On the one-dimensional continuity equation with a nearly incompressible vector field

2016/10/27 by Nikolay A. Gusev, Gusev, Nikolay A.
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1610.08848

11 pages

arxiv created 2016/10/27 · openalex publication_date 2016/10/27 · arxiv updated 2016/10/28 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

We consider the Cauchy problem for the continuity equation with a bounded nearly incompressible vector field b\colon (0,T) × \mathbb R^d → \mathbb R^d, T>0. This class of vector fields arises in the context of hyperbolic conservation laws (in particular, the Keyfitz-Kranzer system). It is well known that in the generic multi-dimensional case (d≥ 1) near incompressibility is sufficient for existence of bounded weak solutions, but uniqueness may fail (even when the vector field is divergence-free), and hence further assumptions on the regularity of b (e.g. Sobolev regularity) are needed in order to obtain uniqueness. We prove that in the one-dimensional case (d=1) near incompressibility is sufficient for existence and uniqueness of locally integrable weak solutions. We also study compactness properties of the associated Lagrangian flows.

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