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 single distinction being that lengths are measured with admire to the metric g. by way of Noether's theorem, symmetries of the Lagrangian indicate conservation legislation for wave maps, akin to conservation of energy.
In coordinates, wave maps are given by way of a process of semilinear wave equations. during the last two decades very important equipment have emerged which handle the matter of neighborhood and international wellposedness of the program. as a result of vulnerable dispersive results, wave maps outlined on Minkowski areas of low dimensions, akin to R2+1t,x, current specific technical problems. This classification of wave maps has the extra vital characteristic of being power severe, which refers back to the indisputable fact that the power scales precisely just like the equation.
Around 2000 Daniel Tataru and Terence Tao, development on prior paintings of Klainerman–Machedon, proved that delicate information of small power result in international delicate suggestions for wave maps from 2+1 dimensions into goal manifolds fulfilling a few usual stipulations. against this, for big info, singularities might happen in finite time for M=S2 as objective. This monograph establishes that for H as objective the wave map evolution of any gentle info exists globally as a gentle function.
While we limit ourselves to the hyperbolic airplane as objective the implementation of the concentration-compactness strategy, the main hard piece of this exposition, yields extra precise info at the resolution. This monograph could be of curiosity to specialists in nonlinear dispersive equations, particularly to these engaged on geometric evolution equations.
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Extra resources for Concentration Compactness for Critical Wave Maps
0 is the tangent plane to the cone which touches the cone along the generator `! 0 . k k 0; 1 2 ;1 XP 0 2 which is obvious from orthogonality of the P0;˙Ä . 15). The energy is controlled via the embedding k kL1 L2 . 2. Finally, the statement concerning the free wave reduces to the case k D 0 for which we need to verify the bound X j j 2 2 kT O T j ˙ j k m 2 j ˙ j jj fO. /kL2 L2 C j 2Z C X 22j kT O T j ˙ j jj m 2 j 2 j ˙ j jj fO. /kL 2 2 L Á 21 . kf k2 j 2Z which are both clear provided T 1 due to the rapid decay of O .
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 ! D C . ; /. 47) into several ! T pieces. Since h. D 0 implies that p C j j D 2 h. / 2j j D e1 DW g. 21 C Ác T. C 2 Ác T g. j j/ 1 ! / ! h. ; Ä/ 1 ! 48) (which gains a factor of 22` . 47) can again be added to «T;Ä;a . Set b D b. / D g. / . Furthermore, set b0 WD b. 21, C Ác T 2 g.
J h. / C. / , one has h. / WD . ; Ä/ by elementary geome- RÄ;! ! /f . ; ! / ; where RÄ;! is a smooth cut-off adapted to the rectangle RÄ;! in the ! -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 .
Concentration Compactness for Critical Wave Maps by Joachim Krieger