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Singular and regular solutions of a nonlinear parabolic system

Petr Plechác1 and Vladimír Sverák2

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Recommended by E S Titi

We study a dissipative nonlinear equation modelling certain features of the Navier–Stokes equations. We prove that the evolution of radially symmetric compactly supported initial data does not lead to singularities in dimensions n≤4. For dimensions n>4, we present strong numerical evidence supporting the existence of blow-up solutions. Moreover, using the same techniques we numerically confirm a conjecture of Lepin regarding the existence of self-similar singular solutions to a semi-linear heat equation.


PACS

02.30.Jr Partial differential equations

47.10.ad Navier-Stokes equations

02.60.Lj Ordinary and partial differential equations; boundary value problems

MSC

35Q30 Stokes and Navier-Stokes equations (See also 76D05, 76D07, 76N10)

34B16 Singular nonlinear boundary value problems

76D05 Navier-Stokes equations (See also 35Q30)

35K05 Heat equation

35K55 Nonlinear PDE of parabolic type

Subjects

Fluid dynamics

Mathematical physics

Computational physics

Dates

Issue 6 (November 2003)

Received 12 February 2003, in final form 1 July 2003

Published 5 September 2003



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