Optimal stability estimates for continuity equations

@article{Seis2018OptimalSE,
  title={Optimal stability estimates for continuity equations},
  author={Christian Seis},
  journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics},
  year={2018},
  volume={148},
  pages={1279 - 1296}
}
  • Christian Seis
  • Published 22 August 2016
  • Mathematics
  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics
This review paper is concerned with the stability analysis of the continuity equation in the DiPerna–Lions setting in which the advecting velocity field is Sobolev regular. Quantitative estimates for the equation were derived only recently, but optimality was not discussed. We revisit the results from our 2017 paper, compare the new estimates with previously known estimates for Lagrangian flows and demonstrate how these can be applied to produce optimal bounds in applications from physics… 
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References

SHOWING 1-10 OF 53 REFERENCES
A quantitative theory for the continuity equation
Transport equation with integral terms
We prove some theorems on the existence, uniqueness, stability and compactness properties of solutions to inhomogeneous transport equations with Sobolev coefficients, where the inhomogeneous term
Continuity equations and ODE flows with non-smooth velocity*
In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and transport equations and for the ordinary differential equation (ODE). In this
ODEs with Sobolev coefficients: The eulerian and the lagrangian approach
In this paper we describe two approaches to the well-posedness of Lagrangian flows of Sobolev vector fields. One is the theory of renormalized solutions which was introduced by DiPerna and Lions
Transport equations with integral terms: existence, uniqueness and stability
We prove some theorems on the existence, uniqueness, stability and compactness properties of solutions to inhomogeneous transport equations with Sobolev coefficients, where the inhomogeneous term
Differential equations with singular fields
Estimates and regularity results for the DiPerna-Lions flow
Abstract In this paper we derive new simple estimates for ordinary differential equations with Sobolev coefficients. These estimates not only allow to recover some old and recent results in a simple
Convergence Rates for Upwind Schemes with Rough Coefficients
TLDR
The case where the advecting velocity field has spatial Sobolev regularity and initial data are merely integrable is interested, and the rate of weak convergence is at least 1/2 in the mesh size.
Error estimate for finite volume scheme
TLDR
A h1/2-error estimate in the L∞(0,t;L1)-norm for BV data is proved, which was expected from numerical experiments and is optimal.
...
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