On one criterion of the uniqueness of generalized solutions for linear transport equations with discontinuous coefficients

2015/04/03 by Evgeny Yu. Panov, Panov, Evgeny Yu. · 1 citation
Engineering · Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1504.00836

arxiv created 2015/04/03 · openalex publication_date 2015/04/03 · arxiv updated 2015/04/06 · openalex created_date 2022/08/30 · openalex updated_date 2026/07/28

Abstract

We study generalized solutions of multidimensional transport equation with bounded measurable solenoidal field of coefficients a(x). It is shown that any generalized solution satisfies the renormalization property if and only if the operator a⋅∇ u, u∈ C₀¹(ℝⁿ) in the Hilbert space L²(ℝⁿ) is an essentially skew-adjoint operator, and this is equivalent to the uniqueness of generalized solutions. We also establish existence of a contractive semigroup, which provides generalized solutions, and give a criterion of its uniqueness.

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