cid:0) − ∂ t φ i + ∂ A ¯ τ Ai − λ δ ij f Aj ∂ A ¯ U (cid:1)(cid:12)(cid:12) t =0 = 0 in B ,∂ ℓt (cid:0) ∆ H ¯ U − J − mχ B (cid:1)(cid:12)(cid:12) t =0 = 0 in R ,∂ ℓt (cid:0) ( ν A ¯ τ Ai ) | ∂ B (cid:1)(cid:12)(cid:12) t =0 = 0 , for ℓ = 0 , , , . . . r where, after formally differentiating, we set ∂ ℓt | t =0 φ i = φ iℓ . Potential theory.
Before we can solve the problem of existence of initial datasatisfying the compatibility conditions and the time evolution of this data, we firstneed to develop some potential theory. To begin, we set B + = B , B − = R \ B , andlet E ( X, Y ) = − π | X − Y | denote the Newton potential so that ∆ E = δ . We also let S and D denote thesingle and double layer potentials S [ f ]( X ) = Z ∂ B E ( X, Y ) f ( Y ) dσ ( Y ) X / ∈ ∂ B and D [ f ]( X ) = Z ∂ B ∂∂ν Y E ( X, Y ) f ( Y ) dσ ( Y ) X / ∈ ∂ B , respectively, where dσ is the induced surface measure on ∂ B , and ν is the outwardpointing normal. Restricting to ∂ B , we have for X ∈ ∂ B that S [ f ] (cid:12)(cid:12) ∂ B ( X ) = Z ∂ B E ( X, Y ) f ( Y ) dσ ( Y ) and D [ f ] (cid:12)(cid:12) ∂ B ± ( X ) = (cid:0) ± I + K (cid:1) [ f ]( X )where K [ f ]( X ) = P.V. Z ∂ B ∂∂ν Y E ( X, Y ) f ( Y ) dσ ( Y ) . Further, ∂∂ν S [ f ] (cid:12)(cid:12) ∂ B ± = (cid:0) ± I + K ∗ (cid:1) [ f ]where K ∗ is the adjoint of K . We recall the following well known relations betweenthe boundary value problems for ∆ in B ± and these potentials:(i) The solution to the Dirichlet problem∆ u = 0 , tr ∂ B u = ψ on B ± is given by u = D [ f ]where f solves (cid:0) ± I + K (cid:1) [ f ] = ψ. (ii) The solution to the Neumann problem∆ u = 0 , tr ∂ B ∂∂ν u = ψ on B ± is given by u = S [ f ] + C where f solves (cid:0) ∓ + K ∗ )[ f ] = ψ, and C is an arbitrary constant. The solution exists if and only if R ∂ B ψ dσ =0. YNAMICAL ELASTIC BODIES IN NEWTONIAN GRAVITY 11
For the moment, we consider the interior Dirichlet and Neumann problems on B + = B . The solution for the Dirichlet problem has the property that u ∈ W s,p ( B ) if tr ∂ B u ∈ B s − /p,p ( ∂ B ) , (3.1)while for the Neumann problem, we have u ∈ W s,p ( B ) if tr ∂ B ∂∂ν u ∈ B s − − /p,p ( ∂ B ) . These results are classical, see [2]. Defining the volume potential of a density f in B to be V [ f ]( X ) = ∆ − ( f χ B )( X ) = Z B E ( X, Y ) f ( Y ) d Y, we abuse notation and say that V [ f ] ∈ W k,p ( B ) if the restriction to B has thisproperty, and similarly for the other potentials. Using the fact that ∂ X A E ( X, Y ) = − ∂ Y A E ( X, Y ), we have ∂ A V [ f ]( X ) = − Z B ∂ Y A E ( X, Y ) f ( Y ) d Y. This gives, after a partial integration, ∂ A V [ f ] = V [ ∂ A f ] − S [tr ∂ B f ν A ] . (3.2)Let u be the solution of the Dirichlet problem∆ u = 0 , tr ∂ B u = ψ. By Green’s theorem, we have Z B (∆ Y u ( Y ) E ( X, Y ) − u ( Y )∆ Y E ( X, Y )) d Y = Z ∂ B (cid:18) ∂∂ν Y u ( Y ) E ( X, Y ) − u ( Y ) ∂∂ν Y E ( X, Y ) (cid:19) dσ ( Y ) . Since u solves the Dirichlet problem with boundary data ψ , and∆ Y E ( X, Y ) = δ ( X − Y ) , (3.3)this gives D [ u ] | B = S [tr ∂ B ∂∂ν u ] + u. (3.4)Upon taking the limit from the interior at ∂ B , we have( I + K )[ u ] | ∂ B = S [tr ∂ Ω ∂∂ν u ] + u. Consider a metric g AB on R with covariant derivative ∇ A . Let { e a } a =1 , be atangential frame on ∂ B , and let h ab denote the induced metric on ∂ B with covariantderivative D . Let a vector field ξ be given. Decompose ξ into tangential and normalcomponent at ∂ B , ξ = P a e a + h ξ, ν i ν. Introduce a Gaussian foliation near ∂ B . Then g takes the form g AB dX A dX B = dr + h ab ( y, r ) dy a dy b where y a are coordinates on ∂ B . Extending ν in a neighborhood of ∂ B using aGaussian foliation, we have ∇ A ξ A = h ab ∇ a ξ b + ( ∇ ξ )( ν, ν ) . The last term vanishes due to ∇ ν ν = 0, which is valid in a Gauss foliation. Thenwe have ∇ A ξ A = h ab ∇ a ξ b = h ab ( D a P b − λ ab h ξ, ν i )where λ ab = h∇ a ν, e b i = L ν h ab is the second fundamental form. Let H = h ab λ ab denote the mean curvature of ∂ B . Then we have D a P a = H h ξ, ν i + ∇ A ξ A . Now specialize to the Euclidean case; let g AB = δ AB be the Euclidean metric andlet ξ = ∂ A for some fixed A . Then ∇ A ξ A = 0, and we have D a P a = Hν A . (3.5)For X ∈ B , we calculate using (3.3) and the divergence theorem that ∂ A S [ f ]( X ) = − Z ∂ B f ( Y ) P a D a E ( X, Y ) dσ ( Y ) − Z ∂ B f ( Y ) h ν, ∂ Y A i ∂∂ν Y E ( X, Y ) dσ ( Y )= Z ∂ B ( D a P a f ( Y ) + P a D a f ) E ( X, Y ) dσ ( Y ) − Z ∂ B f ( Y ) h ν, ∂ y A i ∂∂ν Y E ( x, y ) dσ ( Y ) . Thus we have by the above result and (3.5) that ∂ A S [ f ] = S [ f Hν A + ∂ k A f ] − D [ f ν A ] . (3.6) Proposition 3.2.
The operators S, I + K have the mapping properties I + K : B k − /p,p ( ∂ Ω) → B k − /p,p ( ∂ Ω) , (3.7a) S : B k − − /p,p ( ∂ Ω) → B k − /p,p ( ∂ Ω) , (3.7b) for s = k − /p , k ≥ , k an integer. The corresponding statements for the singleand double layer potentials are D : B k − /p,p ( ∂ Ω) → W k,p (Ω) (3.8a) S : B k − − /p,p ( ∂ Ω) → W k,p (Ω) (3.8b) for k ≥ , k integer.Proof. We use induction to reduce the statement to the case k = 1. This casefollows from [15], see also [26]. Suppose then that we have proved the statementfor k −
1. To do the induction, assume f ∈ B k − /p,p ( ∂ Ω). Let u be the solutionto the Dirichlet problem with boundary data f . Then u ∈ W k,p (Ω), and setting ψ = tr ∂ Ω ∂∂ν u , we have ψ ∈ B k − − /p,p . Equations (3.4) and (3.6) give ∂ x i D [ f ] = ∂ x i S [ ψ ] + ∂ x i u = S [ ψHν i + ∂ k x i ψ ] − D [ ψν i ] + ∂ x i u which by the induction assumption is in W k − ,p . It follows that D [ f ] ∈ W k,p (Ω)and hence ( I + K ) f ∈ B k − /p,p ( ∂ Ω). This proves the statement for for D and( I + K ) at regularity k . YNAMICAL ELASTIC BODIES IN NEWTONIAN GRAVITY 13
Next, for S , we use the equation (3.6) for f ∈ B k − − /p,p ( ∂ Ω). Using theinduction assumption and the statement for D and ( I + K ) just proved, we have ∂ x i S [ f ] ∈ W k,p , which gives S [ f ] ∈ W k,p ( ∂ Ω), and S [ f ] ∈ B k − /p,p ( ∂ Ω). This completes the induc-tion and the result follows. (cid:3)
Example . Let f ∈ W ,p (Ω). Then tr ∂ Ω f ∈ B − /p,p and V [ ∂ x i f ] ∈ W ,p (Ω).Further, S [tr ∂ Ω f ν i ] solves a Dirichlet problem with boundary data S [tr ∂ Ω f ν i ] ∈ B − /p,p ( ∂ Ω), and hence S [tr ∂ Ω f ν i ] ∈ W ,p . It follows, in view of (3.2), that( V f ) χ Ω ∈ W ,p (Ω). Example . Let f ∈ W ,p ( B ). We have ∂ x i V [ f ] = V [ ∂ x i f ] + S [tr ∂ Ω f ν i ] . Since ∂ x i f ∈ W ,p ( B ), we have from Example 3.3 that V [ ∂ x i f ] ∈ W ,p ( B ). Further,tr ∂ Ω f ∈ B − /p,p ( ∂ Ω) and hence S [tr ∂ Ω f ν i ] ∈ W ,p ( B ). Therefore V [ f ] ∈ W ,p ( B ). Proposition 3.5.
