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

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 $a·∇u$, $u∈C_0^1(\mathbb{R}^n)$ in the Hilbert space $L^2(\mathbb{R}^n)$ 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.

Paper:
Available as PDF (245 Kbytes).
Author(s):
E. Yu. Panov,
Submitted by:
; 2015-01-28.