By Joachim Krieger
Wave maps are the best 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 appreciate to the metric g. via Noether's theorem, symmetries of the Lagrangian indicate conservation legislation for wave maps, akin to conservation of energy.
In coordinates, wave maps are given via a method of semilinear wave equations. during the last two decades vital 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 category of wave maps has the extra vital characteristic of being power serious, which refers back to the proven fact that the strength scales precisely just like the equation.
Around 2000 Daniel Tataru and Terence Tao, development on past paintings of Klainerman–Machedon, proved that delicate info of small power result in international soft options for wave maps from 2+1 dimensions into aim manifolds pleasant a few normal stipulations. by contrast, for big facts, singularities could take place in finite time for M=S2 as goal. This monograph establishes that for H as goal the wave map evolution of any soft info exists globally as a tender function.
While we limit ourselves to the hyperbolic airplane as aim the implementation of the concentration-compactness procedure, the main demanding piece of this exposition, yields extra targeted info at the answer. This monograph might be of curiosity to specialists in nonlinear dispersive equations, particularly to these engaged on geometric evolution equations.
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Extra info for Concentration Compactness for Critical Wave Maps
Plane. Furthermore, we set h i C QÄj;! /. h. // f . ; ! / : C By construction, PÄ;! QÄ2`;! 6. In fact, one has e Ä D PÄ;! QC e F F Ä2`;! Ä C and PÄ;! QÄ2`;! is disposable. t/ WD Z C PÄ;! QÄ2`;! s! a/fa ds: C Then ˚T;Ä;a D P0;C Ä QÄ2`CC ˚T;Ä;a and C ˚T;Ä;a D PÄ;! QÄ2`;! Ác T. a fOa . j j C /j . 47) can be added 1 to «T;Ä;a . In fact, C QÄ2`CC F . 1/e 2` C . ; Ä/ 1 . a j j/ kfa kL2 ; O RÄ;! ! /fa . / O RÄ;! ! /fa . 48) L2 L2 48 2 The spaces S Œk and N Œk which is better than needed. 27); the latter estimate can be applied for fixed , since then !
23. Another property which the stronger norm inherits is that it is finite on free wave, cf. 4. k k 0; 1 2 ;1 XP k : As in , one needs to allow for time-localized versions of S Œk, both relative to the original k kS Œk , as well as the stronger jjj jjj-norm. 14) on some time interval Œ T; T . Since the norms of the S Œk and N Œk spaces are defined in phase space, one cannot simply define these norms by time truncations. 68) jjjPk Q jjjS Œk : inf jŒ T;T It is easy to see that the triangle inequality holds for these expressions and that they are actually norms.
7. S Let F 0 C` be a collection of disjoint caps and let Ä 2 C` be a Ä . Suppose further that for every Ä 2 F there is a Schwartz cap with F Ä function Ä adapted to k 2 Z and which is supported on n o 0 TÄ;k WD WD sign. /b 2 Ä; jj j j jj . 2`C` Ck with some k 2 Z. Then X Ä S Œk;Ä 0 Ä2F with some absolute constant C . ÄC X Ä2F k 2 Ä kS Œk;Ä Á 21 36 2 The spaces S Œk and N Œk Proof. First, one may take k D 0 and > 0 (the latter by conjugation symmetry). 12) satisfies the required property due to orthogonality, whereas the PWŒÄ-component is reduced to Cauchy–Schwarz (via 1 2 the jÄj 2 -factor).
Concentration Compactness for Critical Wave Maps by Joachim Krieger