×

An analysis to a model of tornado. (English) Zbl 07447198

Summary: Tornado is a destructive catastrophe. We use compressible isentropic Euler equations to describe this problem. A cylindrically symmetric special solution moving with a constant velocity in \(\mathbb{R}^3\) is given. It depicts how the vorticity function of the flow evolves. Even if the initial inward velocity and acceleration are both very small, the inward velocity could become very large and the vorticity could increase drastically in later time, and most of mass concentrates on a neighborhood of the moving center axis at this time. For this solution, cases when \(\gamma \neq 2\) and when \(\gamma =2\) (shallow water) have some differences, while their evolution dynamics are basically the same. When \(\gamma =2\), the initial vorticity could depend on the space variables.

MSC:

35Qxx Partial differential equations of mathematical physics and other areas of application
35L60 First-order nonlinear hyperbolic equations
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35L67 Shocks and singularities for hyperbolic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Deng, Y.; Liu, T.; Yang, T.; Yao, Z., Solutions of Euler-Poisson equations for gaseous stars, Arch. Ration. Mech. Anal., 164, 3, 261-285 (2002) · Zbl 1038.76036
[2] Deng, Y.; Xiang, J.; Yang, T., Blowup phenomena of solutions to Euler-Possion equations, J. Math. Anal. Appl., 286, 295-306 (2003) · Zbl 1032.35023
[3] Fu, C.; Lin, S., On the critical mass of the collapse of a gaseous star in spherically symmetric and isentropic motion, Jpn. J. Ind. Appl. Math., 15, 3, 461-469 (1998) · Zbl 0913.35108
[4] Gilliam, D., Shubov, V., Bakker, J., Mickel, C., Vugrin, E.: Generalized Donaldson-Sullivan model of a vortex flow
[5] Guo, Y., Hadzic, M., Jang, J.: Continued Gravitational Collapse for Newtonian Stars (arXiv:1811.01616.) · Zbl 07298829
[6] Hadzic, M.; Jang, J., Expanding large global solutions of the equations of compressible fluid mechanics, Invent. Math., 214, 3, 1205-1266 (2018) · Zbl 1426.65152
[7] Li, T., Some special solutions of the Euler equation in \({\mathbb{R}}^N\), Commun. Pure Appl. Anal., 4, 757-762 (2005) · Zbl 1083.35058
[8] Li, T.; Wang, D., Blowup phenomena of solutions to the Euler equations for compressible fluid flow, J. Differ. Equ., 221, 91-101 (2006) · Zbl 1083.76051
[9] Makino, T.: Blowing up solutions of the Euler-Poisson equation for the evolution of gaseous stars. In: Proceedings of the Fourth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Kyoto, 1991). Transport Theory Statist. Phys. 21 , no. 4-6, 615-624 (1992) · Zbl 0793.76069
[10] Mickel, C.: Donaldson-Sullivan tornado model, a thesis in mathematics
[11] Oertel, H.: Prandtl’s essentials of fluid mechanics Springer, 2004. Chinese version translated by Z. Q. Zhu, Y. S. Qian, Z. R. Li (2008) · Zbl 1073.76001
[12] Sideris, T., Global existence and asymptotic behavior of affine motion of 3D ideal fluids surrounded by vacuum, Arch. Ration. Mech. Anal., 225, 1, 141-176 (2017) · Zbl 1367.35115
[13] Torrisi, S.: Dynamical modeling of a tornado
[14] Yuan, D.: Global solutions to rotating motion of isentropic flows with cylindrical symmetry. Commun. Math. Sci. 19 (2021), no. 7, 2019-2034. · Zbl 1479.35702
[15] Yuen, M., Self-similar solutions with elliptic symmetry for the compressible Euler and Navier-Stokes equations in \(R^N\), Commun. Nonlinear Sci. Numer. Simul., 17, 4524-4528 (2012) · Zbl 1431.35143
[16] Yuen, M., Vortical and self-similar flows of 2D compressible Euler equations, Commun. Nonlinear Sci. Numer. Simul., 19, 2172-2180 (2014) · Zbl 1457.76184
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.