Let k ≥ , and assume f ∈ W k,p ( B ) . Then V [ f ] ∈ W k +2 ,p ( B ) .Proof. The proof proceeds by induction, with base case k = 1. For this case,the statement follows by the argument in Example 3.3. Suppose we have provedthe statement for k −
1. We will make use of the identity (3.2). By induction, V [ ∂ x i f ] ∈ W k +1 ,p (Ω). Further, tr ∂ Ω f ∈ B k − /p and hence by (3.7), S [tr ∂ Ω f ν i ] ∈ B k +1 − /p ( ∂ Ω). It follows by (3.1) that S [tr ∂ Ω f ν i ] ∈ W k +1 ,p ( B ). This shows that ∂ x i V [ f ] ∈ W k +1 ,p ( B ) and hence V [ f ] ∈ W k +2 ,p ( B ). (cid:3) Similar arguments combined with the mapping properties (A.7)-(A.8) of theLaplacian on the weighted Sobolev spaces can be used to establish the followingproposition for the volume potential on B − . Proposition 3.6.
Let k ≥ , − < δ < , and assume f ∈ W k,pδ − ( B − ) . Then V [ f ] ∈ W k +2 ,pδ ( B − ) . The Poisson equation in the material frame.
The next step in solvingthe problems of the existence of initial data satisfying the compatibility conditionsand the time evolution of this data is to establish a number of smoothness propertiesfor solutions to the Poisson equations in the material frame. We begin by definingthe spaces W k,s,pδ ( R ) = (cid:8) u ∈ W k,pδ ( R , V ) (cid:12)(cid:12) u | B ∈ W k + s,p ( B ) and u | B − ∈ W k + s,pδ ( B − ) (cid:9) for 1 < p < ∞ , k ∈ Z and δ ∈ R . It is not difficult to verify that these spaces arecomplete with respect to the norm k u k W k,s,pδ ( R ) = k u | B k W k + s,p ( B ) + k u | B − k W k + s,pδ ( B − ) + k u k W k,pδ ( R ) , (3.9)and hence Banach spaces. Theorem 3.7.
Suppose < p < ∞ , s ∈ Z ≥ , s + 1 > /p , and − < δ < . Thenthere exist an open neighborhood e O s +2 ,p ⊂ W s +2 ,p ( B , R ) of ψ , and an analyticmap ¯ U : e O s +2 ,p −→ W ,s,pδ ( R ) : φ ¯ U ( φ ) such that (i) ¯ U ( φ ) satisfies the Poisson equation (2.23b) on R , (ii) for each φ ∈ e O s +2 ,p , the map ˜ φ = ψ + E B (cid:0) φ − ψ |B (cid:1) is a C diffeomorphismon R that satisfies ˜ φ − ψ ∈ W s,p − ( R , R ) ⊂ C − ( R , R ) and ˜ φ − − ψ ∈ W s,p − ( R , R ) , (iii) U = ¯ U ( φ ) ◦ ˜ φ − ∈ W ,pδ ( R ) satisfies the Poisson equation ∆ U = mρχ φ ( B ) on R and U ( x ) = o( | x | δ ) as | x | → ∞ , and (iv) for k ∈ Z ≥ and k ≤ s + 1 , the derivative of ¯ U can be extended to act on W k,p ( B ) , and moreover, the map e O s +2 ,p ∋ φ D ¯ U ( φ ) ∈ L ( W k,p ( B )) is well defined and analytic .Proof. (i) Fix 1 < p < ∞ , s + 1 > /p , and − < δ <
0. Given ψ ∈ W s,p ( B ), wedefine ˜ φ = ψ + E B ( ψ ) , φ iA = ∂ A ˜ φ i , and ( f Ai ) = ( φ iA ) − . Since matrix inversion and the determinant both define analytic maps in a neigh-borhood of the identity, it follows from (A.6), proposition 3.6 of [16], the continuityof extension and differentiation, the analyticity of continuous linear maps, and theproperty that the composition of analytic maps are again analytic that there existsa
R > B R ( W s,p ( B , R )) ∋ ψ det( φ iA ) − ∈ W s,p − ( R ) (3.10)and B R ( W s,p ( B , R )) ∋ ψ ( f Ai − δ Ai ) ∈ W s,p − ( R , M × ) (3.11)are well defined and analytic. Recalling that H AB = f Ai δ ij f Bj , we see from thesame arguments that the map B R ( W s,p ( B , R )) ∋ ψ JH AB − δ AB ∈ W s,p − ( R , M × ) (3.12)is analytic where J = det( φ iA ). Using the multiplication inequalities (A.2) and(A.6), we find that k ∂ A (cid:0) JH AB ∂ B ¯ U (cid:1)(cid:12)(cid:12) B k W s,p ( B ) . k ψ k W s,p ( B , R ) k ¯ U | B k W s,p ( B ) , (3.13) k ∂ A (cid:0) JH AB ∂ B ¯ U (cid:1)(cid:12)(cid:12) B − k W s,pδ ( B − ) . k ψ k W s,p ( B , R ) k ¯ U | B − k W s,pδ ( B − ) (3.14)and k ∂ A (cid:0) JH AB ∂ B ¯ U (cid:1) k W ,pδ − ( R ) . k ψ k W s,p ( B , R ) k ¯ U k W ,pδ ( R ) . (3.15)From the analyticity of the map (3.12), the bilinear estimates (3.13)-(3.15), theanalyticity of continuous bilinear maps, and the property that the composition ofanalytic maps are again analytic, it follows that the B R ( W s,p ( B , R )) × W ,s,pδ ( R ) ∋ ( ψ, ¯ U ) ∂ A (cid:0) JH AB ∂ B ¯ U (cid:1) ∈ W ,s,pδ − ( R )(3.16)is well defined and analytic. Together, (A.7), and Propositions 3.5 and 3.6 implythat ∆ − ( χ B ) ∈ W ,s,pδ ( R ) , (3.17)and the Laplacian ∆ : W ,s,pδ ( R ) −→ W ,s,pδ − ( R ) (3.18) For a Banach space X , L ( X ) denotes the set of continuous linear operators on X YNAMICAL ELASTIC BODIES IN NEWTONIAN GRAVITY 15 is an isomorphism with inverse given by (A.8).From (3.16), (3.17), and (3.18), we see that F : B R ( W s,p ( B , R )) × W ,s,pδ ( R ) −→ W ,s,pδ ( R ) , ( ψ, ¯ U ) ∆ − (cid:16) ∂ A (cid:0) JH AB ∂ B ¯ U (cid:1) − mχ B (cid:17) is well defined and analytic. Evaluating F at ψ = 0 gives F (0 , ¯ U ) = ∆ − (cid:16) ∆ ¯ U − mχ B (cid:17) , which shows that ¯ U = m ∆ − ( χ B ) ∈ W ,s,pδ ( R ) (3.19)satisfies F (0 , ¯ U ) = 0 . (3.20)Also, by the linearity of F in its second argument and the invertibility of theLaplacian, it is clear that D F (0 , ¯ U ) · δ ¯ U = ∆ − (cid:16) ∆ δ ¯ U (cid:17) = δ ¯ U . (3.21)Results (3.20) and (3.21) allow us to apply an analytic version of the implicitfunction theorem (see [13], theorem 15.3) to conclude the existence of a uniqueanalytic map, shrinking R if necessary,¯ U : B R ( W s,p ( B , R )) −→ W ,s,pδ ( R ) (3.22)that satisfies ¯ U (0) = ¯ U , and F ( ψ, ¯ U ( ψ )) = 0 ∀ ψ ∈ B R ( W s,p ( B , R )) . From the definition of F and the invertibility of ∆ − , it then follows that ¯ U ( ψ )satisfies ∂ A (cid:0) JH AB ∂ B ¯ U ( ψ ) (cid:1) = mχ B . (3.23) (ii) & (iii) Following Cantor [9], we consider the following group of diffeomorphismson R D s,qδ ( R ) := (cid:8) φ : R → R | φ − ψ ∈ W s,qη ( R , R ), and φ − − ψ ∈ W s,qη ( R , R ) (cid:9) where s > /q + 1 and η ≤
0. Fixing ψ ∈ B R ( W s,p ( B , R )), we get from (3.10)and (3.11) that ˜ φ = φ ( ψ ) ∈ D s,p − ( R ) . Defining, U := U ( ψ ) ◦ ˜ φ − , (3.24)we can apply Corollary 1.6 of [9] to get U ∈ W ,pδ ( R ) . (3.25)A straightforward calculation using the chain rule and (3.23), (3.24), and (3.25)then shows that ∆ U = m det( D ( ˜ φ − )) χ ˜ φ ( B ) , In [9], Cantor required that δ ≤ − / W k,pδ ⊂ C b . Consequently, the only restriction on δ is that δ ≤ while the fall off condition U ( x ) = o( | x | δ ) as | x | → ∞ follows from the weightedSobolev inequality (A.5). (iv) To begin, we assume that k = 1 and observe that for θ i ∈ W ,p ( B ) and¯ U ∈ W ,s,p ( R ) k E B ( ∂ C θ i ) ∂ B ¯ U k L pδ − ( R ) ≤ k ∂ C θ i ∂ B ¯ U k L p ( B ) + k E B ( ∂ C θ i ) ∂ B ¯ U k L pδ − ( B − ) . k θ k W ,p ( B ) k ¯ U | B k W s +2 ,pδ ( B ) + k θ k W ,p ( B ) k ¯ U | B − k W s +2 ,pδ ( B − ) . k θ k W ,p ( B ) k ¯ U k W ,s,p ( R ) (3.26)where in deriving the result we have used property (A.9) of the extension operator E B , the multiplication inequalities (A.2) and (A.6), and the assumption 1+ s > /p .Letting H ABCi = ∂JH AB ∂φ iC , we get, using the estimate (3.26) and the same arguments as above, that for R small enough the map G A : B R ( W s,p ( B , R )) × W ,p ( B , R ) × W ,s,pδ ( R ) × W ,pδ ( R ) −→ W − ,pδ − ( R ): ( ψ i , θ i , ¯ U , ¯ V ) JH AB ∂ B ¯ V + H ABCi E B ( ∂ C θ i ) ∂ B ( ¯ U )is analytic. From the continuity of differentiation and the trace map, we then havethat the map G : B R ( W s,p ( B , R )) × W ,p ( B , R ) × W ,s,pδ ( R ) × W ,pδ ( R ) −→ W − ,pδ − ( R )defined by G ( ψ i , θ i , ¯ U , ¯ V ) = ∂ A G A ( ψ i , θ iB , ¯ U , ¯ V )is analytic. Taking ¯ U as defined by (3.19), a straightforward calculation and theinvertibility of the Laplacian show that G (0 , , ¯ U ,
0) = 0 and D G (0 , , ¯ U , · δ ¯ V = δ ¯ V .
