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A Cauchy problem for the Tartar equation

Published online by Cambridge University Press:  12 July 2007

Sergey Sazhenkov
Sergey Sazhenkov
Affiliation:
Lavrentyev Institute of Hydrodynamics, Siberian Division of Russian Academy of Sciences, Lavrentyeva pr. 15, Novosibirsk 630090, Russia ([email protected])
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Abstract

In order to study weak limits of quadratic expressions of oscillatory solutions of partial differential equations, there was proposed a construction of H-measures defined on the space of positions and frequencies. The present paper is devoted to the investigation of the Tartar equation which describes the evolution of the H-measure μt associated with a sequence of oscillatory solutions of the linear transport equation in cases when a given solenoidal velocity field v(x, t) is sufficiently smooth. Here, (t, x, y) ∈ (0, T) × Ω × S1, 0 < T < +∞, Ω is a bounded open subset of R2 and S1 is the unit circle in R2, given coefficients Yij = Yij(y) are infinitely smooth.

Assuming that v belongs to , we establish the well posedness of Cauchy problem for the Tartar equation in the same measure class as the H-measures are in. For this purpose, we develop and use an extension of the theory of Lagrange coordinates for a case of non-smooth solenoidal velocity fields.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 132 , Issue 2 , April 2002 , pp. 395 - 418
Copyright
Copyright © Royal Society of Edinburgh 2002

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