Abstract
This paper discusses the Cauchy problem for the fractional Navier–Stokes–Coriolis equation (FNSC). The FNSC equation refers to that obtained by replacing the Laplacian in the Navier–Stokes–Coriolis equation by the more general operator \((-\Delta )^{\alpha }\) with \(\alpha >0\). We prove the time-local existence and uniqueness of the mild solution for every \(\varOmega \in \mathbb {R}\backslash \{0\}\) and \(u_0\in \dot{H}^s(\mathbb {R}^3)^3\) with \(1/4<\alpha \leqslant 3/2\), \(3/2-\alpha<\,s\, <5/4\). Furthermore, we give a lower bound for the time interval of its local existence in terms of \(|\varOmega |\) and \(\Vert u_0\Vert _{\dot{H}^s}\). It follows from our characterization that the existence time T of the solution can be arbitrarily large provided the speed of rotation \(|\varOmega |\) is sufficiently fast.
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References
Babin A., Mahalov, A. Nicolaenko, B.: Regularity and integrability of 3D Euler and Navier-Stokes equations for rotating fluids. Asymptot. Anal., 15:103–150 (1997).
Babin, A., Mahalov, A., Nicolaenko, B.: 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity. Indiana Univ. Math. J., 50:1–35 (2001).
Cannone, M.: Harmonic analysis tools for solving the incompressible Navier-Stokes equations. Handbook of mathematical fluid dynamics, III:161–244 (2004).
Cannone M., Karch, G.: Incompressible Navier-Stokes equations in abstract Banach spaces. Sūrikaisekikenkyūsho Kōkyūroku, 1234:27–41 (2001).
Cao, C., Titi, Edriss S.: Global well-posedness of the three dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. Ann. of Math., 166:245–267 (2007).
Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Anisotropy and dispersion in rotating fluids. Nonlinear partial differential equations and their applications, North Holland, Stud. Math. Appl., 31 (2002).
Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Mathematical Geophysics: An introduction to rotating fluids and the Navier-Stokes equations. Oxford University Press, 1–272 (2006).
Chen, Q., Miao, C., Zhang, Z.: Global well-posedness for the 3D rotating Navier-Stokes equations with highly osillating initial data. Pacific J. Math., 262:263–283 (2013).
Ding, Y., Sun, X.: Uniqueness of weak solutions for fractional Navier-Stokes equations. Front. Math. China, 10: 33–51 (2015).
Ding, Y., Sun, X.: Strichartz estimates for parabolic equations with higher order differential operators. Sci. China Math., 58:1047–1062 (2015).
Fujita, H., Kato, T.: On the Navier-Stokes initial value problem, I. Arch. Rational Mech. Anal., 16:269–315 (1964).
Giga, Y., Inui, K., Mahalov, A., Matsui, S.: Navier-Stokes equations in a rotating frame in \({\mathbb{R}}^{3}\) with initial data nondecreasing at infinity. Hokkaido Math. J., 35:321–364 (2006).
Giga, Y., Inui, K., Mahalov, A., Matsui, S., Saal, J.: Rotating Navier-Stokes equations in \(\mathbb{R}_{+}^{3}\) with initial data nondecreasing at infinity: the Ekman boundary layer problem. Arch. Ration. Mech. Anal., 186:177–224 (2007).
Giga, Y., Inui, K., Mahalov, A., Saal, J.: Uniform global solvability of the rotating Navier-Stokes equations for nondecaying initial data. Indiana Univ. Math. J., 57:2775–2791 (2008).
Hieber, M., Shibata, Y.: The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework. Math. Z., 265:481–491 (2010).
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).
Konieczny, P., Yoneda, T.: On dispersive effect of the Coriolis force for the stationary Navier-Stokes equations. J. Differential Equations, 250:3859–3873 (2011).
Kozono, H., Ogawa, T., Taniuchi, Y.: Navier-Stokes equations in the Besov space near \(L^{\infty }\) and \(BMO\). Kyushu J. Math., 57:303–324 (2003).
Iwabuchi, T., Takada, R.: Global solutions for the Navier-Stokes equations in the rotational framework. Math. Ann., 357:727–741 (2013).
Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math., 63:193–248 (1934).
J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod; Gauthier-Villars, Paris (1969).
Miao, C., Yuan, B., Zhang, B.: Well-posedness of the Cauchy problem for the fractional power dissipative equations. Nonlinear Anal., 68(3):461–484 (2008).
Stein, Elias M.: Singular integrals and differentiability properties of functions. Princeton University Press, Princeton (1970).
Wu, J.: Generalized MHD equations. J. Differential Equations, 195:284–312 (2003).
Wu H., Fan, J.: Weak-strong uniqueness for the generalized Navier-Stokes equations. Appl. Math. Lett., 25:423–428 (2012).
Zhai, Z.: Strichartz type estimates for fractional heat equations. J. Math. Anal. Appl., 356:642–658 (2009).
Zhou, Y.: Regularity criteria for the generalized viscous MHD equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 24:491–505 (2007).
Acknowledgements
The authors would like to express their deep gratitude to the referee for giving many valuable suggestions. X. Sun is supported by NSF of China (Grant: 11461059, 11561062, 11601434), SRPNWNU (Grant: NWNU-LKQW-14-2). Y. Ding is supported by NSF of China (Grant: 11371057) and SRFDP of China (Grant: 20130003110003).
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Sun, X., Ding, Y. Dispersive effect of the Coriolis force and the local well-posedness for the fractional Navier–Stokes–Coriolis system. J. Evol. Equ. 20, 335–354 (2020). https://doi.org/10.1007/s00028-019-00531-7
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DOI: https://doi.org/10.1007/s00028-019-00531-7