Abstract
In this paper, we consider the problem of the existence and regularity of mild solutions to the Navier–Stokes equation with hereditary viscosity in the context of initial data in Besov–Morrey spaces. We employ the theory of resolvent families to prove our main results. We also study the stability of globally defined mild solutions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Azevedo, J., Cuevas, C., Dantas, J., Silva, C.: On the fractional chemotaxis Navier-Stokes system in the critical spaces. Discrete Contin. Dyn. Syst. Ser. B 28(1), 538 (2023)
Bazhlekova, E., Clément, P.: Global smooth solutions for a quasilinear fractional evolution equation. J. Evol. Equ. 3, 237–246 (2003)
Barbu, V., Sritharan, S.S.: Navier-Stokes equation with hereditary viscosity. Z. Angew. Math. Phys. 54(3), 449–461 (2003)
Bennett, C., Sharpley, R.: Interpolation of operators, Pure and Applied Mathematics, vol. 129. Academic Press, Inc., New York (1988)
Chhabra, R.P., Richardson, J.F.: Non-Newtonian Flow in the Process Industries. Oxford (1999)
de Almeida, M.F., Precioso, J.C.: Existence and symme- tries of solutions in Besov–Morrey spaces for a semilinear heat-wave type equation. J. Math. Anal. Appl. 432, 338–355 (2015)
de Andrade, B.: On the well-posedness of a Volterra equation with applications in the Navier–Stokes problem. Math. Methods Appl. Sci. 41, 750–768 (2018)
de Andrade, B., Viana, A., Silva, C.: \(L^q\)-solvability for an equation of viscoelasticity in power type material. Z. Angew. Math. Phys. 72, 1–20 (2021)
de Carvalho-Neto, P.M., Planas, G.: Mild solutions to the time fractional Navier–Stokes equations in \(\mathbb{R} ^n\). J. Differ. Equ. 259, 2948–2980 (2015)
Engler, H.: Global smooth solutions for a class of parabolic integrodifferential equations. Trans. Am. Math. Soc. 348, 267–290 (1996)
Ferreira, J.A., Oliveira, P., Pena, G.: On the exponential decay of waves with memory. J. Comput. Appl. Math. 318, 460–478 (2017)
Ferreira, J.A., Oliveira, P., Pena, G.: Decay of solutions of wave equations with memory. In: Proceedings of the 14th International Conference on Computational and Mathematical Methods in Science and Engineering (2014)
Ferreira, L.C., Postigo, M.: Global well-posedness and asymptotic behavior in Besov–Morrey spaces for Chemotaxis-Navier–Stokes fluids. J. Math. Phys. 60, 061502 (2019)
Ferreira, L.C.F., Precioso, J.C.: Existence and asymptotic behaviour for the parabolic-parabolic Keller–Segel system with singular data. Nonlinearity 24, 1433–1449 (2011)
Kato, T.: Strong \(L^p\)-solutions of the Navier-Stokes equation in \(\mathbb{R} ^m\), with applications to weak solutions. Math. Z. 187, 471–480 (1984)
Kato, T.: Strong solutions of the Navier–Stokes equations in Morrey spaces. Bull. Braz. Math. Soc. (N.S.) 22(2), 127–155 (1992)
Kozono, H., Yamazaki, M.: Semilinear heat equations and the Navier–Stokes equation with distributions in new function spaces as initial data. Commun. Partial Differ. Equ. 19(5–6), 959–1014 (1994)
Mazzucato, A.L.: Besov–Morrey spaces: function space theory and applications to non-linear PDE. Trans. Am. Math. Soc. 355, 1297–364 (2003)
Mohan, M.T., Sritharan, S.S.: Stochastic Navier–Stokes equations perturbed by Lévy noise with hereditary viscosity. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 22(1), 32 (2019)
Miyakawa, T.: On Morrey spaces of measures: basic properties and potential estimates. Hiroshima Math. J. 20(1), 213–222 (1990)
Nohel, J.A.: A nonlinear hyperbolic Volterra equation. In: Londen, S.-O., Staffans, O. (eds.) Volterra Equations, in: Lecture Notes in Mathematics, vol. 737, pp. 220–235. Springer, Berlin (1979)
Prüss, J.: Evolutionary Integral Equations and Applications, vol. 87. Birkhäuser, Basel (2013)
Yang, M., Fu, Z., Sun, J.: Existence and large time behavior to coupled chemotaxis-fluid equations in Besov–Morrey spaces. J. Differ. Equ. 266(9), 5867–5894 (2019)
Funding
Bruno de Andrade is partially supported by CNPQ/Brazil under Grant 310384/2022-2.
Author information
Authors and Affiliations
Contributions
All authors contributed to the implementation of the research, the analysis of the results, and the writing of the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors have no relevant financial or non-financial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
de Andrade, B., Cuevas, C. & Dantas, J. Navier–Stokes equation with hereditary viscosity and initial data in Besov–Morrey spaces. Z. Angew. Math. Phys. 75, 11 (2024). https://doi.org/10.1007/s00033-023-02151-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00033-023-02151-1