Abstract
In the first part of this paper, we prove a new theorem concerning non-smooth solutions of nonlinear Navier–Stokes type PDE as they arise in atmospheric and fluid dynamics, but here in arbitrary dimension. In its simplest form, the theorem states that the velocity field must be tangent to the hypersurface on which it has a jump discontinuity, i.e., its singular support. The theorem is proved using Colombeau algebras of generalized functions, providing yet another example of the fruitfulness of this concept for nonlinear problems with singularities, ill posed in distribution theory.
In the second part, we discuss a numerical and analytic study of a two-dimensional model which, in spite of its simplicity, predicts remarkably correctly the “wall of the eye” of a hurricane and allows us to get analytic expressions for the asymptotic behaviour of radial and tangential wind field near this wall. These results are consistent with and confirm the theoretical results of the first part.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
J.-F. Colombeau, Elementary Introduction to New Generalized Functions. North-Holland Mathematical Studies, vol. 113 (North-Holland, Amsterdam, 1985)
V.G. Danilov, V.P. Maslov, V.M. Shelkovich, Algebras of singularities of generalized solutions of strictly hyperbolic systems of quasilinear first-order systems. Teor. Mat. Fiz. 114(1), 3–55 (1998)
A. Delcroix, D. Scarpalezos, Sharp topologies on asymptotic algebras. Monatshefte Math. 129, 1–14 (2000)
S.Y. Dobrokhotov, Hugoniót-Maslov chains for the trajectories of point vortex singularities of the shallow water equations and the Hill equation. Dokl. Akad. Nauk Ross. Akad. Nauk 354(5), 600–603 (1997). S.Y. Dobrokhotov, Reduction to the Hill equation of the Hugóniot-Maslov chain for the trajectories of solitary vortices of shallow wave equations. Teor. Mat. Fiz. 112(1), 47–66 (1997). S.Y. Dobrokhotov, Hugoniót-Maslov chains for solitary vortices of the shallow water equations, II. Russ. J. Math. Phys. 6(3), 282–313 (1999)
S.Y. Dobrokhotov, K.V. Pankrashkin, E.S. Semenov, Proof of Maslovs conjecture about the structure of weak point singular solutions of the shallow water equations. Russ. J. Math. Phys. 8(1), 25–54 (2001). S.Y. Dobrokhotov, K.V. Pankrashkin, E.S. Semenov, On Maslovs conjecture about the structure of weak point singularities of the shallow-water equations. Dokl. Math. 64(1), 127–130 (2001)
M.Grosser, M. Kunzinger, M. Oberguggenberger, R. Steinbauer, Geometric Theory of Generalized Functions with Applications to General Relativity (Kluwer Academic, Dordrecht, 2001)
M.F. Hasler, Asymptotic extension of topological modules and algebras. Integral Transforms Special Funct. 20(4), 291–299 (2009)
V.P. Maslov, Three algebras corresponding to nonsmooth solutions of systems of quasilinear hyperbolic equations. Uspekhi Matem. Nauk 35(2), 252–253 (1980)
H.K. Moffatt, Singularities in fluid mechanics. Phys. Rev. Fluids 4(11), 110502 (2019)
D. Scarpalezos, Some remarks on functoriality of colombeau’s construction: topological and microlocal aspects and applications. Integral Transforms Spec. Funct. 6, 295–307 (1998); D. Scarpalezos, Colombeau’s generalized functions: topological structures; microlocal properties. A simplied point of view, I. Bull. Cl. Sci. Math. Nat. Sci. Math. 25, 89–114 (2000); D. Scarpalezos, Colombeau’s generalized functions: topological structures; microlocal properties. A simplied point of view, II. Publ. Inst. Math. (Beograd) (N.S.) 76(90), 111–125 (2004)
T. Tao, Searching for singularities in the Navier-Stokes equations. Nat. Rev. Phys. 1, 418–419 (2019)
V.N. Zhikharev, On necessary conditions for the existence and uniqueness of the type of solutions with propagating weak point singularity for the 2D hydrodynamic equations (in Russian). VINITI no. B86-8148, Moscow (1986)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Hasler, M.F. (2022). Singular Solutions to Equations of Fluid Mechanics and Dynamics Near a Hurricane’s Eye. In: Cerejeiras, P., Reissig, M., Sabadini, I., Toft, J. (eds) Current Trends in Analysis, its Applications and Computation. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-87502-2_38
Download citation
DOI: https://doi.org/10.1007/978-3-030-87502-2_38
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-87501-5
Online ISBN: 978-3-030-87502-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)