By Joachim Krieger
Wave maps are the easiest wave equations taking their values in a Riemannian manifold (M,g). Their Lagrangian is equal to for the scalar equation, the one distinction being that lengths are measured with recognize to the metric g. via Noether's theorem, symmetries of the Lagrangian indicate conservation legislation for wave maps, similar to conservation of energy.
In coordinates, wave maps are given by way of a procedure of semilinear wave equations. during the last two decades very important equipment have emerged which deal with the matter of neighborhood and worldwide wellposedness of the program. as a result of susceptible dispersive results, wave maps outlined on Minkowski areas of low dimensions, similar to R2+1t,x, current specific technical problems. This classification of wave maps has the extra vital function of being strength severe, which refers back to the undeniable fact that the power scales precisely just like the equation.
Around 2000 Daniel Tataru and Terence Tao, construction on past paintings of Klainerman–Machedon, proved that tender information of small strength result in international tender strategies for wave maps from 2+1 dimensions into aim manifolds enjoyable a few average stipulations. against this, for big info, singularities may possibly ensue in finite time for M=S2 as aim. This monograph establishes that for H as goal the wave map evolution of any soft info exists globally as a gentle function.
While we limit ourselves to the hyperbolic aircraft as goal the implementation of the concentration-compactness procedure, the main difficult piece of this exposition, yields extra distinctive info at the answer. This monograph can be of curiosity to specialists in nonlinear dispersive equations, specifically to these engaged on geometric evolution equations.
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Extra info for Concentration Compactness for Critical Wave Maps
Next, we assume that 0 D 1 D 0. s/. s/ ds S Œ0 . t/ on the left-hand side. This is where we use that F D IF , but after this point we may no longer assume that F D IF since RC F loses this property. t S Œ0 1 . 21. t Then the space-time Fourier transform of C b. ; / D Ác T. 21, ˇ ˇ jb. ; /j . 56) equals (up to a multiplicative constant) C b Ác T . C j j/ F . ; /; 2 C Œj j 10 Á b . 58) and thus also kQÄ0 k 1 ;1 0; 2 XP 0 C kQ>0 kXP 0;1 0 ";2 . 25. 4 it suffices to assume that F is either 1 an energy or a wave-packet atom.
8. One has the estimates 1 k0 k F kNFŒÄ jÄ 0 j 2 2 2 . Ä; Ä 0 / k jÄj 2 2 2 . 30) For the final two bounds we require that 2Ä \ 2Ä 0 D ;. Proof. 25). Note that both of these estimates have a dispersive character, as they involve space-time integrals. 5. Next, we define the spaces which will hold the nonlinearities. These spaces differ from those used for example in  as far as the “elliptic norm” k k 1 C"; 1 ";2 is concerned. Here the extra XP k 2 " ensures that we achieve exponential gains in the maximal frequencies for certain high-high-low interactions.
Given T > 0, let j"j be very small and set WD TTC" . t; x/ WD . t; x/, and similarly for jjj jjj. Œ T;T R2 / ˇ C kPkC . Œ T;T R2 / : By the energy estimate, kPkC . Œ T;T R2 / . kPkC . k. /kS ŒkC /Œ0kL2 HP C k PkC . k. /Œ0kL2 C k PkC . R1C2 / ! 1. Œ T;T R2 / as claimed. The case of T D 0 follows directly from the energy estimate. The case of jjj jjj is essentially the same. Œ T;T R2 / WD kPk Q kN Œk inf Q jŒ T;T D jŒ T;T for Schwartz functions. Œ T;T R2 / : Furthermore, later we will also need localized norms on asymmetric time intervals Œ T 0 ; T for which the results here of course continue to hold.