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

Nikolay A. Gusev

doi:10.3934/cpaa.2019028

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2019, Volume 18, Issue 2: 559-568. Doi: 10.3934/cpaa.2019028

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

Nikolay A. Gusev^1,2,3,

1.
Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russia
2.
RUDN University, 6 Miklukho-Maklay St, Moscow, 117198, Russia
3.
Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St, Moscow, 119991

Received: May 31, 2017

Revised: June 30, 2018

Published: October 2018

The publication was prepared with the support of the "RUDN University Program 5-100"

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Abstract

We consider the Cauchy problem for the continuity equation with a bounded nearly incompressible vector field $b\colon (0,T) × \mathbb{R}^d \to \mathbb{R}^d$, $T>0$. This class of vector fields arises in the context of hyperbolic conservation laws (in particular, the Keyfitz-Kranzer system, which has applications in nonlinear elasticity theory).

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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Mathematics Subject Classification: Primary: 35D30, 34A12; Secondary: 34A36.

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