Stability in Exponential Time of Minkowski Space Time with a Translation Space-Like Killing Field

Transcription

1 Ann. PDE 6 :7 DOI.7/s Stability in Exponential Time of Minkowski Space Time with a Translation Space-Like Killing Field Cécile Huneau Received: 6 February 5 / Accepted: 6 April 6 / Published online: May 6 Springer International Publishing AG 6 Abstract In this paper, we prove the nonlinear stability in exponential time of Minkowki space-time with a translation space-like Killing field. In the presence of such a symmetry, the 3 + vacuum Einstein equations reduce to the + Einstein equations with a scalar field. We work in generalised wave coordinates. In this gauge the Einstein equations can be written as a system of quasilinear quadratic wave equations. The main difficulty in this paper is due to the decay in t of free solutions to the wave equation in dimensions, which is weaker than in 3 dimensions. As in [], we have to rely on the particular structure of the Einstein equations in wave coordinates. We also have to carefully choose the behaviour of our metric in the exterior region to enforce convergence to Minkowski space-time at time-like infinity. Introduction In this paper, we address the quasi stability of the Minkowski solution to the Einstein vacuum equations with a translation space-like Killing field. In the presence of a translation space-like Killing field, the 3 + Einstein vacuum equations reduces to the following system in the polarized case see Appendix A { g φ =, R μν = μ φ ν φ.. This system has been studied by Choquet-Bruhat and Moncrief in [7] see also [6] in the case of a space-time of the form S R, where is a compact B Cécile Huneau cecile.huneau@univ-grenoble-alpes.fr Institut Fourier, Grenoble, France 3

2 7 Page of 5 C. Huneau two dimensional manifold of genus G, and R is the time axis, with a spacetime metric independent of the coordinate on S. They prove the existence of global solutions corresponding to the perturbations of a particular expanding universe. This symmetry has also been studied in [3], with an additional rotation symmetry. In this paper, we consider a space-time of the form R R x 3 R t,forwhich 3 is a Killing vector field. Minkowski space-time can be seen as a trivial solution of Einstein vacuum equations with this symmetry. The question we address in this paper is the stability of the Minkowski solution in this framework. In the 3 + vacuum case, the stability of Minkowski space-time has been proven in the celebrated work of Christodoulou and Klainerman in [8] in the maximal foliation. It has then been proven by Lindblad and Rodnianski using the wave coordinates in []. Their proof extends also to the Einstein equations coupled to a scalar field. In this work we will use wave coordinates.. Einstein Equations in Wave Coordinates Wave coordinates x α are required to satisfy g x α =. In these coordinates. reduces to the following system of quasilinear wave equations { g φ =, g g μν = μ φ ν φ + P μν g, g,. where P μν is a quadratic form. To understand the difficulty, let us first recall known results in 3 + dimensions. In 3 + dimensions, a semi linear system of wave equations of the form u i = P i u j, u k is critical in the sense that if there isn t enough structure, the solutions might blow up in finite time see the counter examples by John [3]. However, if the righthand side satisfies the null condition, introduced by Klainerman in [4], the system admits global solutions. This condition requires that P i be linear combinations of the following forms Q u,v= t u t v u. v, Q αβ u,v= α u β v α v β u. In three dimensions, the Einstein equations written in wave coordinates do not satisfy the null condition. However, this is not a necessary condition to obtain global existence. An example is provided by the system { φ =, φ = t φ..3 The non-linearity does not have the null structure, but thanks to the decoupling there is nevertheless global existence. In [], Lindblad and Rodnianski showed that the 3

3 Stability in Exponential Time of Minkowski Space Time... Page 3 of 5 7 non linear terms in the Einstein equations in wave coordinates consist of a linear combination of null forms with an underlying structure of the form.3. They used the wave condition to obtain better decay for some coefficients of the metric. However the decay is slower than for the solution of the wave equation. An example of a quasilinear scalar wave equation admitting global existence without the null condition, but with a slower decay is also studied by Lindblad in [8] in the radial case, and by Alinhac in [] and Lindblad in [9] in the general case. In [], Lindblad and Rodnianski introduced the notion of weak-null structure, which gathers all these examples. In + dimensions, to show global existence, one has to be careful with both quadratic and cubic terms. Quasilinear scalar wave equations in 3 + dimensions have been studied by Alinhac in []. He shows global existence for a quasilinear equation of the form u = g αβ u α β u, if the quadratic and cubic terms in the right-hand side satisfy the null condition. Global existence for a semi-linear wave equation with the quadratic and cubic terms satisfying the null condition has been shown by Godin in [9] using an algebraic trick to remove the quadratic terms, which does however not extend to systems. The global existence in the case of systems of semi-linear wave equations with the null structure has been shown by Hoshiga in []. It requires the use of L L estimates for the inhomogeneous wave equations, introduced in [6]. To show the quasi global existence for our system in wave coordinates, it will therefore be necessary to exhibit structure in the quadratic and cubic terms. However, as for the vacuum Einstein equations in 3 + dimension in wave coordinates, our system does not satisfy the null structure. It will in particular be important to understand what happens for a system of the form.3 in+ dimensions. For such a system, standard estimates only give an L bound for φ, without decay. Moreover, the growth of the energy of φ is like t. One can easily imagine that with more intricate a coupling than for.3, it willbe very difficult to prove stability without decay for φ. To obtain a more useful estimate, the idea will be to exploit more precisely the fact that φ also satisfies a wave equation. To understand how this might help, we will look at special solutions of the vacuum Einstein equations with a translation space-like Killing field: Einstein Rosen waves. These solutions have been discovered by Beck see [4], and also [3] and [5] fora mathematical description.. Einstein Rosen Waves Einstein Rosen waves are solutions of the vacuum Einstein equations with two spacelike orthogonal Killing fields: 3 and θ.the3 + metric can be written g = e φ dx 3 + e a φ dt + dr + r e φ r dθ. 3

4 7 Page 4 of 5 C. Huneau The reduced equations { Rμν = μ φ ν φ, g φ =, can be written in this setting R tt = r a t a + r ra = t φ,.4 R rr = r a + t a + r ra = r φ, R tr = r ta = t φ r φ. The equation for φ can be written, since φ is radial e a g φ = t φ + r φ + r r φ =, where g is the metric g = e a dt + dr + r dθ. The equation for φ decouples from the equations for the metric. Therefore we can solve the flat wave equation φ =, with initial data φ, t φ t= = φ,φ and then solve the Einstein equations, which reduce to r a = r r φ + t φ,.5 with the boundary condition φ r= = in order to have a smooth solution. Since φ =, if φ,φ have enough decay, we have the following decay estimate for φ φr, t + t + r + t r 3. Therefore since we have 3 a = R r r φ + t φ dr a, for r < t, + r t a Eφ, for r > t, + r t