Therefore, we can again apply the analytic version of the implicit function theo-rem (see [13], theorem 15.3) to conclude the existence of a unique analytic map,shrinking R if necessary,¯ V : B R ( W s,p ( B , R )) × W ,p ( B , R ) × (cid:0) ¯ U + B R ( W ,s,pδ ) (cid:1) −→ W ,pδ ( R ) (3.27)that satisfies ¯ V (0 , , ¯ U ,
0) = 0 , and G ( ψ i , θ iB , ¯ U , ¯ V ( ψ i , θ iB , ¯ U )) = 0for all ( ψ i , θ i , ¯ U ) ∈ B R ( W s,p ( B , R )) × W ,p ( B , R ) × (cid:0) ¯ U + B R ( W ,s,pδ ( R )) (cid:1) .From the construction of G , and the uniqueness of the maps (3.22) and (3.27),it is not difficult to verify that¯ V ( ψ i , ∂ A δψ i , ¯ U ( ψ )) = D ¯ U ( ψ ) · δψ ∀ ( ψ, δψ ) ∈ B R ( W s,p ( B , R )) × W s,p ( B , R ) . From this and the density of W s +1 ,p ( B ) in W ,p ( B ), it follows that the derivativeof ¯ U can be extended act on W ,p ( B ), and moreover, that the map B R ( W s,p ( B , R )) ∋ ψ D ¯ U ( ψ ) ∈ L ( W ,p ( B )) (3.28) YNAMICAL ELASTIC BODIES IN NEWTONIAN GRAVITY 17 is well defined and analytic. By (3.22) above, we also have that the map B R ( W s,p ( B , R )) ∋ ψ D ¯ U ( ψ ) ∈ L ( W s,p ( B )) (3.29)is well defined and analytic. Together, the maps (3.28)-(3.29) and interpolationimply that the map B R ( W s,p ( B , R )) ∋ ψ D ¯ U ( ψ ) ∈ L ( W k,p ( B ))is well defined and analytic for k ∈ Z and 1 < k < s + 2. (cid:3) Corollary 3.8.
The map
Λ : e O s +2 ,p −→ W s +1 ,p ( B , R ) defined by Λ i ( φ ) = − δ ij f Ai ∂ A ¯ U ( φ ) (cid:0) ( f Ai ) = ( ∂ A φ i ) − (cid:1) is analytic. Moreover, for k ∈ Z ≥ and k ≤ s + 1 , the derivative of Λ can beextended act on W k,p ( B ) , and the map e O s +2 ,p ∋ φ D Λ( φ ) ∈ L ( W k,p ( B ) , W k − ,p ( B )) is well defined and analytic.Proof. This follows directly from theorem 3.7, the multiplication inequality (A.6),proposition 3.6 of [16], and the fact that compositions of analytic maps are againanalytic. (cid:3)
The linearized elasticity operator.
We define the operator linearized elas-ticity and boundary operators by A ( φ ) i := ∂ B (cid:0) a iBjD ∂ D φ j (cid:1) (3.30)and A ∂ ( φ ) i := ν B a iBjD ∂ D φ j , (3.31)respectively. We also define Y s,p = W s,p ( B , R ) × B s +1 − /p,p ( ∂ B , R ) , and for each φ ∈ W s +2 ,p ( B , R ), Y s,pφ = { ( b, t ) ∈ Y s,p | C ( b, t ) = 0 , C ( φ, b, t ) = 0 } where C ( b, t ) = Z B b + Z ∂ B t, (3.32)and C ( φ, b, t ) = Z B b × φ + Z ∂ B t × φ. (3.33)Here, we are using the notation( b × φ ) i = ǫ ijk b j φ k , (3.34) Z B b = Z B b d X, (3.35) and Z ∂ B t = Z ∂ B t dσ. (3.36)For later use, we recall the following theorem from [24] concerning the surjectivityof the linearized elasticity operator. Theorem 3.9. [[24], theorem 1.11, Section 6.1]
Suppose < p < ∞ , s ≥ , and P : Y s,p −→ Y s,pψ is any projection map. Then the map A : W s +2 ,p ( B , R ) −→ Y s,pψ : φ P ( A ( φ ) , A ∂ ( φ )) is surjective and ker A = { a + b × X | a = ( a i ) , b = ( b i ) ∈ R } where ( b × X ) i = ǫ ijA b j X A .Remark . Letting ˜ X j = 1Vol( B ) Z B X j d X denote the center of B , a short calculation using the change of coordinates ¯ X j = X j − ˜ X j and ¯ B = B − ˜ X shows that Z ¯ B ¯ X j d ¯ X = Z B X j − ˜ X j d X. Therefore, we can always arrange that Z B X j d X = 0 (3.37)by translating the domain B . For the remainder of this article, we will alwaysassume that the condition (3.37) holds.Next, for p > /
2, we define the spaces X s,p = n φ ∈ W s,p ( B , R ) (cid:12)(cid:12)(cid:12) Z B φ = 0 o , and observe, using Sobolev’s and H¨older’s inequalities, that (cid:12)(cid:12)(cid:12)Z B φ × ψ (cid:12)(cid:12)(cid:12) ≤ k φ × ψ k L ( B , R ) . k φ k L ∞ ( B , R ) k ψ k L ( B , R ) . k φ k W ,p ( B , R ) k ψ k L p ( B , R ) . k φ k W s,p ( B , R ) k ψ k W s,p ( B , R ) from which the continuity of the bilinear map B : X s +2 ,p × X s −→ R : ( φ, ψ ) Z B φ × ψ follows. Setting B ψ ( φ ) := B ( φ, ψ ) , we define for p > / U s,p = { ψ ∈ X s,p | B ψ : ker A ∩ X s +2 ,p → R is an isomorphism } . Lemma 3.11. ψ ∈ U s,p for all s ≥ and / < p < ∞ . YNAMICAL ELASTIC BODIES IN NEWTONIAN GRAVITY 19
Proof.
