Nonlinear interpolation and the flow map of quasilinear equations
5 affiliations
- 1École Polytechnique, Palaiseau, France
- 2CMLS - Centre de Mathématiques Laurent Schwartz (Route de Saclay, 91128 Palaiseau Cedex - France)
- 3Université Paris Saclay, Faculté des Sciences d’Orsay, France
- 4University of Wisconsin, Madison, USA
- 5University of California, Berkeley, USA
Abstract
We prove an interpolation theorem for nonlinear functionals defined on scales of Banach spaces that generalize Besov spaces. It applies to functionals defined only locally, requiring only some weak Lipschitz conditions, extending those introduced by Lions and Peetre. Our analysis is self-contained and independent of any previous results in interpolation theory. It depends solely on the concepts of Friedrichs’ mollifiers, seen through the formalism introduced by Hamilton, combined with the frequency envelopes introduced by Tao and used recently by two of the present authors and others to study the Cauchy problem for various quasilinear evolutions in partial differential equations. Inspired by this latter work, our main application states that, for an abstract flow map of a quasilinear problem, both the continuity of the flow as a function of time and the continuity of the data-to-solution map follow automatically from the estimates that are usually proven when establishing the existence of solutions: propagation of regularity via tame a priori estimates for higher regularities and contraction for weaker norms.
Related
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BibTeX
@article{alazard2026nonlinear,
title = {{Nonlinear interpolation and the flow map of quasilinear equations}},
author = {Thomas Alazard and Nicolas Burq and Mihaela Ifrim and Daniel Tataru and Claude Zuily},
journal = {Journal of the European Mathematical Society},
year = {2026},
month = oct,
doi = {10.4171/jems/1827},
url = {https://doi.org/10.4171/jems/1827},
note = {CC BY},
}
RIS
TY - JOUR
TI - Nonlinear interpolation and the flow map of quasilinear equations
AU - Thomas Alazard
AU - Nicolas Burq
AU - Mihaela Ifrim
AU - Daniel Tataru
AU - Claude Zuily
PY - 2026
DA - 2026/10/15
JO - Journal of the European Mathematical Society
DO - 10.4171/jems/1827
UR - https://doi.org/10.4171/jems/1827
AB - We prove an interpolation theorem for nonlinear functionals defined on scales of Banach spaces that generalize Besov spaces. It applies to functionals defined only locally, requiring only some weak Lipschitz conditions, extending those introduced by Lions and Peetre. Our analysis is self-contained and independent of any previous results in interpolation theory. It depends solely on the concepts of Friedrichs’ mollifiers, seen through the formalism introduced by Hamilton, combined with the frequency envelopes introduced by Tao and used recently by two of the present authors and others to study the Cauchy problem for various quasilinear evolutions in partial differential equations. Inspired by this latter work, our main application states that, for an abstract flow map of a quasilinear problem, both the continuity of the flow as a function of time and the continuity of the data-to-solution map follow automatically from the estimates that are usually proven when establishing the existence of solutions: propagation of regularity via tame a priori estimates for higher regularities and contraction for weaker norms.
SN - 1435-9855
SN - 1435-9863
ER -