Linear Vortex Symmetrization: The Spectral Density Function

Archive for Rational Mechanics and Analysis - Tập 246 - Trang 61-137 - 2022
Alexandru D. Ionescu1, Hao Jia2
1Princeton University, Princeton, USA
2University of Minnesota, Minneapolis, USA

Tóm tắt

We investigate solutions of the 2d incompressible Euler equations, linearized around steady states which are radially decreasing vortices. Our main goal is to understand the smoothness of what we call the spectral density function associated with the linearized operator, which we hope will be a step towards proving full nonlinear asymptotic stability of radially decreasing vortices. The motivation for considering the spectral density function is that it is not possible to describe the vorticity or the stream function in terms of one modulated profile. There are in fact two profiles, both at the level of the physical vorticity and at the level of the stream function. The spectral density function allows us to identify these profiles, and its smoothness leads to pointwise decay of the stream function which is consistent with the decay estimates first proved in Bedrossian–Coti Zelati–Vicol (Ann PDE 5(4):1–192, 2019).

Tài liệu tham khảo

Arnold, V., Khesin, B.: Topological Methods in Hydrodynamics. Springer, New York (1998) Bassom, A.P., Gilbert, A.D.: The spiral wind-up of vorticity in an inviscid planar vortex. J. Fluid Mech. 371, 109–140, 1998 Bassom, A.P., Gilbert, A.D.: The relaxation of vorticity fluctuations in approximately elliptical streamlines. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456, 295–314, 2000 Bedrossian, J., Masmoudi, N.: Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations. Publ. Math. Inst. Hautes Etudes Sci. 122, 195–300, 2015 Bedrossian, J., Coti Zelati, M., Vicol, V.: Vortex axisymmetrization, inviscid damping, and vorticity depletion in the linearized 2D Euler equations. Ann. PDE 5(4), 1–192, 2019 Benzi, R., Paladin, G., Patarnello, S., Santangelo, P., Vulpiani, A.: Intermittency and coherent structures in two-dimensional turbulence. J. Phys. A Math. Gen. 19, 3771–3784, 1986 Brachet, M., Meneguzzi, M., Politano, H., Sulem, P.: The dynamics of freely decaying two-dimensional turbulence. J. Fluid Mech. 194, 333–349, 1988 Bouchet, F., Morita, H.: Large time behavior and asymptotic stability of the 2D Euler and linearized Euler equations. Phys. D Nonlinear Phenom. 239, 948–966, 2010 Briggs, R.J., Daugherty, J.D., Levy, R.H.: Role of landau damping in crossed field electron beams and inviscid shear flow. Phys. Fluids 13, 421, 1970. https://doi.org/10.1063/1.1692936. Choi, K., Lim, D.: Stability of radially symmetric, monotone vorticities of 2D Euler equations. Calc. Var. 61, 120, 2022. https://doi.org/10.1007/s00526-022-02231-6. Deng, Y., Masmoudi, N.: Long time instability of the Couette flow in low Gevrey spaces. arXiv:1803.01246 Gallay, T.: Enhanced dissipation and axisymmetrization of two-dimensional viscous vortices. Arch. Ration. Mech. Anal. 230, 939–975, 2018 Gallay, T., Sverak, V.: Arnold’s variational principle and its application to the stability of planar vortices. Preprint. arXiv:2110.13739 Gallay, T., Wayne, E.: Global stability of vortex solutions of the two dimensional Navier–Stokes equation. Commun. Math. Phys. 255, 97–129, 2005 Grenier, E., Nguyen, T., Rousset, F., Soffer, A.: Linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method. J. Funct. Anal. 278(3), 108339, 2020 Hall, I., Bassom, A., Gilbert, A.: The effect of fine structures on the stability of planar vortices. Eur. J. Mech. B Fluids 22(2), 179–198, 2003 Ionescu, A., Jia, H.: Nonlinear inviscid damping near monotonic shear flows. Acta Math. See also arXiv:2001.03087(to appear) Ionescu, A., Jia, H.: Inviscid damping near the Couette flow in a channel. Commun. Math. Phys. 374(3), 2015–2096, 2020 Ionescu, A., Jia, H.: Axi-symmetrization near point vortex solutions for the 2D Euler equation. Commun. Pure Appl. Math. 75(4), 818–891, 2022. https://doi.org/10.1002/cpa.21974. Jia, H.: Linear inviscid damping near monotone shear flows. SIAM J. Math. Anal. 52(1), 623–652, 2020 Jia, H.: Linear inviscid damping in Gevrey spaces. Arch. Ration. Mech. Anal. 235(2), 1327–1355, 2020 Kelvin, L.: Stability of fluid motion-rectilinear motion of viscous fluid between two plates. Philos. Mag. 24, 188, 1887 Masmoudi, N., Zhao, W.: Nonlinear inviscid damping for a class of monotone shear flows in finite channel. Preprint. arXiv:2001.08564, 2020 McWilliams, J.: The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 21–43, 1984 McWilliams, J.: The vortices of two-dimensional turbulence. J. Fluid Mech. 219, 361–385, 1990 Orr, W.: The stability or instability of steady motions of a perfect liquid and of a viscous liquid, Part I: a perfect liquid. Proc. R. Ir. Acad. A Math. Phys. Sci. 27, 9–68, 1907 Rayleigh, L.: On the stability or instability of certain fluid motions. Proc. Lond. Math. Soc. S1–11, 57, 1880 Rodino, L.: Linear Partial Differential Operators in Gevrey Spaces. World Scientific Publishing Company, Singapore (1993) Schecter, D., Dubin, D., Cass, A., Driscoll, C., Lansky, I., O’Neil, T.: Inviscid damping of asymmetries on a two-dimensional vortex. Phys. Fluids 2397, 12, 2002 Sverak, V.: Lecture notes. http://www-users.math.umn.edu/~sverak/course-notes2011.pdf Yamanaka, T.: A new higher order chain rule and Gevrey class. Ann. Glob. Anal. Geom. 7, 179–203, 1989 Yudovich, V.: Non-stationary flows of an ideal incompressible fluid (Russian). Z. Vycisl. Mat. i Mat. Fiz. 3, 1032–1066, 1963 Yudovich, V.: Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid. Math. Res. Lett. 2, 27–38, 1995 Wei, D., Zhang, Z., Zhao, W.: Linear inviscid damping for a class of monotone shear flow in Sobolev spaces. Commun. Pure Appl. Math. 71, 617–687, 2018 Wei, D., Zhang, Z., Zhao, W.: Linear inviscid damping and vorticity depletion for shear flows. Ann. PDE 5(3), 2019. See also arXiv:1704.00428 Wolibner, W.: Un theorème sur l’existence du mouvement plan d’un fluide parfait, homogene, incompressible, pendant un temps infiniment long. Math. Z. 37, 698–726, 1933 Zillinger, C.: Linear inviscid damping for monotone shear flows. Trans. Am. Math. Soc. 369, 8799–8855, 2017 Zillinger, C.: Linear inviscid damping for monotone shear flows in a finite periodic channel, boundary effects, blow-up and critical Sobolev regularity. Arch. Ration. Mech. Anal. 221, 1449–1509, 2016