By (3.37), we have that Z Bc ψ i d X = Z B X i d X = 0 , and Z B a + b × X = Z B a + b × Z B X = Vol( B )for all a, b ∈ R . Consequently, ψ ∈ X s,p , (3.38)and ker A ∩ X s +2 ,p = { b × X | b ∈ R } (3.39)by theorem 3.9. Next, B ψ ( b × X ) = Z B ( b × X ) × ψ = Z B ( b × X ) × X = Z B ( X · b ) X − b | X | , which, after taking the innerproduct with a ∈ R , yields b · B ψ ( b × X ) = Z B ( X · b )( X · a ) − a · b | X | . (3.40)The Cauchy-Schwartz inequality shows that( X · b ) − | b | | X | ≤ X · b ) − | b | | X | = 0 ∀ X ∈ B ⇐⇒ b = 0 . (3.42)Combining (3.40)-(3.42), we arrive at b · B ψ ( b × X ) = 0 ⇐⇒ b = 0 . (3.43)By way of contradiction, suppose that the mapker A ∩ X s +2 ,p −→ R (3.44)is not surjective. Then the image B ψ (ker A ∩ X s +2 ,p ) is contained in a two dimen-sional subspace, and therefore, there exists a non zero a ∈ R such that a · B ψ ( b × X ) = 0 ∀ b ∈ R . But this is impossible by (3.43), and hence the map (3.44) is surjective. Sincedim ker A ∩ X s +2 ,p = 3, the map (3.44) must, in fact, be an isomorphism. (cid:3) Existence of initial data satisfying the compatibility conditions.Lemma 3.12.
Suppose that Y , Y , . . . , Y r , and Z are Banach spaces with contin-uous (linear) embeddings ι i,j : Y i −→ Y j i, j ∈ { , , . . . , r } , i < j, U ⊂ Y r is open, and F ∈ C r +1 ( U , Z ) . Then the map defined by F r ( y , y , . . . , y r ) := d r dt r (cid:12)(cid:12)(cid:12) t =0 F ( c ( t )) where c ( t ) = r X j =0 t j ι j,r ( y j ) is in C ( ι − ,r ( U ) × Q rj =1 Y j , Z ) . Proof.
Since
U ∈ Y r is open, it follow from the continuity of the map ι ,r that ι − ,r ( U ) ⊂ Y is open. Next, fix y ∈ ι − ,r ( U ) and y j ∈ Y j for j = 1 , , . . . , r . Thenthe continuity of the maps ι j,r : Y j → Y r and ι ,r ( y ) ∈ U guarantees the existenceof a δ > c ( t ) = r X j =0 t j ι j,r ( y j ) ∈ U ∀ t ∈ ( − δ, δ ) . Clearly, this implies that c ∈ C ∞ (( − δ, δ ) , U ), and hence, that the map ι − ,r ( U ) × r Y j =1 Y j ∋ ( y , . . . , y r ) d r dt r (cid:12)(cid:12)(cid:12) t =0 F ( c ( t )) ∈ Z is well defined and continuously differentiable. (cid:3) Proposition 3.13.
Suppose < p < ∞ , s ∈ Z ≥ , and e O s +2 ,p ⊂ W s +2 ,p ( B , R ) isthe open neighborhood of ψ from theorem 3.7. Then the maps (see (2.27) ) E : e O s +2 ,p −→ W s,p ( B , R ) ,E ∂ : e O s +2 ,p −→ B s +1 − /p,p ( ∂ B ) are C ∞ .Proof. First, we recall that, by assumption ¯ τ iA , is a smooth function of its argu-ments ∂ A φ i in the neighborhood of the identity map ψ i . Since p > s ≥ s + 1 > /p , and it follows from theorem 1, Section 5.5.2, of [30], andthe continuity of differentiation (cf. (A.1)) that the map e O s +2 ,p ∋ φ τ iA ∈ W s +1 ,p ( B , R ) is C ∞ . The proof then follows directly from the continuity of thetrace map (A.3). (cid:3) Defining O s +2 ,p = e O s +2 ,p ∩ X s +2 ,p , we have that ψ ∈ O s +2 ,p by (3.38). We also define F : O s +2 ,p × X s,p × R −→ Y s,p × R : ( φ , φ , λ ) (cid:0) F ( φ , φ , λ ) , B ( φ , ψ ) (cid:1) where F ( φ , φ , λ ) = (cid:0) E ( φ ) + λ Λ( φ ) − φ , E ∂ ( φ ) (cid:1) , and we let P : O s +2 ,p → L ( Y s,p , Y s,p ) (3.45)denote any C ∞ map for which P ψ coincides with the projection operator fromtheorem 3.9. Furthermore, we assume that for each φ ∈ O s +2 ,p , the linear operator P ( φ ) | Y s,pφ : Y s,pφ −→ Y s,pψ (3.46)is an isomorphism and Y s,p = Y s,pφ ⊕ ker P ( φ ) . (3.47)The existence of a map (3.45) satisfying (3.46) and (3.47) can be found in [20]. For two Banach spaces X and Y , L ( X, Y ) denotes the set of continous linear maps from X to Y . YNAMICAL ELASTIC BODIES IN NEWTONIAN GRAVITY 21
For r ∈ Z ≥ , we define φ r = (cid:0) φ , φ , . . . , φ r (cid:1) T ,φ r ( t ) = φ + r X j =1 λ t j j ! φ j ,φ r, ( t ) = r X j =0 λ t j j ! φ j +2 , and F ( φ , φ , λ ) = F ( φ , φ , λ ) ,F ( φ , φ , λ ) = 1 λ ( F ( φ , φ , λ ) , φ , ( t ) , λ ) , λB ( ψ , φ )) ,F r +2 ( φ r , φ r +2 , λ ) = 1 λ (cid:18) F r +2 ( φ r +2 , φ r +4 , λ ) , d r dt r (cid:12)(cid:12)(cid:12) t =0 B ( φ ( t ) , φ , ( t )) (cid:19) ( r ≥ F r ( φ r , φ r +2 , λ ) = d r dt r (cid:12)(cid:12)(cid:12) t =0 F ( φ r ( t ) , φ r, ( t ) , λ ) . Remark . Under the identification φ r = (cid:0) ∂ rt φ (cid:1)(cid:12)(cid:12) t =0 , it follows from the definition of φ r ( t ) and φ r, ( t ) above that d r dt r (cid:12)(cid:12)(cid:12) t =0 φ r ( t ) = λ (cid:0) ∂ rt φ (cid:1)(cid:12)(cid:12) t =0 and d r dt r (cid:12)(cid:12)(cid:12) t =0 φ r, ( t ) = λ (cid:0) ∂ r +2 t φ (cid:1)(cid:12)(cid:12) t =0 . Moreover, under this identification, we have that F ( φ , φ , λ ) = 0if and only if (cid:0) ∂ A ¯ τ iA ( ∂φ ) − λ δ ij f Aj ∂ A ¯ U ( φ ) − ∂ t φ i (cid:1) | t =0 = 0 and (cid:0) ν A ¯ τ iA ( ∂φ ) | ∂ B (cid:1) | t =0 = 0 . Also, by repeatedly differentiating the equations of motion (2.23a)-(2.23c), it is notdifficult to see that F r ( φ r , φ r +2 , λ ) = 0 (3.48)if and only if ∂ rt (cid:0) ∂ A τ iA ( ∂φ ) − λ δ ij f Aj ∂ A ¯ U ( φ ) − ∂ t φ i (cid:1) | t =0 = 0 and ∂ rt (cid:0) ν A ¯ τ iA ( ∂φ ) | ∂ B (cid:1) | t =0 = 0 . This shows that solving (3.48) for r = 0 , , . . . , ℓ will produce initial data thatsatisfies the compatibility conditions to order ℓ .In order to use the implicit function theorem to solve the equations (3.48), weneed to introduce the following maps which are a closely related to the F r and F r maps introduced above: G ( φ , φ , λ ) = P ( φ ) F ( φ , φ , λ ) ,G ( φ , φ , λ ) = ( G ( φ , φ , λ ) , B ( ψ , φ )) ,G ( φ , φ , λ ) = 1 λ ( G ( φ , φ , λ ) , λB ( ψ , φ )) ,G r +2 ( φ r +2 , φ r +4 , λ )= 1 λ (cid:18) d r +2 dt r +2 (cid:12)(cid:12)(cid:12) t =0 G r +2 ( φ r +2 ( t ) , φ r +4 , λ ) , d r dt r (cid:12)(cid:12)(cid:12) t =0 B ( φ ( t ) , φ , ( t )) (cid:19) , ( r ≥ G r ( φ r , φ r +1 , φ r +2 , λ ) = G ( φ , φ , λ ) G ( φ , φ , λ )... G r − ( φ r − , φ r +1 , λ ) G r ( φ r , φ r +2 , λ ) where G r ( φ r , φ r +2 , λ ) = d r dt r (cid:12)(cid:12)(cid:12) t =0 G ( φ r ( t ) , φ r, ( t ) , λ ) . Proposition 3.15.
