We give a new proof of the global stability of Minkowski space originally established in the vacuum case by Christodoulou and Klainerman. The new approach, which relies on the classical harmonic gauge, shows that the Einstein-vacuum and the Einstein-scalar field equations with asymptotically flat initial data satisfying a global smallness condition produce global (causally geodesically complete) solutions asymptotically convergent to the Minkowski space-time.

No abstract

No abstract

We study the motion of an incompressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler's equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a "physical condition", related to the fact that the pressure of a fluid has to be positive.

We prove global stability of Minkowski space for the Einstein vacuum equations in harmonic (wave) coordinate gauge for the set of restricted data coinciding with the Schwarzschild solution in the neighborhood of space-like infinity. The result contradicts previous beliefs that wave coordinates are "unstable in the large" and provides an alternative approach to the stability problem originally solved ( for unrestricted data, in a different gauge and with a precise description of the asymptotic behavior at null infinity) by D. Christodoulou and S. Klainerman.Using the wave coordinate gauge we recast the Einstein equations as a system of quasilinear wave equations and, in absence of the classical null condition, establish a small data global existence result. In our previous work we introduced the notion of a weak null condition and showed that the Einstein equations in harmonic coordinates satisfy this condition.The result of this paper relies on this observation and combines it with the vector field method based on the symmetries of the standard Minkowski space.In a forthcoming paper we will address the question of stability of Minkowski space for the Einstein vacuum equations in wave coordinates for all "small" asymptotically flat data and the case of the Einstein equations coupled to a scalar field. We shall use below the standard convention of summing over repeated indices and the notation ∂α = ∂/∂x α 2 For the definitions of global hyperbolicity and maximal Cauchy development see [H-E], [Wa] 3 The stability result of [C-K] was proved for strongly asymptotically flat data g0 ij = (1 + 2M/r)δij + o(r −3/2 ),where c α J γ are constants such thatThe last identity is a consequence of the relation between c I γ J α and the commutator constants c αβ = [∂ α , Z] β for which we have established that c LL = 0. It therefore follows that 41Decomposing relative to the null frame (L, L, S 1 , S 2 ) we obtainWe now contract the above identity with one of the tangential vector fields T ν , T ∈ {L, S 1 , S 2 } to obtain|Z Im H| · · · |Z I 2 H||∂Z I 1 H|, and (12.11) |∂Z I H| LL |J|≤|I| |∂Z J H| + |J|≤|I|−1 |∂Z J H| LT + |K|≤|I|−2 |∂Z J H| + |I 1 |+...+|Im|≤|I|, m≥2|Z Im H|· · · |Z I 2 H||∂Z I 1 H|.The same estimates also hold for H replaced by h.Proof. This follows directly by the previous lemma with the help of the identities m LT = 0 and c I L J L = 0.

In this paper we prove a sharp global existence theorem in all dimensions for nonlinear wave equations with power-type nonlinearities. The proof is based on a weighted Strichartz estimate involving powers of the Lorentz distance.

We study the motion of a compressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler's equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a "physical condition", related to the fact that the pressure of a fluid has to be positive.

a αβω αωβ ,ω = (−1, ω). Here we have introduced polar coordinates x = rω, ω ∈ S 2 . The classical null condition introduced by Klainerman [K1] is that A nm ≡ 0 under which Klainerman [K2] and Christodoulou [C] proved global existence. In [L2] it was observed that the asymptotic equation corresponding to (1.1) has global solution 1 , contrary to other cases likewhere solutions are known to blow up for all small data, see John [J1, J2]. However, unlike for the classical null condition, the solution of (1.1) do not behave asymptotically like a solution of a free linear wave equation.The method of proof of [L2] is integration along characteristics so it does not directly generalize to the non-symmetric case. However, as observed in [L1], the method of integration along characteristics 1 In [L-R1] we in general say that (1.3) satisfy the weak null condition, if (1.4) has global solution with some decay.