By Joachim Krieger

ISBN-10: 3037191066

ISBN-13: 9783037191064

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 admire to the metric g. via Noether's theorem, symmetries of the Lagrangian suggest conservation legislation for wave maps, corresponding to conservation of energy.

In coordinates, wave maps are given through a procedure of semilinear wave equations. over the last twenty years very important equipment have emerged which deal with the matter of neighborhood and worldwide wellposedness of the program. as a result of vulnerable dispersive results, wave maps outlined on Minkowski areas of low dimensions, reminiscent of R2+1t,x, current specific technical problems. This type of wave maps has the extra vital characteristic of being strength serious, which refers back to the undeniable fact that the strength scales precisely just like the equation.

Around 2000 Daniel Tataru and Terence Tao, development on previous paintings of Klainerman–Machedon, proved that delicate information of small strength bring about international soft options for wave maps from 2+1 dimensions into goal manifolds pleasant a few average stipulations. against this, for big facts, singularities may well happen in finite time for M=S2 as objective. This monograph establishes that for H as aim the wave map evolution of any delicate info exists globally as a tender function.

While we limit ourselves to the hyperbolic airplane as objective the implementation of the concentration-compactness process, the main not easy piece of this exposition, yields extra precise details at the answer. This monograph could 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**

**Sample text**

2 j=2 kP

Later we will prove bi- and trilinear estimates involving S and N space. The following bilinear bounds will be a basic ingredient in that context. 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 [23] as far as the “elliptic norm” k k 1 C"; 1 ";2 is concerned.

Ij R2 / Ä N X j, we have cj j D1 where the implied constant is universal (independent of the decomposition of I or N ). Ij R2 / . Proof. Chose a partition of unity f j g subordinate to the cover fIj g, such that supp j Ij . We shall select the j in such fashion that jsupp j0 j jIj \ Ik j, provided the latter is non-zero (which happens only for at most two other k). 4 A stronger S Œk-norm, and time localizations first deal with the k kS Œk -norms. R2C1 / Ä 2cj . R2C1 / . cj . For simplicity, consider a single interval half-infinite I1 with neighboring half-infinite I2 , and the corresponding expression 1 Q 1 C 2 Q 2: Note that 01 C 02 D 0 on the overlap of the intervals.

### Concentration Compactness for Critical Wave Maps by Joachim Krieger

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