5 Stability in Exponential Time of Minkowski Space Time... Page 5 of 5 7 where the energy Eφ = r r φ + t φ dr does not depend on t. Forr > t, wehavea Eγ and hence is only bounded. In particular, the metric e a dr + r dθ exhibits an angle at space-like infinity, that is to say the circles of radius r have a perimeter growth of e Eφ πr instead of πr. However, in the interior, the decay we get is far better than the one we could have found with standard estimates, if we had used.4 instead of.5..3 The Background Metric We would like to adapt the analysis of Section. in the case where we only assume one Killing field i.e. in the case where 3 is Killing but not θ. Assume that a = R r r φ + t φ dr is still an approximate solution of.3, which will appear to be true in Section 7.As in this case φ also depends on θ, we will have lim at, R,θ= R r r φ + t φ dr = bt,θ. Note that we have to be careful with the dependence on θ. The metric e bθ dt + dr + r dθ is no longer a Ricci flat metric when b depends on θ. Consequently it is not a good guess for the behavior at infinity of our metric solution g. A good candidate should be Ricci flat in the region r > t. Indeed if we considered compactly supported initial data for φ, by finite speed propagation, φ should intuitively be supported in the region r < t. Consequently, the equation R μν = μ φ ν φ implies that g should be Ricci flat for r > t. Consequently, we are yield to consider the following family of space-time metrics g b = dt + dr + r + χqbθq dθ + Jθχqdqdθ,.6 3

6 7 Page 6 of 5 C. Huneau where r,θ are polar coordinates, q = r t and χ is a cut-off function such that χq = forq < and χq = forq >. In the coordinates s = r + t, q,θ,a tedious calculation yields that all the Ricci coefficients are zero except R b qq = bθ q qχq r + bθqχq + qχqχ qjθ θ b r + bθqχq 3 + Jθ χqχ q 4r + bθqχq 3 χ q θ Jθ r + bθqχq, = bθ q qχq + O r Jθχ q R b qθ = r + bθqχq = O Cb, b, J, J <q< r Cb, J<q< r,.7..8 Therefore, the metrics g b are Ricci flat in the region r > t +. We will see in the next section that they are compatible with the initial data for g given by the constraint equations. This choice of background metric will force us to work in generalized wave coordinates, instead of usual wave coordinates. Indeed, for the metric g b defined by.6, the coordinates t, x, x are not wave coordinates, not even asymptotically. The generalized wave coordinate condition reads, for g of the form g = g b + g g λβ Ɣ α λβ = H α b where Hb α is defined by Hb α = H b α + Fα,.9 where H b α is defined by H b α = gλβ b Ɣ b α λβ. and F α is defined by the sum of the crossed terms of the form g θ r g b in g λβ Ɣλβ α H b α. The reason of this choice for F α will be explained in next section, in the proof of Theorem.3. The form of. in generalized wave coordinates is given by...4 The Initial Data In this section, we will explain how to choose the initial data for φ and g. We will note i, j the space-like indices and α, β the space-time indices. We will work in weighted Sobolev spaces. Definition. Let m N and δ R. The weighted Sobolev space Hδ mrn is the completion of C for the norm 3 u H m δ = β m + x δ+ β D β u L.

7 Stability in Exponential Time of Minkowski Space Time... Page 7 of 5 7 is the complete space of m-times continuously differ- The weighted Hölder space Cδ m entiable functions with norm u C m δ = β m + x δ+ β D β u L. Let < α <. The Hölder space Cδ m+α is the the complete space of m-times continuously differentiable functions with norm u C m+α δ m ux m uy + x δ = u C m δ + sup x =y, x y x y α. We recall the Sobolev embedding with weights see for example [6], Appendix I. Proposition. Let s, m N. We assume s >. Let β δ + and <α< min, s. Then, we have the continuous embedding Hδ s+m R Cβ m+α R. Let < δ <. The initial data φ,φ for φ, t φ t= are freely given in Hδ N+ Hδ+ N with <δ<. However the initial data for g μν, t g μν cannot be chosen arbitrarily. The induced metric and second fundamental form ḡ, K must satisfy the constraint equations. The generalized wave coordinates condition must be satisfied at t =. Moreover, we want to prescribe the asymptotic behaviour for g: we want it to be asymptotic to g b, where bθ is arbitrarily prescribed, except for its components in, cosθ and sinθ. We recall the constraint equations. First we write the metric g in the form g = N dt +ḡ ij dx i + β i dt dx j + β j dt, where the scalar function N is called the lapse, the vector field β is called the shift and ḡ is a Riemannian metric on R. We consider the initial space-like surface R ={t = }. We will use the notation = t L β, where L β is the Lie derivative associated to the vector field β. With this notation, we have the following expression for the second fundamental form of R K ij = N g ij. 3

8 7 Page 8 of 5 C. Huneau We will use the notation τ = g ij K ij for the mean curvature. We also introduce the Einstein tensor G αβ = R αβ Rg αβ, where R is the scalar curvature R = g αβ R αβ. The constraint equations are given by G j N j τ D i K ij = φ j φ, j =,,. G N R K + τ = φ g g αβ α φ β φ,. where D and R are respectively the covariant derivative and the scalar curvature associated to ḡ. The following result, proven in Appendix B, gives us the initial data we need. Theorem.3 Let <δ<. Let φ,φ Hδ N+ R Hδ+ N R and bθ W N, S such that bdθ = b cosθdθ = b sinθdθ =. We assume φ H N+ δ + φ H N δ+, b W N,. If > is small enough, there exists b, b, b R R S,J W N, S and g αβ,g αβ H N+ δ such that the initial data for g given by where g b is defined by.6 with are such that H N δ+ g = g b + g, t g = t g b + g, bθ = b + b cosθ + b sinθ + bθ, g ij, K ij = L β g ij satisfy the constraint equations. and.. 3

