Advances in Differential Equations

Nonuniqueness of the heat flow of director fields

I. Primi

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We give a new example of nonuniqueness of weak solutions of the initial-boundary-value problem for the heat flow of director fields in an infinitely long cylinder in $\mathbb R^{3}$. The example confirms the connection between nonuniqueness of axially symmetric solutions for the harmonic map heat flow and the occurrence of point singularities in the solutions. The result is compared with earlier nonuniqueness results. Traveling wave solutions are used as barrier functions.

Article information

Adv. Differential Equations, Volume 15, Number 3/4 (2010), 349-380.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K51: Initial-boundary value problems for second-order parabolic systems 35D30: Weak solutions 35B07: Axially symmetric solutions 35K91: Semilinear parabolic equations with Laplacian, bi-Laplacian or poly- Laplacian 35A02: Uniqueness problems: global uniqueness, local uniqueness, non- uniqueness 35B51: Comparison principles 35B45: A priori estimates 35A21: Propagation of singularities


Primi, I. Nonuniqueness of the heat flow of director fields. Adv. Differential Equations 15 (2010), no. 3/4, 349--380.

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