Abstract
Given initial data u0 ∈ Lp (ℝ3) for some p in \(\left[ {3,{{18} \over 5}} \right[\), the auhtors first prove that 3D incompressible Navier-Stokes system has a unique solution u = uL+v with \({u_L}\mathop = \limits^{{\rm{def}}} \,{{\rm{e}}^{t\Delta }}{u_0}\) and \(v \in {{\tilde L}^\infty }\left( {\left[ {0,T} \right];{{\dot H}^{{5 \over 2} - {6 \over p}}}} \right) \cap {{\tilde L}^1}\left( {\left] {0,T} \right[;{{\dot H}^{{9 \over 2} - {6 \over p}}}} \right)\) for some positive time T. Then they derive an explicit lower bound for the radius of space analyticity of v, which in particular extends the corresponding results in [Chemin, J.-Y., Gallagher, I. and Zhang, P., On the radius of analyticity of solutions to semi-linear parabolic system, Math. Res. Lett., 27, 2020, 1631–1643, Herbst, I. and Skibsted, E., Analyticity estimates for the Navier-Stokes equations, Adv. in Math., 228, 2011, 1990–2033] with initial data in Ḣs(ℝ3) for \(s \in \left[ {{1 \over 2},{3 \over 2}} \right[\).
Similar content being viewed by others
References
Bahouri, H., Chemin, J.-Y. and Danchin, R., Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343, Springer-Verlag, Berlin, Heidelberg, 2011.
Bony, J. M., Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., 14, 1981, 209–246.
Chemin, J.-Y., Le système de Navier-Stokes incompressible soixante dix ans après Jean Leray, Actes des Journées Mathématiques à la Mémoire de Jean Leray, 99–123, Séminaire et Congrès, 9, Société Mathématique de France, Paris, 2004.
Chemin, J.-Y. and Gallagher, I., Wellposedness and stability results for the Navier-Stokes equations in ℝ3, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26, 2009, 599–624.
Chemin, J.-Y., Gallagher, I. and Zhang, P., On the radius of analyticity of solutions to semi-linear parabolic system, Math. Res. Lett., 27, 2020, 1631–1643.
Foias, C. and Temam, R., Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87, 1989, 359–369.
Giga, Y., Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62, 1986, 186–212.
Grafakos, L., Classical Fourier Analysis, Springer-Verlag, New York, 3rd ed., 2014.
Grujič, Z. and Kukavica, I., Space analyticity for the Navier-Stokes and related equations with initial data in Lp, J. Funct. Anal., 152, 1998, 447–466.
Herbst, I. and Skibsted, E., Analyticity estimates for the Navier-Stokes equations, Adv. in Math., 228, 2011, 1990–2033.
Kato, T., Strong Lp-solutions of the Navier-Stokes equation in ℝm with applications to weak solutions, Math. Z., 187, 1984, 471–480.
Kato, T. and Masuda, K., Nonlinear evolution equations and analyticity, I, Annales de l’IHP section C, 3, 1986, 455–467.
Lemarié-Rieusset, P.-G., Nouvelles remarques sur l’analyticité des solutions milds des équations de Navier-Stokes dans ℝ3, C. R. Math. Acad. Sci. Paris, 338, 2004, 443–446.
Weissler, F. B., The Navier-Stokes initial value problem in Lp, Arch. Rational Mech. Anal., 74, 1980, 219–230.
Zhang, P., Remark on the regularities of Kato’s solutions to Navier-Stokes equations with initial data in Ld(ℝd), Chin. Ann. Math. Ser. B, 29, 2008, 265–272.
Author information
Authors and Affiliations
Corresponding authors
Additional information
This work was supported by the National Natural Science Foundation of China (Nos. 11731007, 12031006, 11688101), the National Key R&D Program of China (No. 2021YFA1000800) and K. C. Wong Education Foundation.
Rights and permissions
About this article
Cite this article
Hu, R., Zhang, P. On the Radius of Analyticity of Solutions to 3D Navier-Stokes System with Initial Data in Lp. Chin. Ann. Math. Ser. B 43, 749–772 (2022). https://doi.org/10.1007/s11401-022-0356-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-022-0356-z