9 Stability in Exponential Time of Minkowski Space Time... Page 9 of 5 7 the following generalized wave coordinates condition is satisfied at t = g λβ Ɣ α λβ = gλβ b Ɣ b α λβ + Fα, where F α is the sum of all the crossed term of the form g θ r g b in g λβ Ɣλβ α g λβ b Ɣ b α λβ. Moreover, we have the estimates J W N, S + g H N+ + g δ H N, δ+ b = φ + φ dx + O 4, 4π b = φ φdx + O 4, π b = φ φdx + O 4. π Let us make a remark on the choice of F. Remark.4 The initial data t g and t g i are constructed so that the generalized wave coordinate condition is satisfied at t =. The choice of F is here to prevent terms of the form g U g b in this gauge condition, and therefore allows us to have t g, t g i H N δ+. Before stating our main result, we will recall some notations and basic tools in the study of wave equations..5 Some Basic Tools Coordinates and frames We note x α the standard space-time coordinates, with t = x. We note r,θthe polar space-like coordinates, and s = t + r, q = r t the null coordinates. The associated one-forms are ds = dt + dr, dq = dr dt, and the associated vector fields are s = t + r, q = r t. We note dx the Lebesgue measure with respect to the space coordinates x i. 3

10 7 Page of 5 C. Huneau We note the space-time derivatives, the space-like derivatives, and by the derivatives tangent to the future directed light-cone in Minkowski, that is to say t + r and θ r. We introduce the null frame L = t + r, L = t r, U = θ r.inthisframe,the Minkowski metric takes the form m LL =, m UU =, m LL = m L L = m LU = m LU =. The collection T ={U, L} denotes the vector fields of the frame tangent to the light-cone, and the collection V ={U, L, L} denotes the full null frame. The flat wave equation Let φ be a solution of { φ =, φ, t φ t= = φ,φ..3 The following proposition establishes decay for the solutions of the + dimensional flat wave equation. Proposition.5 Proposition. in [7] Let μ>. We have the estimate + t r [ μ] + φx, t M μ φ,φ + t + r + t r where M μ φ,φ = sup y R + y μ φ y + + y μ+ φ y + φ y and where we used the notation A [α] + = A maxα, if α = and A [] + = lna. Minkowski vector fields We will rely in a crucial way on the Klainerman vector field method. We introduce the following family of vector fields Z = { α, αβ = x α β + x β α, S = t t + r r }, where x α = m αβ x β. These vector fields satisfy the commutation property [, Z] =CZ, where CZ =, Z = S, CS =. 3

11 Stability in Exponential Time of Minkowski Space Time... Page of 5 7 Moreover some easy calculations give t + r = S + cosθ, + sinθ,, t + r r θ =, = cosθ, sinθ,, r t t r = S cosθ, sinθ,. t r With this calculations, and the commutations properties in the region t r t [Z, ], [Z, ], we obtain k l u + q k + s l Z k+l u,.4 where here and in the rest of the paper, Z I u denotes any product of I of the vector fields of Z. Estimates.4 and Proposition.5 yield Corollary.6 Let φ be a solution of.3. We have the estimate where k l φx, t Mμ k+l φ + t r [ μ] +,φ + t + r l+ + t r k+ Mμ j φ,φ = sup + y μ+ j s φ y + + y μ++ j s φ y y R + + j φ y. Weighted energy estimate We consider a weight function wq, where q = r t, such that w q > and for some <μ<. wq + q +μ w q wq + q, Proposition.7 We assume that φ = f. Then we have t wq t φ + φ dx + w q s φ θ u + dx r wq f t φ dx. 3

12 7 Page of 5 C. Huneau For the proof of Proposition.7, we refer to the proof of Proposition 9. which is the quasilinear equivalent of Proposition.7. Weighted Klainerman-Sobolev inequality The following proposition allows us to obtain L estimates from the energy estimates. It is proved in Appendix F. The proof is inspired from the corresponding 3 + dimensional proposition Proposition 4. in []. Proposition.8 We denote by v any of our weight functions. We have the inequality f t, xv x t + t + x + x t I v. tz I f L. Weighted Hardy inequality If u is solution of u = f, the energy estimate allows us to estimate the L norm of u. To estimate the L norm of u, we will use a weighted Hardy inequality. Proposition.9 Let α<and β>. We have, with q = r t vq + q f vq r f L, where L vq = + q α, for q <, vq = + q β, for q >. This is proven in Appendix E. The proof is inspired from the 3 + dimensional analogue Lemma 3. in []. L L estimate With the condition w q > for the energy inequality, we are not allowed to take weights of the form + q α, with α>inthe region q <. Therefore, the Klainerman-Sobolev inequality cannot give us more than the estimate u + q + s, in the region q <, for a solution of u = f. However, we know that for suitable initial data, the solution of the wave equation u = satisfies u + q + s, u + q 3. + s To recover some of this decay we will use the following proposition 3

13 Stability in Exponential Time of Minkowski Space Time... Page 3 of 5 7 Proposition. Let u be a solution of { u = F, u, t u t= =,. For μ> 3,ν > we have the following L L estimate where ut, x + t + x Cμ, νmμ,ν F + t x +[ μ] +, M μ,ν F = sup + y +s μ + s y ν Fy, s, and where we used the convention A [α] + = A maxα, if α = and A [] + = lna. This is proven in Appendix D. This inequality has been introduced by Kubo and Kubota in [6]. An integration lemma The following lemma will be used many times in the proof of Theorem., to obtain estimates for u when we only have estimates for u. Lemma. Let α, β, γ R with β<. We assume that the function u : R + R satisfies u + s γ + q α, for q <, u + s γ + q β for q >, and for t = Then we have the following estimates u + r γ +β. u + s γ max, + q α+, for q <, u + s γ + q β+ for q >. Proof We assume first q >. We integrate the estimate q u + s γ + q β, from t =. We obtain, since β<, for q > u + s γ + q β+. Consequently, we have, for q =, u + s γ. We now assume q <. We integrate q u + s γ + q α, 3

