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Concentration compactness for critical wave maps. (English) Zbl 1387.35006
EMS Monographs in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-106-4/hbk). vi, 484 p. (2012).
This book is devoted on the modified Bahouri-Gérard method and Kenig-Merle method which are adapted to wave maps.
The authors define the spaces $$S$$ and $$N$$ and deduct some their properties. After which the authors introduce the actual system of wave equations for which the $$S$$ and $$N$$ spaces allow to be extracted a priori estimates.
In the book are presented some bounds from $$S\times S$$ into $$L_{tx}^2$$ and it is given a standard bilinear $$L^2$$-bound for free waves, it is shown a bilinear bound which expresses something close to an algebra property $$N\times S\hookrightarrow N$$. The authors give definition for the basic null forms and they are given null-form bounds according to high-high vs. high-low and low-high interactions.
The authors derive the estimates on the trilinear nonlinearities which govern the wave-map system. Basing on modulation vs. frequency the authors split their arguments in two cases which depend on whether all inputs are “hyperbolic” or not. It is proved trilinear estimates in the ”hyperbolic” case. The authors consider the errors generated by repeated application of Hodge decompositions which are of higher than quintic degree.
The authors introduce a norm locally on some time interval $$(-T_0, T_1)$$ with the property that its finiteness insures that the gauged wave map can be continued outside of that time interval.
In the book is discussed the issue of defining wave maps with data which are merely of energy class. Using approximations by smooth wave maps each of which can be continued canonically, the authors propagate such data under wave map evolution.
The authors invoke methods involving BMO and the closely related $$A_p$$-classes, also they give a weighted version of the Coifman-Rochberg-Weiss communicator theorem with the weights belonging to the $$A_p$$-class. After which in the book is considered Bahouri-Gérard theory for wave maps. The authors construct frequency decomposition of initial data, frequency localized approximations to the data, using low-frequency approximations they are obtained a priori bound on the lowest frequency non-atomic components, the authors construct the profile decomposition for the lowest frequency above-threshold energy frequency atoms, it is completed the approximate solutions which is given as a sum of the profiles and the low-frequency term to an exact solution and in the book is given explanation how to add the remaining frequency atoms. In the end of the book authors adapt the method of Kenig-Merle to the context of wave maps.

MSC:
 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 35L05 Wave equation 35L52 Initial value problems for second-order hyperbolic systems 53Z05 Applications of differential geometry to physics
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