Transport equations with partially BV velocities

• Nicolas Lerner ^[1]
  1. [1] Université de Rennes 1, France
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 3, Nº 4, 2004, págs. 681-703
• Idioma: inglés
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• Resumen

  • We prove the uniqueness of weak solutions for the Cauchy problem for a class of transport equations whose velocities are partially with bounded variation.

    Our result deals with the initial value problem ∂t u + Xu = f, u|t=0 = g, where X is the vector field a1(x1) · ∂x1 + a2(x1, x2) · ∂x2 , a1 ∈ BV(R N1 x1 ), a2 ∈ L1x 1
    BV(R N2 x2 )  , with a boundedness condition on the divergence of each vector field a1, a2. This model was studied in the paper [LL] with a W1,1 regularity assumption replacing our BV hypothesis. This settles partly a question raised in the paper [Am]. We examine the details of the argument of [Am] and we combine some consequences of the Alberti rank-one structure theorem for BV vector fields with a regularization procedure. Our regularization kernel is not restricted to be a convolution and is introduced as an unknown function. Our method amounts to commute a pseudodifferential operator with a BV function.