14 7 Page 4 of 5 C. Huneau from q =. We obtain u + s γ max, + q α+. This concludes the proof of Lemma...6 Main Result We introduce an other cut-off function ϒ : R + R + such that ϒρ = forρ and ρ and ϒ = for 4 3 ρ 3. Theorem. is our main result, in which we prove stability of Minkowski space-time with a translational symmetry in exponential time T exp where >isthe size of the small initial data. Theorem. Let < <. Let Hδ N+ R Hδ+ N R. We assume < δ < and N 4. Let φ,φ φ H N+ δ + φ H N δ+. Let T exp. Let ρ σ μ δ. If is small enough, there exists bθ, Jθ W N, S and there exists a global coordinate chart t, x, x such that, for t T, there exists a solution φ, g of. that we can write r gl L g = g b + ϒ t 4 dq + g UL rdqdθ + g such that we have the estimates α w q Z I φ L + α w + t 3 q Z I g L L L I N + α w + t 3 q Z I g LU L + α w q Z I g L + t C. with 3 { w q = + q +δ, q > w q = +, q <, + q { μ w q = + q +δ, q > w q =, q <, + q { +μ w3 q = + q 3+δ, q > w 3 q = +, q <, + q { μ α q = + q σ, q > α q =, q <,

15 Stability in Exponential Time of Minkowski Space Time... Page 5 of 5 7 Moreover, for all ρ>, we have the L estimate, for I N + and r < t Z I Cρ φx, t, + t + r + t r 4ρ Z I Cρ g L L, + t r ρ Z I g LU + Z I g Cρ. + t + r ρ and we have the estimate for b bθ + q φ rdr T,θ where we have used the notation q φ rdr = T,θ Comments on Theorem. T, q φt, r,θ rdr..5 We consider perturbations of 3 + dimensional Minkowski space-time with a translational space-like Killing field. These perturbations are not asymptotically flat in 3 + dimensions, therefore the result of Theorem. does not follow from the stability of Minkowski space-time by Christodoulou and Klainerman [8]. As our gauge, we choose the generalized wave coordinates, which are picked such that the generalized wave coordinates condition is satisfied by g b. Therefore, the method we use has a lot in common with the method of Lindblad and Rodnianski in [] where they proved the stability of Minkowski space-time in harmonic gauge. It is an interesting problem to investigate the stability of Minkowski with a translation symmetry using a strategy in the spirit of [8] or[5]. The function Jθ, and the quantities bθdθ, bθ cosθdθ, bθ sinθdθ are imposed by the constraint equations for the initial data see Theorem.3. The quantity bθdθ is called deficit angle, and the vector bθ cosθdθ, bθ sinθdθ is called linear momentum. We can make a rapprochement of these quantities with the ADM mass and linear momentum. The remaining Fourier coefficients of b are chosen to ensure the convergence to Minkowski in the direction of time-like infinity, and is an essential element in the proof of the quasi stability. In the subsequent paper [], it is shown that these remaining Fourier coefficient 3

16 7 Page 6 of 5 C. Huneau correspond actually to a gauge choice. This remark is the key ingredient in []to show the full stability. The logarithmic growth of w q Z N φ L, and the condition bθ q φ rdr,.6 T,θ give the estimate N b + T C. To avoid factors of the form + T C in all our estimate, we are forced to assume + T C. This is the only place where we need + T C, and this is what prevents us to prove the stability. The condition.6 is not necessary to control the metric in the exterior region r > t. For this reason we believe that the stability holds in the exterir region, without the condition T exp. As we said in the second comment, we use a method similar than Lindblad and Rodnianski method in []. Let us list some of the similarities and differences with their method. Similarities with [] We use the vector field method. The vector fields we use are the Klainerman vector fields of Minkowski space-time. We use the wave coordinate condition to obtain more decay on the coefficients g TT of the metric. We exhibit the structure corresponding to the model problem.3. Differences with [] The asymptotic behaviour given by the solutions of the constraint equations prevent us to work in wave coordinates. Instead we work in generalised wave coordinates. In the exterior region, our solution does not converge to Minkowski, but to a family of Ricci flat metrics g b. The decay of the free wave is weaker in + dimension. Consequently, the coefficient g L L of the metric does not have any decay near the light cone. We have to rely on the null decomposition at all steps in our proof to isolate this behaviour, even in the L estimates. We have to fit bθ so that the condition.6 is satisfied. This leads to regularity issues for b, which prevent us from proving the global existence. The structure of the paper is as followed. In Section we describe the structure of the equations. in generalized wave coordinates. We exhibit the structure of our system in Section. We also describe the interactions between g b and g. In Section.3 we outline the main issues of the proof by discussing some model problems. In section 4 we give our bootstrap assumptions. In section 5 we derive preliminaries estimates thanks to the wave coordinate condition. In section 6 we derive preliminaries estimate for the angle and the linear momentum. In section 7, we will exploit the analysis begun in section.. In section 8.4 we will improve the L estimate. In section 9 we will derive the weighted energy estimate. In section we will improve the L estimates and in section we will adjust the parameter bθ. 3

17 Stability in Exponential Time of Minkowski Space Time... Page 7 of 5 7 Structure of the Equations In this section, we provide a discussion of the specific features of the structure of the equations, which will be relevant for the proof of Theorem.. For sake of simplicity, we assume the result of Proposition 6., and we will go backward in Section 6.. The Generalized Wave Coordinates Wave coordinates allow to recast Einstein equations as a system of non-linear wave equations. The wave coordinates condition, which consists in choosing coordinates such that g x α = can be rewritten as g λβ Ɣ α λβ =. However, for the metric g b defined by.6, the coordinates t, x, x are not wave coordinates, not even asymptotically. We will therefore work with generalized wave coordinates. We will impose that our metric satisfies g λβ Ɣ α λβ = H α b where H α b is defined by.9 H α b = g b λβ Ɣ b α λβ + Fα, with F α of the form g qχq θ b r. TheroleofF α was explained in section.4. In generalized wave coordinates, the expression C.4 of Appendix C allows us to write the system. under the form { g φ = g g μν = μ φ ν φ + P μν g, g + g μρ ν H ρ + g νρ μ H ρ,. where P μν g g, g = gαρ g βσ μ g ρσ α g βν + ν g ρσ α g βμ β g μρ α g νσ μg αβ ν g ρσ. + gαβ g λρ α g νρ β g μρ. Remark. In generalized wave coordinates, the wave operator can be expressed as g = g αρ α ρ H ρ b ρ. 3

