A quantitative theory for the continuity equation
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In this paper, stability estimates for the continuity equation with Sobolev vector fields were derived from contraction estimates for certain logarithmic Kantorovich-Rubinstein distances, and a new proof of uniqueness in the DiPerna-Lions setting was obtained.About:
This article is published in Annales de l'Institut Henri Poincaré C, Analyse non linéaire.The article was published on 2017-12-01 and is currently open access. It has received 43 citations till now. The article focuses on the topics: Uniqueness & Sobolev space.read more
Citations
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Convex integration solutions to the transport equation with full dimensional concentration
Stefano Modena,Gabriel Sattig +1 more
TL;DR: In this paper, the authors constructed infinitely many incompressible Sobolev vector fields u ∈ C t W x 1, p ˜ on the periodic domain T d for which uniqueness of solutions to the transport equation fails in the class of densities ρ ∈ c t L x p, provided 1 / p + 1/p ˜ > 1 + 1 / d.
Posted Content
Quantitative estimates for regular Lagrangian flows with $BV$ vector fields
TL;DR: In this article, the authors prove the well-posedness of regular Lagrangian flows associated to vector fields, and show that regular flow associated to non-smooth vector fields are not unique.
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Convergence rates for upwind schemes with rough coefficients
André Schlichting,Christian Seis +1 more
TL;DR: 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.
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On the vanishing viscosity limit for 2D incompressible flows with unbounded vorticity
TL;DR: For the Navier-Stokes equations on the two-dimensional torus, this article showed strong convergence of the vorticities in the vanishing viscosity limit for the Euler equations.
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Advection Diffusion Equations with Sobolev Velocity Field
Elia Bruè,Quoc-Hung Nguyen +1 more
TL;DR: In this article, the energy dissipation rate of advection-diffusion equations associated with incompressible velocity fields was studied and upper and lower bounds on the enhanced and vanishing viscosity estimates were derived.
References
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Singular Integrals and Differentiability Properties of Functions.
TL;DR: Stein's seminal work Real Analysis as mentioned in this paper is considered the most influential mathematics text in the last thirty-five years and has been widely used as a reference for many applications in the field of analysis.
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Measure theory and fine properties of functions
TL;DR: In this article, the authors define and define elementary properties of BV functions, including the following: Sobolev Inequalities Compactness Capacity Quasicontinuity Precise Representations of Soboleve Functions Differentiability on Lines BV Function Differentiability and Structure Theorem Approximation and Compactness Traces Extensions Coarea Formula for BV Functions isoperimetric inequalities The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem Pointwise Properties this article.
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Topics in Optimal Transportation
TL;DR: In this paper, the metric side of optimal transportation is considered from a differential point of view on optimal transportation, and the Kantorovich duality of the optimal transportation problem is investigated.
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Ordinary differential equations, transport theory and Sobolev spaces.
TL;DR: In this paper, the existence, uniqueness and stability results for ordinary differential equations with coefficients in Sobolev spaces were derived from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions.