INTRODUCTION ix In this monograph we will concentrate on the energy critical nonlinear wave equation in 3d, (NLW±) ∂2u t Δu = ∓u5,x R3,t R u|t=0 = u0 ˙ 1 (R3) ∂tu|t=0 = u1 L2 ( R3 ) . The hope is that the results obtained for (NLW±) will be a model for what to strive for in other critical dispersive problems. In (NLW+) we have the “defocusing case”, while in (NLW−) we have the “focusing case”. (NLW±) are “energy critical” because if u is a solution, so is 1 λ 1 2 u ( x λ , t λ ) , and the scaling leaves invariant the norm of the Cauchy data in ˙ 1 (R3) × L2(R3). Both problems have energies that are constant in time (u(t),∂tu(t)) = 1 2 |∇u(t)|2 + (∂tu(t))2 ± 1 6 u6(t). where + on the right hand side, corresponds to the defocusing case and on the right hand side corresponds to the focusing case. We now summarize the “local theory of the Cauchy problem”, for equations (NLW±). This is described in detail in Chapter 1. If (u0,u1) ˙ 1 ×L2 is small, ∃! solution u, defined for all time, such that u C (−∞, +∞) ˙ 1 × L2 Lxt, 8 which scatters, i.e., (u(t),∂tu(t)) S(t) ( u±,u± 0 1 ) ˙ 1 ×L2 t→±∞ −→ 0, for some ( u0 ± , u1 ± ) ˙ 1 × L2. Moreover, for any data (u0,u1) ˙ 1 × L2, we have short time existence and hence there exists a maximal interval of existence I = (T−(u),T+(u)). Here, S(t) (u0,u1) is the solution of the linear wave equation ∂2 t Δ, with initial Cauchy data (u0,u1). Also, the meaning of, say, T+(u) ∞, is that if {tn} is a sequence of times converging to T+(u), (u(tn),∂tu(tn)) has no convergent subsequence in ˙ 1 × L2. Question: What about large data? We first turn to the defocusing case, which was studied in works of Struwe [97], Grillakis [46], [47], Shatah-Struwe [92],[93], Kapitanski [43], Bahouri-Shatah [5] (80’s and 90’s). They established: (+) Global regularity and well-posedness conjecture (For critical defocusing problems): There is global in time well-posedness and scat- tering for arbitrary data in ˙ 1 ×L2. Moreover more regular data keep this regularity for all time. This closes the study of the dynamics in (NLW+). For the focusing problem, (+) fails. In fact, H. Levine (1974) [73] showed that if (u0,u1) H1 × L2, E(u0,u1) 0, (u0,u1) = (0, 0), ( ˙ 1 × L2 in the radial case), then |T±(u0,u1)| ∞. Levine’s proof is of the “obstruction” type. He shows that there is an obstruc- tion for the global existence, but does not give information on the nature of the “blow-up”. Moreover, u(x, t) = ( 3 4 ) 1 4 (1 t)− 1 2 is a solution. It is not in ˙ 1 × L2, but we can truncate it and use finite speed of propagation to find data in ˙ 1 × L2
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