18 7 Page 8 of 5 C. Huneau The expression C.4 yields also R b μν = g b g b μν + P μνg b g b, g b + gb μρ ν H ρ b + g b μρ μ H ρ b..3 Therefore, subtracting twice the equation.3 to the second equation of. we obtain { g φ =, g g μν = μ φ ν φ + R b μν + P μν g g, g + P μν g, g b,.4 where P μν g g, g is defined by. and P μν g, g b = g αβ b g αβ α β g b μν + F ρ ρ g b μν + P μν g g, g P μν g g, g P μν g b g b, g b.5 + g b μρ ν F ρ + g b νρ μ F ρ + g μρ ν H ρ b + g νρ μ H ρ b. Let us note that P μν g, g b contains only crossed terms between g b and g.. The Weak Null Structure To exhibit the main terms in the structure of.4, let us neglect for a moment P μν, P μν, H b. We will see in the next section that this approximation is relevant. Let us also neglect the nonlinear terms involving derivatives. Then we obtain the following approximate system φ + g LL q φ =, g TV + g LL q g TV =, g L L + g LL q g L L = 4 q φ bθ q χqq, r where we also have used the approximation R b qq bθ q qχq r Cb, b, J, J <q< + O as shown in.7. In + dimensions, a term of the form g LL q φ is impossible to handle if one only relies on the decay for g LL provided by the fact of being a solution of a wave equation. However, as in [], we can exploit the wave condition to obtain better decay for some coefficients of the metric. More precisely, we have roughly 3 g TT g. r,

19 Stability in Exponential Time of Minkowski Space Time... Page 9 of 5 7 This is done properly in Proposition 5. for the coefficient g LL and in Proposition 5. for the coefficients g LU and g UU. Therefore, the g TT coefficients have a better decay in t than the solutions of the wave equation the challenges of the quasilinear terms of the form g LL q φ, g LL q g TV are presented in Section 3.4. Remark. The other quasilinear terms are of the form g TV T V φ, g TV T V g. Consequently, they involved at least one good derivative of φ, g. Thus, they are easier to estimate, and we can always focus on the terms g LL q φ, g LL q g. Assuming that we can also neglect the terms involving g LL, we are reduced to the following system φ =, g L L = 4 q φ bθ q χqq r,.6 which is a system of the form.3 and displays the weak null structure. The second component of the solution of.3 does not have any decay near the light cone in + dimensions see Section. for the radial case. Therefore, the coefficient g L L will not display any decay at all near the light cone see the estimates of Theorem.. To obtain decay for g L L in the q variable, we will approximate g L L 4 by the solution h of the following transport equation q h = r q φ bθ q qχq. The ideas of this approximation are presented in Section 3., and are exploited in Section 7..3 Non-commutation of the Wave Operator with the Null Frame The structure of Einstein equations can only be seen in the null frame. However it is well known that the wave operator does not commute with the null frame. In Theorem. we have decomposed our metric in the following way r g = g b + g + ϒ t gl L 4 dq + g UL rdqdθ. The problems of non-commutation induced by g L L and g UL are totally similar. Consequently, we can neglect the second one. We expressed the -forms dq in the coordinate t, x, x 3

20 7 Page of 5 C. Huneau dq = dr dt = cosθdx + sinθdx dt Therefore, we will have, in the coordinates x, x r ϒ g L L dq r ϒ g L L dq μν t μν t r = ϒ t r u μν θg L L + u μν θ θ g L L.7 where u μν and u μν are some trigonometric functions. The challenges of the terms involving u μν and u μν are explained in Section The Semi Linear Term P μν g g, g Recall the form of the term P μν g g, g. P μν g g, g = gαρ g βσ μ g ρσ α g βν + ν g ρσ α g βμ β g μρ α g νσ μ g αβ ν g ρσ + gαβ g λρ α g νρ β g μρ. The quadratic terms In the null frame L, L, U the only non zero coefficients of the Minkowski metric are m LL = and muu =. Thanks to this remark, we can describe the terms appearing in the different components of P μν. In P TT g g, g, there can not be strictly more than occurrences of the vector field L. Therefore, the quadratic terms are of one of these form V g VT T g TT, T g VV T g TT,.8 where we have used the fact, proved in Section 5 that V g TT T g VV. These terms all have the classical null structure. How this structure can be used to show global existence is explained in Section 3.. Since they are by far easier to handle than the one we will describe in the following, they will be neglected in the proof of Theorem.. In P TV g g, g, there can not be strictly more than 3 occurrences of the vector field L. Therefore, the quadratic terms are of one of these form 3 V g TV T g TV, V g VV T g TT, T g VV V g TT, T g TV T g VV

21 Stability in Exponential Time of Minkowski Space Time... Page of 5 7 where we have used the fact, proved in Section 5 that V g TT T g VV. These terms all have the null structure. However, since g L L does not decay at all in t see the estimates of Theorem., one has to be more careful with the terms of the form T g TT L g L L These terms have a good structure since T g TT is a good derivative of a good component. However, one needs two steps to exploit this structure, which can be difficult to achieve if there is no regularity left. Thankfully, these terms have three occurrences of L, therefore they can only intervene in P T L. In P L L we will have to be careful with L g LL L g L L. This term can be converted in L g LL L g L L with the help of the algebraic trick uv = u v + v u + L u L v + L v L u + U u U v. This fact will be used only in the proof of Lemma.6. In P LU we will have to be careful with U g LL L g L L. This term can not be removed with the previous trick. We will have to single out its influence thanks to the decomposition g = g b + χ r r hdq + χ krdqdθ + g 4, t t where k satisfies g k = U g LL L g L L. This will also be used only in the proof of Lemma.6. The terms in P L L which are not of the previous form can be written L g LL L g L L, L g LL L g L L..9 We note the crucial cancellation of terms of the form L g LL in P L L. The contributions.9 will be singled out in.. 3