Suppose s ∈ Z ≥ r and < p < ∞ . Then the maps F r : e O s +2 ,p × (cid:16) r +2 Y j =1 W s +2 − j,p ( B , R ) (cid:17) × R Y s − r,p × R and G r : e O s +2 ,p × (cid:16) r +2 Y j =1 W s +2 − j,p ( B , R ) (cid:17) × R Y s − r,p × R are C .Proof. First we note that the maps F and G are C ∞ which follows from Corollary3.8, proposition 3.13, and the smoothness of the map (3.45). The proof then followsimmediately from lemma 3.12 and the definition of F r and G r . (cid:3) Introducing E ( φ, λ ) = (cid:0) E ( φ ) + λ Λ( φ ) , E ∂ ( φ ) (cid:1) , we get that F ( φ , φ , λ ) = (cid:0) E ( φ , λ ) − ( φ , , B ( ψ , φ ) (cid:1) , and it follows easily from the definition of the F r maps above that F ( φ , φ , λ ) = (cid:0) D φ E ( φ , λ ) · φ − ( φ , , B ( ψ , φ ) (cid:1) , and F ( φ , φ , λ ) = (cid:0) D φ E ( φ , λ ) · φ − ( φ ,
0) + λD φ E ( φ , λ ) · ( φ , φ ) , B ( φ , φ ) (cid:1) . Proceeding inductively, we obtain for r ≥ F r ( φ r , φ r +2 , λ ) = (cid:0) D φ E ( φ , λ ) · φ r − ( φ r +2 , , B ( φ , φ r ) (cid:1) + λ ˜ F r ( φ r +1 , λ ) (3.49)where the map ˜ F r is C . Setting, P r ( φ r ) = d r dt r (cid:12)(cid:12)(cid:12) t =0 P ( φ r ( t )) , YNAMICAL ELASTIC BODIES IN NEWTONIAN GRAVITY 23 the product rule shows that G r ( φ r , φ r +2 , λ ) = r X j =0 (cid:18) rs (cid:19) P r − j ( φ r − j ) F r ( φ j , φ j +2 , λ ) . (3.50)This and (3.49), in turn, give G r ( φ r , φ r +2 , λ ) = (cid:0) P ( φ ) (cid:2) D φ E ( φ , λ ) · φ r − ( φ r +2 , (cid:3) , B ( φ , φ r ) (cid:1) + λ ˜ G r ( φ r +1 , λ ) r ≥ G r is C .Letting ψ r = ( ψ , , . . . , , it follows directly from (3.51) that the derivative of G r evaluated at ( φ r , φ r +1 , φ r +2 , λ ) =( ψ r , , ,
0) is D φ r G r ( ψ v , , , · δ φ r = A · δ φ r (3.52)where A = L ( ψ ) 0 M ( ψ ) 0 · · · L ( ψ ) 0 M ( ψ ) ...0 0 L ( ψ ) 0 . . . 0... ... 0 L ( ψ ) M ( ψ ). . . 00 0 0 · · · L ( ψ ) (3.53) L ( ψ ) · δψ = (cid:0) P ( ψ ) A δψ, B ( ψ , δψ ) (cid:1) , (3.54)and M ( ψ ) · δψ = (cid:0) − P ( ψ )( δψ, , (cid:1) . We note that in deriving (3.52)-(3.53), we have used E ( ψ ,
0) = 0 and D φ E ( ψ , · δψ = A δψ. Although our main objective is to solve the equations (3.48), we first solve G r = 0and later show that this implies that compatibility conditions are satisfied to order r . To solve G r = 0, we use the implicit function theorem. The proof we present ismodeled on the existence proof for static, self-gravitating elastic bodies presentedin [7]. Proposition 3.16.
There exists an λ > , open neighborhoods N r +1 ⊂ X s +1 − r,p and N r +2 ⊂ X s − r,p both containing , and C maps Φ j : N r +1 ×N r +2 × ( − λ , λ ) −→ X s +2 − j,p : ( φ r +1 , φ r +2 , λ ) Φ j ( φ r +1 , φ r +2 , λ ) , for j = 0 , , . . . , r , such that Φ (0 , ,
0) = ψ , Φ j (0 , ,
0) = 0 , j = 1 , , . . . , r and G r (cid:0) Φ r ( φ r +1 , φ r +2 ) , φ r +1 , φ r +2 , λ (cid:1) = 0 ∀ ( φ r +1 , φ r +2 , λ ) ∈ N r +1 ×N r +2 × ( − λ , λ ) where Φ r = (Φ , Φ , . . . , Φ r ) .Proof. We begin by verifying the the linear operator (3.53) is an isomorphism.
Lemma 3.17.
For s ∈ Z ≥ r and < p < ∞ , the map A : r Y j =0 X s +2 − j,p −→ r Y j =0 (cid:0) Y s − j,p × R (cid:1) is a linear isomorphism.Proof. By lemma 3.11, ψ ∈ U s,p and so it follows from the definition of L ( ψ ) (see(3.54)) and theorem 3.9 that L ( ψ ) : X s +2 − j,p −→ Y s − j,p × R is an isomorphism for j = 0 , , . . . r . The proof now follows immediately from upper triangular structureof the map A (see (3.53)). (cid:3) Recalling that E ( ψ ) = 0, it is clear from the definition of F that F ( ψ , ,
0) = 0 , and hence, by the antisymmetry of the map B , that G ( ψ , ,
0) = 0 . Furthermore, it clear from (3.51) that G j ( ψ j , ,
0) = 0 j = 1 , , . . . , r, which, in turn, shows that G r ( ψ r , , ,
0) = 0 . (3.55)The proof of the proposition now follows proposition 3.15, lemma 3.17, and theimplicit function theorem. (cid:3) Theorem 3.18.
The maps Φ j j = 0 , , . . . , r from proposition 3.16 satisfy F j (cid:0) Φ j ( φ r +1 , φ r +2 ) , Φ j +2 ( φ r +1 , φ r +2 ) , λ (cid:1) = 0 j = 0 , , . . . , r − , F r − (cid:0) Φ r − ( φ r +1 , φ r +2 ) , φ r +1 , λ (cid:1) = 0 and F r (cid:0) Φ r ( φ r +1 , φ r +2 ) , φ r +2 , λ (cid:1) = 0 for all ( φ r +1 , φ r +2 , λ ) ∈ N r +1 × N r +2 × ( − λ , λ ) .Proof. It is shown in [7] that C (cid:0) E ( φ, λ ) (cid:1) = 0 and C (cid:0) φ, E ( φ, λ ) (cid:1) = 0 (3.56)are automatically satisfied for all φ ∈ O s +2 ,p and − λ < λ < λ . Setting E j ( φ j , λ ) = d j dt j (cid:12)(cid:12)(cid:12) t =0 E ( φ r ( t ) , λ ) j = 0 , , . . . , r, the formulas C (cid:0) E j ( φ j , λ ) (cid:1) = 0 and j X k =0 (cid:18) jk (cid:19) C (cid:0) φ k , E j − k ( φ j − k , λ ) (cid:1) = 0 j = 0 , , . . . , r follow from differentiating (3.56). In particular, this implies that the maps Φ j fromtheorem 3.16 satisfy C (cid:0) E j ( Φ j (cid:0) φ r +1 , φ r +2 , λ )) , λ (cid:1)(cid:1) = 0 for j = 0 , , . . . , r (3.57)and for j = 0 , , . . . , r − j X k =0 (cid:18) jk (cid:19) C (cid:0) Φ k (cid:0) φ r +1 , φ r +2 , λ (cid:1) , E j − k (cid:0) Φ j − k (cid:0) φ r +1 , φ r +2 , λ (cid:1) , λ (cid:1)(cid:1) = 0 , (3.58)for all ( φ r +1 , φ r +2 , λ ) ∈ N r +1 × N r +2 × ( − λ , λ ) YNAMICAL ELASTIC BODIES IN NEWTONIAN GRAVITY 25
Next, we note that C (( φ j , , ∀ φ r ∈ X s +2 − j,p , (3.59)and d j dt j (cid:12)(cid:12)(cid:12) t =0 B (cid:0) φ j ( t ) , φ ,j ( t ) (cid:1) = 0 ⇐⇒ d j dt j (cid:12)(cid:12)(cid:12) t =0 C (cid:0) φ j ( t ) , ( φ ,r ( t ) , (cid:1) = 0 ⇐⇒ j X k =0 (cid:18) jk (cid:19) C (cid:0) φ k , ( φ j +2 − k , (cid:1) = 0 . (3.60)If we defineΦ r +1 ( φ r +1 , φ r +2 , λ ) = φ r +1 and Φ r +2 ( φ r +1 , φ r +2 , λ ) = φ r +2 , then it follows from (3.59), (3.60), and the definition of G r that the maps Φ j satisfy C (cid:0) Φ j ( φ r +1 , φ r +2 , λ ) , (cid:1) = 0 j = 0 , , . . . , r + 2 , (3.61)and j X k =0 (cid:18) jk (cid:19) C (cid:0) Φ k ( φ r +1 , φ r +2 , λ ) , (cid:0) Φ j +2 − k ( φ r +1 , φ r +2 , λ ) , (cid:1)(cid:1) = 0 j = 0 , , . . . , r (3.62)for all ( φ r +1 , φ r +2 , λ ) ∈ N r +1 × N r +2 × ( − λ , λ ).Fixing ( φ r +1 , φ r +2 , λ ) ∈ N r +1 × N r +2 × ( − λ , λ ), the identities (3.57), (3.58),(3.61), (3.62) and the definition of the maps F j show that C (cid:0) F j ( Φ j ( φ r +1 , φ r +2 , λ ) , Φ j +2 ( φ r +1 , φ r +2 , λ ) , λ (cid:1)(cid:1) = 0 , (3.63)and j X k =0 (cid:18) jk (cid:19) C (cid:0) Φ k ( φ r +1 , φ r +2 , λ ) , F j − k ( Φ j − k ( φ r +1 , φ r +2 , λ ) , Φ j +2 − k ( φ r +1 , φ r +2 , λ ) , λ (cid:1)(cid:1) = 0(3.64)for 0 ≤ j ≤ r , while G (cid:0) Φ ( φ r +1 , φ r +2 , λ ) , Φ ( φ r +1 , φ r +2 , λ ) , λ (cid:1) = 0follows from theorem 3.16, or equivalently, by the definition of G , and P (cid:0) Φ ( φ k +1 , φ r +2 , λ ) (cid:1) F (cid:0) Φ ( φ r +1 , φ r +2 , λ ) , Φ ( φ r +1 , φ r +2 , λ ) , λ (cid:1) = 0 . (3.65)But, we also have that F (cid:0) Φ ( φ r +1 , φ r +2 , λ ) , Φ (cid:0) φ r +1 , φ r +2 , λ (cid:1) , λ (cid:1) ∈ Y s,p Φ ( φ k +1 ,φ r +2 ,λ ) by (3.63)-(3.64), and thus, F (cid:0) Φ ( φ r +1 , φ r +2 , λ ) , Φ ( φ r +1 , φ r +2 , λ ) , λ (cid:1) = 0by (3.47) and (3.65).To finish the proof, we proceed by induction. So, we assume that F (cid:0) Φ j ( φ r +1 , φ r +2 , λ ) , Φ j +2 ( φ r +1 , φ r +2 , λ ) , λ (cid:1) = 0 (3.66)for 0 ≤ k ≤ j < r . Then G k (cid:0) Φ k ( φ r +1 , φ r +2 , λ ) , Φ k +2 ( φ r +1 , φ r +2 , λ ) (cid:1) = 0 (3.67) for 0 ≤ k ≤ j by theorem 3.16. Clearly, (3.50), (3.66), and (3.67) imply that P (cid:0) Φ ( φ k +1 , φ r +2 , λ ) (cid:1) F j +1 (cid:0) Φ j +1 ( φ r +1 , φ r +2 , λ ) , Φ j +2 ( φ r +1 , φ r +2 , λ ) , λ (cid:1) = 0 . (3.68)But, since F j +1 (cid:0) Φ j +1 ( φ r +1 , φ r +2 , λ ) , Φ j +3 (cid:0) φ r +1 , φ r +2 , λ (cid:1) , λ (cid:1) ∈ Y s,p Φ ( φ k +1 ,φ r +2 ,λ ) by (3.63)-(3.64), we must, in fact, have F j +1 (cid:0) Φ j +1 ( φ r +1 , φ r +2 , λ ) , Φ j +3 ( φ r +1 , φ r +2 , λ ) , λ (cid:1) = 0by (3.47) and (3.68), and the proof is complete. (cid:3) In light of remark 3.14, the following corollary is a direct consequence of theabove theorem.
Corollary 3.19.
Suppose s ∈ Z ≥ , < p < ∞ , and the maps Φ j , the neighborhoods N s +1 ⊂ X ,p ⊂ W ,p ( B , R ) and N s +2 ⊂ X ,p ⊂ W ,p ( B , R ) , and λ are as inproposition 3.16. Then for each ( φ s +1 , φ s +2 , λ ) ∈ N s +1 × N s +2 × ( − λ , λ ) , theinitial data ( φ | t =0 , ∂ t | t =0 φ ) = (cid:0) Φ ( φ r +1 , φ r +2 , λ ) , Φ ( φ r +1 , φ r +2 , λ ) (cid:1) ∈ e O s +2 ,p × e O s +1 ,p satisfy the compatibility conditions to order s . Local well-posedness
In this section, we prove local well-posedness for the system (2.23a)-(2.23c) usingthe approach of [19]. The system under consideration is an initial-boundary valueproblem of elliptic-hyperbolic type, due to the presence of the equation (2.23b)in the system, and hence the results of [19] do not apply directly. However, thetechniques of [19] are readily adapted to include non-local terms, and we will presentan outline of the proof of this fact below. To conclude this section, we will applythe resulting existence theorem to establish the existence of dynamical solutions to(2.23a)-(2.23c).4.1.
Setup and notation.
For ease of reference, we adopt the index and othernotational conventions of [19], with some exceptions, as pointed out below. We areinterested only in the case n = 3, but it is convenient to treat the case of general n ≥
3. The number of components of the system of equations is in the case ofelasticity equal to N = 3, but the treatment below applies to general N .Let 0 ≤ i, k ≤ n , 1 ≤ α, β ≤ n , 1 ≤ j, l ≤ N . We work in a coordinatesystem ( x i ) and let t = x . The summation convention is used. Further, we shalldenote the unknown field by u rather than φ . Let Du = ( ∂ t u, ∂ x u, . . . , ∂ x n u ),and D x u = ( ∂ x u, . . . , ∂ x n u ). Fix some s > n/ s an integer, and considerthe system in the domain Ω with boundary Γ of regularity class C s +2 , and denoteΩ T = [0 , T ) × Ω and Γ T = [0 , T ) × Γ where 0 < T ≤ ∞ .Following [19], we consider equations of the form ∂ x i F ij ( t, x, u, Du ) = w j [ t, x, u, Du ] in Ω T , (4.1a) v j F ij ( t, x, u, Du ) = g j ( t, x, u, Du ) in Γ T , (4.1b) u = u , ∂ t u = u in { } × Ω (4.1c)where, in contrast to [19], w j [ t, x, u, Du ] is a functional of ( u, Du ) that we allow tobe non-local. The properties required of w are specified below in Assumption w . YNAMICAL ELASTIC BODIES IN NEWTONIAN GRAVITY 27
Assumption 1.
We assume u ∈ W s +1 (Ω), u ∈ W s (Ω). We assume F, g ∈ C s +1 ( U )where U is a neighborhood of the graph of u , u , D x u as in [19, § a ikjl = ∂F ij ∂ ( ∂ x k u l ) , h kjl = ∂g j ∂ ( ∂ x k u l ) . We decompose h kjl = h skjl + h ukjl where h skjl = h sk ( jl ) is the symmetric part and h ukjl = h uk [ jl ] is the antisymmetric part. We assume that the symmetric part h sk ( jl ) is of theform h sk ( jl ) = θ k h jl where θ ( t, x, u, Du ) is a vector field which is tangential to Γ T and satisfies θ = 1, and h jl is symmetric.For the elastic body, we have h kjl = 0 (4.2)while a ikjl can be calculated in terms of the elasticity tensor L , cf. (2.16). Assumptions 2-5.
The structural relations of the elastic body imply the hyperbolic-ity of the system. In particular, for the self-gravitating elastic body, the symmetryand coerciveness assumptions, Assumptions 2 and 3 of [19], hold. Further, Assump-tion 4 of [19] on the time components a jl follow directly from the structure of theelastic system. For a discussion of the compatibility conditions on the initial data,see Assumption 5 in [19].Following [19, p. 25], we introduce the spaces E s , G s , and F s with norms || u || E s ( t ) = s X i =0 || ∂ it u ( t ) || W s +1 − i ! / , || u || G st ,t = sup t ≤ t ≤ t || u || E s ( t ) , and || u || F st ,t = Z t t || u || E s ( t ) dt, respectively. Assumption w . For non-local w , we make the following further assumptions. Weassume w to be well defined if the graph of ( u, Du ) lies in a suitable subset of U ,where U as in Assumption 1 above. There, the following conditions are imposed.(1) If u ∈ ∩ ≤ j ≤ s C j ([0 , T ] , W s +1 − j (Ω)) , we have w [ u, Du ] ∈ ∩ ≤ j ≤ s C j ([0 , T ] , W s − j (Ω)) . (2) We require the map u w [ u, Du ] to be Lipschitz in the above topology.In particular, see section 4.3.1 below, we shall make use of the estimate || ∂ t ( w ( v ) − w ( v )) || L ≤ c || ( ∂ t ( v − v ) || E . (4.3)(3) Finally, we require the following uniform estimate || w || E s ≤ c (1 + || D u || L ∞ )(1 + || u || E s +1 )where c is a constant depending on || Du || L ∞ as well as the coercivity con-stants κ, µ for the system.We have the following result, which is the analog of [19, theorem 1.1]. Theorem 4.1. (1)
Existence, regularity: There exists a unique < t ≤ T , and a uniqueclassical solution u ∈ C (Ω t ∪ Γ t ) of (4.1) with D σ u ( t ) ∈ L (Ω) if ≤ σ ≤ s + 1 . Here D σ u denotes all derivatives of order σ . (2) Continuous dependence on initial data. (3)
Blow up: t is characterized by the two alternatives: either the graph of ( u, Du ) is not precompact in U or Z t || D u ( τ ) || L ∞ (Ω) dτ → ∞ , as t → t . Linear systems.
