In a seminal work in 1934, J. Leray constructed solutions of the Navier–Stokes equations for arbitrary initial data and left it open whether would necessarily tend to zero as . This question was answered positively fifty years later by T. Kato, using a different approach. Here, we reexamine Leray's problem and solve this and other important related questions using Leray's original ideas and some standard tools (Fourier transform, Duhamel's principle, heat kernel estimates) already in use in his time.
En 1934, J. Leray a construit des solutions faibles pour les équations de Navier–Stokes avec des données initiales arbitraires, où il a laissé non résolue la question de savoir si tendrait toujours vers zéro quand , à laquelle a été répondu positivement en 1984 par T. Kato, au moyen d'une autre approche. Ici, on reconsidère le problème de Leray et quelques-unes de ses extensions, qui sont résolus en n'employant que des idées développées par Leray en 1934 et des techniques classiques, très utilisées déjà à cette époque.
Accepted:
Published online:
Joyce C. Rigelo 1; Lineia Schütz 2; Janaína P. Zingano 2; Paulo R. Zingano 2
@article{CRMATH_2016__354_5_503_0, author = {Joyce C. Rigelo and Lineia Sch\"utz and Jana{\'\i}na P. Zingano and Paulo R. Zingano}, title = {Leray's problem for the {Navier{\textendash}Stokes} equations revisited}, journal = {Comptes Rendus. Math\'ematique}, pages = {503--509}, publisher = {Elsevier}, volume = {354}, number = {5}, year = {2016}, doi = {10.1016/j.crma.2016.02.008}, language = {en}, }
TY - JOUR AU - Joyce C. Rigelo AU - Lineia Schütz AU - Janaína P. Zingano AU - Paulo R. Zingano TI - Leray's problem for the Navier–Stokes equations revisited JO - Comptes Rendus. Mathématique PY - 2016 SP - 503 EP - 509 VL - 354 IS - 5 PB - Elsevier DO - 10.1016/j.crma.2016.02.008 LA - en ID - CRMATH_2016__354_5_503_0 ER -
Joyce C. Rigelo; Lineia Schütz; Janaína P. Zingano; Paulo R. Zingano. Leray's problem for the Navier–Stokes equations revisited. Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 503-509. doi : 10.1016/j.crma.2016.02.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.02.008/
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