22 7 Page of 5 C. Huneau The cubic terms In two dimensions, cubic terms could be troublesome. However, in the form P VT, if there are 4 occurrences of the vector field L, orinp L L if there are 5 occurrences of the vector field L, then we have a factor g L L, which has a decay equivalent to g LL. Therefore we can neglect the cubic terms in this nonlinearity..5 The Crossed Terms In this section, we discuss the structure of the crossed terms between b and g,φ. The crossed terms involving two derivatives of b are absent In the expression g g μν g μρ ν H ρ b + g νρ μ H ρ b, there could be terms involving two derivatives of bθ, which would be troublesome since they would lead to a loss of a derivative recall that we only have the regularity b W N,. However, the terms involving two derivatives of b in this expression, are the same than the terms involving two derivatives of b in R μν g. Thus, these terms cancel in the expression g αβ b g αβ α β g b μν + g b μρ ν F ρ + g b νρ μ F ρ + g μρ ν H ρ b + g νρ μ H ρ b, which appears in P μν g, g b defined by.5. These cancellations can be checked for example with Mathematica. Thecrossedtermsin P μν We recall from.6 that g b = dsdq + r + χqqbθ dθ + Jθχqdqdθ. Therefore in P μν we can find terms involving g b UU = + χqqbθ and g b UL = Jθχq. r r Since g b UL decays faster than g b UU let us focus on the crossed terms between g b UU and g. The problem with the term g b UU is that far from the light cone, it does not decay at all. This is one of the causes of the logarithmic growth of the energy in the statement of Theorem.. However, these terms are present only in the exterior region. Moreover they display also a special structure. Since the terms involving two derivatives of b are absent, and the terms involving two derivatives of g are only present in g g,thetermsin P μν are of the form g g b UU g. In P TV the crossed terms involving L g b UU can not contain more than two occurrences of L. They must be of the following form 3 L g b UU T g TV, T g b UU V g TV, T g b UU T g VV,

23 Stability in Exponential Time of Minkowski Space Time... Page 3 of 5 7 where we have used the wave coordinates condition V g TT T g TV. We have the following inequalities, thanks to.4 L g b UU T g TV q> b + θ b T g TV q> b + θ b + r + r Z g TV, T g b UU V g TV q> + q b + θ b + r V g TV q> b + θ b + r Z g TV. These two contributions are therefore quite similar. In the following, it will be sufficient to study the term L g b UU T g TV.. The challenges of this term will be discussed in Section 3.5 In P L L, we may have three occurrences of L. Therefore there are terms of the form L g b UU T g L L, L g b UU L g L L, T g UU L g L L. We have the following inequalities, thanks to.4 L g b UU T g L L q> b + θ b T g L L q> b + θ b + r + r Z g L L L g b UU L g L L q> b + θ b L g L q> b + θ b + r + r + q Z g LL T g b UU L g L L q> + q b + θ b + r L g L L q> b + θ b + r Z g L L. Consequently, the worst term is L g b UU L g L L.. We introduce the following notation, to single out the contributions of. and.9 Q L L h, g = L g LL L h + L g LL L h + L g b UU L g L L.. The crossed terms involving two derivatives of g With our choice of coordinates, these terms only appear in g g. They are of the form b + q q> + r U g. 3

24 7 Page 4 of 5 C. Huneau Their contribution is most of the time similar than the one of., except in the energy estimate, where they require a special treatment because of their lack of decay far from the light cone see Section 9. Thecrossedtermsin g φ The crossed terms between g b and φ are of the form g g b UU φ. Consequently, they must be of the following form V g b UU T φ, T g b UU V φ. Like for P VT, it will be sufficient to study The crossed terms between g b and φ are of the form V g b UU T φ..3 b + q q> U + r φ. As for g, their contribution is most of the time similar than the one of.3, except in the energy estimate, where they require a special treatment because of their lack of decay far from the light cone see Section 9. Remark.3 In the region q > it is generally sufficient to study the crossed terms. Indeed, the crossed terms are the one presenting the less decay far from light cone. 3 Model Problems The proof relies on a bootstrap scheme. Roughly speaking, we will assume some estimates on the coefficients Z I φ, Z I g L L and Z I g TV : L estimates for I N, L estimates for I N. We rewrite the bootstrap assumptions in the condensed form φ X C, g X C, where C is a constant depending only on the quantities ρ,σ,μ,δ, N introduced in the statement of Theorem. and such that at t = φ X C, g X C. Thanks to the L L estimate and the energy estimate, we will be able to prove 3 φ X C + C, g X C + C.

25 Stability in Exponential Time of Minkowski Space Time... Page 5 of 5 7 Therefore, for chosen small enough so that C C, this improves the bootstrap assumptions. We will first consider a toy model, which exhibits some of the mechanisms involved in the proof. 3. Global Well Posedness for a Semi Linear Wave Equation with the Null Structure We consider the following + dimensional semi-linear wave equation { u = u u, u, t u t= = u, u. 3. Note that the nonlinearity satisfies the null condition. Consequently, this model will show us how to treat the terms of the form.8. The following result is proved in []. We will give a proof of it for sake of completeness, and because it exhibits some of the mechanisms involved in the proof of Theorem.. Proposition 3. Let <δ<. Let N 8. Let u, u H N+ H N such that +δ δ+ u H N+ +δ + u H N δ+. If >is small enough, the equation 3. has a global solution u. Proof Let <μ< 4. We introduce the weight function { wq = +, q <, + q μ wq = + q +δ q >. Let <ρ< δ. To prove global existence for equation 3., we consider a time T > such that, on t T Z I u C + s + q δ, I N, 3. Z I u C + s + q δ, I N +, 3.3 w Z I u L C + t ρ, I N. 3.4 Thanks to the Klainerman-Sobolev inequality, the assumption 3.4 yields, for I N Z I u Z I u + t ρ + s + q, for q <, t ρ, for q >. + s + q +δ 3

26 7 Page 6 of 5 C. Huneau and consequently, thanks to Lemma. Z I u + q, for q <, Z I u, for q >. + s ρ + s ρ + q δ 3.6 We use the L L estimate to improve the estimates 3. and 3.3. We write Z I u = I +I I Z I u Z I u. 3.7 We first treat the case I N. We assume I N 4 the case I N 4 the same way. Therefore, we can estimate thanks to.4 can be treated in Z I u + q Z I + u. Since N 4 + N we obtain thanks to 3. Z I u + q +δ + s. To estimate Z I u we use.4 and the bootstrap assumption 3.3 to obtain This yields Z I u + s Z I + u Z I u. + s 3 + q δ + s 3δ + + q We can now use the L L estimate of Proposition., together with the estimate of Proposition.5 and the Sobolev injection of Proposition., which gives Z I u C + s + q δ + C ln + q. + s + q. This implies, since ln + q + q δ Z I u C + s + q δ + C + s + q δ. 3.8 We now treat the case I = N +. We assume I N+ 4 N so we have the same estimate as before for Z I u. To estimate Z I u, since N + N weuse3.6. We obtain 3

27 Stability in Exponential Time of Minkowski Space Time... Page 7 of 5 7 Therefore we obtain Z I u Z I u + s Z I+ u + q. + s 3 ρ. + s ρ + q +δ + s 3 + δ + q + δ ρ Therefore, like for 3.8, the L L estimate yields Z I u C + s + q δ + C. 3.9 δ + s + q We now use the weighted energy estimate to imrpove 3.4. Let I N. Inviewof 3.7, it implies d dt wq Z I u L + w q Z I u L w Z I u Z I u L w Z I u L. 3. I +I I We first assume I N. Then we estimate Z I u. + s 3 + q δ This yields w Z I u Z I u L w + t 3 Z I u L. We now assume I N. Then, we estimate Z I u. + s + q + δ Therefore we obtain Since w Z I u Z I u L + t w + q + δ Z I u L. w + q + δ w q, 3

28 7 Page 8 of 5 C. Huneau we infer w Z I u Z I u L w Z I u L Therefore 3. writes + t w Z I u L + w q Z I u L. d dt wq Z I u L + w q Z I u L + t w Z I u L + w q Z I u L, so for small enough d dt wq Z I u L + w q Z I u L + t w Z I u L. We obtain wq Z I u L C + t C. 3. For small enough so that C C, + tc 3 + tρ, we have proved, in view of 3.8, 3.9 and 3. that for t T we have Z I u 3 C + s + q δ, I N, Z I u 3 C, I N δ + s + q +, w Z I u L 3 C + t ρ, I N, which concludes the proof. Remark 3. Actually, only the highest order energy w Z N u L grows in t.tosee this, we estimate for I N and I N. Since w Z I u Z I u L Z I u + s Z I +, we obtain, together with the weighted Hardy inequality w Z I u Z I w u L +t 3 + q Z I+ u 3 L w + t 3 Z I + u L.

29 Stability in Exponential Time of Minkowski Space Time... Page 9 of 5 7 Therefore, the weighted energy estimate yields, for I N and hence d dt w Z I u L, + t 3 C w Z I u L. Remark 3.3 Theuseoftheterm w q Z I u L to exploit the structure in the energy estimate has been introduced by Alinhac in [] and is sometimes called Alinhac ghost weight method. It has also been used in the case of Einstein equations in wave coordinates in []. Unfortunately, the Einstein equations in wave coordinates do not have the null structure, but only a weak form of it. In the next sections, we will see what problems this creates and the method we used to tackle them. We will be less precise than in this first example, since full details will be provided when we proceed with the proof of Theorem.. 3. The Coefficient g LL To understand how to deal with g L L, let us consider the question of global existence for the following system, which is of the form.6 { φ =, h = q φ bθ q qχq r. 3. with initial data for φ of size and zero initial data for h. We recall b L S. We have the following estimates for φ w φ L, φ + s + q +δ. Therefore, the energy estimate for h can be written d dt w h L w q φ L + w bθ q qχq r L w h L, and thus d dt w h L w φ L + + t + t + t. 3

30 7 Page 3 of 5 C. Huneau We infer w h L + t. 3.3 This estimate is not sufficient. To obtain more information on h, we will approximate it by the solution h of the following transport equation this procedure will be made more precise in Section 7 q h = r q φ bθ q qχq, 3.4 with initial data h = att =. The L estimate for φ, and the fact that χ is supported in [, ] yield To estimate h we write h Q, s,θ= q h Q s + q +δ. q φ r bθ q qχq dq, so we obtain h s, Q,θ= O s h s, Q,θ= s + O, Q >, + Q +δ q φ r bθ q qχq dq + Q +δ, Q <. Therefore, since s s q qχqdq =, for s to maximize the decay in q for h and hence for h, provided one has a suitable control over h h we will choose b such that bθ s s q φ rdq. 3.5 Remark 3.4 bθ is a free parameter, except from bθ, bθ cosθ and bθ sinθ which are prescribed by the resolution of the constraint equations, and correspond intuitively to the ADM angle energy and linear momentum. Let be the projection defined by : W,N S {u W,N S, u = cosθu = sinθu = }

31 Stability in Exponential Time of Minkowski Space Time... Page 3 of 5 7 Then s bθ q φ rdq, s will be forced in the course of the bootstrap procedure. On the other hand, the fact that s bθ q φ rdqdθ, s s bθ cosθ q φ cosθrdqdθ, s s bθ sinθ q φ sinθrdqdθ, s will be obtained by integrating the constraint equations at any time t see Section Non Commutation of the Wave Operator with the Null Frame In this section, we will discuss the influence of the terms appearing in.7. We have seen in the previous section that h does not decay at all with respect to the s variable. In turn, we will show that this is also the case for h, and finally for the coefficient g L L. We do not want this behavior to propagate to the other coefficients of the metric. To this end, we will rely on a decomposition of the type r g = g b + ϒ t gl L 4 dq + g i. However, since the wave operator does not commute with the null decomposition, we have to control the solution g i of an equation of the form r g i = ϒ t h r, where h is the solution of 3.. The term ϒ r h t r has the form of the terms appearing in.7. Provided that we can approximate h by the solution h of the transport equation 3.4, we obtain decay with respect to q for h. The decay we will be able to get is h + q. With this decay we infer g i + s + q, 3

32 7 Page 3 of 5 C. Huneau and therefore, with the L L estimate, we deduce g i + s ρ, for all ρ>. On the other hand, assume we are only allowed to use the energy estimate for h, which is the case when deriving L type estimates for g i at the level of the highest energy. When applying the weighted energy estimate for g i, we obtain d dt wq gi L wq r ϒ t h r L wq gi. We estimate wq r ϒ t h r L + t wq h L + t, 3.7 where we have used the estimate 3.3 of the previous section for h. This yields So d dt wq gi L. + t wq gi L + t, which is precisely the behaviour we are trying to avoid with this decomposition! However we have not been able to exploit all the structure in 3.7. In order to do so, we will use different weight functions for g i and for h.ifweset wq = + q +μ wq, with <μ 4 and we assume that we have then we can estimate wq r ϒ t We write 3 h r wq h L + t, L + t wq r h ϒ t + q +μ h + s Zh Zh, + s +μ + q μ L.