For a solution to the system (analogous to [19, (2.10)]) ∂ x i ( a ik ∂ x k u ) = w + ∂ x i f i in Ω T , (4.4a) v α ( a αk ∂ x k u − f α ) = h sk ∂ x k u in Γ T (4.4b)with the coercivity and structure conditions as in [19, § a, h ∈ G s ∩ C (Ω T ) ∂ t a, ∂ t h ∈ F s , (4.5a) w ∈ G s − ∩ W s, ([0 , T ] , L (Ω)) ∩ W s − , ([0 , T ] , W (Ω)) , (4.5b) f i ∈ G s , Df i ∈ W s, ([0 , T ] , L (Ω)) , (4.5c) u ∈ W s +1 (Ω) u ∈ W s (Ω) , (4.5d)we have the estimate || u || E s +1 ( t ) ≤ ˜ c (cid:18) || u ( t ) || E s +1 + || w || G s − + || f || G s + Z t t || ∂ st w ( t ) || L (Ω) + || ∂ s +1 t f i ( t ) || L (Ω) dt (cid:19) (4.6)where ˜ c = ˜ c ( κ, µ, || a || G s ∩ C (Ω T ) , || ∂ t a || F s , || h s || G s ∩ C (Ω T ) , || ∂ t h s || F s ) . Note that the system given in (4.4) is of a restricted form with g = 0, h u = 0.These terms can be absorbed into the others, cf. the discussion in [19, § a ik , ¯ f i , ¯ w as in [19, p. 31], seealso (4.12) below.Applying the above estimate to a system of the form ∂ x i ( a ik ∂ x k u ) = w + ∂ x i f i in Ω T (4.7a) v α ( a αk ∂ x k u − f α ) = h k ∂ x k u + g in Γ T , (4.7b)we get, in view of the above discussion, the inequality || u || E s +1 ( t ) ≤ ˜ c (cid:18) || u ( t ) || E s +1 + || w || G s − + || f || G s + || g || G s + Z t t || ∂ st w ( t ) || L (Ω) + || ∂ s +1 t f i ( t ) || L (Ω) dt + Z t t || ∂ st g || W (Ω) + || ∂ s +1 t g || L (Ω) ds (cid:19) (4.8) YNAMICAL ELASTIC BODIES IN NEWTONIAN GRAVITY 29
Proof of theorem 4.1.
In this section, we discuss the main steps in the proofof theorem 4.1. In the following calculations, we suppress the indices u j on u andthe corresponding indices on a ikjl , w j , etc. First, we apply a time derivative to (4.1)which gives ∂ x i ( a ik ∂ x k ,t u ) = ∂ t w + ∂ x i f i in Ω T , (4.9a) v α ( a αk ∂ x k ,t u − f α ) = h k ∂ x k ,t + ¯ g in Γ T , (4.9b)where f i ( t, x, u, Du ) = a ik ∂ x k ,t u − ∂ t F i , ¯ g = ∂ t g − h k ∂ x k ,t u. For the elastic system, the coefficients have no explicit time dependence, and theboundary condition is homogenous. So we have that f i = 0 , ¯ g = 0 . (4.10)Next, the system is rewritten in the form ∂ x i (¯ a ik ∂ x k ,t u ) = ¯ w + ∂ x i ¯ f i in Ω T , (4.11a) v α (¯ a αk ∂ x k ,t u − ¯ f α ) = h sk ∂ x k ,t u in Γ T , (4.11b)where¯ f i = f + v i ¯ g, ¯ a ik = a ik − v i h uk + v k h ui (4.12a)and¯ w = ∂ t w − (( ∂ v + div v ) h uk ) ∂ x k ,t u + ( ∂ x i v β h ui ) ∂ x β ,t u − ∂ v ¯ g − (div v )¯ g. (4.12b) Remark . By introducing the modifications ¯ a ik , ¯ f i , ¯ w of a ik , f i , w as in (4.12),the resulting system (4.11) has no term g and also h u = 0. Thus it is of the formof the system (4.4) considered in [19, theorem 2.4].For the elastic system, we have f = g = h = 0 and hence¯ a ik = a ik , ¯ w = ∂ t w. (4.13)We have that ¯ a, ¯ f depend on ( t, x, u, Du ) and ¯ w depends linearly on D u .For technical reasons, we assume s > n/ s .This is one more degree of smoothness than one would normally expect to requirefor a solution of a non-linear wave equation using energy estimates. However, thisassumption reflects the use of v = ∂ t u as the main variable in Koch’s approach [19],which has one less degree of differentiability compared to u . The stated result forinitial data with s > n/ Y τ,R be the subset of H τ = ∩ ≤ i ≤ s +1 W i, ∞ ([0 , τ ] , W s +1 − i (Ω))of functions v that satisfy ∂ it v (0) = u i , where u i are the formal time derivatives of u for 0 ≤ i ≤ s at t = 0, and || v || H τ ≤ R . By choosing R sufficiently large, we canmake sure this set is non-empty.The construction of solutions for the system (4.1) makes use of a standard fixedpoint argument, where one proves boundedness in a high norm and contraction in We write ∂ x k ,t where ∂ x k t is used in [19] a low norm. The high norm in this case is W s +1 , which for the linearized (time-differentiated) system corresponds to W s . The low norm for the time-differentiatedsystem is W .This type of argument has been carried out for a quasi-linear elliptic-hyperbolicsystem with no boundary conditions in [4]. The difference between that system andthe present situation is that we have neumann-type boundary conditions, and thesystem has symbol depending on Du , i.e. it is fully non-linear.In the rest of this section, we consider the details of the contraction estimateand the continuation property. The proof of the continuous dependence given in[19] can be readily adapted to the present case with the details given below.4.3.1. Contraction estimate.
Define J ∈ C ( Y τ,R , H τ ) as the map v u , where u solves the linear system ∂ x i (¯ a ik ( v ) ∂ x k ,t u ) = ¯ w ( v ) + ∂ x i ¯ f i in Ω T , (4.14a) v α (¯ a αk ( v ) ∂ x k ,t u − ¯ f α ( v )) = h k ( v ) ∂ x k ,t u in Γ T , (4.14b) ∂ t u = u , ∂ t u = u , in { } × Ω (4.14c)where we have denoted a ( v ) = a ( t, x, v, Dv ) etc.For the case of the elastic system, we again have¯ a ik = a ik , ¯ w = ∂ t w, ¯ f = 0 , h = 0 , and in particular, ¯ w = D u w.∂ t u + D D x u w.D x ∂ t u where we have used D to denote the Frechet derivative. There is no explicit ( t, x )dependence for this case.Next, we observe that ∂ t w ∈ G s − , ∂ s +1 t w ∈ F by Assumption w . From this, we see that for large R , the image of J lies in Y provided τ is chosen small enough.We have Y ⊂ C ([0 , τ ] , L ) ∩ C ([0 , τ ] , W ) ∩ C ([0 , τ ] , W )One shows that the set Y is compact in the topology of the space defined by theright hand side of this expression.Let u = J ( v ), u = J ( v ). In order to derive the system solved by u − u , welet Z ( u ) = ∂ x i ¯ a ik ( t, x, u, Du ) ∂ x k ,t u , and note that Z ( u ) − Z ( u ) = R ddλ Z ( u + λ ( u − u )). Expanding this out gives Z ( u ) − Z ( u ) = ∂ x i ([ Z ¯ a ikλ dλ ] ∂ x k ,t ( u − u )+ ∂ x i ( Z [ e a ikλ,u . ( u − u )] ∂ x k ,t ( u + λ ( u − u ))) dλ + ∂ x i ( Z [ e a ikλ,Du . ( Du − Du )] ∂ x k ,t ( u + λ ( u − u ))) dλ A typo in the boundary condition in [19, eq. (3.2)] is corrected here
YNAMICAL ELASTIC BODIES IN NEWTONIAN GRAVITY 31 where ¯ a ikλ = ¯ a ik ( t, x, u + λ ( u − u ) , Du + λ ( Du − Du )) , ¯ a ikλ,u = ∂ ¯ a∂u ( t, x, u + λ ( u − u ) , Du + λ ( Du − Du )) , ¯ a ikλ,Du = ∂ ¯ a∂Du ( t, x, u + λ ( u − u ) , Du + λ ( Du − Du )) . Now, let e a ik = Z ¯ a ikλ dλ e w = ¯ w ( v ) − ¯ w ( v ) − (cid:20) ∂ x i ( Z [¯ a ikλ,u . ( u − u )] ∂ x k ,t ( u + λ ( u − u ))) dλ + ∂ x i ( Z [¯ a ikλ,Du . ( Du − Du )] ∂ x k ,t ( u + λ ( u − u ))) dλ (cid:21)e f = ¯ f ( v ) − ¯ f ( v ) . Then u − u solves ∂ x i ( e a ik ∂ x k ,t ( u − u )) = e w + ∂ x i e f i , (4.15a) v α ( e a αk ∂ x k ,t ( u − u ) − e f α ) = e h k ∂ x k ,t ( u − u ) + e g (4.15b)where e h, e g can be calculated along the same lines as above.In particular, for the elastic system, we may calculate e w using ¯ w = ∂ t w , ¯ h =¯ f = ¯ g = 0. Note that e g is non-vanishing in general due to contributions from ¯ a .We have the estimate || ∂ t e w ( t ) || L + || ∂ t e f || E ( t ) + || ∂ t e g || E ( t ) ≤ c || ∂ t ( v − v ) || E (4.16)where we made use of e w = ¯ w ( v ) − ¯ w ( v ) + terms involving ¯ a. As discussed above, ¯ w can be estimated given estimates for ∂ t w , and hence ∂ t e w canbe estimate in terms of ∂ t ( w ( v ) − w ( v ). The terms involving ¯ a can be estimatedin terms of u − u and hence can be absorbed when applying Gronwall.From Assumption w , (4.3), we have || ∂ t ( w ( v ) − w ( v )) || L ≤ c || ∂ t ( v − v ) || E . The required contraction estimate is obtained by applying the estimate for the linearsystem given in [19, theorem 2.4], as discussed in section 4.2, to the system (4.15).One checks that after suitable modifications, cf. section 4.2, the assumptions of[19, theorem 2.4] holds for this system, and this provides the needed contractionestimates. A typo in [19, p. 32] is corrected here. This corrects a typo in [19, p. 32]. This corrects a typo in [19, p. 32]., which leads to an incorrect estimate for ∂ t ( u − u ). Applying the inequality (4.8) with s = 1 to the system (4.15), we get the in-equality || ∂ t ( u − u ) || E ≤ c (cid:18) || u (0) − u (0) || E + || e w || G + || e f || G + || e g || G + Z t || ∂ t e w ( σ ) || L + || ∂ t e f i || E + || ∂ t e g ( σ ) || E dσ (cid:19) use that u , u have the same initial data, and that e w, e f , e g vanish at t = 0 ≤ c (cid:18)Z t || ∂ t e w ( σ ) || L + || ∂ t e f i || E + || ∂ t e g ( σ ) || E dσ (cid:19) use (4.16) ≤ c || ∂ t ( v − v ) || E where we made use of the fact that e w, e f , e g all vanish at t = 0.4.3.2. Continuation principle.
