Multi-Page Printing

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 E± (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