33 Stability in Exponential Time of Minkowski Space Time... Page 33 of 5 7 so we obtain wq r ϒ t h r L wq Zh + t 3 +μ + q wq + t 3 +μ Zh L, L where we used the weighted Hardy inequality. Consequently, the energy inequality for g i yields and therefore d dt wq gi L + t +μ, wq gi L. Recall that the weighted energy inequality forbids weights of the form + q α with α> in the region q <. Therefore we are forced to make the following choice in the region q < wq = O, wq = + q +μ. Thus, for g i,the t loss has been replaced by a loss in + q +μ. 3.4 The Quasilinear Structure In this section we will discuss the challenges of the quasilinear structure. We will take as an example the equation for φ, g φ =. Following Remark., we can focus on the terms of the form g LL q φ. The wave coordinates condition yields g LL g. If g satisfied g =, the L estimates for g given by Corollary.6 for suitable initial data would imply g LL + s 3, + q We would like to keep this decay in after integrating with respect to q. However, +s 3 we are not in the range of application of Lemma.. To this end, we will assume more decay on the initial data. As stated in Theorem.,wetakeg, t g Hδ N+ Hδ+ N 3

34 7 Page 34 of 5 C. Huneau with <δ<. Then, with the weight w stated in Theorem., the weighted energy inequality yields w q Zg L, and consequently, for q >, the weighted Klainerman-Sobolev inequality yields Zg + s + q 3 +δ. If we integrate from t =, we obtain for q > Zg + s + q +δ. By writing g +s Zg, we obtain g LL g LL, for q <, + s 3 + q, for q >. + s 3 + q +δ Since δ> we can apply Lemma., which yields g LL + q, for q <, g + s 3 LL Consequently we easily estimate, for q >. + s 3 + q δ w gll q Z I φ L w + t 3 q Z I + φ L. This strong decay in the region q > is also needed when estimating w Z I g LL q φ L. The idea will be first to use the weighted Hardy inequality to derive 3 w Z I g LL q φ L + t + t w + q Z I g LL w + q Z I g LL L L.

35 Stability in Exponential Time of Minkowski Space Time... Page 35 of 5 7 Then we rely on the wave coordinates condition, which yields Z I g LL Z I g + s Z I + g, and then use the weighted Hardy inequality again. However, one has to be careful when using the weighted Hardy inequality. In the region q > the weight must be sufficiently large to allow to perform it twice. This is an other reason why we work with initial data in Hδ N with δ>, which is more than the decay which is necessary to prove the global well posedness of a semi linear wave equation with null structure. 3.5 Interaction with the Metric g b In this section we want to discuss the influence of the crossed terms between g b and φ, g. We will take as an example the equation for φ, g φ =. As discussed in Section.5, we can focus on the term.3. We may look at the following model problem φ = r χq φ. If we perform the weighted energy estimate, we obtain Therefore d dt w q Z I φ + w q Z I φ L + t w Z I φ L. w q Z I φ L C + t C and for all σ> T + t σ w q Z I φ L dt. 3.8 To avoid this logarithmic loss, we need to exploit more the structure of the equation. To this end we introduce the weight modulator { αq = + q σ, q >, αq =, q <, for <σ <. Then the energy inequality yields d dt αw q Z I φ L αw q> + s Z I φ L αw q Z I φ L. 3

36 7 Page 36 of 5 C. Huneau We estimate, for q > And therefore, we obtain d dt αw q Z I φ L αq + s. + t +σ + q + t +σ w q> Z I φ + q and consequently in view of 3.8 we obtain L αw q Z I φ L + t σ w q Z I φ L + + t +σ αw q Z I φ L. αw q Z I φ L C + C. With this technique, the logarithmic loss in t has been replaced by a small loss in q. 4 Bootstrap Assumptions and Proof of Theorem. 4. Bootstrap Assumptions Let <δ<. In view of the assumptions of Theorem., the initial data φ,φ for φ are given in Hδ N+ and For b W,N such that bdθ = S R H N δ+ R. b cosθdθ = b sinθdθ =, S S b W,N C, Theorem.3 allows us to find initial data g and t g such that g ij, K ij satisfy the constraint equations, g and t g are compatible with the decomposition g = g b + g, where bθ = bθ + b + b cosθ + b sinθ 4. with b, b, b, Jθ given by Theorem.3, the generalized wave coordinate condition given by H b is satisfied at t =. The system. being a standard quasilinear system of wave equations, we know that there exists a solution until a time T. Moreover with our conditions on the initial data, 3

37 Stability in Exponential Time of Minkowski Space Time... Page 37 of 5 7 our solution g,φis solution of the Einstein equations., and the wave coordinate condition is satisfied for t T see Appendix C. Remark 4. Our choice of generalized wave coordinates does not change the hyperbolic structure because H b does not contain derivatives of g. We take three parameters ρ,σ,μ such that ρ σ μ δ, 4. σ + ρ<δ. 4.3 We consider a time T > such that there exists bθ W N, S and a solution φ, g of.4on[, T ], associated to initial data for g. We assume that on [, T ],the following estimates hold. Bootstrap assumptions for b θ I bθ + q φ rdq B, for I N T,θ L S T θ I bθ L S B, for I N 4.5 where is the projection defined by 3.6, T,θ is defined by.5 and B is a constant depending on ρ,σ,μ,δ, N. We introduce four decomposition of the metric g r g = g b + ϒ h dq + g, 4.6 t r g = g b + ϒ h + hdq + g, 4.7 t r g = g b + ϒ hdq + g 3, 4.8 t r r g = g b + ϒ hdq + ϒ krdqdθ + g 4, 4.9 t t where h is the solution of the transport equation { q h = r q φ bθ q χqq, h t= =, 4. h is solution of the linear wave equation { h = ϒ r t h + ϒ rt gll q h + ϒ r t q φ R b qq + ϒ r t Q L L h, g, h, t h t= =,, 4. 3

38 7 Page 38 of 5 C. Huneau where Q L L h, g = L g LL L h + L g b UU L g L L. 4. { g h = q φ + R b qq + Q L L h, g, 4.3 k, t k t= =,, and k is the solution of { g k = U g LL q h, h, t h t= =,. 4.4 L -based bootstrap assumptions For I N 4 we assume Z I φ C + s + q 4ρ, 4.5 Z I g C, s ρ where here and in the following, C is a constant depending on ρ,σ,μ,δ, N such that the inequalities are satisfied at t = with C replaced by C.ForI N we assume Z I C φ, + s ρ 4.7 Z I g C. + s ρ 4.8 We assume the following estimate for h for I N 7 and q < Z I h C + s + C q 4ρ and for q > Z I h C + q +δ σ. 4. We also assume the following for h and I N 7 3 Z I h C q ρ