Renormalization of active scalar equations

Akramov I, Wiedemann E (2019)

Publication Type: Journal article

Publication year: 2019

Journal

Nonlinear Analysis - Theory Methods & Applications Elsevier

Book Volume: 179

Pages Range: 254-269

DOI: 10.1016/j.na.2018.08.018

Abstract

We consider transport equations with an incompressible transporting vector field. Whereas smooth solutions of such equations conserve every Lp norm simply by the chain rule, the question arises how regular a weak solution needs to be to guarantee this conservation property. Whereas the classical DiPerna–Lions theory gives sufficient conditions in terms of the regularity of the coefficients, with no regularity requirement on the transported scalar, we give here sufficient conditions in terms of the combined regularities of the coefficients and the scalar. This is motivated by the case of active scalar equations, where the transporting vector field has the same regularity as the transported scalar. We use commutator estimates similar to those of Constantin–E–Titi in the context of Onsager's conjecture, but we require novel arguments to handle the case of Lp norms when p≠2.

Authors with CRIS profile

Emil Wiedemann

Involved external institutions

Gottfried Wilhelm Leibniz Universität Hannover DE Germany (DE) Universität Ulm DE Germany (DE)

How to cite

APA:

Akramov, I., & Wiedemann, E. (2019). Renormalization of active scalar equations. Nonlinear Analysis - Theory Methods & Applications, 179, 254-269. https://dx.doi.org/10.1016/j.na.2018.08.018

MLA:

Akramov, Ibrokhimbek, and Emil Wiedemann. "Renormalization of active scalar equations." Nonlinear Analysis - Theory Methods & Applications 179 (2019): 254-269.

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