We next consider the proof of the continuation prin-ciple. Suppose the graph of ( u, Du ) lies in a compact set U . We need an estimateof the following form, cf. [19, eq. (3.3)] || u ( t ) || E s +1 ≤ ˜ c (cid:18) U, Z t || D u ( τ ) || L ∞ dτ (cid:19) (1 + || u || W s +1 + || u || W s ) . (4.17)This estimate is proved by applying operators D sP of order s , tangent to Γ τ , toboth sides of the equation (4.1), and applying the estimate for the linear system.Let s > n/ u ∈ G s +1 be a solution of the system (4.1). We then have u ∈ ∩ ≤ j ≤ s +1 C j ([0 , T ] , W s +1 − j ). One finds, cf. [19, p. 34], that D sP u solves anequation of the form ∂ x i (¯ a ik ∂ x k D sP u ) = b w + ∂ x i b f i in Ω T , (4.18a) v i (¯ a ik ∂ x k D sP u − b f i ) = h sk ∂ x k D sP u in Γ T , (4.18b) D sP u (0) = u s , ∂ t D sP u = u s (4.18c)where u s , u s are obtained by formal calculations from the initial data u , u . Herea transformation which absorbs the terms g and h uk has been applied, along thelines discussed in section 4.2, see [19, p.34] for details.The basic energy estimate for systems of this type, see [19, theorem 2.2], givesan estimate of the form || u || E s +1 ≤ c (cid:18) || u || E s +1 (0) + Z t || b w || L ( τ ) + || ∂ t b f i || L ( τ ) dτ (cid:19) (4.19)As shown in [19], the L norms in the right hand side of (4.19) that involve localexpressions can be estimated at a fixed time in terms of(1 + || D u || L ∞ )(1 + || u ( t ) || E s +1 ) The norms || · || in [19, p. 35] should be || · || L . YNAMICAL ELASTIC BODIES IN NEWTONIAN GRAVITY 33 with a constant depending on || Du || L ∞ as well as the coercivity constants κ, µ . Theterm b w contains D sP w , and thus we need the nonlocal term w to satisfy at a fixedtime an estimate of precisely this form, namely || D sP w || L ≤ c (1 + || D u || L ∞ )(1 + || u || E s +1 )which we can state as || w || E s ≤ c (1 + || D u || L ∞ )(1 + || u || E s +1 ) . This estimate holds by Assumption w .4.4. Application to the elastic system.
We are now ready to apply the localexistence theorem 4.1 to our system (2.23a)-(2.23c).
Theorem 4.3.
Suppose s ∈ Z ≥ < p < ∞ , and let ( φ i | t =0 , ∂ t | t =0 φ i ) = ( φ i , φ i ) ∈ e O s +2 ,p × e O s +1 ,p ⊂ W s +2 ( B ) × W s +1 ( B ) be the initial data from Corollary 3.19.Then there exists a t > and a unique classical solution φ i ∈ C ( B t ∪ ∂ B t ) of (2.23a) - (2.23c) with D σ φ i ( t ) ∈ L ( B ) for all ≤ | σ | ≤ s + 1 and ≤ t < t .Proof. First, we observe by Corollary 3.8 that non-local function Λ i ( φ ) = − δ ij f Ai ∂ A ¯ U ( φ )satisfies Assumption w of section 4.1 for φ i in the open set e O s +1 ⊂ W s +1 ( B ). Sincethe initial data( φ i | t =0 , ∂ t | t =0 φ i ) = ( φ i , φ i ) ∈ e O s +2 ,p × e O s +1 ,p ⊂ e O s +2 , × e O s +1 , ⊂ W s +2 ( B ) × W s +1 ( B )from Corollary 3.19 satisfies the compatibility conditions to order s , the proof thenfollows directly from theorem 4.1. (cid:3) Acknowledgements.
Part of this work was completed during visits of the authorsT.A.O. and B.G.S. to the Albert Einstein Institute. We are grateful to the Instituteits support and hospitality during these visits.
Appendix A. Function Spaces
A.1. W k,p spaces. Give a finite dimensional vector space V , an open subset Ω ⊂ R with a C ∞ boundary, k ∈ Z , and 1 ≤ p ≤ ∞ , we let W k,p (Ω , V ) denote thestandard Sobolev space for maps u : Ω → V . If V = R , then we will just write W k,p ( B ), while if p = 2 we set W k (Ω , V ) = W k, (Ω , V ).For these spaces, we recall the following results:(i) Differentiation ∂ A = ∂∂X A : W k,p (Ω) −→ W k − ,p (Ω) A = 1 , , ≤ p < ∞ , k + k > k , k ≥ k , k
0, and k ∈ R , the Laplacian∆ = δ AB ∂ A ∂ B : W k +2 ,pδ ( R ) −→ W k,pδ − ( R ) (A.7)is a linear isomorphism with inverse given by the formula[∆ − ( u )]( X ) = − π Z R u ( Y ) | X − Y | d Y. (A.8)(The proof of this statement follows from using the fact that ∆ : W k +2 ,pδ ( R ) → W k,pδ − ( R ) is an isomorphism for k ∈ Z ≥ and − < δ < .)For bounded Ω ⊂ R , we let E Ω denote an extension operator that satisfies k E Ω ( u ) k W k,p − ( R ,V ) ≤ C k,p k u k W k,p (Ω ,V ) ∀ u ∈ W k,p (Ω , V ) , (A.9)and (cid:2) ∂ αA E Ω ( u ) (cid:3) | Ω = ∂ αA u ∀ u ∈ W k,p (Ω , V ) , | α | ≤ k. (A.10) Here | X | = p δ AB X A X B . As noted by Maxwell [25], the weighted spaces W k,pδ ( R ) and their fractional extensionscorrespond to the spaces h kp,p ( k − δ ) − in [34, 35] (cf. remark 2 and theorem 2 in [35]). Thefollowing duality and interpolation results follow from remark 2, and theorems 2 and 3 in [35]:(a) For 1 < p < ∞ and q = p/ ( p − W − k,q − − δ ( R ) is the dual of W k,pδ ( R ).(b) If 1 < p < ∞ , 1 < p < ∞ < θ < k = (1 − θ ) k + θk , δ = (1 − θ ) δ + θδ , and1 /p = (1 − θ ) /p + θ/P , then W k,pδ ( R ) is the interpolation space [ W k ,pδ ( R ) , W k ,pδ ( R )] θ . YNAMICAL ELASTIC BODIES IN NEWTONIAN GRAVITY 35
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E-mail address : [email protected] E-mail address : [email protected]@aei.mpg.de